Algebraic Methods in Game Theory

by

Ruchira Sreemati Datta

B.S. (California Institute of Technology) 1991M.S. (University of California at Berkeley) 2002

A dissertation submitted in partial satisfaction of therequirements for the degree of

Doctor of Philosophy

in

Mathematics

in the

GRADUATE DIVISIONof the

UNIVERSITY of CALIFORNIA at BERKELEY

Committee in charge:

Professor Bernd Sturmfels, ChairProfessor Alberto GrunbaumProfessor Laurent El Ghaoui

Fall 2003

Algebraic Methods in Game Theory

Copyright 2003by

Ruchira Sreemati Datta

1

Abstract

Algebraic Methods in Game Theory

by

Ruchira Sreemati DattaDoctor of Philosophy in Mathematics

University of California at Berkeley

Professor Bernd Sturmfels, Chair

In this dissertation we apply algebraic methods to game theory. The central objects ofstudy in game theory, Nash equilibria, can be characterized in several ways. We focuson their characterization as the solutions to certain systems of polynomial equationsand inequalities. Thus we bring to bear the techniques of commutative algebra,algebraic geometry, and combinatorics used in solving polynomial systems. In Chapter1 we give a brief overview of the game theory we need. We restrict attention togames with a �nite number of players each with a �nite number of pure strategies.We mostly consider noncooperative normal form games, although we discuss �nite-horizon extensive form games brie�y and also mention cooperative games.

In Chapter 2 we prove the universality of Nash equilibria. Every real algebraicvariety is isomorphic to the set of totally mixed Nash equilibria of some game with3 players, and also of some game with N players in which each player has two purestrategies. Our proof is constructive. The numbers of pure strategies in the gamewith 3 players, and the number of players in the game with two pure strategies each,are polynomial in the degrees of the equations. Thus the problem of computing Nashequilibria in general is equivalent to the problem of �nding the real roots of a systemof polynomial equations.

In Chapter 3 we prove a theorem computing the number of solutions to a systemof equations which is generic subject to the sparsity conditions embodied in a graph.We apply this theorem to games obeying graphical models and to extensive-form

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games. We de�ne emergent-node tree structures as additional structures which normalform games may have. We apply our theorem to games having such structures. Webrie�y discuss how emergent node tree structures relate to cooperative games.

In Chapter 4 we discuss how to compute all Nash equilibria of a game using com-puter algebra. We �nd that polyhedral homotopy continuation is the most ef�cientavailable method in practice. It also has the advantage of being naturally paralleliz-able. We discuss further directions for developing algebraic algorithms for computingNash equilibria.

Professor Bernd SturmfelsDissertation Committee Chair

i

DedicationTo Sr�la Bhakti Pramod Pur� Maharaj,

by whose grace it was possible.namo gurave namah.

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Contents

List of Figures iii

1 Introduction 11.1 Solving Polynomial Systems . . . . . . . . . . . . . . . . . . . . . . . 21.2 Concepts From Combinatorics . . . . . . . . . . . . . . . . . . . . . 31.3 Game Theory Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Universality of Nash Equilibria 122.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Games and Graphs 343.1 Generic Number of Quasiequilibria . . . . . . . . . . . . . . . . . . . 353.2 Graphical Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Extensive-Form Games . . . . . . . . . . . . . . . . . . . . . . . . . 433.4 Games With Emergent Node Tree Structure . . . . . . . . . . . . . . 52

4 Tools For Computing Nash Equilibria 614.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 The Status Quo: Gambit . . . . . . . . . . . . . . . . . . . . . . . . 624.3 Pure Algebra: Grobner Bases . . . . . . . . . . . . . . . . . . . . . . 644.4 Polyhedral Homotopy Continuation: PHC . . . . . . . . . . . . . . . 664.5 Other Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Bibliography 71

iii

List of Figures

3.1 Graphical game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Associated polynomial graph for graphical game . . . . . . . . . . . . 423.3 An Extensive Form Game . . . . . . . . . . . . . . . . . . . . . . . . 493.4 Associated Polynomial Graph For An Extensive Form Game . . . . . 503.5 Emergent Node Structure For The Saboteur Game . . . . . . . . . . 553.6 Graphical Model For The Saboteur Game . . . . . . . . . . . . . . . 58

iv

Acknowledgements

As I'm about to �le my dissertation, I'm bursting with more gratitude than an Oscar-winning actress, and in the same way I have many more people to thank than Ipossibly can. In the interests of brevity, I can only mention by name the people whohelped me most directly with this dissertation, but there have been many others alongthe way whose friendly support I appreciate. I thank them all.

I thank INFORMS, the Institute for Operations Research and the ManagementSciences, for their permission to present material similar to [Dat03a] (for which theircopyright is hereby noted) as Chapter 2 of this dissertation, and the ACM, the Asso-ciation for Computing Machinery, for their permission to present much of [Dat03b](for which their copyright is hereby noted) as Chapter 4 of this dissertation.

Firstly I must thank Professor Michael Aschbacher, who introduced me to thebeautiful world of abstract algebra in his crystal-clear lectures and who was my un-dergraduate adviser at Caltech. Towards the end of my senior year, he met with mein his of�ce and encouraged me to pursue a career in mathematics. His words havehelped sustain me through my doctoral studies, and I am sincerely grateful.

Next I must sincerely thank Professor Robion Kirby who, as Graduate Vice Chair,encouraged me to come to Berkeley. (I also remember his differential topology classwith great fondness!) Many people in the Berkeley mathematics department havehelped create a friendlier environment. In particular, among my fellow graduatestudents I would like to thank Alex Gottlieb for being such a great of�cemate andBenson Farb for making exam grading go faster; among the faculty I would like tothank Professor George Bergman for frequently asking how I was doing; and amongthe staff I would like to thank Dave Hernes, the building manager, and Janet Yonan,Marsha Snow, Catalina Cordoba and Thomas Brown in the graduate of�ce for theirgreat service. Really there are too many people to list; I wish I could thank them all.

I would like to acknowledge support by National Need Fellowships from the U.S.Department of Education, and I would like to thank Dr. Robert Sonderegger andSRC Systems Inc. for their �nancial support, during portions of my doctoral studies.

I was a Graduate Student Instructor for two years, and I would like to thank the

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professors, my fellow GSIs, and the students in the classes I taught, from all of whomI learned a lot.

I would especially like to thank Professor Ken Ribet. As Graduate Vice Chair, hisencouragement was extremely helpful at a crucial juncture, and I am deeply grateful.I would also like to thank Professor David Eisenbud for the wonderful course hetaught, along with Greg Smith, from his book on commutative algebra.

I would like to thank Professor Pravin Varaiya in the Electrical Engineering de-partment as well as Duke Lee and the rest of the gang with whom I worked on theWireless Token Ring Project. We had a great time! I also would like to thank Profes-sor Raja Sengupta in the Civil & Transportation Engineering department and the restof the Systems Theoretical & Applied Research group, as well as the Mixed InitiativeControl for Automata project. I really enjoyed learning about and working in systemstheory, and I hope to return to the subject in the very near future. I received supportfrom both these projects during my doctoral studies.

I would like to thank Professor Laurent El Ghaoui for his great class on convexoptimization (and especially for inviting me to take one class period to give a pre-sentation on polynomial optimization), for many friendly discussions of interestingproblems, and especially for being on my dissertation committee (about which morebelow).

I would like to thank Professor Mark Haiman. He discussed beautiful mathemat-ical problems with me and encouraged me at a critical time. He was also on myqualifying exam committee.

I would like to thank Professor Alberto Grunbaum. He has known me throughoutmy doctoral studies, from my �rst year when I took his differential equations class, andhis sincere good wishes have meant a lot to me. He has also supported me �nancially.It was partially through his support that I purchased the laptop, described in Chapter4, which I used in researching and writing this dissertation. He also supported meduring the research that led to Chapter 2 and my paper [Dat03a]. Finally I thank himfor being on my dissertation committee.

I would like to thank Professor Reinhard Laubenbacher of the Virginia Bioin-formatics Institute at Virginia Tech. We've had many interesting discussions about

vi

computational biology. He also supported me during the research that led to Chapter4 and my paper [Dat03b]. I would also like to thank Professor Richard Fateman in theComputer Science Department here at Berkeley, my advisor for my master's degreein computer science. It was he who suggested I submit the paper [Dat03b] to ISSAC2003, and he supported my attendance there to present it.

I would like to thank Professor Lior Pachter. I enjoyed the research seminar oncomputational biology he ran with Professor Dick Karp. We've had many interestingdiscussions and he's been a great friend. I'm also learning a lot in the seminar on theMathematics of Phylogenetic Trees he's running now with Bernd. Lior supported meduring the research that led to Chapter 3.

I would like to thank my mathematical siblings. Of those who came before me, Iknow Rekha Thomas, John Dalbec, Jesus De Loera, Serkan Hosten, Diane Maclagan,Ezra Miller, Laura Matusevich, Amit Khetan, and Mike Develin. I've interactedwith them to varying degrees, and their friendly advice, encouragement, and help inmany substantial ways has meant a lot to me. I especially remember a conversationwith Ezra during the banquet at the AMS meeting, Mathematical Challenges ofthe 21st Century, at UCLA in 2000 (before he became my mathematical sibling!),which had a critical in�uence on my subsequent path. Serkan suggested I submitmy paper [Dat03a] to Mathematics of Operations Research, and Jesus suggested thelectureship at UC Davis where I'm going next. I would also like to thank myyounger mathematical siblings, from whom I've learned a lot: my of�cemate SethSullivant, Chris Hillar, Dave Speyer, Nick Eriksson, and Josephine Yu, as well as mymathematical �nieces� Maya Ahmed and Ruriko Yoshida. I'm privileged to be intheir company, and I'm sure there are many more to come!

At last I would like to thank my thesis advisor, Professor Bernd Sturmfels. Hetook a chance on me and I will always strive to justify his faith. Bernd has helped mein more ways than it ever even occurred to me to expect. He has completely changedmy idea (and set the gold standard) of what a thesis advisor should be, and I think Iappreciate him all the more deeply for that reason. How can I possibly thank Berndenough?

Bernd �rst agreed to let me work with him in the fall of 2001. Shortly afterward,

vii

at a post-seminar dinner, he regaled us with the tale of his recent visit to NewYork. His friend, Professor Dave Bayer of Columbia, was then acting as mathematicaladvisor during the making of the �lm A Beautiful Mind. Dave had taken Bernd to visitthe set, where he had met Russell Crowe and Ron Howard. I was duly awed. (ThusDave was responsible for introducing Bernd, and indirectly me, to game theory, forwhich I must thank him!)

One Saturday morning in late March of 2002, Bernd emailed me very excitedly.Could I go to campus and pick up the copy of John Nash's 1951 paper �Non-cooperative games� he had put in my cubbyhole? Bernd was in the midst of preparinga series of lectures on solving systems of polynomial equations, in which he wantedto include game theory, and he asked me to write up Nash's somewhat terse exampleof the Three-Man Poker Game. As Bernd said, �First the book, then the movie, nowthe Grobner bases!� This is just one example of the energy and enthusiasm Berndbrings to all of his myriad mathematical interests, and conveys to those around him.

Shortly afterwards we met with Professor El Ghaoui and his student Aswin Gane-san, who were also studying game theory. Professor Emeritus David Gale graciouslyagreed to meet with all of us, and we had a very interesting discussion, for which Ithank him.

At this time Bernd suggested to me the problem of proving the universality the-orem, the main subject of Chapter 2 and [Dat03a]. Bernd presented his lectures onpolynomial equations in a course (one of several great courses I took from him),and later that spring at Texas A&M (as part of the Conference Board MathematicalSeries), where he took me along. That summer Bernd gave me the privilege of help-ing him with the resulting book [Stu02]. (This partially funded the purchase of theabovementioned laptop.)

I showed an early draft of Chapter 2 to Professor Andrew Maclennan in theeconomics department at the University of Minnesota. His enthusiastic receptionwas extremely encouraging. He suggested that I apply Nash's beautiful theorem of[Nas52]. During and after my visit with him that September, we had very valuable andextensive discussions on the presentation of [Dat03a]. I thank him for his generoushelp.

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Bernd also suggested that I experiment with computing Nash equilibria, whichled to Chapter 4. I would like to express my gratitude to Andrew McLennan and TedTurocy, Mike Stillman, Gert-Martin Greuel, and Jan Verschelde for generously takingthe time to personally discuss with me the use of their software packages Gambit[MMT], Macaulay2 [GS], Singular [GPS01], and PHC [Ver99] respectively.

Bernd also suggested that I prove a Bernstein bound on the number of totallymixed Nash equilibria of graphical games, which led to Chapter 3. During thefall of 2003 I have also sometimes attended Professor Shachar Kariv's course onnoncooperative game theory in the economics department, which led me to includethe treatment of extensive-form games in Chapter 3. Shachar also very generouslyagreed to take time out of his schedule to read the draft of my dissertation, for whichI sincerely thank him.

Finally I must thank my family: my mother Debbie Datta, my father Robin Datta,and my sister Saheli S. R. Datta, for their unwavering support, morally, �nancially,physically, spiritually, and in every way through all these years. They have neverdoubted that I had a contribution to make as a mathematician. Without their faithand love I could never have completed my studies.

Most of all I must thank my Gurudev, my spiritual teacher, Sr�la Bhakti PramodPur� Maharaj. I feel his grace upon me every day of my life, which makes possibleeverything that I achieve.Colophon. This dissertation was written almost entirely with free software (themajor exception being the commercial fonts). I have to thank the many peopleresponsible for Debian Gnu/Linux (including TEX tools); Bram Moolenaar for Vim;Donald Knuth for TEX (I'm a big fan of the TeXbook!) and Leslie Lamport forLATEX; and the former Berkeley graduate students who created the ucthesis style,for making typesetting this dissertation as painless as possible. I especially must thankAlan Hoenig for his beautiful book TEX Unbound and the vfinst and mathinstfont utilities. I used mathinst to make a mathematical font family of MonotypeBembo Semibold (from Agfa-Monotype), MathTime (from Y&Y), Chantilly (fromSoftmaker, similar to Gill Sans), Typewriter (from the Electronic Font Foundry), anda few others, with which I typeset this dissertation.

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Chapter 1

Introduction

In 1654, a gambler, the Chevalier de Mere, posed two questions on games. Inthe ensuing exchange of letters between Blaise Pascal and Pierre Fermat, probabilitytheory�or as Pascal more evocatively called it, la geometrie du hasard (�the geometryof chance�)�was born [Pas63]. Returning to the roots of probability theory, EmileBorel turned his attention to games, and between 1921 and 1927 wrote a seminalseries of papers in which he introduced the abstract formalism which we now callgame theory [Bor21], [Bor24], [Bor27].

Decision theory, a special case of game theory, was used by Abraham Wald tocreate a �rm foundation for mathematical statistics [Wal50], as described by Black-well and Girshick [BG54]. Game theory also provides a unifying framework forapproaching arti�cial intelligence [Wel95]. Since John von Neumann and OskarMorgenstern published their in�uential Theory of Games and Economic Behavior, gametheory has taken on an increasingly central role in mathematical economics; indeedin his textbook [Kre90] David Kreps describes microeconomics through the lens ofgame theory. Maynard Smith applied game theory to evolutionary biology to createevolutionary game theory [Smi68]. Thus, game theory is a powerful abstraction whichilluminates many �elds.

So what is game theory? We summarize the concepts we need below, but �rst wewill very brie�y go over a few concepts from algebra and combinatorics that we willneed.

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1.1 Solving Polynomial Systems

A monomial in n variables x1; : : : ; xn is an expression of the form xd11 : : : xdn

n for somenonnegative integers d1; : : : ; dn. It is squarefree if di � 1 for all i. The degree in xi of thismonomial is di, and its total degree is

PniD1 di. The product of two monomials xd1

1 : : : xdnn

and xe11 : : : xen

n is xd1Ce11 : : : xdnCen

n .A polynomial in n variables x1; : : : ; xn with coef�cients in a �eld K is a �nite sum

of terms. Each term is of the form cm for some c 2 K and some monomial m inx1; : : : ; xn. The set of polynomials in n variables is a vector space whose basis is theset of monomials in n variables. Since we can also multiply monomials together, wecan extend this to de�ne a product on the space of polynomials, using commutativity,associativity, and distributivity. This makes the set of polynomials in n variables withcoef�cients in K into a commutative ring, and the study of such objects is the subjectof commutative algebra.

We can evaluate a polynomial f DP cd1:::dnxd11 � � � xdn

n at a point .a1; : : : ; an/ 2 Kn bysubstituting a1; : : : ; an for the variables x1; : : : ; xn respectively, obtaining an expressionP

cd1:::dnad11 � � � adn

n and carrying out all the multiplications and additions in the �eld K .We denote the resulting element of K as f .a1; : : : ; an/.

A polynomial equation is an expression f D g for some polynomials f and g. Apoint x 2 Kn satis�es this equation if f .x/ D g.x/. Since this is equivalent tof .x/ � g.x/ D 0, we can always write a polynomial equation as p D 0 for somepolynomial p. A point satisfying this equation is called a root of p. A polynomial inone variable with coef�cients in K need not have any roots in K. But there is alwaysa �eld containing K , called the algebraic closure NK of K, such that every nonconstantunivariate polynomial with coef�cients in NK has a root in NK. The �eld of complexnumbers is the algebraic closure of the �eld of real numbers.

A system of polynomial equations in n variables over K is a �nite set of polynomialequations in n variables over K, and a point in Kn satis�es, or is a root of, thissystem if it satis�es all the constituent equations. The study of solution sets ofpolynomial equations is called algebraic geometry. Recent years have seen a renaissancein computational algebraic geometry; the interested reader is referred for example to

3

[CLO97].

1.2 Concepts From Combinatorics

A graph is given by a set of elements, called nodes or vertices, together with a set ofpairs of nodes, called edges. We will only consider �nite graphs, that is, graphs witha �nite set of vertices. If the edges are ordered pairs, we say the graph is directed,and each edge goes from its �rst element (or source, or tail) to its second element (ortarget, or head). If the edges are unordered pairs (i.e., sets of two nodes), we say thegraph is undirected, and each edge goes between its elements. We will only considerdirected graphs. An edge from a node to itself is called a self-loop. The graphs wewill consider will all be simple, i.e., they will have no self-loops and will not havemultiple edges in the same direction between the same pair of nodes. A path in agraph is a sequence �0�1 : : : �n of nodes such that .�i; �iC1/ is an edge in the graph foreach i D 0; : : : ; n � 1. We say the path goes from �0 to �n, or between �0 and �n.

A (rooted) tree is a special kind of directed graph. It has a distinguished node calledthe root, and its edges are called branches. No edge goes to the root. Every other nodev has a unique edge going to it. The node w at the tail of this edge is called v's parent,and v is said to be w's child. For each node v, there is a unique path in the tree fromthe root to v. We can �nd this path by going backward and �nding v's parent, thenthe parent of its parent, and so forth, until we reach the node with no parent�theroot. A node with no children is called a leaf. If there is a (directed) path in the treefrom a node v to a node w, we say v is an ancestor of w and w is a descendant of v.Each node is an ancestor and a descendant of itself, and also a descendant of the root.For each node v in a tree, the subtree below v has the descendants of v as vertices andthe same edges between these vertices as were in the original tree. The node v itselfis the root of this subtree.

A polytope is the set of solutions to some system of (nonstrict) linear inequalities inEuclidean space. If this system is irredundant (that is, no inequalities can be droppedwithout changing the polytope) and there are points in the polytope which satisfyone of the inequalities strictly, then the set of points in the polytope where it holds

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with equality is called a facet of the polytope. One can consider facets of the facets,facets of the facets of the facets, and so forth; all of these are called faces. Each suchdescending chain of faces, where each face is a facet of the previous one, eventuallyterminates in a face which is a single point, or vertex.

Given a �nite set of points v0; : : : ; vd in a real vector space such that v1�v0; : : : ; vd�v0 are linearly independent, the simplex with those points as vertices is the set(

nXiD0

�ivi : 0 < �i 2 R for i D 0; : : : ; n andPn

iD0 �i D 1):

The (d-dimensional) probability simplex over a �nite set of events v0; : : : ; vd is obtainedby identifying v0 with the origin and v1; : : : ; vd with the unit coordinate vectors in Rd.It is the polytope f.�1; : : : ; �d/ 2 Rd : �i � 0 for all i and

PdiD1 �d � 1g.

The notion of the determinant of a matrix M is familiar from linear algebra: it isthe sum of certain signed products of entries of M. We will also need the notion ofthe permanent of M. It is the sum of the same products of entries of M, but withoutthe additional sign factors.

1.3 Game Theory Concepts

The concepts we describe in this section can be found in a standard game theory textsuch as [OR94]. However, in some cases we use simpli�ed notation for the restrictedsituations we will consider.

Game theory is the study of strategic interaction. Such interaction takes placebetween multiple agents in a single setting, or environment. An agent is an entitywhich can receive information about the state of the environment (including itself andother agents), take actions which may alter that state, and express preferences amongthe various possible states. These preferences are encoded for each agent by a utilityfunction, a mapping from the set of all states to R. Its value for a particular state isthe utility of that state for the agent. The agent prefers one state to another if itsutility is greater, and is indifferent between them if their utilities are equal.1 Changes

1Instead of specifying the utility of each state for each agent, one might specify the change in utility,

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in the state of the environment may also occur spontaneously (i.e., not due to theactions of any of the agents). A strategy is a (possibly stochastic) rule for an agentto choose an action at every point when the agent may act, given the availableinformation. A rational agent is one whose strategy maximizes its expected utilityunder the circumstances.

We will restrict attention to games which take place in a �nite number of timesteps between a �nite number of agents, each of which has a �nite number of possibleactions. The agents are called players, and whenever they take an action they are saidto move. A spontaneous change in the state of the environment is called a move bynature. The game is over when no player (including nature) has any possible actions.The state of the environment at such a terminal stage is called an outcome. Generallypreferences are speci�ed only over outcomes, not at intermediate stages of the game.

The �rst type of game we will consider is the normal-form game. In a normal-formgame, there is only one time step, at which all the players move simultaneously. Wedenote the set of players by I D f1; : : : ;Ng. The actions a player can take are calledpure strategies. (Thus a pure strategy is the simplest possible rule: it always choosesone particular action.) We associate to the players �nite disjoint sets of pure strategiesS1; : : : ;SN . For each i let di D jSij � 1. We write the set Si as fsi0; : : : ; sidi g. Wewrite S D Qi2I Si. Game play consists of the collective choice of an element of S bythe players: each player i moves by choosing an element of Si. We identify S as theset of possible outcomes. We denote by ui.s/ the utility for player i of the outcomes 2 S. Thus, the game is completely speci�ed by the number N of players, the sets Si

of pure strategies, and the utility functions (or payoff functions) ui : S 7! R.A player may move stochastically rather than deterministically. In that case the

player is said to execute a mixed strategy. The mixed strategy speci�es the probabilitywith which the player chooses each possible action. The set 6i of mixed strategies

or marginal utility, which accrues to each agent upon each transition between states. Clearly any utilityfunction induces a marginal utility function, but unless one imposes additional conditions a marginalutility function may not induce a utility function. Such a marginal utility function, which one mightcall intransitive, could still be a useful model of reality. For example, one wouldn't necessarily feel thesame about being laid off and then immediately rehired as if one had simply continued in the sameposition. However, we will not consider such intransitive marginal utility functions any further.

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of player i is the set of all functions �i : Si 7! [0; 1] withP

si j2Si�i.si j/ D 1. That

is, it is the di-dimensional probability simplex. We write 6 D Qi2I 6i. An element

� of 6, which speci�es the strategies executed by all the players, is called a strategypro�le. If the players execute the strategy pro�le � , then the probability of outcome sis �.s/ D QN

iD1 �i.si/. The expected utility for player i of the strategy pro�le � is givenby multilinearity as ui. � / DPs2S ui.s/ � .s/.

When considering how agent i should behave, it will be convenient to separateout i's own strategy, over which i has control, from the strategies of all the otherplayers. We write 6�i D Q j2I�fig6 j, and we write ��i for the image of � 2 6 underthe projection ��i from 6 onto 6�i. By abuse of notation, we write ui.�i; ��i/ forthe ith player's expected payoff from the strategy � whose ith component is �i andwhose other components are de�ned by ��i. � / D ��i.

We assume perfect information: each player knows the complete speci�cation ofthe game, knows that every player knows, knows that every player knows that everyplayer knows, ad in�nitum. That is, the speci�cation of the game is common knowledge.Under these circumstances, what is rational behavior? In his landmark paper [Nas50],John Nash answered this question in terms of what is now called best response. Abest response of player i to the strategy pro�le � is a mixed strategy � �i such thatui. � �i ; ��i/ � ui. � 0i ; ��i/ for any other mixed strategy � 0i of player i. That is, giventhat all the other players execute the strategy pro�le ��i, the mixed strategy � �imaximizes player i's expected utility. A Nash equilibrium is a strategy pro�le which isa best response to itself for all the players. That is, it is a strategy pro�le � � such thatfor each player i, we have ui. � �/ � ui. � 0i ; � ��i/ for every other mixed strategy � 0i ofplayer i. Nash proved that such an equilibrium always exists.

How can we compute the Nash equilibria of a given game? We need to search theset 6 of strategy pro�les, which is a polytope: the product of probability simplices.We can decompose the problem by stratifying this polytope: �rst we look for Nashequilibria in its interior, then in the interiors of its facets, then in the interiors of thefacets of those facets, and so forth, until �nally we look for Nash equilibria at thevertices of the polytope (that is, pure strategy Nash equilibria). A strategy pro�le �

lies in the interior of this polytope if �i.si j/ > 0 for every si j 2 Si, for every i. Such a

7

strategy pro�le is called totally mixed. Note that a totally mixed Nash equilibrium neednot exist.

Suppose we had a procedure to �nd the totally mixed Nash equilibria, that is,those lying in the interior of the polytope, for any game. Then we could also �ndthe Nash equilibria lying in the interior of any facet of the polytope. Such a facet isde�ned by an equation �i.si j/ D 0 for some player i and some pure strategy si j of thatplayer. We apply our procedure to �nd the totally mixed Nash equilibria of the gameinduced by restricting to the pure strategy sets S1; : : : ;Si � fsi jg; : : : ;SN . For eachstrategy pro�le � we obtain in this way, we check whether ui. �i; ��i/ � ui.si j; ��i/;if so, then � is a Nash equilibrium of the original game. Thus, once we have thisprocedure we can �nd all Nash equilibria by a divide-and-conquer approach. Ofcourse, here we have not dealt with ef�ciency questions, but only the existence of a�nite algorithm. Indeed, even when the players only have two pure strategies each,just the number of vertices of the polytope, 2N , is exponential in the number Nof players, let alone the number of faces of all dimensions. Thus the combinatorialexplosion keeps this algorithm from being tractable.

So we concentrate our attention on the totally mixed Nash equilibria. We observethat for a totally mixed strategy pro�le � to be a Nash equilibrium, it is necessaryand suf�cient that for each player i we have ui.si j; ��i/ D ui.sik; ��i/ for any purestrategies si j; sik 2 6i. These equations are called the indifference equations for player i.The suf�ciency is clear. For the necessity, suppose to the contrary that ui.si j; ��i/ >

ui.sik; ��i/. De�ne � 0i by

� 0i .sil/ D

8>>>><>>>>:�i.si j/ C �i.sik/; l D j

0; l D k

�i.sil/; otherwise

:

Then since �i.sik/ > 0, we have ui. � 0i ; ��i/ D ui. � /C �i.sik/�ui.si j; ��i/ � ui.sik; ��i/

�>

ui. � /, a contradiction.So we have a system of

PNiD1 di equations, ui.si j; ��i/ D ui.si0; ��i/ for j D

1; : : : ; di, for i D 1; : : : ;N, inPN

iD1 di unknowns �i.si j/ for j D 1; : : : ; di, for i D

8

1; : : : ;N. (Here we have dehomogenized, that is, we have eliminated �i.si0/ by sub-stituting 1 �Pdi

jD1 �i.si j/). What we are equating are the expressions ui.si j; ��i/ DPs�i2S�i

ui.si j; s�i/ �1.s1/ � � � �i�1.si�1/ �iC1.siC1/ � � � �N.sN /, which are multilinear poly-nomials whose coef�cients are the real numbers ui.s/. The (possibly complex) rootsof this system are called quasiequilibria, and those roots which are totally mixed strat-egy pro�les (that is, which are real with �i.si j/ > 0 and

PdijD1 �i.si j/ < 1) are the

totally mixed Nash equilibria. The study of systems of polynomial equations is the�eld of algebraic geometry, and the subject of this dissertation is algebraic methods ingame theory.

If si j and sik are pure strategies of player i, then si j strictly dominates sik if for alls�i 2 S�i, we have ui.si j; s�i/ > ui.sik; s�i/. (Then of course no totally mixed Nashequilibria can exist in the game as speci�ed.) In this case player i, being rational,should never play sik, so we might as well eliminate it from Si. After we do this somemore pure strategies of some other players may become strictly dominated, and so weiterate. We will always assume that we have already carried out iterated elimination ofstrictly dominated strategies on any game under consideration.

We will also brie�y consider �nite horizon extensive-form games. (See for example[OR94], Chapter 6.) Such a game takes place in a �nite number of time steps, ateach of which only a single player (possibly nature) may move. (Which player moves,and what actions the player is allowed to take, may depend on what moves weremade previously.) Such a game is completely speci�ed as follows. We specify a set ofplayers I D f1; : : : ;Ng, and we specify a game tree T : a �nitely branching tree of �nitedepth in which each non-leaf node is labelled by a number in 0; : : : ;N, each leaf islabelled by an N-tuple of real numbers, and each branch emanating from a (non-leaf)node labelled 0 is assigned a positive real weight, so that the total weight emanatingfrom such a node is 1. (We consider the branches of this tree to be directed awayfrom the root.)

Game play proceeds as follows. Each node of the tree represents a state of theenvironment. At each time step, if we are at a non-leaf node labelled by i in 1; : : : ;N,then player i acts by choosing one of the branches emanating from that node. Thenthe environment undergoes the transition to the node at the end of that branch, and

9

we advance to the next time step. If we are at a non-leaf node labelled by 0, then theenvironment instead makes a random transition along one of the branches emanatingfrom that node; the probability of each branch is given by its weight. If we are at aleaf node � labelled by .u1.�/; : : : ; uN.�//, then the game is over, and each player iaccrues utility ui.�/. Thus, the leaf nodes are the outcomes of the game.

Unless otherwise speci�ed, we will assume perfect information. Not only do allplayers have common knowledge of the speci�cation of the game, but whenever aplayer is about to move, that player knows what moves have been made by all theother players (including nature) up to that point.

Every extensive-form game is equivalent to a normal-form game. For each node� of the game tree, we write E.�/ for the set of edges emanating from �. Then theset of pure strategies of player i is

Si DY�2T

label.�/Di

E.�/:

Given a pure strategy pro�le s 2 S D Qi2I Si, we can compute the probability of eachleaf node � of the game tree. A unique path �0�1 : : : �m D � leads from the root �0 ofT to �. Then Pr[�js] D Qm�1

jD0 Pr[� j ! � jC1js], where

Pr[� j ! � jC1js] D

8>>>><>>>>:1; � j is labelled by i 2 I and si� j D .� j ! � jC1/

0; � j is labelled by i 2 I and si� j 6D .� j ! � jC1/

wt.� j ! � jC1/; � j is labelled by 0

and so the utility functions of the normal-form game are given by

ui.s/ DX�2T�leaf

ui.�/ Pr[�js]:

We note that the game speci�cation implies certain equalities among the numbersui.s/. If we consider the set of normal-form games with a �xed set of players I andoutcomes S to be a linear space with basis fui.s/ : s 2 Sg, then the extensive-formgames with the same set of players I and a �xed game tree having S as the set ofoutcomes lie in a linear subspace of this space, given by these equalities. Let A be the

10

set of non-leaf nodes of the tree which are not labelled by 0. Then we can identify Swith

Q�2A E.�/. For any s 2 S and � 2 A, we write s� D si�, where i is the label of

�. Suppose � 2 A is an ancestor of � 2 A. Then � has a unique child � that is alsoan ancestor of � (possibly � itself). Let � be any other child of �. If s; s0 2 A withs� D .� ! �/ and

s0� D8><>:e; � D �

s�; otherwise

for some edge e 2 E.�/, then ui.s/ D ui.s0/. This is because Pr[�js] D Pr[�js0] D 0unless � is a descendant of � or � is not a descendant of �, and in either case �

cannot be a descendant of �. In short, the node � is never reached, so it doesn'tmatter which action is chosen there.

If different players act at � and �, then there is no way to eliminate this redun-dancy, but when the same player i acts at � and �, we can do so. In this case wereplace all the pure strategies which are forced to be equal by a single pure strategy,called a reduced pure strategy. See for example [OR94], p. 94.

We note that after iterated elimination of strictly dominated pure strategies, forany node all of whose children are leaves, the payoffs to the player who acts atthat node must be equal at all these child leaves. If nature acts at such a node �

whose children are leaves �1; : : : ; �k, then we can replace � by a leaf with utilitiesui.�/ DPk

lD1 wt.� ! �l/ui.�/ for each i 2 I. So we assume nature never acts at suchnodes.

For extensive-form games, the equilibrium concept can be re�ned. Each subtreeof the game tree induces a new extensive-form game, called a subgame. Each purestrategy of the original game induces a pure strategy of each subgame by restrictionto that subtree, and thus each strategy pro�le of the original game induces a strategypro�le of each subgame. A strategy pro�le is a subgame perfect Nash equilibrium of anextensive-form game if it induces a Nash equilibrium of each subgame.

We can �nd a subgame perfect pure strategy Nash equilibrium by backwards in-duction. We construct the pure strategy pro�le as follows. We perform iteratedelimination of strictly dominated strategies. Then at each node all of whose children

11

are leaves, we choose one leaf (recall that the payoffs of all leaves for the player whoacts at that node will be the same). We assign this branch to the correspondingcomponent of the pure strategy pro�le, replace this node by this leaf, and repeat theprocedure on the resulting subtree.

So far we have been discussing noncooperative game theory. Cooperative gametheory deals with groups, or coalitions of players. When a coalition forms, the playersmaking up the coalition draw up an agreement binding themselves to act so as tomaximize the gain of the coalition. This gain is then split up among the constituentsin accordance with the agreement.

1.4 Results

In Chapter 2 we prove the universality of Nash equilibria. Every real algebraic varietyis isomorphic to the set of totally mixed Nash equilibria of some game with 3 players,and also of some game with N players in which each player has two pure strategies.Our proof is constructive. The numbers of pure strategies in the game with 3 players,and the number of players in the game with two pure strategies each, are polynomialin the degrees of the equations. Thus the problem of computing Nash equilibria ingeneral is equivalent to the problem of �nding the real roots of a system of polynomialequations.

In Chapter 3 we prove a theorem computing the number of solutions to a systemof equations which is generic subject to the sparsity conditions embodied in a graph.We apply this theorem to games obeying graphical models and to extensive-formgames. We de�ne emergent-node tree structures as additional structures which normalform games may have. We apply our theorem to games having such structures. Webrie�y discuss how emergent node tree structures relate to cooperative games.

In Chapter 4 we discuss how to compute all Nash equilibria of a game using com-puter algebra. We �nd that polyhedral homotopy continuation is the most ef�cientavailable method in practice. It also has the advantage of being naturally paralleliz-able. We discuss further directions for developing algebraic algorithms for computingNash equilibria.

12

Chapter 2

Universality of Nash Equilibria

2.1 Introduction

We consider the set of Nash equilibria of an N-person normal form noncooperativegame with perfect information, viewed as the set of solutions to a �nite system ofpolynomial equations and inequalities. The unknowns in this system are the com-ponents of the mixed strategy selected by each player. A set of real points given bya system of polynomial equations and inequalities is called a semialgebraic variety, andthe special case when the system does not involve inequalities is called a real algebraicvariety. Thus the set of Nash equilibria of a game is a semialgebraic variety.

The generic �niteness result of Harsanyi [Har73] states that for each assignmentof generic payoffs to the normal form of a game, the number of Nash equilibria is�nite and odd. In fact, McKelvey and McLennan [MM97] have computed the exactmaximal number of totally mixed Nash equilibria in the generic case. Our results arecomplementary; they describe how complex the non-generic case can be. We showthat every real algebraic variety is isomorphic (in a sense to be speci�ed) to the set oftotally mixed Nash equilibria of some game:

Theorem 1. Every real algebraic variety is isomorphic to the set of totally mixed Nashequilibria of a 3-person game, and also of an N-person game in which each player has two purestrategies.

13

The theorem of Nash [Nas52] and Tognoli [Tog73] states that every compactdifferentiable manifold is diffeomorphic to some (nonsingular) real algebraic variety.So, since the isomorphism above is also a diffeomorphism when the variety is non-singular, it follows from our result that for any compact differentiable manifold M,there is some game whose set of totally mixed Nash equilibria is diffeomorphic toeither M or a tubular neighborhood of M. Similarly, the theorem of Akbulut andKing [AK92] shows that every piecewise linear manifold is homeomorphic to somereal algebraic variety. So for every piecewise linear manifold M, there is some gamewhose set of totally mixed Nash equilibria is homeomorphic to either M or a tubularneighborhood of M.

Theorem 1 derives from the following more speci�c results:

Theorem 2. Let S � Rn be a real algebraic variety given by m polynomial equations inn unknowns x1; : : : ; xn, such that each point .x1; : : : ; xn/ 2 S satis�es 0 < xi < 1 fori D 1; : : : ; n and

PniD1 xi < 1. Let d be the highest power to which any unknown xi occurs in

any of the m equations. Set D D m ..1C d/n � 1/ and N D nd C m.

(a) there is a 3-person game in which the �rst player has n C 1 pure strategies, the secondplayer has D � m C 1 pure strategies, and the third player has D C 1 pure strategies,whose set of totally mixed Nash equilibria is isomorphic to S.

(b) there is an N-person game in which each player has two pure strategies, whose set oftotally mixed Nash equilibria is isomorphic to S.

Theorem 1 will follow since any real algebraic variety is isomorphic to one sat-isfying the hypotheses of Theorem 2. Note that specifying particular values of n, m,and d, and/or giving more detailed information about the form of the equations, mayallow using games of smaller formats (fewer pure strategies in (a), fewer players in(b)). For example,

Theorem 3. Suppose S is the set of those roots of a polynomial of degree d in one unknownwhich are real and lie in the open interval .0; 1/. Then there is a 3-person game in which the�rst player has two pure strategies and the other two players each have dd=2eC1 pure strategies,

14

such that the projection of the set E of totally mixed Nash equilibria of this game onto its �rstcomponent (the probability that the �rst player picks her �rst pure strategy) is S, and #E D #S.

The notion of isomorphism being used in this chapter is that of stable isomorphismin the category of semialgebraic varieties. Semialgebraic varieties are the subject ofstudy in real algebraic geometry. Two semialgebraic varieties are (semialgebraically)isomorphic if there exists a homeomorphism between them whose graph is a semial-gebraic set. They are stably isomorphic if they are in the same equivalence class underthe equivalence relation generated by semialgebraic isomorphisms and the (canonical)projections V � Rk ! V for any k. Intuitively, the word �stable� here means that weare allowed to thicken the objects before mapping them isomorphically to each other.

The result in this chapter is one of a series of �universality theorems� in combi-natorics. Another example is the theorem of Richter-Gebert and Ziegler [RGZ95],that realization spaces of 4-polytopes are universal. A polytope has a combinatorialdescription as a collection of faces of smaller dimensions, together with the inclusionsbetween them (which vertices lie in which edges, etc.) The realization space of thepolytope is the set of all geometric polytopes for a given combinatorial polytope, andthe result states that an arbitrary primary semialgebraic set is stably equivalent to therealization space of some 4-polytope. Other universality theorems were proved byMnev [Mne88] and Shor [Sho91].

2.2 Preliminaries

At a totally mixed Nash equilibrium, for any given player, if the other players' mixedstrategies are kept �xed then the payoffs at each of that player's pure strategies areequal. These conditions can be expressed as a system of polynomial equations andinequalities. To simplify notation, we now give the systems for the two cases dealtwith in this chapter: a 3-person game, and an N-person game in which each playerhas two pure strategies.

The players in the 3-person game will be called Alice, Bob, and Critter. Theyhave da C 1, db C 1, and dc C 1 pure strategies respectively. Their payoff matrices

15

will be denoted .Ai jk/, .Bi jk/, and .Ci jk/ respectively, where Xi jk indicates the payoffreceived by player X when Alice chooses her ith pure strategy, Bob chooses his jthpure strategy, and Critter chooses its kth pure strategy.

A mixed strategy of Alice is described by a daC1-tuple of numbers Na D .a0; : : : ; ada /

with 0 � ai for all i and a0 C � � � C ada D 1. Here ai signi�es the fraction player A allo-cates to her ith pure strategy to make up this mixed strategy. Similarly, Nb D .b0; : : : ; bdb /

and Nc D .c0; : : : ; cdc / denote mixed strategies of Bob and Critter, respectively. Now theexpected payoff received by Alice for a particular choice . Na; Nb; Nc/ of mixed strategies is

daXiD0

dbXjD0

dcXkD0

Ai jkaib jck;

and similarly for the other two players. A totally mixed strategy . Na; Nb; Nc/ is one in whichai > 0, b j > 0, and ck > 0 for all i, j, and k.

The condition that . Na; Nb; Nc/ be a totally mixed Nash equilibrium is precisely thesystem of equations

dbXjD0

dcXkD0

Ai jkb jck DdbXjD0

dcXkD0

A0 jkb jck; for i D 1; : : : ; da; (2.1)

daXiD0

dcXkD0

Bi jkaick DdaX

iD0

dcXkD0

Bi0kaick; for j D 1; : : : ; db; (2.2)

daXiD0

dbXjD0

Ci jkaib j DdaX

iD0

dbXjD0

Ci j0aib j; for k D 1; : : : ; dc. (2.3)

together with the inequalities ai > 0, b j > 0, and ck > 0 for all i, j, and k.The players in the N-person game will be denoted X1; : : : ;XN . They each have

two strategies. Now a mixed strategy of player Xl is described by a pair of numbers.p.0/l ; p.1/l /, where 0 � p.0/l ; p.1/l � 1 and p.0/l C p.1/l D 1. Here p. j/l signi�es the fractionplayer Xl allocates to her jth pure strategy to make up this mixed strategy. Wewrite pl D p.1/l . For the strategy to be totally mixed, we must have 0 < pl < 1 forl D 1; : : : ;N. The X.l/

i1:::iN entry of the lth player's payoff matrix indicates the payoffreceived by player Xl when player X1 chooses her i1th pure strategy, player X2 chooseshis i2th pure strategy, and so forth. This event has probability p.i1/1 p.i2/2 � � � p.iN /N . Now

16

holding the strategies of the other N�1 players �xed, one can compute the expectedpayoff to player Xl of choosing her �rst pure strategy. For example, for player XN ,this is

X.N/0:::00p

.0/1 � � � p.0/N�2p

.0/N�1 CX.N/

0:::010p.0/1 � � � p.0/N�2p

.1/N�1 C � � � C X.N/

1:::110p.1/1 � � � p.1/N�2p

.1/N�1

D X.N/0:::00.1� p1/ � � � .1� pN�1/ C X.N/

0:::010.1� p1/ � � � .1� pN�2/pN�1 C � � �� � � C X.N/

1:::10p1 � � � pN�1

Observe that this expression is multilinear in the pl's and that pN does not occur(because we are conditioning on the event that XN chooses her �rst pure strategy).At a totally mixed Nash equilibrium, this must equal the expected payoff to player XN

of choosing her second pure strategy. In this way, a totally mixed Nash equilibriumcorresponds to a solution to a system of N multilinear equations, where the lthunknown pl does not occur in the lth equation:

X.1/00:::0.1� p2/ � � � .1� pN / C X.1/

00:::01.1� p2/ � � � .1� pN�1/pN C � � �� � � C X.1/

01:::1p2 � � � pN

D X.1/10:::0.1� p2/ � � � .1� pN / C X.1/

10:::01.1� p2/ � � � .1� pN�1/pN C � � �� � � C X.1/

11:::1p2 � � � pN;

...

X.N/0:::00.1� p1/ � � � .1� pN�1/ C X.N/

0:::010.1� p1/ � � � .1� pN�2/pN�1 C � � �� � � C X.N/

1:::10p1 � � � pN�1

D X.N/0:::01.1� p1/ � � � .1� pN�1/ C X.N/

0:::011.1� p1/ � � � .1� pN�2/pN�1 C � � �� � � C X.N/

1:::11p1 � � � pN�1:

See for example McKelvey and McLennan [MM96] or Sturmfels [Stu02], Chapter 6.

2.3 Examples

We illustrate our method of encoding real varieties with a few numerical examples.In the �rst two examples, the set of totally mixed Nash equilibria is isomorphic to a

17

circle. In the last example, there is a unique totally mixed Nash equilibrium, and itsdegree is 2. The same method allows us to construct a game with a unique totallymixed Nash equilibrium of any given topological degree.

We de�ne a 3-person game in which each player has three pure strategies asfollows. If a player picks their �rst pure strategy, then all that player's payoffs are zero.If Alice picks her second pure strategy, then her payoff matrix is

.A1 jk/ D0BB@

c0 c1 c2b0 0 �1 0b1 1 0 1b2 0 �1 0

1CCA: (2.4)

If she picks her third pure strategy, her payoff is always zero. If Bob picks his secondpure strategy, his payoff matrix is

.Bi1k/ D0BB@

c0 c1 c2a0

�12

�12

12

a1�12

�12

12

a212

12

32

1CCA; (2.5)

and if he picks his third pure strategy, his payoff matrix is

.Bi2k/ D0BB@

c0 c1 c2a0 0 1 0a1 0 1 0a2 0 1 �1

1CCA: (2.6)

If Critter picks its second pure strategy, its payoff matrix is

.Ci j1/ D0BB@

b0 b1 b2

a0�12

�12

12

a112

12

32

a2�12

�12

12

1CCA; (2.7)

18

and if it picks its third pure strategy, its payoff matrix is

.Ci j2/ D0BB@

b0 b1 b2

a0764

�5764

764

a1764

�5764

�5764

a2764

�5764

764

1CCA: (2.8)

Suppose . Na; Nb; Nc/ is a totally mixed Nash equilibrium. Then in particular, Alice's payofffrom her second pure strategy, which from equation (2.4) is

�b0c1 C b1c0 C b1c2 � b2c1 D �.1� b1 � b2/c1 C b1.1� c1 � c2/ C b1c2 � b2c2

D b1 � c1;

must be equal to her payoff from her �rst pure strategy, which is zero. In this way onecan verify that when a totally mixed Nash equilibrium occurs, b1 D c1, a2C c2� 1

2 D 0,c1 � a2c2 D 0, a1 C b2 � 1

2 D 0, and �a1b2 � b1 C 764 D 0. Substituting into the last

equation yieldsa1

�a1 � 1

2

�C a2

�a2 � 1

2

�C 7

64D 0

which can be rewritten as �a1 � 1

4

�2C�a2 � 1

4

�2D 1

64:

This equation describes a circle in the a1a2 plane, which lies completely in the interiorof the coordinate simplex 0 < a1, 0 < a2, a1 C a2 < 1. One can check that for thesevalues of .a1; a2/, the values of .b1; b2/ and .c1; c2/ given by the above equationsalso lie in the interiors of their respective coordinate simplices. So the set of totallymixed Nash equilibria is exactly the set of solutions to the above equations, which isisomorphic to a circle.

We can obtain the same circle as the set of totally mixed Nash equilibria of a5-person game in which each player has two pure strategies. We will call the playersAlice, Bob, Critter, Deus, and Elizabeth, with payoff matrices A, B, C, D, and Erespectively. Alice's mixed strategy is denoted .a0; a1/ D .1 � a; a/, and likewise forthe other players. If a player picks their �rst pure strategy, then all that player's payoffs

19

are zero. If Alice picks her second pure strategy, her payoff depends only on whatBob and Deus do. Her payoff matrix is

.A1i� j�/ D0@

d0 d1

b0 0 �1b1 1 0

1A: (2.9)

If Bob picks his second pure strategy, his payoff depends only on what Alice andCritter do. His payoff matrix is

.Bi1 j��/ D0@

c0 c1a0 0 �1a1 1 0

1A: (2.10)

If Critter picks its second pure strategy, its payoff depends only on what Elizabethdoes. Its payoff matrix is

.C��1�i/ D� e0 e1�1

212

�: (2.11)

Deus's payoff is always zero. Finally, if Elizabeth picks her second pure strategy, herpayoff is given by

Ei jkl1 D

0BBBBB@c0d0 c0d1 c1d0 c1d1

a0b0764

764

764

764

a0b1�2564

3964

�2564

3964

a1b0�2564

�2564

3964

3964

a1b1�5764

764

764

7164

1CCCCCA (2.12)

One can verify that when a totally mixed Nash equilibrium occurs, b�d D 0, a�c D 0,e � 1

2 D 0, and ac � 12aC bd � 1

2b C 764 D 0. Substituting into this last equation yields

a2 � 12aC b2 � 1

2b C 7

64D 0;

the same equation as before. It is clear that for a and b satisfying this equation, thevalues of c, d, and e given by the above equations satisfy 0 < c; d; e < 1. So again theset of totally mixed Nash equilibria is isomorphic to a circle.

20

We next de�ne another 5-person game in which each player has two pure strate-gies. Again, if a player picks their �rst pure strategy, then all that player's payoffs arezero. Alice's, Bob's, and Critter's payoffs are the same as in the previous example.Now Deus's payoff for picking his second pure strategy depends only on what Aliceand Bob do, and his payoff matrix is

.Di j�1�/ D0@

b0 b1

a018

�38

a1�38

98

1A: (2.13)

This time if Elizabeth picks her second pure strategy, her payoff is given by

Ei jkl1 D

0BBBBB@c0d0 c0d1 c1d0 c1d1

a0b0 0 0 0 0a0b1

12

�12

12

�12

a1b0�12

�12

12

12

a1b1 0 �1 1 0

1CCCCCA (2.14)

Now when a totally mixed Nash equilibrium occurs, b�d D 0, a� c D 0, and e� 12 D 0

as before, and one can verify that 2ab � 12a� 1

2b C 18 D 0 and ac� bd � 1

2aC 12b D 0 as

well. The fourth equation may be rewritten as 2.a � 14 /.b � 1

4 / D 0. Substituting the�rst two equations into the �fth yields a2 � 1

2a� b2 C 12b D 0, which may be rewritten

as .a � 14 /

2 � .b � 14 /

2 D 0. Clearly the unique totally mixed Nash equilibrium occurswhen a D 1

4 , b D 14 . Putting x D a � 1

4 and y D b � 14 , we see that the polynomials

de�ning the totally mixed Nash equilibrium are .x2� y2; 2xy/, which is the canonicalexample of a map from the plane to itself of degree 2; considering the plane as havingcomplex coordinate z D x C iy, it is given by z 7! z2. Every map z 7! zn for n 2 Nsimilarly gives rise to a system of two polynomial equations in two variables, andso we can obtain a unique totally mixed Nash equilibrium of any given topologicaldegree.

21

2.4 Proofs

We begin with some simple observations. In the equations (2.1) for Alice to be at atotally mixed Nash equilibrium in the 3-player case, if we substitute b0 D 1�b1�� � ��bdb and c0 D 1 � c1 � � � � � cdc , and subtract each right-hand side from each left-handside, we get da equations

�.i/11b1c1 C �.i/12b1c2 C � � � �.i/1dcb1cdc C �.i/21b2c1 C � � � C �.i/.db /.dc/

bdb cdc

C�.i/10b1 C � � � �.i/.db /0bdb C �.i/01 c1 C � � � C �.i/0.dc/cdc C �.i/00 D 0 .A/

for i D 1; : : : ; da. Similarly we get db equations

�. j/11 a1c1 C �. j/12 a1c2 C � � � �. j/1dca1cdc C �. j/21 a2c1 C � � � C �. j/.da/.dc/

ada cdc

C�. j/10 a1 C � � � �. j/.da/0ada C �. j/01 c1 C � � � C �. j/0.dc/cdc C �. j/00 D 0 .B/

for j D 1; : : : ; db, and dc equations

�.k/11 a1b1 C �.k/12 a1b2 C � � � �.k/1dba1bdb C �.k/21 a2b1 C � � � C �.k/.da/.db /

adabdb

C�.k/10 a1 C � � � �.k/.da/0ada C �.k/01 b1 C � � � C �.k/0.db /bdb C �.k/00 D 0 .C /

for k D 1; : : : ; dc.

Lemma 4. If we are given any arbitrary coef�cients �.i/jk , �. j/ik , and �.k/i j , we can choose payoffmatrices .Ai jk/, .Bi jk/, and .Ci jk/ so that the above equations have the prescribed coef�cients.

Proof. We �rst set each player's payoff equal to zero whenever they choose their �rstpure strategy, no matter what the other players do. Then the equations (2.1) implythat, for example,

PdbjD1Pdc

kD1 A1 jkb jck D 0. Now we show how to obtain the �rst ofthe above equations,

�.1/11 b1c1 C �.1/12 b1c2 C � � � �.1/1dcb1cdc C �.1/21 b2c1 C � � � C �.1/.db /.dc/

bdb cdc

C�.1/10 b1 C � � � �.1/.db /0bdb C �.1/01 c1 C � � � C �.1/0.dc/cdc C �.1/00 D 0;

for any prescribed �.1/jk by appropriately choosing the matrix .A1 jk/; the other equa-tions are obtained completely analogously. First, we set A100 D �.1/00 . Then for j D

22

1; : : : ; db, we set A1 j0 D �.1/j0 C �.1/00 , and for k D 1; : : : ; dc� 1, we set A10k D �.1/0k C �.1/00 .Finally, for j D 1; : : : ; db and k D 1; : : : ; dc, we set A1 jk D �.1/jk C �.1/j0 C �.1/0k C �.1/00 . Thisyields

dbXjD0

dcX0D1

A1 jkb jck D �.1/00

1�

dbXjD1

b j

! 1�

dcXkD1

ck

!C

dbXjD1

��.1/j0 C �.1/00

�b j

1�

dcXkD1

ck

!C.1�

dbXjD1

b j/dcX

kD1.�.1/0k C �.1/00 /ck

CdbXjD1

dcXkD1.�.1/jk C �.1/j0 C �.1/0k C �.1/00 /b jck

D �.1/00 CdbXjD1

�.1/j0 b j CdcX

kD1�.1/0k ck C

dbXjD1

dcXkD1

�.1/jk b jck

as desired.

Similarly, for the N-person game we have N equations, one for each player l,where the lth equation is of the formX

"2f0;1gN�1

�.l/" p"11 � � � p"l�1l�1 p"llC1 � � � p"N�1

N D 0:

Lemma 5. If we are given any arbitrary coef�cients �.l/" , we can choose payoff matrices X.l/i1:::iN

so that the above equations have the prescribed coef�cients.

Proof. As before, we set all players' payoffs from their �rst pure strategies equal tozero, regardless of what the other players do. We show how to obtain the equationX

"2f0;1gN�1

�.N/" p"11 � � � p"N�1N�1 D 0

for any given �.N/" by appropriately choosing payoff X.N/i1:::iN�11; again, the other equa-

tions are obtained analogously. We set

X.N/i1:::iN�11 D

i1X"1D0

i2X"2D0� � �

iN�1X"N�1D0

�.N/" :

23

We show by induction on N that

X.N/0:::01.1� p1/ � � � .1� pN�1/ C � � �X.N/

1:::11p1 � � � pN�1 DX

"2f0;1gN�1

�.N/" p"11 � � � p"N�1N�1

For N D 2, we have

X.2/01 .1� p1/ C X.2/

11 p1 D �.2/0 .1� p1/ C .�.2/0 C �.2/1 /p1 D �.2/0 C �.2/1 p1

as desired. Now suppose we have shown the identity for N � 1. We observe that

X.N/i1:::iN�201 D

i1X"1D0

i2X"2D0� � �

iN�2X"N�2D0

�.N/"1:::"N�20:

and

X.N/i1:::iN�211 D

i1X"1D0

i2X"2D0� � �

iN�2X"N�2D0

.�.N/"1:::"N�20 C �.N/"1:::"N�21/

Then

X.N/0:::01.1� p1/ � � � .1� pN�1/ C � � �X.N/

1:::11p1 � � � pN�1

D�X.N/

0:::001.1� p1/ � � � .1� pN�2/ C � � � C X.N/1:::101p1 � � � pN�2

�.1� pN�1/

C�X.N/

0:::011.1� p1/ � � � .1� pN�2/ C � � �� � � C X.N/

1:::111p1 � � � pN�2

�pN�1

D0@ X"02f0;1gN�2

�.N/"00 p"01

1 � � � p"0N�2

N�2

1A .1� pN�1/

C0@ X"02f0;1gN�2

.�.N/"00 C �.N/"01 /p"011 � � � p"

0N�2

N�2

1A pN�1

D X"02f0;1gN�2

.�.N/"00 C �.N/"01 pN�1/p"01

1 � � � p"0N�2

N�2

D X"2f0;1gN�1

�.N/" p"11 � � � p"N�1N�1

24

In fact Lemma 4 and Lemma 5 are instances of a more general lemma, which isalso proved in [Dat03a].

This lemma, which also appears for example in McKelvey and McLennan [MM97]or Sturmfels [Stu02], Chapter 6, is crucial to the proofs of the following theorems,from which Theorem 2 follows:

Theorem 6. Let S � Rn be a real algebraic variety given by m equations in n unknownsx1; : : : ; xn, such that each point .x1; : : : ; xn/ 2 S satis�es 0 < xi for i D 1; : : : ; n andPn

iD1 xi < 1, and suppose the highest power of xi in equation j is xdi ji . Set

D D �1CmXjD1.1C d1 j/.1C d2 j/ � � � .1C dn j/:

Then there is a 3-person game in which Alice has n C 1 pure strategies, Bob has D � m C 1pure strategies, and Critter has DC1 pure strategies, whose set of totally mixed Nash equilibriais isomorphic to S.

Proof. We suppose S to be given by the m equations

Fj.x1; : : : ; xn/ D 0

for j D 1; : : : ;m. We now consider the equations (C) associated with Critter's payoffs;recall that these are equations involving only the ai's and bi's. We will show how anarbitrary system of polynomial equations Fj.x1; : : : ; xn/ D 0 can be encoded in thissystem. (We will consider the ci's and the equations (A) and (B) later.) The variablesa1; : : : ; an will take the roles of x1; : : : ; xn.

We will repeatedly use the following observation. Suppose we have a system ofpolynomial equations

f1.x1; : : : ; xi/ D 0;...

fk.x1; : : : ; xi/ D 0

such that fk.x1; : : : ; xi/ D �xi C g.x1; : : : ; xi�1/ where � is some nonzero constantcoef�cient and g is a polynomial in the remaining variables other than xi. Then our

25

system is logically equivalent to (i.e., it implies and is implied by) the system

f1.x1; : : : ; xi�1;���1 g.x1; : : : ; xi�1; // D 0;...;

fk�1.x1; : : : ; xi�1;���1 g.x1; : : : ; xi�1// D 0;

xi D ���1 g.x1; : : : ; xi�1/:

Effectively we have substituted the value of xi given by the last equation into the otherequations. Notice that the variable xi no longer appears in the �rst i � 1 equations.In our construction we will actually be going the other way: we will be starting witha system of equations in fewer variables and adding a new variable xi as above. Theold system de�ned a variety V lying in Ri�1, and the new system de�nes a varietyV 0 lying in Ri. The two varieties are isomorphic, with isomorphism given by theembedding .x1; : : : ; xi�1/ 7! .x1; : : : ; xi�1;���1 g.x1; : : : ; xi�1/.

Most of the equations in our system will be of the form

b0i D �alb0j C �0

for some constants � and �0, where b0i D sibi C �i for some constants si and �i. Weimagine that we are computing with a device which allows us to multiply a previousresult b0j by one of the coordinates al and a constant �, add another constant �0, andstore the result in b0i. By multiplying and adding in this way, we will eventually be ableto evaluate the polynomial Fj.a1; : : : ; an/, and then we �nally use another equationto express the constraint that Fj.a1; : : : ; an/ D 0. The si's and �i's are constants withsi 6D 0, which we choose so that for any point .a1; : : : ; an/ 2 S, we will have 0 < bi forall i and

PD�miD1 bi < 1. This is possible since the set S �ts inside .0; 1/n.

Write Fj in recursive form as

Fj.x1; : : : ; xn/ D xd1 j1 Fjd1 j .x2; : : : ; xn/ C � � � C Fj0.x2; : : : ; xn/

D � � � DD xd1 j

1 .xd2 j2 � � � .xdn j

n Fjd1 j :::dn j C � � � / � � � / C � � � C Fj0:::0

26

where the Fji1:::in are constants, the Fji1:::in�1 are polynomials in an, the Fji1:::in�2 are poly-nomials in an�1 and an, and so forth. To evaluate Fj.a1; : : : ; an/, �rst we will evaluatethe polynomials Fji1:::in�1.an/. Then we will evaluate the polynomials Fji1:::in�2.an�1; an/,which are polynomials in an�1 with the Fji1:::in�1.an/ we computed previously as coef-�cients; and so forth.

At each stage, we will be evaluating a polynomial in one of the ai's, whose coef-�cients are some of our previous results. To evaluate this univariate polynomial, wewill use Horner's rule, which states that a univariate polynomial

�dxd C �d�1xd�1 C � � � C �1x C �0can be evaluated as

.� � � ..�dx C �d�1/x C �d�2/x C � � � C �1/x C �0:

Our �rst equation is

s1b1 C �1 D anF1d11:::dn1 C F1d11:::d.n�1/1.dn1�1/:

Our second equation is

s2b2 C �2 D an.s1b1 C �1/ C F1d11:::d.n�1/1.dn1�2/:

Continuing in this way, our dn1th equation is

sdn1bdn1 C �dn1 D an.s.dn1�1/b.dn1�1/ C �.dn1�1// C F1d11:::d.n�1/10:

Observe that the righthand side of this last equation is F1d11:::d.n�1/1.an/. In the same way,we obtain all the polynomials F1i1:::in�1.an/ for i1 D 0; : : : ; d11, . . . , in�1 D 0; : : : ; d.n�1/1,setting up dn1 equations for each. This takes care of the �rst k D .1 C d11/.1 Cd21/ � � � .1C d.n�1/1/dn1 equations. Now we start building up the bivariate polynomials.We begin by constructing d.n�1/1 equations starting with

skC1bkC1 C �kC1 D an�1.sdn1bdn1 C �dn1 / C .s2dn1b2dn1 C �2dn1 /;

27

and end up with F1d11:::d.n�2/1.an�1; an/ on the righthand side. In this way we use.1 C d11/.1 C d21/ � � � .1 C d.n�2/1/d.n�1/1 more equations to obtain all the polynomi-als F1i1:::in�2.an�1; an/ for i1 D 0; : : : ; d11,. . . ,in�2 D 0; : : : ; d.n�2/1. Continuing in thismanner, we at last end up with the equation 0 D F1.a1; : : : ; an/. We have used

d11 C .1C d11/d21 C � � � C .1C d11/.1C d21/ � � � .1C d.n�1/1/dn1

D .1C d11/.1C d21/ � � � .1C dn1/ � 1

equations. In this way we construct D equations to encode all the m equations0 D Fj.a1; : : : ; an/. The lefthand sides of each of these equations contains a distinctbi, except for the m equations 0 D Fj.a1; : : : ; an/ themselves. Thus we have made theset of totally mixed Nash equilibria consist exactly of those points .a1; : : : ; an/ in theset S, and for each such point we have set the values of all D � m C 1 bi's (the lastequation is

Pbi D 1).

It remains to set the values of the D ci's. We have n equations (A) and D � mequations (B) left, each of which we can use to set some ci equal to 1

D . If m > nthere will be m � n ci's left over. These are unconstrained except that 0 < ci < 1 andP

ci D 1. Thus the set of totally mixed Nash equilibria will be a Cartesian product ofS and a product of open simplices, which is stably isomorphic to S.

Theorem 7. Let S � Rn be a real algebraic variety given by m equations in n unknownsx1; : : : ; xn, such that each point .x1; : : : ; xn/ 2 S satis�es 0 < xi for i D 1; : : : ; n andPn

iD1 xi < 1, and suppose the highest power of xi in equation j is xdi ji . Set

D 0 DnX

iD1max

jdi j:

Then there is a game with .D 0 C m/ players in which each player has 2 pure strategies, whoseset of totally mixed Nash equilibria is isomorphic to S.

Proof. We �rst give a game with D 0 Cmaxfm; ng players. We take the �rst n variablesp1; : : : ; pn to represent x1; : : : ; xn. Let di D max j di j, and rename the last D 0 variables

28

as p11; : : : ; p1d1; : : : ; pn1; : : : ; pndn . Then the last D 0 equations are

p11 D p1; p12 D p1p11; : : : ; p1d1 D p1p1.d1�1/I...

pn1 D pn; pn2 D pnpn1; : : : ; pndn D pnpn.dn�1/:

The �rst m equations are Fj.x1; : : : ; xn/ D 0 for j D 1; : : : ;m, with xki replaced by

pik. Any remaining equations can be 0 D 0.Note that this means the �rst n variables p1; : : : ; pn do not occur in the �rst n

equations. If m > n, the next m� n variables do not occur in any equations. We mustarrange the last D 0 equations such that pi j does not occur in the .i; j/th equation. Wecould show that such an arrangement exists using Philip Hall's Marriage Theorem,but for concreteness we instead construct one such arrangement explicitly. If di � 3,we arrange the equations involving pi as follows:

.i; 1/ pi3 D pipi2;...

.i; di � 2/ pidi D pi pi.di�1/;

.i; di � 1/ pi1 D pi;

.i; di/ pi2 D pi pi1:

So it remains to consider those i for which di D 1 or di D 2. To simplify notation,assume that di D 1 for i D 1; : : : ; u, that di D 2 for i D uC 1; : : : ; uC v, and that di > 2for i > u C v.

If u D v D 0, we're done. If u � 1 and v � 1, then we can arrange the remaining

29

equations as follows:

.1; 1/ p21 D p2;...

.u � 1; 1/ pu1 D pu;

.u; 1/ p.uC1/2 D p.uC1/p.uC1/1;

.u C 1; 1/ p.uC2/1 D p.uC2/;

.u C 1; 2/ p.uC2/2 D p.uC2/p.uC2/1;...

.u C v � 1; 1/ p.uCv/1 D p.uCv/;

.u C v � 1; 2/ p.uCv/2 D p.uCv/p.uCv/1;

.u C v; 1/ p11 D p1;

.u C v; 2/ p.uC1/1 D p.uC1/:

If v D 0 and u � 2, we instead arrange the u remaining equations as follows:

.1; 1/ p21 D p2;...

.u � 1; 1/ pu1 D pu;

.u; 1/ p11 D p1:

If u D 0 and v � 2, we instead arrange the v remaining equations as follows:

.1; 1/ p21 D p2;

.1; 2/ p22 D p2p21;...

.v � 1; 1/ pv1 D pv;

.v � 1; 2/ pv2 D pvpv1;

.v; 1/ p11 D p1;

.v; 2/ p12 D p1p11:

If u D 1 and v D 0, we do not actually need the equation p11 D p1. Recall that wehad replaced x1 by p11 in the �rst m equations, Fj.x1; : : : ; xn/ D 0 for j D 1; : : : ;m.Instead, we put p11 D 1

2 for the very �rst equation, replace x1 by p1 in the next m� 1

30

equations F2.x1; : : : ; xn/ D 0; : : : ;Fm.x1; : : : ; xn/ D 0, and also replace x1 by p1 in the.1; 1/th equation F1.x1; : : : ; xn/ D 0.

It remains to consider the case where u D 0; v D 1. If there is at least one morevariable, then we have d2 � 3, so the �rst d2C 2 of the D 0 remaining equations can be

.1; 1/ p21 D p2;

.1; 2/ p22 D p2p21;

.2; 1/ p23 D p2p22;...

.2; d2 � 2/ p2d2 D p2p2.d2�1/;

.2; d2 � 1/ p11 D p1;

.2; d2/ p12 D p1p11:

So now suppose there is only one variable x D x1. We have n D 1 and D 0 D 2; ourgame has mC 2 players. Write the polynomials Fj.x1/ as Fj.x1/ D a jx2

1 C b jx1 C c j forj D 1; : : : ;m. Then the mC 2 equations are

.1/ a1p.mC1/p.mC2/ C b1p.mC1/ C c1 D 0;

.2/ a2p1p.mC1/ C b2p1 C c2 D 0;...

.m/ amp1p.mC1/ C bmp1 C cm D 0;

.mC 1/ p.mC2/ D p1;

.mC 2/ p.mC1/ D p1:

Now we have encoded S in a game with D 0 C maxfm; ng players. It remainsto show that we only need D 0 C m players. So suppose n > m. Above, we haveused di equations pi1 D pi; : : : ; pidi D pi pi.di�1/, but we could have gotten away withdi � 1 equations instead if we were willing to use pi in the equations encoding thepolynomials Fj.x1; : : : ; xn/ D 0. In that case all the variables p1; : : : ; pn and all thepi j's might occur in each of the equations Fj.x1; : : : ; xn/ D 0, so they could notbe associated with any of the players associated with those variables. Instead onemight have to introduce m new players with whom to associate these equations. (Forexample, this was the role of Elizabeth in the circle example.) Now the equations

31

associated with the players whose variables are p1; : : : ; pn are free; since n > m thesecan be used to �x the variables associated with the m new players at 1

2 (as was doneto the variable e in the circle example).

The values of D and D 0 in Theorem 2 are obtained by setting di j D d for all iand j. Theorem 1 follows since R is semialgebraically isomorphic to .�1; 1/ by thechange of variables t 7! t=.1� t2/ and .�1; 1/ is isomorphic to .0; 1/ by the change ofvariables t 7! .tC1/=2; then since the new xi's take values in .0; 1/, their sum

PniD1 xi

takes values in some interval .0; �/, and dividing them all by � lets us achieve thehypotheses of Theorems 6 and 7. Now the map t 7! t=.1� t2/, when considered as amap from R to the whole line R, is not one-to-one but one-to-two. So the image ofour real algebraic variety under this map will have several pieces, but the piece lying inthe interior of the n-cube .�1; 1/� � � � � .�1; 1/ will be semialgebraically isomorphicto the original variety (and will not be connected to any other piece). Note thatwhen the real algebraic variety is given by no more equations than unknowns, theisomorphism we exhibit in Theorems 6 and 7 is a homeomorphism.

In the game constructed in the proof of Theorem 7, the payoffs for many of theplayers depend only on the mixed strategies chosen by two or three of the otherplayers. This is why the game is not generic. At the same time, this can happen verynaturally in situations where the players interact locally. For example, a manufacturermaking a particular product interacts with various suppliers who make the compo-nents that go into that product. These local interactions can be described by a graph.Such graphical models are studied by Kearns, Littman, and Singh [KLS01].

As mentioned before, in many cases games of smaller formats can be used. Inparticular, we restate Theorem 3 and prove it here:

Theorem 8. Let S be the set of those roots of one polynomial equation �dad C � � � C �0 D 0in one unknown a which are real and lie in the interval .0; 1/. Then S is the set of �rstcoordinates of the totally mixed Nash equilibria of a 3-person game in which Alice has two purestrategies and the Bob and Critter each have dd=2e C 1 pure strategies.

Proof. Suppose d is even, say d D 2e. We set Alice's payoffs so that equating them

32

yields ce D b1. We set Bob's payoffs so that equating them yields the e equations

0 D �0 C a.s1c1 C �1/;s1c1 C �1 D �1 C a.s2c2 C �2/;

...

se�1ce�1 C �e�1 D �e�1 C a.sece C �e/Iand we set Critter's payoffs so that equating them yields the e equations

seb1 C �e D �e C a.seC1b2 C �eC1/;

seC1b2 C �eC1 D �eC1 C a.seC2b3 C �eC2/;...

s2e�2be�1 C �2e�2 D �2e�2 C a.s2e�1be C �2e�1/;

s2e�1be C �2e�1 D �2e�1 C a�2e:

As in the proof of Theorem 6, the si's and �i's are constants with si 6D 0 chosen sothat 0 < bi, 0 < ci,

PeiD1 bi < 1, and

PeiD1 ci < 1, for all a 2 S.

Suppose d is odd, say d D 2e � 1. Then we replace the last of Critter's equationsby

be D 12� 1

2.b1 C � � � C be�1/:

In the proof of Theorem 6, Alice's and Bob's equations were essentially wasted.On the other hand, in the proof of Theorem 8, we started the same way, multiplyingout the polynomial according to Horner's rule and accumulating the result in one ofthe ci's. But then we used one of Alice's equations to transfer the result to the bi's, sowe could continue the calculation using these variables. This can be done in manycases (although in general there may be more results that would have to be transferredthan Alice's equations could accomodate). For instance, this is one reason why wewere able to encode the circle in a 3-person game in which each player has threepure strategies; Theorem 6 would have predicted that Alice would need three purestrategies, Bob would need eight, and Critter would need nine. The other reason

33

is that the polynomial a1�a1 � 1

2� C a2

�a2 � 1

2� C 7

64 can be written as the sum of apolynomial in a1 and a polynomial in a2. So we don't need to evaluate a polynomialof degree d2 D 2 in a2 for each of the d1 D 2 powers of a1; we just need to evaluate apolynomial in a1 and a polynomial in a2 separately.

Although we have stated our results in the geometric language of varieties, ourproofs are purely algebraic and require few assumptions. Experts may note that thismeans our results are actually more general than what we have stated: they concernschemes, which generalize varieties.

2.5 Conclusion

Although the set of totally mixed Nash equilibria might comprise an arbitrary realalgebraic variety, this does not mean it cannot be computed. As mentioned before,generically this set is �nite. It is the set of solutions to a system of polynomialequations, which can be found, for instance, using polyhedral homotopy continuationsoftware such as PHCpack by Verschelde [Ver99], as we discuss in Chapter 4 and in[Dat03b]. Sommese and Verschelde [SV00] have extended these methods to posi-tive dimensional algebraic sets (e.g., curves and surfaces rather than isolated points).Indeed, how to solve systems of polynomial equations is a very active area of research.

Since Nash equilibria are usually not unique, the way that players approach equi-librium dynamically in repeated games under assumptions of imperfect informationand/or bounded rationality has been studied both theoretically and experimentally.For example, Kalai and Lehrer [KL93] showed that under certain assumptions, �ra-tional learning leads to Nash equilibrium.� The existence of equilibrium sets withvarying geometry and topology suggests that in these same dynamical models, inter-esting phenomena might continue to occur after equilibrium has been reached.

34

Chapter 3

Games and Graphs

The set of Nash equilibria for a game with generic payoff functions is �nite[Har73]. This implies that the set of totally mixed Nash equilibria for a game withgeneric payoff functions is also �nite. These are the real solutions to a system of poly-nomial equations and inequalities. The complex solutions to the system of equationsare called quasiequilibria. Thus, the set of totally mixed Nash equilibria is a subset ofthe set of quasiequilibria. In fact, the set of quasiequilibria is also �nite in the mostgeneric case, and its cardinality can be computed as a function of the numbers of purestrategies of the players. Thus, this is an upper bound on the number of totally mixedNash equilibria. Even in a nongeneric case, as long as the set of quasiequilibria is�nite, its cardinality will be bounded above by the number in the generic case.

For the main theorem of this chapter, Theorem 9, we hypothesize a set of technicalconditions that a system of polynomial equations may satisfy, which are encoded in anassociated graph, the polynomial graph, and we prove a formula describing the numberof solutions in this case. We then show how to associate such a graph to threespecial classes of games. The �rst two are graphical games and extensive-form games.The last is games with emergent node tree structure, a new model for games in whichthe players can be hierarchically decomposed into groups. Usually such hierarchicaldecomposition is modelled by cooperative games, and we brie�y discuss how our modelis related to, yet differs from, the cooperative framework.

35

3.1 Generic Number of Quasiequilibria

McKelvey and McLennan [MM97] have computed the exact number of quasiequilib-ria for games in the most generic case. The following theorem generalizes theirs tothe situation in which the payoff matrices have more structure.

Theorem 9. Suppose that 0 < d 2 N and that we are given a partition f1; : : : ; dg D `NiD1 Ti

of f1; : : : ; dg. Write di D jTij. Suppose further that we are given a directed graph G, thepolynomial graph, on d vertices, denoted v1; : : : ; vd, without self-loops and with the propertythat for any v j and Ti, if there is some k 2 Ti such that there is an edge from v j to vk in G,then for every k 2 Ti there is an edge from v j to vk in G. Let

f1. �1; : : : ; �d/ D 0;

f2. �1; : : : ; �d/ D 0;...

fd. �1; : : : ; �d/ D 0

be a system (3.1) of d polynomial equations in d variables �1; : : : ; �d with the followingproperties:

1. All monomials occuring in the fi's are squarefree.

2. If � j; �k 2 Ti with j 6D k then � j and �k do not both occur in any monomial of any ofthe fi's.

3. If there is no edge from v j to vk in G then the variable �k does not occur in f j.

Thus, the equations are multilinear, and they are linear over the variables from each Ti. Constructa d � d matrix M as follows: If variable �k occurs in the polynomial f j, with Ti the subsetcontaining vk, then

M jk D 1.di!/1=di

;

otherwise M jk D 0. If the system (3.1) is 0-dimensional, then the number of its solutions in.C�/d (i.e. such that �k 6D 0 for all k) is bounded above by the permanent of M, and is equalto the permanent of M for generic coef�cients.

36

Proof. Without loss of generality, assume

Ti D(1C

i�1XlD1

dl; 2Ci�1XlD1

dl; : : : ; di Ci�1XlD1

dl

);

that is, that the Ti's are contiguous.Let ai j D 1 if there is an edge in G from v j to vk for k 2 Ti, and ai j D 0 otherwise.

Then the Newton polytope Pj of f j is the Cartesian product P1 j � P2 j � � � � � PN j,where Pi j is the convex hull of the scaled coordinate vectors

�ai jek j k 2 Ti

and the

origin. For i with ai j D 1, Pi j is the di-dimensional unit simplex, and for i with ai j D 0,Pi j degenerates to the di-dimensional origin (which is a 0-dimensional simplex). Bythe Bernstein-Kouchnirenko Theorem [Ber75] [Kou76], it suf�ces to show that themixed volume of the polytopes P1; : : : ;Pd is given by the permanent of M.

Let Q j D �1P1 C � � � C � jPj, where C denotes Minkowski addition and the scalefactors �1; : : : ; � j are parameters. We show by induction on j that Q j D Q1 j �Q2 j �� � � �QN j, where Qi j is the convex hull of�

.ai1�1 C ai2�2 C � � � C ai j� j/ek j k 2 Ti

and the origin. (If ai1�1Cai2�2C� � �Cai j� j D 0 then Qi j degenerates to the origin.) Thebase case follows from our characterization of Pj above. Now consider the Minkowskisum of Q j D Q1 j�� � ��QN j and � jC1PjC1 D .� jC1P1. jC1//�� � ��.� jC1PN. jC1//. It followsfrom the de�nition of Minkowski sum that this is .Q1 j C � jC1P1. jC1// � � � � � .QN j C� jC1PN. jC1//, and (using the induction hypothesis) that each factor Qi j C � jC1Pi. jC1/ isequal to the convex hull of�

.ai1�1 C ai2�2 C � � � C ai j� j C ai. jC1/e jC1/ek j k 2 Ti

and the origin.The di-dimensional volume of the di-dimensional unit simplex scaled by � in each

dimension is�di

.di/!:

We are interested in the d-dimensional volume of Qd. If ai1 D ai2 D � � � D aid D 0for some i, then this volume vanishes, and hence the mixed volume also vanishes. In

37

this case the kth column of the matrix M will be all zeroes for any k 2 Ti, so thepermanent of M also vanishes, and the theorem holds. So assume that for each i,there is some j with ai j D 1. Then the volume of Qd is

NYiD1

.ai1�1 C � � � C aid�d/di

di!:

Let . g jk/ be the adjacency matrix of G, that is, g jk D 1 if there is an edge in G fromv j to vk and g jk D 0 otherwise. Then ai j D g jk for all k 2 Ti. So the volume of Qd isQd

kD1�g1k�1 C � � � C gdk�d

�QNiD1 di!

:

The mixed volume of P1; : : : ;Pd is the coef�cient of �1�2 � � � �d in the above expres-sion, which is the permanent of . g jk/ divided by

QNiD1 di!.

It remains to show that the permanent of M is the permanent of . g jk/ divided byQNiD1 di!. Note that M jk 6D 0 exactly when g jk 6D 0. We induct on N. For the base

case, d1 D d, and each nonzero entry of M is .1=d!/1=d. A term in the permanent of Mis the product of d entries from M, so if it is nonzero it is 1=d!. Thus the permanentof M is 1=d! times the permanent of . g jk/, as required. Now partition the matrix Mand the matrix . g jk/ into two vertical bands corresponding to the subsets [N�1

iD1 Ti andTN . The permanent can be computed as the sum of a term for each choice of dN

rows 1 � j1 < � � � < jdN � d: compute the .d � dN / � .d � dN / subpermanent of theleft band obtained by crossing out those rows, compute the dN � dN subpermanentof the right band corresponding to those rows, and multiply them together. By theinductive hypothesis, the left subpermanent of M is the left subpermanent of . g jk/

divided byQN�1

iD1 di!. For the right subpermanent, every row is either all nonzero orall zero. If any row is all zero, both right subpermanents vanish. If every entry isnonzero, then all the entries are the same: g jk D 1 and M jk D .1=dN !/1=dN . The rightsubpermanent of M is dN !

�.1=dN !/1=dN

�dN D 1, and the right subpermanent of . g jk/ isdN !. So the whole term for M is the whole term for . g jk/ divided by

QNiD1 di!.

We note that if the coef�cients are generic subject to the conditions given inTheorem 9, all the solutions to the system will lie in the torus .C�/d. In what follows

38

we will refer to �the number of solutions in the torus .C�/d� as �the number ofsolutions� by abuse of language.

Corollary 10. Convert the directed graph G of Theorem 9 into a bipartite graph on 2d vertices,with the source of every edge on the left side and the target of every edge on the right side. Ifthe system in Theorem 9 is 0-dimensional with generic coef�cients, then it has a solution if andonly if this bipartite graph has a perfect matching.

Proof. From the proof of Theorem 9, we see that the number of solutions is nonzeroif and only if the permanent of the adjacency matrix is nonzero. It is a well-knownfact that this is equivalent to the existence of a perfect matching: any permutation� which contributes a nonvanishing term

QdjD1 g j�. j/ to the permanent corresponds

to a perfect matching, where vertex j on the left is matched to vertex �. j/ on theright.

In fact, we could have used the bipartite graph in Theorem 9. However, wede�ned the polynomial graph to be the directed graph to remain consistent with theusual de�nition of graphical models of games.

Corollary 11. If the system in Theorem 9 is 0-dimensional and has a solution, then everynode in the graph G lies on a directed cycle.

Proof. As in the proof of the previous corollary, a permutation � must exist such thatj has an edge to �. j/ for every j D 1; : : : ; d. This permutation can be expressed asa product of disjoint cycles. Each node lies in one of these cycles, and a cycle of thepermutation corresponds to a directed cycle in the graph.

We should note carefully that the Bernstein-Kouchnirenko theorem gives thenumber of solutions to a 0-dimensional polynomial system. So when the number givenby that theorem�in particular, the permanent of the matrix in Theorem 9�vanishes,either the polynomial system has no solution, or its solution set has positive dimension.

Note that the conditions on G imply that the matrix M has a di � di block ofzeroes along its diagonal for i D 1; : : : ;N. This is because G has no self-loops, and

39

if it had an edge from an element v j of Ti to any other element vk of Ti, then therewould have to be an edge from v j to every element of Ti including itself.

Our theorem applies to games. In this case, each Ti corresponds to the set ofstrategies of player i. The blocks of zeroes along the diagonal imply that a player'sexpected payoffs from their own pure strategies do not depend on the probabilitiesthey have assigned to their own pure strategies, so these polynomial systems do indeedcorrespond to the equations for totally mixed Nash equilibria of games.

Corollary 12. Consider a normal form game between players I D f1; : : : ;Ng with purestrategy sets Si for each i and generic utility functions ui :

Qi2I Si ! R. Construct a graph

G with nodes`

i2I.Si � fsi0g/ such that there is an edge from sik to s jl in G if and only ifi 6D j. Let the variable corresponding to sik be �i.sik/ and the equation corresponding to sik bethe indifference equation ui.sik; ��i/ D ui.si0; ��i/. Then this system of equations obeys theconditions of Theorem 9, so the number of solutions in the generic case is given by that theorem.

This special case was proved as Theorem 2 in [McL99], so our Theorem 9 is ageneralization of that theorem.

3.2 Graphical Games

Kearns, Littman, and Singh [KLS01] de�ned the concept of graphical games, or gamesobeying graphical models. (That paper considers undirected graphs, but the extensionto directed graphs which we will use is straightforward.) A game between players1; : : : ;N obeys a directed graphical model, if the payoffs to player i1 only depend onthe actions of those players i2 6D i1 for which there is an edge from i1 to i2 in thegraphical model.

Our theorem applies in particular to graphical games. As in Corollary 12, we takethe pure strategy sets Si to be the sets Ti of Theorem 9. Given a polynomial graph Gas in Theorem 9, we draw an edge from i1 to i2 in the graphical model if there is anyj 2 Ti1 with edges to the vertices in Ti2 in the polynomial graph G. The polynomialgraph G may not represent the most generic case of the graphical model, however.If we are given a graphical model, then to construct its polynomial graph G, for any

40

edge from i1 to i2, we draw edges in G from every vertex j 2 Ti1 to every vertex inTi2 .

Corollary 13. Suppose a normal form game between players I D 1; : : : ;N with pure strategysets Si for each i and utility functions ui :

Qi2I Si ! R obeys a directed graphical model

with nodes 1; : : : ;N. Construct a graph G with nodes`

i2I Si such that there is an edge fromsik to s jl in G if and only if there is an edge from i to j in . Then the system of equationsde�ning the quasiequilibria of G satis�es the hypotheses of Theorem 9, so the number of suchquasiequilibria in the generic case is given by the permanental formula.

For example, consider a game with 4 players, each with 3 pure strategies. Gener-ically, such a game has

per

0BBBBBBBBBBBBBBBBB@

0 0 1p2

1p2

1p2

1p2

1p2

1p2

0 0 1p2

1p2

1p2

1p2

1p2

1p2

1p2

1p2 0 0 1p

21p2

1p2

1p2

1p2

1p2 0 0 1p

21p2

1p2

1p2

1p2

1p2

1p2

1p2 0 0 1p

21p2

1p2

1p2

1p2

1p2 0 0 1p

21p2

1p2

1p2

1p2

1p2

1p2

1p2 0 0

1p2

1p2

1p2

1p2

1p2

1p2 0 0

1CCCCCCCCCCCCCCCCCAD 297

quasiequilibria.But suppose now that game obeys a graphical model as in Figure 3.1. The

nodes in the graphical model refer to the players, and the edges specify that thepayoff to the source player depends on the actions of the target player. For brevity,write a D �1.s11/, b D �2.s12/, c D �2.s21/, d D �2.s22/, e D �3.s31/, f D �3.s32/,g D �4.s41/, and h D �4.s42/. Since the payoff to player 1 depends only on the actionsof player 2, equating the payoff to player 1 from pure strategies s10 and s11 gives

u1.s10; s20; �/ �2.s20/ C u1.s10; s21; �/ �2.s21/ C u1.s10; s22; �/ �2.s22/

D u1.s11; s20; �/ �2.s20/ C u1.s11; s21; �/ �2.s21/ C u1.s11; s22; �/ �2.s22/

41

1 2

4 3Figure 3.1: Graphical game

or

.u1.s11; s20; �/ � u1.s10; s20; �// .1� c � d/CC .u1.s11; s21; �/ � u1.s10; s21; �// c C .u1.s11; s22; �/ � u1.s10; s22; �// d D 0:

Thus for player 1 we have two equations of the form

�c C �d C � D 0;

for player 2 we have two equations of the form

�e C � f C � D 0;

for player 3 we have two equations of the form

�g C �h C � D 0;

and for player 4 we have two equations of the form

�aC �b C � D 0:

Then the associated polynomial graph is depicted in Figure 3.2. The equation asso-

42

2

44

1 2

33

1a

b c

d

e

fg

h

Figure 3.2: Associated polynomial graph for graphical game

43

ciated with the node labelled 1a equates the payoffs to player 1 from choosing s11

(which 1 does with probability a) or choosing s10. The game has

per

0BBBBBBBBBBBBBBBBB@

0 0 1p2

1p2 0 0 0 0

0 0 1p2

1p2 0 0 0 0

0 0 0 0 1p2

1p2 0 0

0 0 0 0 1p2

1p2 0 0

0 0 0 0 0 0 1p2

1p2

0 0 0 0 0 0 1p2

1p2

1p2

1p2 0 0 0 0 0 0

1p2

1p2 0 0 0 0 0 0

1CCCCCCCCCCCCCCCCCAD 1

quasiequilibrium. Indeed, this will always be the case for a graphical model which is adirected cycle, where each player has the same number of pure strategies. The reasonis that the indifference equations in this case are linear, as we saw in this example.

The polynomial graph G as de�ned in Theorem 9 contains more re�ned informa-tion than the graphical model. The partition into the Ti's also can be more re�nedthan the partition of the set of all pure strategies into the sets of pure strategies foreach player. Next we will see an example of such a re�nement when consideringthe reduction of extensive-form games to normal-form, where actions correspond tobranches of the game tree.

3.3 Extensive-Form Games

Now we consider extensive-form games. We begin by noting the following:

Theorem 14. All totally mixed Nash equilibria of an extensive form game are subgame perfect.

Proof. Let � be a totally mixed Nash equilibrium of an extensive form game with Nplayers de�ned by game tree T . Note that the strategy pro�le induced by � on everysubgame is also totally mixed. Let � be a non-leaf node of T . Let Q� be the strategypro�le induced by � in the subgame induced by �. Let Qs j and Qt j be pure strategiesof player j in this subgame. Choose an action for j at each node � that is not a

44

descendant of � where j acts, such that if � is an ancestor of � then j chooses thebranch leading towards �, and use this choice to extend Qs j and Qt j to pure strategies s j

and t j of player j in the original game. (So, s j and t j specify the same actions outsidethe subtree.) Let �0 : : : �m D � be the unique path from the root �0 of T to �. Wehave u j.s j; �� j/ D u j.t j; �� j/. Let L be the set of all leaves of T under � and L0 bethe set of all other leaves. Then

u j.s j; �� j/ DP�2L u j.�/ Pr[�j.s j; �� j/]CP�2L0 u j.�/ Pr[�j.s j; �� j/]

DP�2L u j.�/ Pr[�j.s j; �� j/]CP�2L0 u j.�/ Pr[�j.t j; �� j/]

since s j and t j choose the same actions outside the subtree. ThusX�2L

u j.�/ Pr[�j.s j; �� j/] DX�2L

u j.�/ Pr[�j.t j; �� j/]: (3.1)

Furthermore, for any � 2 L, we have

Pr[�j.s j; �� j/] D Pr[�j. Qs j; Q�� j/]m�1YkD0

Pr[�k ! �kC1j.s j; �� j/]

D Pr[�j. Qs j; Q�� j/]m�1YkD0

Pr[�k ! �kC1j.t j; �� j/]:

Noting that the common factorQm�1

kD0 Pr[�k ! �kC1j.t j; �� j/] in equation (3.1) ispositive by our choice of s j; t j and because � is totally mixed, we have that

u j. Qs j; Q�� j/ DX�2L

u j.�/Pr[�j. Qs j; Q�� j/]

D X�2L

u j.�/Pr[�j. Qt j; Q�� j/]

D u j. Qt j; Q�� j/:

Thus Q� is a (totally mixed) Nash equilibrium of the subgame induced by �.

In light of this observation, the divide-and-conquer approach to �nding all Nashequilibria of a normal form game can be modi�ed in the spirit of backwards inductionto �nding all subgame perfect equilibria (including mixed ones) of an extensive formgame. Recall that in a normal form game, we would consider subproblems in which

45

one pure strategy of one player i was removed. Now we instead consider subproblemsin which, for some edge � ! � where i acts at �, we delete that edge and the entiresubtree below �. We compute the normal form for the game described by this prunedtree and recursively �nd all its subgame perfect equilibria. Each such equilibrium �

induces an equilibrium Q� in the subgame under � in the pruned tree. To checkwhether � is an equilibrium of the original game, we recursively compute all theequilibria of the subgame under � (where i does not act), and check that for eachsuch equilibrium � , we have ui. Q� / � ui.� /.

We saw during the above proof that for a totally mixed strategy pro�le � , theequations u j.s j; �� j/ D u j.t j; �� j/ for all pure strategies s j; t j of j imply the corre-sponding equations for each subtree. The converse implication also clearly holds.

We will now associate a polynomial graph to a system of equations for thequasiequilibria of an extensive-form game, so that we can apply Theorem 9. Foreach node in the game tree where a player acts, we will have a variable for everyedge emanating from that node except one distinguished edge. This is because thesum of the probabilities of choosing each of those edges must be 1, so we elimi-nate one variable. Thus, we compare the payoffs between choosing the distinguishededge and choosing any other edge. The equations will be indifference equations forsubgames of the extensive-form game.

Theorem 15. The set of quasiequilibria of a generic extensive-form game is either empty orhas positive dimension.

Proof. Consider an extensive form game with players I D 1; : : : ;N and game tree T .Let A be the set of non-leaf nodes in T not labelled by 0. For each � 2 A, let E.�/be the set of edges emanating from �. For each � 2 A, let i be the player whichacts at � and pick an element ei� 2 E.�/. Let d D P�2A jE.�/ � 1j and partition d as`

�2A .E.�/ � fei�g/. De�ne a directed graph G on a set of d vertices[�2Afne j e 2 E.�/ � fei�gg

as follows: there is an edge from ne with e 2 E.�/ � fei�g to ne0 with e0 2 E.�/ � fe j�gif i 6D j, � is an ancestor of �, either e or ei� lies on the path from � to �, and if i

46

acts at some node � between � and �, then the edge ei � lies on the path from � to�. We will de�ne a system of equations equivalent to the equations de�ning totallymixed Nash equilibria of the extensive form game and satisfying conditions 1 to 3 ofTheorem 9. The polynomial graph G is acyclic, so Corollary 11 implies our assertion.

First we must state what the equations are. Fix a node � 2 A and let i be the playerwhich acts at �. Then jE.�/j�1 equations refer to the subgame induced by this node.For each e 2 E.�/, de�ne the pure strategy sie of i in this subgame by sie.�/ D e andsie.�/ D ei� for any node � below � where i acts. Writing Q� for the strategy pro�leinduced by � in the subgame under �, the jE.�/j � 1 equations are the equationsui.se; Q��i/ D ui.sei� ; Q��i/ for e 2 E.�/ � fei�g. In these equations we eliminated �.e j�/

for every � below � where i does not act, by substituting 1 �Pe2 QE.�/�ef j�g � j.e/ for� j.e j�/.

These are some of the indifference equations for the subtree below �, which aswe saw in the previous theorem are implied by the indifference equations for thewhole tree. We show by induction that these equations also imply all the indifferenceequations for the subtree below �. (Thus we will have the indifference equationsfor every subtree, and hence the whole tree, i.e., the original game.) Firstly, i isindifferent between all i's pure strategies in the subgame below �, because althoughwe �xed i's pure strategies at nodes � below � where i acts to be ei�, we also have thati is indifferent between i's pure strategies in the subgame below � by the inductionhypothesis. Secondly, consider any other player j. Let �1; : : : ; �m be the nodes below� where j acts, such that j does not act at any node between � and �k for any k.Let Qs j; Qt j be pure strategies of j in the subgame below �, and write Qs jk; Qt jk for therespective induced pure strategies of j in the subgame below �k. So Qs j D . Qs j1; : : : ; Qs jm/

and Qt j D . Qt j1; : : : ; Qt jm/. Write the set L of leaves below � as L D L0 [SmkD1 Lk, where

L0 is the set of leaves � such that j does not act between � and � and Lk is the set of

47

leaves below �k for k D 1; : : : ;m. Then

u j. Qs j; Q�� j/ DX�2L

u j.�j Qs j; Q�� j/

D X�2L0

u j.�j Q�� j/ CmX

kD1

X�2Lk

u j.�j Qs jk; Q�� j/

D X�2L0

u j.�j Q�� j/ CmX

kD1

X�2Lk

u j.�j Qt jk; Q�� j/

D u j. Qt j; Q�� j/

since for each k,P

�2Lku j.�j Qs jk; Q�� j/ D P�2Lk

u j.�j Qt jk; Q�� j/ by the induction hypoth-esis.

We can already see that the set of solutions to these equations, if nonempty, ispositive-dimensional. If player i acts at the root �, then for any edge e emerging from�, �i.e/ does not appear in any of the equations.

All the monomials occurring in these equations are squarefree. For each leaf �under �, let the path from � to � be � D �1 : : : �k D �. Then for any player j with purestrategy Qs j, we have Pr[�j Qs j; Q�� j] D Qk�1

lD1 Pr[�l ! �lC1j Qs j; Q�� j], and each nonconstantterm in the product is �n.�l ! �lC1/ for some player n 6D j. So for any edge e wheren acts, the variable �n.e/ occurs at most once in such a product. In fact �n.e/ occursin such a product for at most one e 2 E.�l/. (That is, if e; e0 2 E.�l/ then �n.e/ and�n.e0/ do not both occur in this monomial. So condition 2 of Theorem 9 holds.)When we eliminate �n.en�l /, we replace it by an af�ne expression, so this remainstrue. Thus condition 1 of Theorem 9 holds.

The equations corresponding to E.�/ � fei�g concern only the subgame below �,so � j.�! �/ occurs in these equations only if � is an ancestor of �. Furthermore, ifi acts at � below �, then �i.e/ does not occur for any edge e 2 E. �/� fei �g, since we�x that i chooses ei � . For the same reason � j.e/ does not occur for e 2 E.�/ � fe j�gfor any � that lies below � but not below ei � . Thus condition 3 holds.

Our result does not contradict Harsanyi's generic �niteness theorem [Har73], be-cause generically, iterated elimination of weakly dominated strategies/backward in-duction will lead to a unique subgame perfect equilibrium (and so indeed there will

48

be no totally mixed Nash equilibria). On the other hand, another way to look at ourresult is that in every interesting extensive-form game�one which is not completelysolved by backward induction, giving a unique equilibrium�the set of totally mixedNash equilibria is also interesting; it has positive dimension.

In particular, if � is a node all of whose children are leaves, the equations corre-sponding to � will be equations between constants, stating that for the player i whoacts at �, the utilities ui.�/ at all the leaves � below � must be equal. This is true ifiterated elimination of strictly dominated pure strategies has already been performedon this game.

It is clear that the system of equations we obtained is not canonical, since we havemade arbitrary choices of the edges ei� and the subtrees below each possible choiceare different. Choosing a different system may make it easier to compute the set ofquasiequilibria.

We now present an example where the set of totally mixed Nash equilibria is apositive-dimensional semialgebraic variety. Consider the extensive form game speci-�ed in Figure 3.3. The polynomial graph associated with this game tree is depicted inFigure 3.4. For brevity, we write for example �1.C / for �1.A! C /. The quasiequi-libria obey a system of 4 equations as in Theorem 15. The equation associated withthe edge E! G equates the payoff to player 3 from choosing this edge with that fromchoosing the edge E ! F, i.e., u3.F/ D u3.G/. No variables occur in this equation,that is, it is an equation between constants. Similarly, the equation associated withthe edge E ! H is u3.F/ D u3.H/. The equation associated with the edge C ! Eis u2.D/ D u2.E/, where we have written u2.E/ for the expected payoff u2.E; ��2/ toplayer 2 for choosing the edge C ! E, given the strategy pro�le of the other players.In this case u2.E/ D u2.F/ �3.F/ C u2.G/ �3.G/ C u2.H/ �3.H/, so

u2.D/ D u2.F/ C .u2.G/ � u2.F// �3.G/ C .u2.H/ � u2.F// �3.H/:

49

BD

A

CE

F G H

2

1

3(3/2,0,2)

(2,2,0)

(0,0,1) (1,0,1) (3,6,1)

Figure 3.3: An Extensive Form Game

50

A C

C E

E G E HFigure 3.4: Associated Polynomial Graph For An Extensive Form Game

51

Finally, the equation associated to the edge A! C is

u1.B/ D u1.C /

D u1.D/ .1� �2.E// C u1.F/ �2.E/ .1� �3.G/ � �3.H//

Cu1.G/ �2.E/ �3.G/ C u1.H/ �2.E/ �3.H/:

Looking at the speci�c payoffs in Figure 3.3, we see that the payoffs to player 3for choosing F, G, or H are equal, as required. Equating the payoffs to player 2 forchoosing D or E, we get 6 �3.H/ D 2, or �3.H/ D 1

3 . This leaves �3.G/ free to varysuch that 0 < �3.G/ < 2

3 . Finally, we must equate the payoffs to player 1 for choosingB or C. This gives

2.1� �2.E// C �2.E/ . �3.G/ C 1/ D 32

or�2.E/.1� �3.G// D 1

2:

Thus the points �3.G/ and �2.E/ lie on a hyperbola. This hyperbola intersects theinterior of the product of simplices. For instance, the point �3.G/ D 5

12 (so �3.F/ D 14)

and �2.E/ D 67 lies in this intersection. So the set of quasiequilibria is a portion of

a hyperbolic cylinder, the product of a segment of a hyperbola with a line segment(since �1.B/ varies freely with 0 < �1.B/ < 1).

We can analyze this game a little further. Player 3 would like player 1 to sometimeschoose B, but cannot force player 1 always to choose B, since if player 2 alwayschooses D then both player 1 and player 2 are better off with player 1 choosing C .The best player 3 can do is make the payoffs to player 1 from choosing B and Cequal. Now if player 3 made player 2 get a greater payoff from choosing D thanE, then player 2 would always choose D, player 1 would always choose C, andplayer 3 would get nothing. So player 3 must make u2.D/ � u2.E/. We analyzedthe case u2.D/ D u2.E/ above. If player 3 makes �3.H/ > 1

3 , then u2.D/ < u2.E/and player 2 will always choose E. Then the payoff to player 1 from choosing C is�3.G/C3 �3.H/. Thus we have �3.G/C3 �3.H/ D 3

2 with 13 < �3.H/ � 1

2 (this makes0 � �3.G/ < 1

2 and 16 < �3.F/ � 1

2). Then �1.C / varies freely with 0 � �1.C / � 1,

52

so we have a rectangle of partially mixed equilibria. Player 3 is better off choosingthese, since then the outcome D where player 3 gets zero payoff is never reached.Along the line �3.G/ C 3 �3.H/ D 3

2 , equilibria with greater �3.H/ Pareto dominatethose with smaller �3.H/, i.e., they make some player better off and no player worseoff. Speci�cally, the payoff to player 2 increases, the payoff to player 1 is always 3

2 ,and the payoff to player 3 stays the same at 2.1 � �1.C // C �1.C / D 2 � �1.C /.Thus the Pareto dominant equilibrium among those on this line is that player 3 has�3.F/ D 1

2 , �3.G/ D 0, and �3.H/ D 12 . On the other hand, at the pure strategy

equilibrium where player 3 always chooses H, we have that player 1 always choosesC, and the payoff to player 3 falls from 2 � �1.C / to 1. Thus player 3 does notprefer this equilibrium, and instead mixes F and H equally to have some chance of ahigher payoff. As �1.C / increases, the payoff to player 3 decreases and the payoff toplayer 2 increases, so the equilibria along this line do not Pareto dominate each other.Thus without introducing other issues (such as risk-aversion) there is no criterionfor predicting which of the equilibria along the line 0 < �1.C / < 1, �2.E/ D 1,�3.F/ D �3.H/ D 1

2 should be chosen.

3.4 Games With Emergent Node Tree Structure

So far we have been discussing normal form games with �nite numbers of players,each with a �nite number of pure strategies. Such a game is de�ned by giving a setof players I D f1; : : : ;Ng, for each player i a �nite set of pure strategies Si, and foreach pure strategy pro�le � (element of the product S D Q

i2I Si) and each player ithe utility ui. � / received by that player when that strategy pro�le is played. Now wewill introduce a particular kind of structure that a normal form game may have.

We now de�ne an emergent node tree structure on a normal form game. This is a newmodel for games in which the players can be hierarchically decomposed into groups.Usually such hierarchical decomposition is discussed in the framework of cooperativegame theory. Instead, we de�ne certain conditions on the payoff functions in anoncooperative game such that a given hierarchical decomposition �makes sense�, ina way that we will de�ne precisely. At the end of this section we brie�y describe how

53

our framework relates to that of cooperative game theory.De�nition. An emergent node tree structure on a normal form game with player

I D f1; : : : ;Ng, pure strategy sets Si for i 2 I, and utility functions ui :Q

i2I Si ! R toconsist of:

� A tree T with N leaves. The leaves are in bijection with the players I Df1; : : : ;Ng. Write Cv for the set of children of a node v 2 T , Bv for the set ofits siblings, and f .v/ for its parent.

� For each non-leaf, non-root node v of the tree (which we call an emergent player),a set Sv of pure strategies, with jSvj � Qw2Cv jSwj.� For each non-leaf, non-root node v, for each element sCv of the product SCv DQ

w2Cv Sw of the pure strategies of its children and each element svk of Sv, anumber pv.k; sCv / signifying the probability that the (emergent) strategy of theemergent player v is svk when the strategies of its children are given by sCv . So ifv has K pure strategies, then

PKkD1 pv.k; sCv / D 1. If the children of v execute a

mixed strategy, then the emergent mixed strategy of v is given by multilinearity.Thus we have de�ned a linear map from the strategy space of the children tothe strategy space of the parent. We require that this map have full rank.

� For each non-root node v (including the leaf nodes), real numbers vw foreach non-root ancestor w of v and real numbers Uv.s/ for each element s 2Sv�Qw2Bv Sw. From these we de�ne a utility function uv, which is a sum of twoterms: Uv. �v;Bv /, a multilinear function of the strategies executed by v and itssiblings in Bv, and

Pnonroot ancestors u vwuw. We require that the utility function uv

at a leaf node v be equal to the utility function ui of the player i correspondingto the leaf node v.

We will refer to an emergent node tree structure as an ENT for short. Note thatfor a given normal form game, we can always de�ne a class of ENTs by de�ninga tree with a single emergent node (the root node), so that all the leaf nodes aresiblings. We call such an ENT trivial. For any given normal form game, there needbe no nontrivial ENT, or there may be many distinct possible ENTs.

54

The behavior of the emergent players is completely determined by the behavior ofthe actual players (the leaf nodes). The emergent strategy �v executed by the emergentplayer v when the actual players execute strategy pro�le � is de�ned recursively bymultilinearity:

�v.svk/ DXs2SCv

pv.k; s/Y

w2Cv

�w.sw/:

So we compute the emergent strategies from the bottom up.From the above de�nition, we see that at a non-root node w of the tree, the utility

function is

uw. � / D Uw. �w/ CX

nonroot ancestor v wvuv. � /

D Xs2Sw�Qx2Bw Sx

Uw.s/ �w.sw/Yx2Bw

�x.sx/ CX

nonroot ancestor v wvuv. � /:

So we compute the utility from the top down.We see that the utilities of each actual player (the leaf nodes) may depend on

the strategies executed by every other actual player. So, the graphical model of theactual game may be the complete graph. Imposing an emergent node tree structure,corresponds to deleting some of these edges and adding more nodes, and edgesconnected to those nodes, to the graph, so that the new graph has a nontrivialstructure. With the addition of the new variables �v.svk/, we get more informationabout the sparsity of our multilinear equations.

In our de�nition, we did not require that the numbers vw have the same sign forall descendants v of a node w. Thus, our de�nition does not require that the emer-gence of a node represent a common interest among its descendant nodes (althoughof course it does cover that situation).

For example, consider a normal form game with the ENT in Figure 3.5 wherethe leaf nodes correspond to

1. An American citizen

2. A Soviet saboteur living in America

3. A Soviet citizen

55

7World

5America

6Soviet Union

1American citizen

2Soviet saboteur

3Soviet citizen

4American saboteur

Figure 3.5: Emergent Node Structure For The Saboteur Game

56

4. An American saboteur living in the USSR

The parent of nodes 1 and 2 is node 5, corresponding to America, the parent ofnodes 3 and 4 is node 6, corresponding to the USSR, and the the root is node 7,corresponding to the world. Then while 15 > 0 and 36 > 0, we have 25 < 0 and 46 < 0.

We now de�ne a natural re�nement of the equilibrium concept for games withan ENTs.

De�nition. If a normal form game has an ENT as de�ned above, then a Nashequilibrium � of that game is hierarchically perfect with respect to this ENT if for everyemergent node v, given the strategies induced on the siblings of v by � , the payoffu.v/ at v cannot be increased by changing only �.v/.

Note that since our de�nition requires the linear map from the strategy space ofthe children of v to the strategy space of v to be full-rank, any strategy � 0.v/ deviatingfrom �.v/ which could result in a higher payoff u.v/ would be achievable by somestrategy pro�le of the descendants of v.

We will also need the following de�nition:De�nition. A strategy pro�le of a normal form game with an ENT is totally

mixed with respect to this ENT if it is totally mixed in the usual sense and the emergentstrategies at each emergent node are also totally mixed.

Theorem 16. For a generic game with an ENT as above, construct a directed graphical modelG whose nodes are the nodes of the tree except the root, with edges as follows: the children inT of a node v form a directed clique in G, and each such child also has a directed edge fromv and each ancestor of v except the root, and from each of their siblings. Then the Bernsteinnumber we obtain by applying Theorem 13 to this directed graphical model is an upper boundon the number of totally mixed Nash equilibria of this game which are hierarchically perfect andtotally mixed with respect to this ENT.

Proof. This is the graphical model we would obtain if all the emergent players wereactual players. That is, we have ignored the equations

�v.svk/ DXs2SCv

pv.k; s/Y

w2Cv

�w.sw/:

57

So the set of totally mixed Nash equilibria of our game which are hierarchicallyperfect with respect to this ENT is a subset of the set of totally mixed Nash equilibriaof the game with this graphical model.

Generically, there may be no hierarchically perfect totally mixed Nash equilibria.If the system of equations de�ning the quasiequilibria of the game with the directedgraphical model is 0-dimensional, then none of the �nitely many solutions to thissystem may satisfy the additional equations

�v.svk/ DXs2SCv

pv.k; s/Y

w2Cv

�w.sw/:

For example, consider a game as in Figure 3.5 in which each actual player has twopure strategies and each emergent player also has two pure strategies. Generically, agame with 4 players, each with 2 pure strategies, would have

per

0BBBBB@0 1 1 11 0 1 11 1 0 11 1 1 0

1CCCCCA D 9

quasiequilibria. On the other hand, if the game has an ENT as in Figure 3.5, thenthe directed graphical model given by the theorem is as in Figure 3.4. Thus there isno more than

per

0BBBBBBBBBBB@

0 1 0 0 1 01 0 0 0 1 00 0 0 1 0 10 0 1 0 0 10 0 0 0 0 10 0 0 0 1 0

1CCCCCCCCCCCAD 1

quasiequilibrium which is hierarchically perfect and totally mixed with respect to thisENT. Indeed this would hold whenever the ENT is a binary tree, that is, each non-leafnode has two children, and all siblings have the same number of pure strategies.

For example, say that if players 1 and 2 either both choose their 0th pure strategyor both choose their 1st pure strategy, then the emergent strategy of node 5 is s51,

58

1 2 3 4

65

Figure 3.6: Graphical Model For The Saboteur Game

otherwise it is s50. Similarly, if players 3 and 4 either both choose their 0th purestrategy or both choose their 1st pure strategy, then the emergent strategy of node 6is s61, otherwise it is s60. Let U5 and U6 be given by

0@s60 s61

s50 0; 0 0;�1s51 7; 0 �5; 1

1A; (3.2)

(where the .i; j/th entry is the pair U5.s5i; s6 j/;U6.s5i; s6 j/). Let 1 D 3 D 1 and 2 D 4 D �1. Let U1 and U2 be given by

0@s20 s21

s10 0; 0 0;�1s11 1; 0 �4; 1

1A; (3.3)

and let U3 and U4 be given by

0@s40 s41

s30 0; 0 0;�1s31 1; 0 �3; 2

1A: (3.4)

We abbreviate �i.si1/ as �i by abuse of notation. At a totally mixed Nash equilibrium� which is hierarchically perfect and totally mixed with respect to the ENT of Figure

59

3.5, we have 0 D U5.s50; �6/ D U5.s51; �6/ D 7.1� �6/� 5 �6 D 7� 12 �6, so �6 D 712 .

Similarly we have 0 D �.1� �5/ C �5 D 2 �5 � 1 so �5 D 12 .

We also have u1.s10; �2; �5; �6/ D U1.s10; �2/ C u5. �5; �6/, which we must equateto u1.s11; �2; �5; �6/ D U1.s11; �2/ C u5. �5; �6/, for hierarchical perfection (here weare ignoring the fact that �5 is a function of �1 and �2). This gives us that0 D U1.s10; �2/ D U1.s11; �2/ D .1 � �2/ � 4 �2 D 1 � 5 �2, so �2 D 1

5 . Simi-larly we have u2.s20; �1; �5; �6/ D U2.s20; �1/ � u5. �5; �6/, which we must equate tou2.s21; �1; �5; �6/ D U2.s21; �1/ � u5. �5; �6/, so U2.s20; �1/ D U2.s21; �1/. This gives0 D �.1� �1/C �1 D 2 �1�1, so �1 D 1

2 . We also have 0 D .1� �4/�3 �4 D 1�4 �4,so �4 D 1

4 , and 0 D �.1� �3/ C 2 �3 D 3 �3 � 1, so �3 D 13 .

Finally, we check that �1 �2 C .1 � �1/.1 � �2/ D 110 C 4

10 D 12 D �5, and �3 �4 C

.1� �3/.1� �4/ D 112 C 6

12 D 712 D �6. Now given ��1, player 1 cannot increase either

U1 or u5 by changing only �1, so player 1 cannot increase u1. Similarly, player 2 canneither increase U1 nor decrease u5 by changing only �2, so player 2 cannot increaseu2. In this way, we see that � is a Nash equilibrium of the actual game.

A strategy pro�le of the actual players is a point in the product of probabilitysimplices corresponding to their actual strategy spaces. When we pass to an emergentplayer one level up, we project the product of simplices for the actual players belowthat emergent player to a smaller dimensional simplex, the space of emergent mixedstrategies of this emergent player. That we are able to do this means that the payoffsto other actual players, not below this emergent player, depend only on the choice ofa point in the smaller dimensional simplex by these actual players.

We can use ENTs to analyze certain cooperative games. We consider each coali-tion to be an emergent player. An actual player's pure strategies specify the highestlevel of coalition to join. So the number of its pure strategies is the number of itsancestors in the tree(including itself). Each coalition forms if all its descendants agreeto join it, otherwise it doesn't form. The number of pure strategies of a coalitionis one more than the number of its ancestors (including itself). Its pure strategiescorrespond either to the highest level of coalition containing this coalition which itsmembers have agreed to form, or to not forming this coalition itself. The functionUv for each coalition v is zero if the coalition forms and is equal to the value of the

60

coalition if it does form; it does not depend on the actions of v's siblings. The number vw represents v's share of the gain from the larger coalition w, if it forms.

Note that a given ENT does not allow all possible subsets of players to formcoalitions, but only certain ones. We could extend the de�nition to all possiblesubsets by positing that for any partition of a coalition into subcoalitions not in thetree, the subcoalitions receives the same utility by joining or not joining the coalition.Thus not all cooperative games correspond to ENTs. Those that do, however, mayoften occur in modeling real situations.

61

Chapter 4

Tools For Computing Nash Equilibria

4.1 Introduction

The main tool for computing Nash equilibria today is the free software package Gambitof McKelvey, McLennan, and Turocy [MMT]. It implements a variety of techniquesfor �nding a single Nash equilibrium of a game, and a single technique for �ndingall the Nash equilibria. Since �nding all the Nash equilibria is dif�cult, they are notoften computed in practice. However, techniques for solving systems of polynomialequations have continued evolving for several years since their implementation inGambit. Here we experiment with the use of various general-purpose polynomialsystem solvers to solve polynomial equations arising from games, and thus �nd all theNash equilibria. Our goal is to determine which of the algebraic techniques todayperforms best for these problems. We �nd that the polyhedral homotopy continuationpackage PHC of Verschelde is robust and able to solve games with thousands of Nashequilibria. Furthermore, polyhedral homotopy continuation clearly lends itself toparallelization, although we have not had the opportunity to experiment with this.

We concentrate on the problem of computing all totally mixed Nash equilibria.Recall that once we have a procedure to do this, it can be used as a subroutine tocompute all Nash equilibria. For every subset of the set of all pure strategies of allplayers not containing all of any particular player's pure strategies, one derives a newnormal form game in which that subset of pure strategies is unavailable, and �nds

62

the totally mixed Nash equilibria for the new game. Then one checks if these wouldstill be Nash equilibria if the deleted pure strategies were available. If so, then theseare partially mixed Nash equilibria of the original game, that is, they are equilibriain which the probabilities allocated to the pure strategies in the subset are zero. Inthe special case when for each player, the subset contains all but one of that player'sstrategies, the resulting point is trivially a Nash equilibrium in the new game, andpotentially a pure strategy Nash equilibrium of the old game. This is in fact howGambit computes all Nash equilibria of a game with more than two players. (Thereare many other techniques to �nd a single Nash equilibrium. However, none ofthese techniques, even when repeated, can be guaranteed to �nd all Nash equilibria.Moreover, the single Nash equilibrium found in this way is not distinguished by anyspecial properties which would make it more signi�cant in the analysis of the game.)

We have seen that the problem of computing totally mixed equilibria reduces tothat of solving a system of polynomial equations subject to some inequality constraints.As we saw in Chapter 2, the solution set can be (stably isomorphic to) any algebraicvariety (i.e., to the solution set of any polynomial system). However, Harsanyi [Har73]showed that the set of payoffs for which there are �nitely many Nash equilibriais a generic set. In his formulation �generic� meant �except on a set of measurezero�. From an algebraic point of view, this implies that for any assignment of payoffvalues outside of an algebraic subset of positive codimension in the space of all suchassignments, the solution set to the polynomial system is a zero-dimensional algebraicvariety. This chapter focuses on applying various techniques to compute the completeset of totally mixed Nash equilibria.

4.2 The Status Quo: Gambit

Gambit, developed by McKelvey, McLennan, and Turocy, is currently the standardsoftware package for computing Nash equilibria. Most of the code focuses on solvingtwo-person games. (Indeed, a large proportion of the game theory literature itselffocuses on two-person games. This may be because the two-person situation isalready quite rich and interesting, but it also may be partially due to the current

63

inability to solve even moderate examples of games with more than three players.)For normal-form games with more than two players, the program can compute allequilibria with a routine called PolEnumSolve. It can also compute single equilibria(or multiple equilibria, one at a time, with no guarantee at any point that all havebeen found) with several other algorithms. However, since the algebraic techniques(including PolEnumSolve) which we are comparing here solve for all equilibria, wewill only consider PolEnumSolve. By default, PolEnumSolve solves for all Nashequilibria by recursing over the possible subsets of used strategies as explained above.But we chose not to recurse for ease of comparison with the other methods, whichwe will use to compute all (and only) the totally mixed Nash equilibria.

PolEnumSolve works by spatial subdivision, a technique often used as well incomputer-aided geometric design. The algorithm starts with a higher-dimensionalcube that contains the entire strategy space. It uses Newton iteration to �nd asolution within this cube. If there is one, it checks whether this solution is within thestrategy space; if not it discards it and starts again. When it �nds a bona �de solution,it checks that it is the unique solution within this cube�in fact, within the spherecircumscribing the cube. Basically, if the system were linear it could have no othersolutions at all (since the solution set is zero-dimensional). Gambit checks that thelinear Taylor approximation is good enough (that is, the nonlinear part of the systemis small enough) within this sphere to guarantee that there is no other root inside. Ifthe solution cannot be guaranteed to be unique, then the cube is subdivided and theprocess is repeated within the smaller cubes.

We generated various games with random entries to try to solve with Gambit.(We used the standard Ocaml library routines to generate random numbers uniformlydistributed within a �xed range.) Unfortunately the current version of Gambit (version0.97.0.3, �Legacy�), is extremely unstable and crashes with a segmentation fault onmany of the simplest games. The only games which it was able to solve with anyconsistency were the smallest case of more than two players, namely three playerseach with two pure strategies. (Even here many segmentation faults occurred.) Thesetook it from 60 to 160 ms to solve. (All these computations, and all others reportedhere, were done on the same machine: a Dell Latitude C840 2.0GHz Mobile Pentium

64

4 laptop with 1GB RAM running Linux kernel version 2.4.19.)

4.3 Pure Algebra: Grobner Bases

The set of solutions to a system of polynomial equations is called an algebraic variety.Conversely, consider the set of polynomials which vanish on a set of a points. Anysum of two such polynomials will also vanish on the same set, and so will any productof such a polynomial with any other polynomial. These two conditions mean that theset of polynomials vanishing on a set is an ideal, i.e., closed under addition and undermultiplication by a polynomial.

A generating set for an ideal is a set of elements such that every other elementis a sum of products of these elements with other polynomials. It so happens thatevery polynomial ideal has a �nite generating set. Given such a generating set, onemight try to determine whether a particular candidate polynomial lies in the idealby dividing by the polynomial generators. However, in general, the remainder is notuniquely determined, and may not be zero even though the candidate polynomiallies in the ideal. This state of affairs is remedied by requiring that the generating setsatisfy certain technical conditions (Buchberger's criterion). If it does, it quali�es as aGrobner basis.

A Grobner basis is de�ned with respect to a particular term order. There is a naturalpartial order on monomials, namely that induced by divisibility (with 1 being the leastmonomial). A term order extends this partial order to a total order, while respectingmultiplication. More precisely, if m1 � m2 for a term order �, then mm1 � mm2 forany monomial m. Every polynomial ideal has a Grobner basis with respect to eachterm order. If we specify that the Grobner basis must be reduced, then it is uniquefor a given term order. (This means that no term in any element of the basis can bedivisible by the leading term of another element of the basis.) Grobner bases can beused to solve many of the fundamental problems of computational algebra.

Perhaps the most intuitive term order is the lexicographic one. One speci�es anordering of the variables. Then in comparing two monomials, one �rst comparesthe powers of the heaviest (greatest) variable. If they are unequal, this is decisive;

65

otherwise one compares the powers of the next heaviest variable, and so on.The reduced Grobner basis with respect to the lexicographic term order almost

always has higher degrees than the Grobner basis with respect to some other termorders, and so the computational complexity of many algorithms is worsened whenusing this order. However, this term order in particular supports solving 0-dimensionalpolynomial systems. It follows from elimination theory that from the reduced Grobnerbasis, a collecton of triangular sets describing the solutions can be computed. Atriangular set consists of a polynomial in which only one variable occurs, one inwhich that and another variable occur, one in which those two and a third variableoccur, and so on. The roots of the �rst polynomial can be found numerically (or bya combination of symbolic and numerical methods). Then each of these values canbe substituted into the second polynomial, making it a polynomial only in the secondvariable. Solving this numerically gives the values of the second variable, and so on.

Two popular software packages for Grobner basis computations are Macaulay2 [GS]and Singular [GPS01]. Unfortunately Macaulay2 is not currently well set up for solvingpolynomial systems, although support is planned for the future. In our tests it tookabout 10 ms to �nd a Grobner basis for the case of 3 players with 2 strategies each,and 2.64 seconds to �nd a Grobner basis for the case of 4 players with 2 strategieseach. On larger instances it exited with a segmentation fault. Of course, this doesnot include the time to actually use the Grobner basis to �nd the roots, which wouldrequire exporting the problem to another numerical solving routine such as one fromthe Netlib repository or in Matlab. This is not trivial, since Macaulay2 computes Grobnerbases with arbitrary-precision coef�cients (and indeed, the coef�cients can rapidlybecome very large), whereas numerical solvers usually compute in �xed-precision.

Singular, on the other hand, did much better. It comes with a standard librarysolve.lib for complex symbolic-numeric polynomial solving. Using the main rou-tine solve from this library, we were able to solve the case of 3 players with 2strategies each in 10ms, and with 3 strategies each in 1150 ms. The case of 4 playerswith 2 strategies each was solved in 70 ms.

While the above results seem promising, they are eclipsed by the performance ofthe polyhedral homotopy continuation method, to which we turn next.

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4.4 Polyhedral Homotopy Continuation: PHC

The polynomial systems we want to solve are very sparse. That is, given the totaldegree of each equation, we don't see all the terms that there could be in an equa-tion of that degree. Speci�cally, these monomials are multilinear. Moreover, in theequation associated with player i, none of the variables associated with i appear, andin each term of such an equation, only one of the variables associated with any otherparticular player j can appear at a time.

The number of solutions to such a sparse 0-dimensional polynomial system isgenerically far smaller than the Bezout number obtained by multiplying total degrees.The sparsity of the system can be described in terms of the exponent vectors occurringin the monomials which occur in each polynomial. For a single polynomial, theconvex hull of these vectors forms a lattice polytope, called its Newton polytope. TheMinkowski sum of two polytopes is obtained by translating one of them by each of thevectors in the other and taking the convex hull of the result. Similarly, the Minkowskisum of any �nite number of polytopes can be de�ned inductively. One can subdividethe Minkowski sum (non-uniquely) into smaller lattice polytopes by following thecourse of the various faces during the translation. This results in a mixed subdivision:each of its elements, called cells, is full-dimensional and is a Minkowski sum of facesfrom all the original polytopes. If a cell is the Minkowski sum of an edge or a vertexfrom each of the original polytopes, then it is a mixed cell. The mixed volume ofthe system is the sum of the normalized (with respect to the integer lattice) volumesof these mixed cells, and it is equal to the number of roots of a generic polynomialsystem with those Newton polytopes. This number is known as the Bernstein numberafter Bernstein, who proved this equality [Ber75].

This number is only a function of the Newton polytopes. The polyhedral homotopycontinuation method, introduced by Huber and Sturmfels [HS95], takes advantage ofthis. In general, one uses homotopy continuation to solve a polynomial system by �rststarting with another system of the same multidegree whose roots are obvious (and alldistinct�roots with multiplicity are generally troublesome for numerical solvers), andgradually perturbing (or �morphing�) the coef�cients towards the system of interest.

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At each step, one �nds all the roots of the intermediate system by iterating from theroots of the system in the previous step. In this way one traces out a path from eachroot of the starting system to each root of the system of interest (which is why thismethod is also called �path-following�). The main practical drawback previously wasthat many of these paths would not lead to roots, because the starting system wasgeneric and dense, whereas polynomial systems arising in practice are usually sparse.Thus the number of paths would explode with the size of the problem. However,with polyhedral homotopy continuation, the starting system is also chosen to besparse in the same way as the system of interest. So only those paths which canlead to actual roots are followed. Indeed, Bernstein originally proved his theoremby exhibiting a similar homotopy [Ber75], and a solver using this homotopy waspreviously implemented by Verschelde, Verlinden, and Cools [VVC94].

Verschelde's software package PHC [Ver99] for polyhedral homotopy continuationis in continuous development yet is very stable. Furthermore, it is well-documentedand very simple to use. We were able to solve the following cases:

� 3 players with 2 pure strategies each: 2 roots found in 20ms.

� 3 players with 3 pure strategies each: 10 roots found in 350ms.

� 3 players with 4 pure strategies each: 56 roots found in 13s280ms.

� 3 players with 5 pure strategies each: 346 roots found in 3m19s540ms.

� 3 players with 6 pure strategies each: 2252 roots found in 48m41s870ms.

� 4 players with 2 pure strategies each: 9 roots found in 260ms.

� 4 players with 3 pure strategies each: 297 roots found in 4m3s220ms.

� 4 players with 4 pure strategies each: 13833 roots found in 7h2m20s780ms.

� 5 players with 2 pure strategies each: 44 roots found in 7s200ms.

� 6 players with 2 pure strategies each: 265 roots found in 7m10s790ms.

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The running time seems to go up somewhat superlinearly with the Bernsteinnumber (which may be considered part of the inherent complexity of the problem).Furthermore, this method is trivially parallelizable, requiring no communication be-tween processors following different paths. (In fact, quite recently an alternativeimplementation of polyhedral homotopy continuation, PHoM, including a parallel im-plementation, has been made available by Gunji, Kim, Kojima, et. al. [GKKC03].)For all but the smallest problems, it is the path-following which takes up most of therunning time. Indeed, these polynomial systems are multilinear, which is a special caseof multihomogeneity, and the mixed volume computations needed to set up the startsystem can be carried out more ef�ciently for multihomogeneous polynomial systemsthan in general. For the smallest games, though, the time used to compute the startsystem is signi�cant. The start system, which depends only on the Newton polytopeand thus can be the same for all games of a given format, could be precomputed.

4.5 Other Directions

An interesting direction for further work is the computation of Nash equilibria underuncertainty. Speci�cally, the payoff functions may not be known exactly, but onlyapproximately. A natural formulation is that each payoff value is known to lie insome interval. This leads to the question of how the set of Nash equilibria varies asthe set of payoff values (now considered as parameters) varies.

Purely symbolically, such variations could be studied either through parametricGrobner bases, as computed for example by Montes [Mon02] or Faug�ere [Fau02], orthrough resultants, as computed for example by Emiris and Canny [EC95]. For para-metric Grobner bases, the basic idea is to carry out the Grobner basis computation,treating the payoff values as parameters and assuming that no cancellations ever occur.In this way one arrives at the generic solution. (A cancellation occurs whenever twoalgebraic expressions involving the parameters are equal. Thus, if desired, one cankeep track of the algebraic equations assumed not to hold along the way, and thusdetermine in the end the algebraic subvariety of the payoff space which is not generic,as a by-product of this computation.) One might de�ne resultants as the end result

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of such computations, but in fact resultants can be expressed much more compactlyusing determinantal formulas. In either case, if such formulas were precomputed forgames of various formats, the Nash equilibria for any speci�c set of payoffs couldbe computed by evaluation of the formula in polynomial time (provided the payoffswere indeed generic; if not, division by zero would occur). Such formulas have beencomputed by Emiris in his thesis [Emi94] for very small cases, but unfortunately forlarger cases, these computations are still intractable for the present.

A recent approach to �nding real roots of polynomial systems is through semidef-inite programming. Semialgebraic constraints can include nonnegativity constraints(such as arise in our problem) as well as equations. These nonnegativity constraintsare relaxed to the (suf�cient) condition that the polynomials in question be the sumsof squares of other polynomials (which of course are of lower degree). This conditioncan be expressed as the positive semide�niteness of a matrix, namely the Gram ma-trix, which represents the quadratic form in the smaller polynomials in the monomialbasis. Unfortunately we were not able to test the primary exemplars of this approach,SOStools and Gloptipoly, which require particular versions of Matlab. However, it isclear that this approach lends itself easily to the formulation of such �robust� com-putations. One uses parametric values for the payoffs and adds the constraints on thepayoffs (for example, that they lie in a certain interval) to the problem.

As was discussed earlier, PHC succeeds at �nding all the �quasi-equilibrium�points, and the result of McKelvey and McLennan [MM97] shows that these mayall be actual Nash equilibria. Thus, there is no way to avoid worst-case complexitygiven by the Bernstein number. However, in practice it will often be the case thatmany of the �quasi-equilibrium� points do not lie in the product of simplices, or arenot even real. Practically, time would be saved by heuristic methods for examiningthe starting system and determining that a signi�cant subset of the paths will notconverge to a solution in the product of simplices and so do not need to be followed.Such heuristic methods have yet to be de�ned.

Game theorists generally would prefer that there be one distinguished equilibriumpoint for any particular game. Not only does this allow the game theorists to predictwhat will happen during the game, it allows the players themselves to predict what

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the other players will do. After all, if different players have different equilibriumpoints in mind when choosing their strategies, then the resulting behavior may noteven be at equilibrium. For this reason various re�nements of the Nash equilibriumconcept have been proposed; these are summarized in [vD87]. In the past algebraictechniques have been used to �nd all Nash equilibria, and other techniques have beenused to try to �nd a single Nash equilibrium (preferably the �best� one in one ofthese senses). It would be interesting to see if the methods we have used can bemodi�ed to compute these more re�ned equilibria.

4.6 Conclusion

Game theory is a mathematical model of strategic interaction. The main computerpackage for studying game theory today is Gambit. Although there are many ways tocharacterize Nash equilibria, the one which lends itself most easily to the computationof all Nash equilibria of a game with more than two players is as solutions to systemsof polynomial equations. However, the algorithm currently implemented in Gambitcould be outperformed by the existing polyhedral homotopy continuation softwarePHC. So hopefully PHC or some similar package will soon be incorporated into Gambit.Furthermore, there are many other promising directions to pursue in applying algebra,and in particular computer algebra, to game theory.

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