icm millennium lectures on games

Springer-Verlag Berlin Heidelberg GmbH
Leon A. Petrosyan David W.K. Yeung Editors
ICM Millennium Lectures on Games With 48 Figures and 27 Tables
, Springer
Professor David W. K. Yeung
Centre of Game Theory Hong Kong Baptist University Kowloon Tong Hong Kong, PR China and
Centre of Game Theory St. Petersburg State University Petrodyvorets 198904 St. Petersburg, Russian Federation
ISBN 978-3-642-05618-5 ISBN 978-3-662-05219-8 (eBook) DOI 10.1007/978-3-662-05219-8
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Preface
Since the first Congress in Zürich in 1897, the ICM has been an eagerly awaited event every four years. Many of these occasions are celebrated for historie developments and seminal contributions to mathematics. 2002 marks the year of the 24th ICM, the first of the new millennium. Also historie is the first ICM Satellite Conference devoted to game theory and applications. It is one of those rare occasions, in which masters of the field are able to meet under congenial surroundings to talk and share their gathered wisdom.
As is usually the case in ICM meetings, participants of the ICM Satellite Conference on Game Theory and Applications (Qingdao, August 2(02) hailed from the four corners of the world. In addition to presentations of high qual ity research, the program also included twelve invited plenary sessions with distinguished speakers. This volume, which gathers together selected papers read at the conference, is divided into four sections:
(I) Foundations, Concepts, and Structure. (II) Equilibrium Properties. (III) Applications to the Natural and Social Sciences. (IV) Computational Aspects of Games.
The papers in Section I explore fundamental ideas, leading to new and analytically interesting analysis of current problems as weIl as new games and new modeling approaches in games. In Section II, seven papers discuss issues in the solution of games, and present a number of potentially very fruitful ideas regarding game equilibrium under different assumptions and conditions. Section III is devoted to applications. In particular, the articles on market structure and game-based computations would be of interest to researehers and practitioners in commerce, industry, banking and finance, and the public sector. Section IV focuses on the computational aspects of games: both computational algorithms and computability of equilibria.
Twelve invited plenary lectures were delivered in a feast of ideas. However, given that the distinguished lecturers were delivered before their audiences and
~ f>reface
that transcriptions from video recordings are likely to be insufficiently ace urate to do justice to the subtle complexities of the reasoning, the editors have reluctantly decided to just list the lectures in this preface. Readers anxious to learn more about the great things discussed during the invited sessions can rest assured that they will be more formally published in good time.
Among the invited lectures, remarlmble new ideas regarding solutions and equilibria in games were presented by Reinhard Selten, "On Two Behavioral Equilibrium Concepts" , Lloyd Shapley, "Convexing the Pareto Set by Individ ual Order-Preserving Transformations", and Dov Samet, "Ordinal Solutions for Bargaining Problems". We can expect many analytically and practically interesting new games to follow from the insights offered by Robert Simon, "Games of Incomplete Information, Ergodie Theory, and the Measurability of Equilibria" , Abraham Neyman, "Repeated Games with Bounded Descriptive Strategie Complexity" , John Forbes Nash, "Projects Studying Cooperation in Games through Modelling in terms of Formally Non-cooperative action in a Repeated Game Context", and David Yeung, "Randomly Furcating Stochas tic Differential Games: A Paradigm for Interactive Decision-Making under Sturcture Uncertainty". Finally, researehers of game modelling will discover a cornucopia of inspiration in the lectures of Robert John Aumann, "Bayes Rational Play", Michael Maschler, "Voting for Voters", Sergiu Hart, "Simple Adaptive Strategies" , Leon Petrosjan, "Bargaining in Dynamic Games" , and Sylvain Sorin, "The Operator Approach to Zero-Sum Repeated Games".
The editors would like to thank Qingdao University and the Organizing Committee of the Conference for hosting this historie event, and the Academic Committee for striving to maintain the great ICM tradition in the presentation of high quality papers and the dissemination of thought-provoking ideas.
We trust that the ICM Millenium Lectures on Games will prove to be a volume that researehers in game theory and applications will treasure and re-read in the years to come.
Leon A. Petrosyan and David W. K. Yeung St. Petersburg, January 2003
Contents
Stable Schedule Matching under Revealed Preference A. Alkan, D. GaZe................................................ 3
Banzhaf Permission Values for Games with aPermission Structure Rene van den Brink ............. ................................. 21
Moral Hazard in Teams Revisited Baomin Dong ................................................... 47
Endogenous Determination of Utility Functions: An Evolutionary Approach Alexander A. Vasin .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75
N-person Prisoner's Dilemma with Mutual Choice by Agent-based Modeling Tomohisa Yamashita, Azuma Ohuchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 89
Randomly-Furcating Stochastic Differential Games David W K. Yeung .............................................. 107
On Modulo 2 Game Chizhong Zhou .................................................. 127
Bargaining in Dynamic Games Leon A. Petrosyan ............................................... 139
VIII Contents
Extensions of Hart and Mas-Colell's Consisteney to Eflieient, Linear, and Symmetrie Values for TU-Games Theo Driessen, Tadeusz Radzik .................................... 147
On the Stablllty of Co operation Struetures Guillaume Haeringer ............................................. 167
Time-Consistent Imputation Distribution Proeedure for Multistage Garne Victor V. Zakharov, Maria B. Dementieva .......................... 185
Rationality of Final Declsions Leads to Sequential Equlllbrium Ryuichiro Ishikawa, Takashi Matsuhisa ............................. 193
The Core in the Presenee of Externalities Ldszl6 A. K6czy ................................................. 205
Network Topology and the Eflicleney of Equilibriurn Igal Milchtaich .................................................. 233
Essential Components of the Set of Weakly Pareto-Nash Equilibria for Multiobjeetive Generalized Garnes Hui Yang ....................................................... 267
Part 111 Applieations to the Natural and Soelal Sclenees
Diseretization of Information Colleeting Situations and Continuity of Compensation Rules R. Branzei, F. Scotti, S. Tijs, A. Torre ............................. 281
Some Variation Caleulus Problems in Dynamie Games on 2D Surfaees Arik Melikyan, Naira Hovakimyan ................................. 287
On The Chinese Postman Game Daniel Granot, Herbert Hamers, Jeroen Kuipers, Michael Maschler . .... 297
Farsighted Stabillty in Duopoly Markets with Produet Differentiation Takeshi Masuda, Shigeo Muto ..................................... 305
International Emissions 'ftading with Agent-Based Simulation and Web-Based Gaming Hideyuki Miruta, Yoshiki Yamagata ................................ 319
Contents IX
Comparlson of two Economic Models for a Business-to Business Exchange Nikolay Zenkevich, Suzhou Huang .................................. 335
Part IV Computational Aspects of Games
Computability of N ash Equilibrium Hidetoshi Tashiro ................................................ 349
Numerical Algorithm for Solving Cross-Coupled Multiparameter Algebraic Riccati Equations ofMultimodeling Systems Related to Nash Games Hiroaki Mukaidani, Tetsu Shimomura, Hua Xu ...................... 359
Effects of Symmetry on Paradoxes in Non-Cooperative Dlstributed Computing Hisao Kameda, Yoshihisa Hosokawa ............................... . 373
Computation of Stackelberg Trajectories in a Class of Linear Differential Games on Plane Sergei Osipov, Anatolii Kleimenov ................................. 391
AGame Theoretic Approach to Value Information in Data Mining Yücel Saygin, Arnold Reisman, Yun Tong Wang ..................... 397
Part I
1 Sabanci University, 81474 Thzla, Istanbul, Thrkey E-mail: alkanesabanciuni v. edu
2 U. C., Berkeley, Ca. 94720, USA E-mail: gale@math. berkeley. edu
Sum.m.ary. In arecent study Baiou and Balinski [3] generalized the notion of two sided matching to that of schedule matching which determines not only what part nerships will form but also how much time the partners will spend together. In particular, it is assumed that each agent has a ranking of the agents on the other side of the market. In this paper we treat the scheduling problem using the more general preference structure introduced by Blair [5] and recently refined by Alkan [1, 2], which allows among other things for diversity to be a motivating factor in the choice of partners. The set of stable matchings for this model turns out to be a lattice with other interesting structural properties.
Key words: Stable Matching, Two-sided Market, Lattice, Diversity, Sched ule JEL Classijication Numbers: C71, C78, D49
1 Introduction
The formulation of the Stable Matching Problem [7J was originally motivated by the real world problem of college admissions. It was an attempt to find a rational criterion for matching students with colleges which respected the preferences of both groups. The original approach was to first consider a spe cial case, the so-called Stable Marriage Problem in which each college could accept only one student. The general case was then reduced to the marriage case hy assuming that each college had a complete preference ordering on the set of students it was willing to admit as weH as a quota giving an upper hound to the numher of students that could be admitted. The model has applications
* Partial support by Thrkish National Academy of Sciences is gratefully acknowl edged as are useful comments from participants at Telaviv, Hebrew, Pennsyl vaina, Northwestern, Columbia University seminars, SAET Ischia meeting, Is tanbul NATO Advanved Research Workshop.
L. A. Petrosyan et al. (eds.), ICM Millennium Lectures on Games © Springer-Verlag Berlin Heidelberg 2003
4 A. Alkan, D. Gale
in other situations. A particularly natural application is the problem of hiring of workers by firms. 3 In general we refer to such a model as a market and the participants on the two sides as agents.
The present paper presents a broad generalization of the original model incorporating extensions in several directions.
(1) Agents on both sides of the market may form multiple partnerships. (2) Preferences of agents over sets of possible partners are given by choice
functions which are more general than those given by complete orderings of individuals. This is especially relevant for the college market where colleges are typically interested in the overall composition of an entering dass, particularly these days as regards diversity.4 A simple example will illustrate the point.
College A can admit two students. The applicants are two men m and m' and two women wand w'. A's first choice is the pair mw but if m (w) is not available the choice is m'w (mw'). One sees at once that these choices are not possible from any strict ordering since A would have to order the pair m', w'. For example if the ordering was m > w > m' > w' then it would mean that mm' was preferred to the diversified pair m'w'. Indeed, as regards diversity, in the algorithm which solves the original college admissions problem, there is nothing to prevent a college from ending up with a dass which is either ninety percent male or female.
The remedy for this via choice functions simply formalizes what happens approximately in actual negotiations between colleges and students or firms and workers. Each agent is assumed to have a choice function C which, given a set P of agents on the other side of the market, picks out the most preferred subset S = C(P) contained in P. S is then said to be revealed preferred to all other subsets of P. The case where colleges rank-order applicants is then a special ease in which C(P) eonsists of the q highest ranked applieants in P, but if, for example, the goal was gender balance one could choose, roughly, the highest ranked q/2 applicants of each sex or if, say there was an insufficient number of male applicants then choose all the men and fill the quota with the highest ranked women.
TOhe choice function approach was first introduced by Blair [5] using some what different terminology, and a very recent variation on this idea has been given by Alkan [1, 2]. From now on we will refer to the case where agents are linearly ordered as the classical caseo
It is worth pointing out that while our preference stucture is mathemat ically equivalent to that of Blair [5] we do not assurne as he does that that
3 See [8, 10]. 4 We quote Mr. Bollinger, the president of the University of Michigan: "Admissions
is not and should not be a linear process of lining up applicants according to their grades and test scores and then drawing a line through the list. It shows the importance of seeing racial and ethnic diversity in a broader context of diversity, which is geographie and international and socio-economic and athletic and all the various forrns of differences, complementary differences, that we draw on to compose classes year after year."
Stahle Schedule Matching under Revealed Preference 5
agents have a complete ordering of subsets of agents on the other side of the market. In our approach there is only a partial order on subsets. For example, if a college with quota 2 in the classical case ranks students a > b > c then by revealed preference the pair ab is prefered to bc and ac, since given the tripie abc the pair ab is chosen, hut the pairs ac and bc are not comparable since neither of these pairs will be chosen from abc.
(3) Recently Baiou and Balinski [3] have generalized the not ion of matching to that of a schedule matching. In the context of a set of workers W with members wand a set of firms F with members f, the idea is that a firm decides not only which workers it will hire but also how many man-hours of employment to give each of them. Similarly, the workers must decide how many of their available hours to allocate to each job. A schedule is then a F x W matrix X whose entries x(Jw) give the amount of time worker w works for firm f. The schedule matching is said to be stable (or pairwise stable) if there is no pair fand w who could make themselves better off by increasing the hours they work together while not increasing (possibly decreasing) the hours they work with their other partners.5 This is the natural generalization of stability for ordinary (college admissions) matchings. In fact matchings correspond to the special case of schedule matchings where all entries of X are either 0 or 1. In [3] it is assumed that we are in the classical case where each agent has a strict ordering of the agents on the other side of the market and preferences on schedules are given by the condition that an agent, say a worker w, is made better off if he can increase the time he works for firm f by reducing the time he works for some less preferred firm f'.
The present paper treats schedules using the more general revealed prefer ences. Our main result shows that under appropriate conditions which include the classical case the set of stable matchings forms a distributive lattice with other interesting structural properties.
In the next section we develop the necessary material on the revealed preference ordering of an individual and show that if the preferences are con sistent and persistent (to be defined) then the set of schedules becomes a (non-distributive) lattice with other important structural properties. These are then used in the following section to prove that stable schedules always exist. The lattice properties are then derived in the final sections.
2 The Revealed Preference Lattice
In the matching theory of later sections we will think of economic agents as firms and workers, or students and colleges, men and women, etc. However, the theory of revealed preference of the individual belongs to the general standard model of consumption or demand theory, and it will be presented in this context here.
5 If coalitions other than pairs can form to hlock a matching, then stahle matchings may not exist. See (11).
6 A. Alkan, D. Gale
An agent (consumer) chooses (demands) amounts of n items (goods) from given availabilities of each item. This is formalized as follows.
Let R+ be the nonnegative orthant, b an upper bound vector and B = {x E R~ I x ~ b}. Let B be a subset of B which is closed under V and 1\ (the standard join and meet in Rn). A choice function is a map C: B ---+ B such that
C(x) ~ x
for all xE B. The elements x = (x(1), . .. , x(n)) of the domain B will be called choice vectors. The range of C is denoted by A and its elements are called (acceptable) schedules.
The most relevant domains for our purposes are the divisible domain B itself and the discrete domain that consists of all the integer vectors in B. When all bounds are equal to 1, the discrete domain corresponds to the case of ordinary multipartner matching as in college admissions.
An important special case of our model is one in which the items can be measured in some common unit, for example, dollars worth for goods, or man-hours for services. In this case we denote the sum of the entries of a vector x by lxi and call it the size of x. In such a model an agent may have a quota q which bounds the size of the schedule he can choose. For the college admissions case q is the maximum number of students a college can admit.
A choice function C is called quota jilling if
IC(x)1 = q if lxi ~ q and C(x) = x otherwise.
Two interesting examples of quota filling choice functions are as follows.
Example 1. The items are ranked so that, say, item i is more desirable than i + 1. Given a choice vector x with lxi> q, let j be the item such that r = Ei x(i) ~ q and r + x(j + 1) > q. Then
C(x) = (x(1), .. . , x(j), q - r, 0, ... ,0) .
Thus, the agent filIs as much of his quota as possible with the most desirable items. We will henceforth refer to this C as the classical choice function.
Example 2. The domain is B. Given a choice vector x with lxi> q, let r be the number such that Ei r 1\ x(i) = q. Then
C(x) = (r 1\ x(1), .. . , r 1\ x(n)) .
In words, the agent tries to use all items as equally as possible. (On the discrete domain, as in college admissions, there may be more than one such schedule hence a tie-breaking criterion is necessary.) We will refer to C as the diversifying choice function.
As an illustration, suppose an agent with quota 5 is given the choice vector (2,1,0,4,2). Then, the classical choice function chooses the schedule (2,1,0,2,0) while the diversifying choice function chooses (4/3,1,0,4/3,4/3).
Definition 1. We say that x E A is revealed preferred to y E A, and write x t y, if C(x vy) = x. We write x >- y if x t y and x =/; y.
We now impose some standard conditions on the choice function C.
Definition 2. C is consistent if C (x) :s:; y :s:; x implies C (y) = C(x).
This is a highly plausible assumption. Applied to college admissions, it says that if some set S of students is chosen for admission from a pool P then the same set will be chosen from any subset of P which contains S.
An immediate consequence of consistency is that C(x) = x if and only if x E A. Without some furt her restrictions, revealed preference will not be transitive, hence not a partial ordering, as shown by the following example for the college admissions case.
Example 3. A college C can admit two students from two men m, m' and two women w, w'. The pair mw is C' s first choice, but if either w or m are not available then
(i) C(mm'w') = mw' ,
(ii) C(m'ww') = m'w' .
(In the case of college admissions, we will use the customary notation and represent a choice vector or schedule x by the set of all students s for whom x(s) = 1.) Transitivity fails because from (i) we have mw' >- m'w' and from (ii) m'w' >- m'w but mw' and m'w are not comparable since C(mw'm'w) = mw.
To avoid this situation, we introduce the following condition of persistence (which extends the well known condition on the discrete domain that has been called substitutability ).
Definition 3. Cispersistent if x ~ y implies C(y) ~ C(x) 1\ y.
For the college admissions problem, persistence means that if a college offers admission to a student from a given pool of applicants then it will also admit hirn if the pool of applicants is reduced. The condition is closely analogous to the gross substitutes condition in production theory where one assurnes that an input which maximizes profits at given prices will remain profit maximizing if the price of some other good is increased [8). It should be pointed out that there are natural choice functions in which persistence fails. Suppose as in Example 3 that mw is a college's first choice but it prefers not to separate m' and w' so that C(mm'w') = C(m'ww') = m'w'. This clearly vio lates persistence. In general, persistence rules out this sort of complementarity between items.
It is easy to verify that the classical and diversifying choice functions satisfy consistency and persistence.
An immediate consequence of persistence is that if x E A and x ~ y then y E A.
8 A. Alkan, D. Gale
Definition 4. Cissubadditive if C (x vy) $ C(x) Vy for all x, y.
Lemma 1. If C is persistent then it is subadditive.
Proo/. Since C(x Vy) $ x Vy, we have
C(x Vy) = C(x Vy) /\ (x Vy) = (C(x Vy) /\ x) V (C(x Vy) /\ y) (1)
by distributivity. Since x $ x V y, we have C(x V y) /\ x $ C(x) by persis tence. Also C(x Vy) /\ Y $ y. Substituting these two inequalities in (1) gives subadditivity. 0
Definition 5. C is stationary ifC(x vy) = C(C(x) Vy) for all x,y.
Lemma 2. If C is subadditive and consistent then it is stationary.
Prao/. By subadditivity C (x vy) $ C(x) V y. Also C(x) V y $ x V y. So C(C (x) Vy) = C (x Vy) by consistency. 0
It will be assumed from here on that all choice functions are consistent and persistent.
Notation. We write x Y y for C(x Vy).
As immediate consequence of stationarity, we have
Corollary 1. The relation!:: is transitive and x Y y is the least upper bound ofx and y.
Proo/. The operation Y is associative: (x Y y) Y z = C(C(x V y) V z) = C((x Vy) V z) = C(x V (y V z)) = C(x V C(y V z)) = x Y (y Y z). Thus, if x!:: y, y !:: z then x Y z = (x Y y) Y z = x Y (y Y z) = x Y Y = x so x !:: z. Also, if z !:: x, z !:: y then z Y (x Y y) = (z Y x) Y Y = z Y Y = z so z !:: x Y y.
o
Thus, the set of schedules A is an upper-semilattice (with join Y) in the partial order given by !::. It is, in fact, a lattice and we will need an expression for its meet .A.. First note, it follows at once from stationarity that if C(x) = z and C(y) = z then C(xVy) = z.
Definition 6. The closure xE B of xE Ais sup{y E B I C(y) = x}.
In the classical college admissions case, x consists of x together with all students ranked below the least desired student in x. We henceforth assume that C is continuous. It then follows that C(x) = x. Define C: A --+ B by
C(x)=x.
Lemma 3 (Isomorphism). The mapping C is a lattice isomorphism from (A,!::) to (B, ~).
Stable Schedule Matching under Revealed Preference 9
Proof. It suffices to prove that Cisorder preserving: suppose x ~ y. Then x = C (x Vy) = C( x V Y) by stationarity. So by definition of closure x V y ~ x thus Y ~ x. Conversely, if x 2 y then x V y = x, so x = C(x) C(xvY) = C (x Vy) by stationarity, that is, x ~ y. 0
Lemma 4. The revealed preference meet is given by x J... Y = C(x /\ Y).
Proof. We must show (i) C(x /\ y) :::S x (and C(x /\ Y) :::S y) and (ii) z :::S x and z :::S y implies z:::s C(x /\ Y).
By definition (i) is true if and only if C(C(x /\ Y) V x) = x. By stationarity this is equivalent to C(x' ) = x where x' = (x /\ y) V x and, since x ~ x' ~ x and C(x) = x, the result follows by consistency.
To prove (ii) we must show that C(C(x /\ Y) V z) = C((x /\ Y) V z) (by stationarity) = C(x /\ y), so note that z :::S x means C (x V z) = x, hence by definition of closure x V z ~ x, so z ~ x and similarly z ~ y so z ~ x /\ Y so (x /\ Y) V z = x /\ y and the result follows. 0
Note that in college admissions, xJ...y may include students who are neither in x nor y: suppose there are four students 1,2,3,4 ranked in that order, and x = {1,3} ,y = {2,3}. Then x J...y = {3,4}.
We will need some furt her properties of the revealed preference lattice.
Lemma 5. x J... Y 2 x /\ y.
Proof. Since x 2 x /\ y, we have from persistence x J... y = C(x /\ y) 2 C(x) /\ x/\y=x/\x/\y=x/\V 0
Lemma 6. (x J... y) /\ (x Y y) ~ x /\y.
Proof. Since x V y 2 x, we have from persistence C( x) = x 2 C( x V y) /\ x = (x Y y)/\ x from stationarity, and similarly y 2 (x Y y)/\ 'fj, so x /\ Y 2 (x Y y) /\ (x /\ Y) 2 (x Y y) /\ C(x /\ Y) = (x Y y) /\ (x J... y) from Corollary 4. 0
2.1 Satiation
The following definition is basic.
Definition 7. A schedule x is i-satiated if Ci (y) ~ x( i) for all y 2 x.
In words, x is i-satiated if the agent would not choose more of item i if it were offered. To illustrate, in the classical case, x is i-satiated if i is the highest ranked item with x(j) = 0 for j > i. For the diversifying choice function, x is i-satiated if x(i) = maxj {x(j)}. The following properties will be needed in the next section.
Lemma 7. x is i-satiated ifthere exists y 2 x, y(i) > x(i) such that C(y) ~ x.
10 A. Alkan, D. Gale
Proof. Suppose z ~ x and z(i) > x(i) (otherwise there is nothing to prove). Let y' = z I\y and note that y'(i) > x(i). Now y ~ y' ~ x so by consistency C(y') = C(y) ::; x. Also z ~ y' so by persistence x ~ C (y') ~ C(z) 1\ y' so x(i) ~ Ci(z) 1\ y'(i) but since Y'(i) > x(i) we have Ci(z) ::; x(i). 0
Lemma 8. x is i-satiated if and only if x(i) = b(i).
Proof. If x(i) = b(i) there is nothing to prove so suppose x(i) < b(i). If x(i) = b(i) then x is i-satiated by the previous Lemma. If xis i-satiated then let y = xVbi where bi is the vector wlth ith entry b(i) and others O. Then from satiation Ci (y) ::; x( i) and since C j (y) ::; x(j) for j =I i we have C(y) ::; x ::; y so by consistency C(y) = C(x) = x so y ::; x so x(i) = b(i). 0
Lemma 9. Suppose x!:::: y. (i) If y is i-satiated then x is i-satiated. (ii) If x(i) > y(i) then y is not i-satiated.
Proof. (i) By the isomorphism lemma, x ~ y so x(i) ~ y(i) = b(i) so x is i-satiated by the previous Lemma. (ii) Since x !:::: y we have x Vy ~ Y and C (x Vy) = x so Ci (x Vy) = x(i) > y(i) so y is not i-satiated. 0
Lemma 10. (i) If x or y is i-satiated then x Y y is i-satiated. (ii) If x and y are i-satiated then x A y is i-satiated.
Proof. (i) Say xis i-satiated. Then since xYy!:::: x the conclusion follows from Lemma 9(i). (ii) We have C((x A y) V bi) = C(C(x I\y) V bi) = C((x I\y) V bi )
(by stationarity) = C((x V bi) 1\ (y V bi)) = C(x 1\ y) (using Lemma 8, since x and y are i-satiated) = x A y. 0
3 Stahle Matchings
We now consider two finite sets of agents which we interpret as firms, F, with members f, and workers, W, with members w, having respectively the choice functions CI,Cw , with ranges A"Aw. We write Y I, A/,!::::I for thejoin, meet, preference ordering for f, and similarly for w.
A matching X is a nonnegative Fx W matrix whose entries, written x(fw) , represent the amount of time w works for f. We write x(f) for the f-row and x( w) for the w-column of X. We assume all matchings X are bounded above by some positive matrix B. The choice functions CF, Cw are defined from CI,Cw in the natural way.
The revealed preference ordering for agents translates in an obvious way to an ordering on matchings.
Definition 8 (Group Preference). The matching X is preferred to Y by F, written X!::::F Y, if x(f) !::::I y(f) for all f in F.
Definition 9 (Aeeeptability). A matching X is F-acceptable if x (f) E A f for alt f, and it is W-acceptable if x (w) E A w for alt w. It is acceptable if it is both Fand W -acceptable.
The fundamental stability not ion is now formalized as folIows.
Definition 10 (Stability). An acceptable matching X is stable iJ, for every pair fw, either x(f) is w-satiated or x(w) is f-satiated (or both).
3.1 Existenee, Polarity, Optimality, Comparative Staties
We will show that stable matchings always ex ist by constructing a sequenee of alternately Fand W -acceptable matchings which converge to a stable match ing.6 The method is the standard one in which the firms make offers of em ployment to the workers who then choose (via their choice functions) their most preferred schedule. Firms whose offers have been declined then make al ternative offers. Of course the proof must make use of persistence of all firms' and workers' choice functions since counterexamples exist if this condition is not satisfied (see Section 4).
Theorem 1 (Existenee). There exists a stable matching.
Proof. Define the sequences (Bk), (Xk), (yk) by the following recursion rule:
BO = B, X k = CF(Bk) , yk = CW(Xk) ,
and B k+1 is obtained from Bk as folIows:
bk+l(fw) = bk(fw) if yk(fw) = xk(fw) ,
bk+l(fw) = yk(fW) if yk(fw) < xk(fw).
The matrices Bk will be called the choice matrix for the firms. Note that (Bk) is a nonincreasing nonnegative sequence and hence converges, so by continuity of CF it follows that (Xk) converges, and hence by continuity of Cw it follows that (yk) converges. Call the limits fj, X, Y. Note that if xn is W -acceptable for some n then the sequence becomes stationary so this will be included as a special case. We will show,
(i) X = Y and hence it is acceptable, (ii) X(= Y) is stable. To prove (i), note that yk:::; X k :::; Bk. If, for some fw, x(fw) -fj(fw) >
10, then xk(fw) - yk(fw) > 10 for infinitely many k and therefore from the recursion rule bk(fw) - bk+l(fw) > 10 which is impossible since Bk converges.
6 In the discreet case the sequence actually terminates after a finite nurnber of iterations. It is not known whether this is also true for the general continuous case treated here.
To prove (ii), we first show that yk+! !::w yk, thus workers are "bett er off" after each step of the recursion. From the recursion rule Y k ::;; Bk+! ::;; Bk, so from persistence we have
Cp(Bk+!) = X k+! ~ Cp(Bk) /\Bk+! ~ Xk /\ yk = yk,
so yk+l = CW(Xk+!) is revealed preferred to yk. It follows by continuity that
~ k Y!::w Y . (2)
Now suppose fj(f) is not w-satiated. Then from Lemma 8 fj(fw) < b(fw) so from the recursion rule, for some k, yk (fw) < xk(Jw) so, since yk (w) = Cw(xk(w)) ::;; xk(w), from Lemma 7 we have yk(w) is f-satiated and from (2) fj(w) !::w yk(w), so from Lemma 9(i) fj(w) is f-satiated. This proves stability ofY. 0
The following are extensions of familiar properties of the marriage market (see [10, 6]).
Lemma 11. Let X be a stable matching and let Y be an F-acceptable match ing such that y!::p X. Then Cw(X V Y) = X.
Prao/. If the conclusion is false, then there is some w such that Cw(x(w) V y(w)) = z(w) =/: x(w). Hence, z(w) >-w x(w), so z(Jw) > x(Jw) for some f, hence from Lemma 9(ii) x(w) is not f-satiated, but z(fw) ::;; y(Jw) so x(Jw) < y(Jw) and by hypothesis y(J) !::I x(J) so again by Lemma 9(ii) x(J) is not w-satiated, contradicting stability of X. 0
Corollary 2 (Polarity). 1f X, Y are stable matchings then X !::p y <==::}
Y!::w X .
Theorem 2 (Optimality). 1f X is the matchinJ!.. given by the Existence The orem and X is any other stable matching then X !::p X.
Prao/. Let X be a stable matching. We will show that X ::;; B where Ei is the matrix given in the Existence Theorem. Since X = Cp(B), the conclusion folIows. So suppose not. Then there is an index k such that Bk ~ X k but bk+!(Jw) < x(Jw) for some fw. From the recursion rules, this means that
(3)
Now X k = Cp(Bk) so Xk!::p X so from Lemma 11 Cw(X V X k) = X. But Xv X k ~ X k so from persistence yk = CW(Xk) ~ X k /\C(XV X k) = XkV X so yk(JW) ~ xk(Jw) /\ x(Jw) for all fw which contradicts (3). 0
Suppose a new firm or a new worker enters the market. The following theorem shows that, in the firm-optimal matching, in the first case no firm is better off and no worker worse off, while in the second case no worker is
better off and no firm worse off. Forma11~et X be the firm-optimal matching
in the original F X W market, and let X'" denote the F x W component of the firm-optimal matching with the additional firm cf> and let Xw be the same for the additional worker w.
Theorem 3 (Comparative Statics).
(i) X ?::F X<P and X<P ?::w X; (ii) Xw ?::F X and X ?::w Xw. Proof. To prove (i), we continue the algorithm of the Existence Theorem. The new firm cf> offers an employment schedule x", which gives a new offer schedule X' to W where X' ~ X and since workers get no worse off with each step of the recursion they are at least we11 off under X'" as under X. The firms are no better off since their choice matix can never exceed jj.
To prove (ii), we suppose the original market includes w but bw = O. We denote by (Bk), (Xk), (yk) and (B'k), (X'k), (y'k), respectively, the sequences in the Existence Theorem recursion for the original and new market. Note B' ~ B. It suffices to show that B,k ~ Bk and x,k (w) ::; xk(w) for a11 k and w #- w'. Assume this is true up to k. Since B,k ~ Bk, we have by persistence X k = CF(Bk) ~ CF(B,k) /\ Bk = X,k /\ Bk so xk(Jw) ~ x'k(Jw) /\ bk(Jw) but for w #- w' we have bk(Jw) = b'k(Jw), hence xk(w) ~ x'k(w). This shows that no W-worker is better off in the new market.
To show that no firm is worse off, we show that B,k ~ Bk for a11 k. Since xk(w) ~ X'k(W) for w #- w', we have by persistence y'k(W) ~ yk (w) /\ x'k(w). There are two cases. If x'k(Jw) ::; yk(Jw) then y'k(Jw) = x'k(Jw) so from the recursion rule b,k+1 = b,k > bk > bk+1. If on the other hand xk(Jw) ~ X'k(Jw) > yk(Jw) then by co~isten~y y'k(Jw) = yk(Jw) so from the recursion rule bk+1(Jw) = yk(Jw) = y'k(JW) ::; b'k+l(JW), completing the proof. 0
3.2 The Stable Matching Lattice
Let X, X' be an arbitrary pair of matchings fixed throughout this section. We writeXF = XYFX' forthe matchingwhose f-row, xF(J), is x(J)Y fx'(J) and write XF = X AF X' for the matching whose f-row, XF(J), is x(J) Af x'(J). We define XW,Xw via w-columns similarly.
Note that if X, X' are acceptable then X F is of course F-acceptable but not in general W -acceptable.
The fo11owing is a key result.
Lemma 12. 1f X and X' are stable matchings then X F ::; XW.
?roof. We must show that xF(Jw) ::; xw(Jw) for a11 fw. Case (i) xF(Jw) ::; x(Jw)/\x'(Jw). Then, since by Lemma 4 x(w)/\x'(w) ::;
x(w) Aw x'(w) = xw(w), the conclusion fo11ows. Case (ii) x(Jw) < xF(Jw) ::; x'(Jw). Then, since xF(J) ?::f x (J), we have
by Lemma 8(ii), x(J) is not w-satiated, so by stability x(w) is f-satiated, so
from Lemma 7 x(w)(f) = b(fw), so xF(fw) ~ x(w)(f) /\ x'(fw) = (x(w) /\ x'(w))(f) ~ (x(w) A w x'(w))(f) (again from Lemma 4) = xw(fw). 0
In order to make the above inequality to an equation, it is necessary to make some further assumption. We will assurne that the entries of a schedule are measured in some common unit so that it makes sense to add them up. The following condition extends the condition of "cardinal monotonicity" for the discrete case introduced in [2J.
Definition 11. The choice function G is size monotone if x ~ y implies IG(x)1 ~ IG(y)1 for all x,y in A.
Remark Note that size monotonicity implies that if x t: y then lxi ~ lyl since x Vy ~ y.
The condition means, for example, that if a worker is forced to cut down on the hours allocated to some firm, then he may choose to work longer for other firms, but he will not increase his total working hours. In the discrete model the condition says that if a firm loses the services of one worker it will replace hirn by at most one worker. Note that if Gis quota filling then it is automatica11y size monotone. From size monotonicity, we get
Theorem 4 (Lattice Polarity). If all choice functions are size monotone thenXF =Xw.
Proof. First, since for a11 w, xW (w) t: xw(w), it fo11ows from the remark above that Ixw(w)1 ~ Ixwl so IXwl = Ew Ixw(w)1 ~ E IxW(w)1 = IXwl, and similarly IXFI ~ IXFI. From the previous Lemma IX'" I ~ IXwl. so now IXFI ~ IXFI ~ IXwl ~ IXwl ~ IXFI. so IXFI = IXwl, so the conclusion follows, and also for any agent, say w,
Ixw (w)1 = Ixw(w)1 . (4)
o Theorem 5. The set of stable matchings is a lattice und er the orderings t:F and t:w.
Proof. It suffices to show that X F is a stable matching. By definition X F is F-acceptable and, since X F = Xw, it follows that X F is also W-acceptable. It remains to show stability, so suppose xF (f) is not w-satiated. Then by Lemma 9(i) x(f), x'(f) are not w-satiated. So by stability x(w),x'(w) are f-satiated, so by Lemma 9(ii) xw(w) is f-satiated, but by Theorem 4 xw(w) is the w-column of X F , so X F is stable. 0
3.3 Properties of the Stahle Matching Lattice
The following property, which says Ix(w)I=lx'(w)1 for all w, generalizes a result for the classical model.
Theorem 6 (Unisize). The schedules that an agent may have in any stable matching all have the same size.
Proof Note xW (w) y w x(w) = xW (w) and XW (w) Aw x(w) = x(w) so from (4) IxW(w}1 = Ix(w)1 and similarly IxW(w)1 = Ix'(w)l. 0
An immediate consequence is the following result which was first shown by Roth and Sotomayor [9] for the classical college admissions model.
Corollary 3. If the choice function of an agent is quota filling and he does not fill his quota in a stable matching then he has the same schedule in all stable matchings.
Proof. Suppose x(J) =I- x'(J) and Ix(J}1 = Ix'(J)1 = c < q. Then Ix(J)Vx'(J)1 > c, so by quota filling IxP (J) I = Ix(J) V x' (I) I > c, contradicting Theorem 6.
o
A striking structural property of stable matchings is that, for all pairs fw, {xP(Jw),xp(Jw)} = {x(Jw),x'(Jw)}, stated equivalently in the following form.
Theorem 7 (Complementarity). If X and X' are stable matchings x P V Xp = XV X' and x P /\Xp = X /\X'.
Proof Let f be any firm. First, from Lemma 6 we have
(5)
Secondly, for all w, by lattice polarity (Theorem 4) xp(Jw) = X W (Jw) = (x(w) Y w x'(w»(J) ~ (x(w) V x'(w»(J) = x(Jw) V x'(Jw), thus xp(J) ~ x(J) V x'(J) so, since xP(J) = x(J) Y f x'(J) ~ x(J) V x'(J), we have
xP(f) V xp(f) ~ x(f) V x'(f) , (6)
so IxP(J) I + Ixp(J)I-lxP(J) /\ xp(J)1 = IxP(J) V xp(J)1 ~ Ix(J) V x'(J) I = Ix(J}I + Ix'(J)I- Ix(J) /\ x'(J)I, but from the unisize property (Theorem 6) IxP(J) I = Ixp(J)1 = Ix(J)1 = Ix'(J)1 so
IxP(J)/\xp(J)I2: Ix(J)/\x'(J)I, (7)
therefore (5) and (7) are equations, hence (6) also is an equation. 0
Complementarity implies that the lattice of stable matchings is distribu tive.
Definition 12. A lattice C, with join Y and meet A, is distributive if z Y
(z' A Zll) = (z Y z') A (z Y Zll) and z A (z' Y Zll) = (z A z') Y (z A Zll) for all z, z', Zll in C.
Remark A standard fact in lattice theory (Corollary to Theorem 11.13 in [4]) is that a lattice (.c, Y, A) is distributive if and only if the following cancellation law holds:
if zYz' = zYz" and ZAZ' = ZAZ" then z' = z" for all z,z',z" in .c.(8)
Theorem 8 (Distributivity). The (YF,AF) and (YW,AW) lattices ofsta ble matchings are distributive.
Proof. Let X,X',X" be any three stable matchings. If X YF X' = X Y F X" and X AF X' = X AF X" then (X Y F X') V (X AF X') = (X Y F X") V (X AF X") and (X YF X') /\ (X AF X') = (X Y F X") /\ (X AF X"), hence by complementarity (Theorem 7) X V X' = X V X" and X /\ X' = X /\ X", so by distributivity of V, /\ using cancellation X' = X". Thus the cancellation law holds for Y F, AF, similarly for Y W , A w, and the theorem follows from the remark above. 0
An important theorem in the classical case asserts that for stable match ings the schedules x(f) and x' (f) are comparable, that is either they are identical or f prefers one to the other. This was proved for college admissions in [10] and for schedules in [3]. This result does not hold in the general case as we show in the next section. However, we will here show that, for classical agents, it is a direct consequence of complementarity and the unisize property.
Corollary 4. In the classical case let x and y be schedules where x ~ y. Then xCi) > 0 implies x(j) ~ y(j) for j < i.
Proof. If y (j) > x(j) then for sorne € > 0 define the schedule x~ ~ x V Y by x~(i) = xCi) - E, x€(j) = x(j) + E, xe(k) = x(k) otherwise. Then x~ ~ x contradicting C (x V y) = x. 0
Theorem 9. In the classical case if X and X' are stable matchings then either x(f) ~f x'(f), x(f) = x'(f), or x(f) -<f x'(f).
Proof. Let y(f) = XF(f) = x(f) Af x'(f). By the unisize property we cannot have y(f) < x(f) or y(f) < x'(f). Therefore, if y(f) is distinct from x(f) and x'(f) then by complementarity there is a w such that y(fw) = x(fw) > x'(fw) and there is a w' such that y(fw') = x'(fw') > x(fw'). But if, say, w' is preferred by f to w then since x(f) ~f y(f) and x(fw) > 0 it follows from Corollary 4 that x(fw') ~ y(fw'), contradiction. 0
4 Examples
In this section we will show by examples the need for our various assumptions. All examples are in the context of the special case of college admissions.
Example 4. If choice functions are consistent and size monotone but not per sistent then stable matchings may not exist.
There are two colleges A and Band four students m,w,m',w'. College A has quota 2 and the choice function as in Section 1 so that
mW»--A mw'»--A m'w'»--A m'w. (9)
College B has quota 1 and prefers m to wand will not admit m' or w'. Student m prefers A to B while student w prefers B to A. In the table below, the preferences of B, m, ware indicated by the arrows.
A m
Am l'
w Aw !
B Bm<--- Bw
For every assignment of students to A there is a blocking as seen from (9) and the table above:
(A,mw) (A,mw') (A,m'w)
(A,m'w') and (B,m) (A,m'w') and (B,w)
is blocked by Band m, " A and w, " A and w', " Band w,
A and m.
Example 5. If preferences are consistent and persistent but not size monotone then stable matchings may not form a lattice. More precisely, the sup of stable matchings may not be stable.
There are colleges A, . .. ,E and students a, . .. ,e. Preferences are given by the table below. Note that the preferences of A and B violate size monotonic ity.
A B a* b# ce# deO
C D E c* d# e
a# b*
abc d e C# D* A# B* A# A* B# C* D# B*
E
One easily verifies that the entries marked * and those marked # cor respond to stable matchings. Namely, in each matching, where a college is matched with its second choice, the preferred student is matched with her first choice. But in the matching which is the college supremum of * and #, both E and e are unmatched, hence they block and the college supremum is therefore unstable. Note that the unisize condition is also violated for A and B in the matchings * and #.
The following two examples show that certain results for the classical model do not generalize to the model with revealed preference (with con sistent, persistent and size monotone choice functions).
Example 6. The college optimal stable matching may not be Pareto optimal for colleges.
There are colleges A, B, Z with quotas 1,1,2, male students m, m' and female students w, w'.
Z chooses mw if all four st udents are available and otherwise chooses the sexually diverse pair.
The other preferences are given by the table below where the left entry in each pair is the college's ranking of the student and the right entry is the student's ranking of the college.
m w m' w' Z (-,2) (-,2)# (-,1)*# (-,1)* A (1,3)# (2,1)* (4,3) (3,2) B (2,1)* (3,3) (4,2) (1,3)#
The <>nly stable matching is the student optimal matching *. One sees this by checking from the algorithm that it is also the college optimal matching. But the matching # makes all colleges (strictly) better off. Of course # is unstable, being blocked by Z and m.
Example 7. As shown in Corollary 5, in the classical model all stable match ings are comparable for each agent. This need not be so in the non-classical model.
Colleges A and B have quota 2. Students are m, w, m', w', and the choice function of A and Bare as in Example 3 except A most prefers mw and B most prefers m'w'. For the students, m and w prefer B to A, m' and w' prefer AtoB.
One easily verifies that all four ways of allocating diverse pairs to A and B are stable and also that mw' and m'w are noncomparable in the preferences of both A and B.
References
1. Alkan, A. (2001): On Preferences over Subsets and the Lattice Structure of Stable Matchings. Review of Economic Design 6, 99-111
2. Alkan, A. (2002): A Class of Multipartner Matching Models with a Strong Lat tice Structure. Economic Theory (to appear)
3. Baiou, M., Balinski, M. (2000): The Stable Scheduling (or Ordinal Transporta tion) Problem. Laboratoire d'Econometrie, Ecole Polytechnique, Paris (mimeo)
4. BirkhofI, G. (1973): Lattice Theory. In: American Mathematical Society Collo quium Publications, Vol. XXV
5. Blair, C. (1988): The Lattice Structure of the Set of Stable Matchings with Multiple Partners. Mathematics of Operations Research 13, 619-628
6. Crawford, V. P. (1991): Comparative Statics in Matching Markets. Journal of Economic Theory 54, 389-400
7. Gale, D., Shapley, L. (1962) College Admissions and the Stability of Marriage. American Mathematical Monthly 69, 9--15
8. Kelso, A. S., Crawford, V. P. (1982): Job Matching, Coalition Formation and Gross Substitutes. Econometrica 50, 1483-1504
9. Roth, A. E., Sotomayor, M. (1989): The College Admissions Problem Revisited. Econometrica 57, 559--570
10. Roth, A. E., Sotomayor, M. (1990): Two-Sided Matching: A Study in Game Theoretic Modeling and Analysis. Cambridge University Press, Cambridge
11. Sotomayor, M. (1999): Three Remarks on the Many-to-Many Stable Matching Problem. Mathematical Social Sciences 38, 55-70
Banzhaf Permission Values for Games with a Permission Structure
Rene van den Brink
Department of Econometrics, Free University, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands ~mail: jrbrinkGfeweb.vu.nl
Summary. Agame with a permission structure describes a situation in a cooper ative game with transferable utility in whlch cooperation possibilities are limited because some players need permission from other players before they are allowed to cooperate. In the conjunctive approach, it is assumed that every player needs pennis sion from oll direct superiors. In the disjunctive approach, it is assumed that each player only needs permission from at least one direct superiors. Both approaches yield modified games which take account of limited cooperation possibilities. Apply ing the BanzhaJ value to these modified games yields allocation rules that distribute the payoffs over the individual players. This paper provides axiomatic characteriza tions of these BanzhaJ permission values.
Key words: cooperative TU-game, Banzhaf value, directed graph, permis sion structure
1 Introduction
A situation in which a finite set of players N c lN can generate certain payoffs by cooperation can be described by a cooperative game urith transferable utility (or simply a TU-game), being a pair (N,v) where v: 2N --+ IR is a charac teristic function on N satisfying v(0) = O. The co11ection of a11 characteristic functions on a particular player set N is denoted by gN.
In a TU-game there are no restrictions on the cooperation possibilities of the players, i. e., every coalition E c N is formable and can generate a payoff. Examples of models in which there are restrictions on the possibilities of cooperation are the games in coalition structure in which it is assumed that the set of players is partitioned into disjoint sets w hich represent social groups such that for a particular player it is more easy to cooperate with players in its own group than to cooperate with players in other groups (see, e. g., [1], [20], [15] and [23]), and the games with limited communication structure in which the edges of an undirected graph on the set of players represent binary
L. A. Petrosyan et al. (eds.), ICM Millennium Lectures on Games © Springer-Verlag Berlin Heidelberg 2003
22 Rene van den Brink
communication links between the players such that players can cooperate only if they are connected (see, e. g. [18], [16], [21] and [3]).
This paper considers games with a permission structure in which it is assumed that players in a TU-game are part of a hierarchical organization in which there are players that need permission from other players before they are allowed to cooperate within a coalition. Thus the possibilities of coalition formation are determined by the positions of the players in this so called permission structure. Various assumptions can be made about how a permission structure affects the cooperation possibilities. In the conjunctive approach, as developed in [12] and [6], it is assumed that every player needs permission from all its direct superiors before it is allowed to cooperate with other players. Alternatively, in the disjunctive approach, as considered in [11] and [4] it is assumed that every player needs permission from at least one of its direct superiors before it is allowed to cooperate with other players.
Given agame and apermission structure a modified game is defined which takes account of the limited cooperation possibilities. The conjunctive and disjunctive approach yield different modified games. An allocation rule for games with apermission structure is a function that assigns to every game with apermission structure a distribution of the payoffs that can be obtained by cooperation over the individual players. Applying solutions for TU-games (being functions that assign a payoff distribution to every TU-game) to these modified games yields allocation rules for games with apermission structure. Applying, for example, the Shapley value [22] to these modified games yields the conjunctive and disjunctive (Shapley) permission values which have been characterized in [6] and [4], respectively. In [5] it has been shown that these two permission values only differ with respect to the fairness axiom that is used. The disjunctive (Shapley) permission value satisfies disjunctive fairness implying that deleting the relation between a player and one of its direct supe riors changes the payoffs of these two players by the same amount (under the condition that the player has at least one other direct superior). Instead, the conjunctive (Shapley) permission value satisfies conjunctive fairness implying that deleting the relation between a player and one of its direct superiors changes the payoffs of this player and each of its other direct superiors by the same amount.
In this paper we study another weIl known solution for TU-games, namely the Banzhaf value which is based on the Banzhaf index for voting games [2], and is generalized to arbitrary games by, e. g., [19] and [10]. A poten tial approach to the Banzhaf value has been given by Dragan [9]. Applying the Banzhaf value to the modified games in the conjunctive and disjunctive approach yields the conjunctive, respectively, disjunctive Banzhaf permission value. In this paper we give axiomatic characterizations of these Banzhaf per mission values.
New properties that are satisfied by both Banzhaf permission values are predecessor fairness and two neutrality properties. Predecessor fairness states that the payoffs of two direct superiors of a player change in opposite direction
Banzhaf Permission Values for Games with aPermission Structure 23
if we delete its relation with one of the two superiors, and moreover, the absolute values of the changes in payoffs of these two direct superiors are the same. The neutmlity properties state that, if a new player enters the game with permission structure and "merges" with a player that is already present, then these two players share the payoff that was earned by the player that was already present in the original game. We distinguish between two such neutrality properties. Vertical neutmlity considers the situation in which the new player enters the game as a null player, and enters the permission structure as a direct superior of the old player. Horizontal neutmlity considers the situation in which the new player enters the game as a veto player for the old player, and enters the permission structure at the same level as the old player.
The paper is organized as folIows. In Section 2 we state some preliminaries on games with apermission structure. In Section 3 we discuss predecessor fair ness. In Section 4 we introduce the neutrality axioms and characterize the two Banzhaf permission values. Finally, in Section 5 we show how predecessor fair ness and the neutrality axioms can be adapted to give new characterizations of the (Shapley) permission values.
2 Preliminaries: Games with aPermission Structure
We assume that players who participate in a TU-game are part of a hierarchi cal organization in which there are players that need permission from certain other players before they are allowed to cooperate. For a finite set of players NeIN such a hierarchical organization is represented by a pair (N, S), where the mapping S: N -+ 2N is called a permission structure on N. (So the set ((i,j) E N X N I i E N, j E S(i)} describes a directed graph on N.) The players in S(i) are called the successors of player i E N in S. The players in S-l(i) := {j E N I i E S(j)} are called the predecessors of i in S. By S we denote the tmnsitive closure of the permission structure S, i. e., j E S(i) if and only if there exists a sequence of players (hI, ... , h t ) such that h 1 = i, hk+1 E S(hk) for all 1:::; k :::; t - 1, and ht = j. The players in S(i) are called the subordinates of i in S, and the players in S-l(i) := {j E N I i E S(j)} are called the superiors of i in S. In this paper we restriet our attention to hiemrchical permission structures being permission structures S: N -+ 2N
that are
1) acyclic, i.e., i ~ S(i) for all i E N, and 2) quasi-strongly connected, i. e., there exists an i E N such that S( i)
N \ {i}.
We denote the collection of all hierarchical permission structures on a particular player set N by S~. A tripIe (N, v, S) with NeIN, v E gN and SE s~ is called agame with a (hiemrchical) permission structure. In a hierarchical
permission structure there exists a unique player io such that S(io) = N\ {io}. Moreover, S-l(io) = 0 for this player. We call this player the top-player in the permission structure. Since we only consider hierarchical permission structures we will often refer to these simply as permission structures. (The results in this paper could be stated for acyclic permission structures that not necessarily are quasi-strongly connected. For notational convenience we restrict attention to permission structures satisfying both conditions (i) and (ii).)
2.1 Disjunctive and Conjunctive Restrictions
In the disjunctive approach, as developed in [l1J and [4], it is assumed that each player needs permission from at least one of its predecessors before it is allowed to cooperate with other players. Consequently, a coalition is formable if and only if every player in the coalition, except the top-player io, has a predecessor who also belongs to the coalition. Thus, the formable coalitions are the ones in the set
iP'Jv,s:= {E C NI S-l(i) nE:f:; 0 or S-l(i) = 0 for aB i E E} . (1)
Note that E E iP'Jv s implies that i o E E. The coalitions in iP'Jv s are called the disjunctive auton~mous coalitions in S. The largest disjunctive autonomous subset of E C N in S E S~ is denoted by a'Jv s(E) = U{F E iP'Jv s I FeE}, and is called the disjunctive sovereign part ~f E in S. It consists of those players in E that can be reached by a directed "permission path" start ing from the top-player such that all players on this path belong to coali tion E. Using this concept we can transform the characteristic function v into a modified characteristic function which takes account of the limited cooperation possibilities as determined by the permission structure S as fol lows. Given agame with apermission structure (N,v,S), the disjunctive re striction of v on S is the characteristic function r'Jv v s: 2N -+ IR given by
r'Jv,v,s(E) = v(a'Jv,s(E)) for all E C N. ' , Alternatively, in the conjunctive approach, as developed in [12J and [6], it
is assumed that each player needs permission from all its predecessors before it is allowed to cooperate. This implies that a coalition E is formable if and only if for every player in the coalition it holds that all its predecessors belong to the coalition. The set of formable coalitions in this approach thus is given by
iP'N,s := {E C N I S-l(i) cE for all i E E} . (2)
The coalitions in the set iP'N s are called the conjunctive autonomous coalitions in S. The largest conjunctiv~ autonomous subset of Eis denoted by a'N s(E) = U{F E iP'N s I FeE}, and is refered to as the conjunctive sovereign p~rt of E in S. It co~sists of all players in E whose superiors all belong to E. Given game with permission struct ure (N, v, S), the conjunctive restriction of v on S is the characteristic function r'N,v,s: 2N -+ IR given by r'N,v,s(E) = v(a'N,s(E)) for all E C N.
Example 1. Consider the game with permission structure (N, v, S) with N = {1, 2, 3, 4}, v E gN given by v(E) = 1 if 4 E E, v(E) = 0 otherwise, and SE sff given by S(l) = {2,3}, S(2) = S(3) = {4}, S(4) = 0.
The disjunctive and conjunctive restrictions of v on S, respectively, are given by
d (E) {1 if E E {{1,2,4}, {1,3,4}, {1,2,3,4}}, rN,v,S = 0 th . o erWlse,
and
2.2 The (Shapley) Permission Values
An allocation rule for games with apermission structure is a function that assigns to every game with apermission structure (N, v, S) a distribution of the payoffs that can be obtained by cooperation according to v taking into account the limited cooperation possibilities determined by S. The disjunctive (Shapley) permission value cpd is obtained by appyling the Shapley value [22] to the disjunctive restricted games, while the conjunctive (Shapley) permis sion value cpc is obtained by applying the Shapley value to the conjunctive restricted games, i. e.,
cpd(N, v, S) = Sh(N, r'Fv,v,s) and cpC(N, v, S) = Sh(N, r'Fv,v,s) ,
where
Shi(N, v) = L (INI-I~~!I~IEI- 1)! (v(E) - v(E \ {i})), for all i E N. EeN iEE
Example 2. The disjunctive and conjunctive (Shapley) permission values of the game with permission structure given in Example 1 are cpd(N, v, S) = l2 (5,1,1,5) and cpC(N, v, S) = t(l, 1, 1, 1).
The disjunctive (Shapley) permission value is axiomatized in [4], while axiomatizations of the conjunctive (Shapley) permission value can be found in [6] and [5]. We refer to these papers for a discussion on the axioms pre sented below. The first two axioms are straightforward generalizations of the efficiency and additivity axioms of solutions for TU-games.
Axiom 1 (Efficiency). For every NeIN, v E gN and SE sff, it holds that LiEN fi(N,v,S) = v(N).
Axiom 2 (Additivity). For every NeIN, v, w E gN and S E sff, it holds that f(N, v + w, S) = f(N, v, S) + f(N, w, S), where (v + w) E gN is defined by (v + w)(E) = v(E) + w(E) for all E c N.
Player i E N is inessential in game with permission structure (N, v, S) if i and an its subordinates are null players in (N, v), i. e., if v (E) = v (E \ {j} ) for an E C N and j E {i} U S(i).
Axiom 3 (Inessential player property). For every NeIN, v E gN and SE sir, ifi E N is an inessential player in (N,v,S) then !i(N,v,S) = o.
The next two axioms are stated for monotone characteristic functions. A characteristic function v E gN is monotone if v(E) ~ v(F) for an E C F C N. The dass of an monotone characleristic functions on N is denoted by gZ. Player i E N is caned necessary in game (N, v) if v(E) = 0 for an E C N\ {i}.
Axiom 4 (Necessary player property). For every NeIN, v E gZ and SE sir, ifi E N is a necessary player in v then fi(N,v,S) ~ !;(N,v,S) for allj E N.
We say that player i E N dominates player JEN "completely" if an directed "permission paths" from the top-player io to player j contain player i. We denote the set of players that player i dominates "completely" by 8(i), i. e.,
. {. ~. i E {io,il, ... ~it:-l} for.every sequence of nodes 8(z)= JES(z) ZO,ZI, ••• ,Zt suchthat
it = j and ik E S(ik-l), k E {I, ... ,t}, } We also define SI(i) = {j E S-I(i) li E B(j)}.
AxlOln 5 (Weak structural Inonotonicity). For every NeIN, v E g~ and SE sir, ifi E N and jE B(i) then !i(N,v, S) 2': !;(N,v,S).
Weak structural monotonicity is a weaker version of structural monotonic ity as introduced in [6] which states that for a monotone game with a per mission structure, if i E N and j E S(i) then !i(N, v, S) 2': !;(N, v, S). (This stronger structural monotonicity is satisfied by the conjunctive (Shapley) per mission value but is not satisfied by the disjunctive (Shapley) permission value.)
The five axioms given above are satisfied by both the disjunctive and conjunctive (Shapley) permission values. These two permission values differ with respect to the fairness axiom that is used. Disjunctive fairness states that deleting the relation between two players hand jE S(h) (with IS-1(j)1 2': 2) changes the payoffs of players hand j by the same amount. Moreover, also the payoffs of an players i that "completely" dominate player h, in the sense that i E 8- 1 (h), change by this same amount.
For SE Sir, h E N and j E S(h) we denote S_(h,j)(h) = S(h) \ {j} and S_(h,j)(i) = S(i) for i E N \ {h}.
Axiom 6 (Disjunctive fairness). For every NeIN, v E gN and SES:, if hE N and jE S(h) with IS- 1 (j)1 ~ 2, then
fi(N,v,S) - fi(N,v,s-Ch,j)) = h(N,v,S) - h(N,v,s-Ch,j))
--1 for all i E {h} U S (h).
Disjunctive fairness is related to fairness as introduced in (18) for games with a limited communication structure. In that model fairness means that deleting a communication relation between two players in an undirected com munication graph has the same effect on both their payoffs. (Note that in our fairness property we require that the successor on the relation to be deleted has at least two predecessors.)
Theorem 7 (van den Brink [4]). An allocation rule fisequal to the dis junctive (Shapley) permission value 'Pd if and only if it satisfies ejficiency, additivity, the inessential player property, the necessary player property, weak structural monotonicity and disjunctive fairness.
The conjunctive (Shapley) permission value does not satisfy disjunctive fairness. However, it satisfies the alternative conjunctive fairness which states that deleting the relation between two players hand jE S(h) (with IS-1(j)1 ~ 2) changes the payoffs of player j and any other predecessor g E S-1 (j) \ {h} by the same amount. Moreover, also the payoffs of a11 players that 'completely' dominate the other predecessor g change by this same amount.
Axiom 8 (Conjunctive fairness). For every NeIN, v E gN and SES:, if h,j, gEN are such that h 'I g and jE S(h) n S(g), then
fi(N,v,S) - fi(N,v,s-Ch,j)) = fj(N,v,S) - fj(N,v,s-Ch,j))
-1 fOT all i E {g} U S (g).
Theorem 9 (van den Brink [5]). An allocation rule fisequal to the con junctive (Shapley) permission value 'Pe if and only if it satisfies ejficiency, additivity, the inessential playeT property, the necessary player property, weak structural monotonicity and conjunctive fairness.
3 Predecessor Fairness and Banzhaf Permission Values
The two fairness axioms that are mentioned in the previous section compare the effects of deleting the relation between players hand j E S(h) on the payoffs of players hand j, and on the payoffs of players g E S-1(j) \ {h} and j, respectively. These properties do not compare the change in payoffs of players hand g E S-1 (j) \ {h} after deleting the relation between hand j. For monotone games the changes in the disjunctive and conjunctive (Shapley) permission values of these players are opposite.
Proposition 1. For every NeIN, v E gN, S E S% and h,g,j E N with h =1= 9 and j E S(h) n S(g):
1) cp1(N,v,S) ~ cp1(N,v,s-(h,j»), cp~(N,v,S) > cp~(N,V,s-(h,j») and
cp~(N,v,S) ::; cp~(N,V,S_(h,j»); 2) cp'j(N,v,S) ::; Cp'j(N,V,s-(h,j»)' cpj.(N,v,S) ~ cpj.(N,V,s-(h,j») and
cp~(N,v,S) ::; cp~(N,V,s-(h,j»).
The proof of this proposition is straightforward and is therefore omitted. As a consequence we have that deleting the relation between player hand jE S(h) (with IS- 1 (j)1 ~ 2) in a monotone game with a hierarchical permis sion structure changes the disjunctive and conjunctive (Shapley) permission values of players hand 9 E S-1 (j) \ {h} in opposite direction. The fairness properties discussed in the previous section hold for aB games with a hier archical permission structure. This is not the case for Proposition 1 and the "opposite change" consequence described here.
Example 3. Consider the game with permission structure (N, v, S) given by N = {1,2,3,4,5}, v = U{4,5} - 1~U{4}' where UT denotes the unanimity game of T C N (i. e., uT(E) = 1 if E :J T, and uT(E) = 0 otherwise) and S(1) = {2, 3, 5}, S(2) = S(3) = {4} and S(4) = S(5) = 0. In this example, cp~(N,v,S) - cp~(N,V,S_(2,4») = -1~O - 0< 0 and cp~(N,v,S) cp~(N, v, S-(2,4») = -1~O -lo = -10 < o. Thus, deleting the relation between players 2 and 4 changes the disjunctive (Shapley) permission values of player 4's predecessors 2 and 3 in the same direction. (By disjunctive fairness the change in the disjunctive (Shapley) permission value of player 4 is the same as for player 2: cp~(N,v,S) - cp~(N,v, S-(2,4») = l~O -lo = -l~O·)
For the conjunctive (Shapley) permission values cp'2(N,v,S) - cp'2(N,v, S-(2,4») = 4~ - 0> 0 and CP3(N, v,s) - CP3(N, V,s-(2,4») = 10 - 6~ = 1~O > O. Thus, deleting the relation between players 2 and 4 also changes the con junctive (Shapley) permission values of player 4's predecessors 2 and 3 in the same direction. (By conjunctive fairness the change in the conjunc tive (Shapley) permission value of player 4 is the same as for player 3: cp4(N,v,S) -cp4(N,V,s-(2,4») = 4~ - 6~ = 1~O·)
It turns out that applying the Banzhaj value to the conjunctive and dis junctive restricted games yields a11ocation rules for games with apermission structure that satisfy the "opposite change" property for every game with a hierarchical permission structure. The Banzhaf value for TU-games is given by
B;(N,v) = 21:1- 1 I)v(E) - v(E\ {i})) , for a11 i E N. ECN iEE
(3)
The disjunctive Banzhaj permission value ßd and the conjunctive Banzhaj permission value ßC are obtained by applying the Banzhaf value to the dis junctive and conjunctive restricted games, respectively, yielding
Banzhaf Permissiün Values für Games with a Permissiün Structure 29
ßd(N,v,S) = B(N,r'fv,1J,s) and ßC(N,v,S) = B(N,r'J.r,1J,s) .
Example 4. The disjunctive and conjunctive Banzhaf permission values of the game with permission structure given in Example 1 are ßd(N,v,S) = k(3, 1, 1,3) and ßC(N, v, S) = k(l, 1, 1, 1).
The disjunctive and conjunctive Banzhaf permission values of players h and g, h i- g, always change in opposite direction after deleting the relation between players hand j E S(h) n S(g). Moreover, the absolute values of the changes in the Banzhaf permission values of these two players are the same.
Axiom 10 (Predecessor fairness). For every NeIN, v E gN and SE SfI, if h, g,j E N are such that h i- g and jE S(h) n S(g), then
fh(N,v,S) - h(N,v,s-(h,j») = fg(N,v,s-(h,j») - fg(N,v,S) .
Theorem 11. The disjunctive and conjunctive Banzhaf permission values satisfy predecessor fairness.
Proo! Let NeIN, v E gN, S E SfI and h,g,j E N be such that j E
S(h) n S(g) and h i- g. Since a'fv,s(E) = a'fv,S_<h,j) (E) if h f/. E or g E E, it follows that
ß~(N, v, S) - ß~(N, v, S-(h,j») = Bh(N, r'fv,1J,s) - Bh(N, r'fv,1J,S_<h,j))
= 2IN~-1 L (v(a'fv,s(E)) - v(a'fv,s(E \ {h})) - v(a'fv,s_<h,)E)) EeN hEE
+v(a'fv,s_<h,j)(E\ {h})))
1
= 21:1- 1 L (v(a'fv,s(E\{g}))-v(a'fv,s_<h,)E\{g}))) EeN gEE
= 21:1- 1 L (v( a'fv,S_<h,j) (E)) - v(a'fv,S_<h,j) (E \ {g})) - v (a'fv,s(E)) EeN gEE
+v(a'fv,s(E \ {g})))
= Bg(N, r'fv,1J,S_<h,j)) - Bg(N, r'fv,1J,s) = ß:(N,v, S-(h,j») - ß:(N, v, S) ,
showing that ßd satisfies predecessor fairness. To show that ßC satisfies predecessor fairness note that a'N s(E) =
a'N S . (E) if h E E or g ~ E. But then, in a similar way as ~bove for , -(h.])
ßd, it follows that
= 2IN~-1 L: (v(a'N,s(E)) - v(a'N,s(E \ {h})) - v(a'iv,s_(h,j) (E)) EeN hEE
+v(a'N,s_(h,j)(E\ {h})))
= 21;1- 1 L: (v(a'N,s_(h,j)(E\ {h})) -v(a'N,s(E\ {h})))
1
= 21;1- 1 L:(v(a'N,s_(h,)E))-v(a'N,s_(h,)E\{g})) EeN gEE
-v(a'N,s(E)) +v(a'iv,s(E\ {g})))
= Bg(N, rN,V,s_(h,) - Bg(N,rN,v,S) = ß;(N,v, S-(h,j)) - ß;(N,v,S) ,
showing that ßC satisfies predecessor fairness.
4 Axiomatizations of Banzhaf Permission Values
Main purpose of this paper is to axiomatize the Banzhaf permission values.
4.1 The Disjunctive Banzhaf Permission Value
o
Compared to the disjunctive (Shapley) permission value the disjunctive Ban zhaf permission value satisfies the axioms stated in Theorem 7 except effi ciency. However, it satisfies efficiency for one player games with apermission structure. Note that S E Sff with INI = 1 implies that S(i) = 0 for i E N, and thus r'Jv v s = v in that case. , ,
Axiom 12 (One player efficiency). For every NeIN with INI = 1, v E gN and S E S~, it holds that fi(N,v,S) = v({i}) fori E N.
Replacing efficiency in Theorem 7 by one player efficiency and predecessor fairness is not sufficient to characterize the disjunctive Banzhaf permission value as can be seen from Example 5.2 and 5.3 at the end of this subsection. To characterize the disjunctive Banzhaf permission value we introduce two more axioms that, in particular, will be applied to permission trees, i. e., hierarchical permission structures in which each player has at most one predecessor. These two axioms are related to amalgamation neutmlity and collusion neutrality as used, respectively, in [17] and [13] to characterize the Banzhaf value for TU-games. A similar neutrality property is used here and is defined in the appendix of this paper.
First, suppose that all subordinates of player JEN are null players (note that this does not imply that j is an inessential player because j need not be a null player itself). Now, let h E :IN \ N be a new player whose only task is to supervise player j, i. e., h will be a null player in the new game and in the permission structure he becomes the only predecessor of j and gets all previous predecessors of j as its predecessors. Then the sum of the payoffs of players hand j in the new game with per mission structure is equal to the payoff of player j in the original game with permission structure.
Axiom 13 (Vertical neutrality). For every N C :IN, v E gN, S E S~ and hE:IN \ N, if JEN and all i E S(j) are null players in (N,v), then
!i(NU {h},v',S') + A(NU {h},v',S') = !i(N,v,S) ,
where v' E gNU{h} is given by v'(E) = v(E n N) for all E c Nu {h}, and S' E S;U{h} is given by
{ {j}, if i = h ,
S'(i) = (S~i)\{j})U{h}, i! ~ E S-l(j) , . S(~), if~EN\S-l(J).
Fig. 1. An illustration of permission structures S and S' as described in Axiom 13
For the second neutrality property, suppose that player JEN, with S(j) = o and S-l(j) #- 0, is split in two in the sense that a new player h enters who
gets the same successors (none) and predecessors as j, and in the new game hand j veto each other. Then, again, the sum of the payoffs of players hand j in the new game with permission structure is equal to the payoff of player j in the original game with permission structure.
Axiom 14 (Horizontal neutrality). For every N C lN, v E gN, 8 E Sjf and h E lN \ N, if 8(j) = 0 and 8- 1 (j) =/;0 for JEN, then
h(N U {h}, v", 8") + fh(N U {h}, v", 8") = h(N, v, 8) ,
where v" E gNU{h} is given by
v"(E) = {V(E\{j}) , if E C N , v(EnN), if ECNU{h} withhEE,
{ 0, if i = h,
and 8" E SZU{h} is given by 8"(i) = 8(~) U {h}, ~f ~ E 8- 1 (j), . 8(z) , ifzEN\8- 1 (J).
h
Fig. 2. An illustration ofpermission structures 8 and 8" as described in Axiom 14
Replacing efficiency in Theorem 7 by one player efficiency, predecessor fair ness, vertical neutrality and horizontal neutrality characterizes the disjunctive Banzhaf permission value ßd. First, we show that ßd satisfies the axioms.
Theorem 15. The disjunctive Banzhaf permission value ßd satisfies one player efficiency, vertical neutrality, horizontal neutrality, additivity, the ines sential player property, the necessary player property, weak structural mono tonicity, disjunctive fairness and predecessor fairness.
Proof. Proving that the disjunctive Banzhaf permission value satisfies addi tivity, the inessential player property, the necessary player property and weak structural monotonicity is along the same lines as this is shown for the dis junctive (Shapley) permission value in [4J. One player efficiency is evident. Predecessor fairness follows from Theorem 11.
Vertical neutrality follows from Proposition 2 (see Appendix) and the fact that r'Jv v' s' = (r'Jv v S)hj where v' and 8 ' are as given in Axiom 13, and Vhj is as gi~e~ in equati~n (16) of the appendix. Similarly, horizontal neutrality
foHows from Proposition 2 and the fact that r'f" v" S" = (r'f" v S)hj where v" and 8" are as given in Axiom 14. ' , , ,
So, we are left to show that ßd satisfies disjunctive fairness. Let 8 E sff and
h,j E N be such that jE 8(h) and 18- l (j)1 ~ 2. Further, let i E {h}US-l(h). Since {i,j} ct. E implies that a'f" seE) = a'f" S . (E), it follows that
, , -(h.,)
ßf(N, v, 8) - ßf(N, V,s-(h,j)) = Bi(N, r'f",v,s) - Bi(N, r'f",v,S_(h,j»)
= 21~-1 L (v(a'f",v,s(E)) - v (a'f",v,s(E \ {i})) - v (a'f",v,S_(h,j) (E)) ECN iEE
+v(a'f",v,s_(h,)E \ {i})))
= _1_ '"' (v(ad (E)) _ v(ad (E))) 21 NI-l ~ N,v,S N,V,S_(h,j) ECN iEE
= _1_ '"' (v(ad (E)) - v(ad (E))) 21 NI-l ~ N,v,S N,V,S_(h,j) ECN
{i,j}CE
= 21;1- 1 L (v(a'f",v,s(E)) - v (a'f",v,s(E \ {j})) - v(a'Fv,V,s_(h,j) (E)) ECN jEE
+v(a'f",v,S_(h,j) (E \ {j}))) = ß1(N, v, 8) - ß'j(N, V,s-(h,j)) ,
showing that ßd satisfies disjunctive fairness. o
Next we show that the axioms mentioned in Theorem 15 characterize the disjunctive Banzhaf permission value. We do this in three steps. First, we prove a lemma for positively scaled unanimity games with apermission tree that has no inessential players. We denote by Sf:.ee = {8 E Sff I for all i E N,18- l (i)l :$ 1} the dass of aH permission trees on N. For T C N, T:j:. 0, and CT > 0, the positively scaled unanimity game WT = CTUT is given by
(E) {CT' if E::>T, WT = 0, otherwise . (4)
For 8 E sff and hE N the permission structure 8_ h E 8 N \{h} is given by
S (') _ {(S(i) \ {h}) u S(h) , if i E S-I(h) , -h Z - S(i) , if iEN\({h}US- 1(h)). (5)
Lemma 1. If the allocation rule f satisfies one player efficiency, vertical neu trality, horizontal neutrality, the necessary player property, weak structural monotonicity and predecessor fairness, then J(N, WT, S) is uniquely deter mined for all S E sf:.ee and WT = CTUT for some CT > 0 and T C N satisfying Tu §-1 (T) = N.
j g
Fig. 3. An illustration of permission structures S-h
Proof. Suppose that the a11ocation rule f satisfies the six axioms. Con sider a hierarchical permission (tree) structure 8 E sf:.ee and the mono tone characteristic function WT = CTUT, T c N, CT > O. Suppose that T U S-1 (T) = N. (So, there are no inessential players.) Clearly, f sat isfying the necessary player property implies that there exists a constant c· E IR such that fi(N,WT,8) = c' for an i E T, and fi(N,WT, 8) ::; c' for an i E N \ T. But then weak structural monotonicity and the fact that S(i) = S(i) for an i E N and 8 E sf:.ee imply that fi(N,WT,S) = c' for an i E TUS- 1(T) = TUS- 1(T) = N. So, we have determined f(N, WT, 8) if we determine C·. We do this by induction on INI.
If INI = 1 then one player efficiency implies that fi(N,WT,8) = CT for iEN.
Proceeding by induction assume that h(N,wf'S) = cf/(2INI - 1) for an - -. - N -(N,wi", 8) wlth 8 E Stree, ci" > 0 and INI < INI.
We distinguish the following three cases (of which at least one must occur).
1. Suppose there exist h,g,j E N such that 8(h) = 8(j) = 0 and {h,j} C 8(g). (Note that this case can only occur if INI ~ 3.) By the assumption that TUS- 1(T) = N, we have {h,j} C T. Horizontal neutrality implies that
/i(N,WT, 8) + A(N,WT, 8) = f;(N \ {h},CTUT\{h},s-h) .
With the induction hypothesis and the fact that /i(N,WT,8) = fh(N, wT,8) = c' this yields that 2c' = cT/(2INI - 2 ), and thus fi(N, WT, 8) = c' = cr/(2INI - 1 ) for a11 i E N.
2. Suppose that there exist JET and hE N \ T with 8(j) = 0 and 8(h) = {j}. Vertical neutrality implies that
/i(N,WT, 8) + A(N,WT, 8) = f;(N\ {h},WT,s-h).
With the induction hypothesis and the fact that f;(N,WT,8) = fh(N, wT,8) = c' this yields that 2c' = cT/(2INI - 2 ), and thus h(N, WT, 8) = c' = cT/(2INI - 1 ) for a11 i E N.
3. Suppose that there exist h,j E T with 8(j) = 0 and 8(h) = {j}. Then take a 9 E 1N \ N, and define 8',8" E SNU{g} by
Banzhaf Permission Values for Games with a Permission Structure 35
{ {h}' if i=g,
8'(i) = (8(i) \ {h}) U {g}, if i E 8- 1 (h) , 8(i), ifiEN\8- 1 (h)
and
{ {h,j}, if i=g,
8"(i) = (8~i) \ {h}) U {g}, ~f ~ E 8- 1(h) , 8(z), If ZEN\8- 1(h).
(Note that 8" rt S::~e{g}.) The necessary player property and weak struc tural monotonicity imply that
fh(NU {g},WT, 8") = h(NU{g},WT,8") = fg(NU{g},WT,8") .(6)
Since «N U {g}) \ {h},CTUT\{h},8~h) E S~~{g})\{h} (with I(N U {g} \ {h} ) I = INI) is as considered in case (ii) it follows from that case that
h«N U {g}) \ {h}, CTUT\{h},s~\)
=fg«NU{g})\{h},CTUT\{h},8~h)= 21~~-1'
But then horizontal neutrality implies that
h(N U {g}, WT, 8~(h,j)) + h(N U {g},WT, 8~(h,j))
= h«N U {g}) \ {h},CTUT\{h}, 8~h) = 21~~-1 .
With the necessary player property and weak structural monotonicity this yields
fh(N U {g},WT, 8~(h,j)) = h(N U {g}, WT, 8~(h,j))
= fg(NU{g},WT,8~(h,j)) = 2~~1 . (7)
Now, predecessor fairness implies that
fh(N U {g}, WT, 8") - fh(N U {g}, WT, 8~(h,j))
= fg(NU {g},WT,s~(h,j)) - fg(NU {g},wT,8"),
which with (6) and (4) gives
h(N U {g}, WT, 8") = fg(N U {g}, wT,8")
= h(NU {g},wT,8") = 2~~1 . (8)
Again applying predecessor fairness yields
fg(N U {g}, WT, 8") - fg(N U {g}, WT, 8')
= fh(N U {g}, WT, 8') - h(N U {g}, WT, 8") ,
which with the necessary player property, weak structural monotonicity and (3) yields
fg(N U {g}, WT, S') = fh(N U {g}, WT, S') = fJ(N U {g},WT, S') = 2~~1 .
Finally, vertical neutrality yields
c* = h(N, WT, S) = fh(NU {g}, WT, S') + Ig(NU {g}, wT, S') = 21~f-l '
and thus fi(N,WT,S) = c* = cT/(2IN1 - 1 ) for all i E N.
So, for S E St:.ee and WT = CTUT with CT > 0 and T U S- 1 (T) = N, we conclude that fi(N,WT,S) = c* = cTj(2 IN I-1) = ßf(N,WT,S) for all i E TUS-1(T) = N. 0
The next step is to show that adding the inessential player property and disjunctive fairness to the axioms implies that 1 is uniquely determined for all positively scaled unanimity games with apermission tree.
Lemma 2. 11 the allocation rule f satisfies one player efficiency, vertical neu trality, horizontal neutrality, the inessential player property, the necessary player property, weak structural monotonicity, disjunctive fairness and prede cessor fairness, then f(N, WT, S) is uniquely determined whenever S E St:.ee and WT = CTUT for some T C N, T =1= 0, and CT > o. Proof Suppose that the allocation rule 1 satisfies the eight axioms. Consider apermission (tree) structure S E St:.ee and monotone characteristic function WT = CTUT as given in (4) for some CT > o. Denote as(T) = TUS-1(T).
Since all players in N \ as(T) are inessential players in (N, wT, S), the inessential player property implies that fi(N, WT, S) = 0 for all i E N \ as (T). Further , f satisfying the necessary player property and weak structural monotonicity and the fact that S(i) = S(i) for all i E N and SE St:.ee imply that there exists a constant c* E IR such that fi(N, WT, S) = c* for all i E as(T). So,
f .(N S) = {c*, if i E as(T) =TUS-1(T) , "WT, 0, if i E N\as(T).
(9)
We prove that c* (and thus f(N, WT, S)) is uniquely determined by induc tion on IN \ as(T)I.
If IN \ as(T)1 = 0 then c* (and thus f(N, WT, S)) is uniquely determined by Lemma l.
Proceeding by induction assume that c* = lio(N, wf' S) is uniquely deter-
mined for all (N, wf' S) with SE sßee> cf> 0 and IN\asCr)1 < IN\as(T)I. Since N \ as(T) =1= 0 there exists a JEN \ as(T) with S(j) = 0. We
distinguish the following three cases (of which at least one must occur).
1. Suppose there exists an h E as(T)nS(io) such that j f/. S(h). (Remember that io denotes the unique top-player in the permission structure.) Define S' E Sff by
SI(.)_{{h}, ifi=j, t - S(i) , otherwise.
(Note that S' f/. sfj.ee') Disjunctive fairness implies that
fio(N,WT, S') - fio(N,WT,S) = /j(N,WT, S') - /j(N,WT,S). (10)
Predecessor fairness implies that
So, from (10) and (11) it foIIows that
fio(N,WT,S) = fiO(N,WT, S') and /j(N,WT,S) = !i(N,WT, S') .(12)
With the inessential player property it then foIIows that
/j(N,WT, S') = !i(N,WT,S) = O. (13)
Predecessor fairness also yields that
fio(N, wT, S') - fio(N, WT, S~(io,h»
= /jeN, WT, S~(io,h» - fj(N, WT, S') . (14)
Since IN \ as, (T)I< IN \ as(T)1 and S~(. h) E sfj.ee' the induction -('O.h) '0,
hypothesis, (13) and (14) yield that lio(N,WT,S')=fio(N,WT,S~(io,h»+ !i(N,WT,S~(io,h» is uniquely determined. So, with (12) also c' = fio(N, WT, S) is uniquely determined.
2. Suppose that there exists an h E as(T) n S(io) such that j E S(h). Take gE S(h) n (S-l(j) U {j}), and define S' E S~ by
S'(') = {S(i)U{g}, if i=io, Z Sei) , otherwise.
(Note that again S' f/. sfj.ee') Disjunctive fairness implies that
fio(N,WT, S') - fio(N,WT,S~(h,g» = h(N, WT,S/) - fh(N,WT,S~(h,g) ,
while predecessor

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