kala i caltech modelling large

7/28/2019 Kala i Caltech Modelling Large

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Modelling Large Games

by

Ehud Kalai

Northwestern University


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Full paper to come

Related past papers:

Kalai, E., Large Robust Games, Econometrica, 72,

No. 6, November 2004, pp 1631-1666.

Kalai, E., Partially-Specified Large Games, Lecture

Notes in Computer Science, Vol.3828, 2005, 3 13.

Kalai, E., Structural Robustness of Large Games,

forthcoming in the new New Palgrave (available by

request).


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In semi anonymous games

many players structural robustness

Lecture plan:

1. Overview and motivating examples (3 slides).

2. Definition of structural robustness (4 slides).

3. Implications of structural robustness (4 slides).

4. Sufficient conditions for structural robustness (3 slides).

5. More formally (4 slides)

6. Future work (1 slide)


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Message and Motivating

Examples


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In Baysian games with many anonymous players

all Nash equilibria are structurally robust.

The equilibria survive changes in the order of play,

information revelation, revisions, communication,

commitment, delegation,

Nash modeling of large economic and political

systems, games on the Web, etc. is (partially) robustin a strong sense.


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Example: Ex-post Nash in Match Pennies

Players: k males and k females.

Strategies: H or T.

Males payoff: The proportion of females his choice

matches.

Females payoff: The proportion of males her choice

mismatches.

The mixed strategy equilibrium becomes

ex-post Nash as k increases.


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Example: Computer choice game

Players:1,2,,n

Strategies: I or M

Players types: I-type or M-type, iid w.p. .50-.50

Individuals payoff:.1 if he chooses his computer type (0 otherwise)

+.9 x (the proportion of opponents he matches).

The favorite computer equilibrium survives

sequential play as n becomes large.

identical payoffs and priors are not needed in the general model


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Definitions


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Want:

A general definition that accommodates

both previous robustness notions and more.

Idea of definition:

An equilibrium s of a one-simultaneous-move Bayesian

game G is structurally robust, if it remains equilibriumin all alterations ofG.

Alterations ofG are described by extensive games, As.

s remains equilibrium in an alterationA, if every

adaptation ofs toA, , is equilibrium in A.As


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G: any n-person one-simultaneous-move Baysian game.

An alteration ofG is any finite extensive game A s.t.

Aincludes the G players: {APlayers} = {G players}

Unaltered G types: initially, the G players are assigned types as in G.

Unaltered payoffs: At every final node of A, z, the G-players payoffs arethe same as in G

Preservation of G choices: Every pure strategy of a G-player i, ,has at least one adaptation, , in A.i

aA

ia

Examples: (1) A game with revision (or one dry run), (2) sequential play

PlayingA

means playing G:with every final node z ofAthere is an

associated profile ofG pure strategies, .)(za

That is: playing leads to final nodes z with (z) = ,

no matter what strategies are used by the opponents.i

ai

aA

ia


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Given an alterationAand a G-pure-strategy .i

a

An adaptation of toAis a strategy of playeriinA,

that leads to a final nodes z withno matter what strategies are used by the opponents.

ia

A

ia

ii

aza

)(

Given a G-strategy-profile s

An adaptation ofs is anA-strategy-profile, ,s.t. for every G playeri, is an adaptation of .

AsAi

si

s

Example: mixed strategies in match pennies.

Given a G-mixed-strategy of player i.i

s

An adaptation of is anA-strategy, , s.t for everyA

i

sis

ia )()( AA

iiiiaa ss

A

iaG-pure-strategy : for some .


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Definition: An equilibrium ofG, s,is structurally robust ifin every alteration Aand in every adaptation ,

every G- playeriis best responding, i.e.

As

A

is

A

isis best response to .

It is (e,r)structurally robust if in every alteration andadaptation as above:

Pr(every G-player is e-optimizing at allhis positive probability information sets) > 1-r.


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Implications of structural

robustness


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1. Play preceded by a dry run:Invariance to revisions, Ex-post Nash and being

information proof.

No revelation of information, even strategic, can

give any player an incentive to revise his choice.

2. Invariance to the order of play in a strong sense.


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3. Revelation and delegation.

Ex: Computer Choice game with delegation.

Players: the original n computer choosers + one outsider, Pl. n+1.

Types: original players are assigned types as in the CC game.

First: simult. play; each original player chooses between(1) self-play, or (2) delegate-the-play and report a type to Pl. n+1.

Next: simultaneously, self-players choose own computers,Pl. n+1 chooses computers for the delegators.

Payoffs of original computer choosers: as in CC.

Payoff of Pl n+1:1 if he chooses the same computer for all, 0 otherwise.

There is a new and more efficient equilibrium, but the old

favorite computer equilibrium survives.


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4. Partially-specified games:

Ex.: Computer Choice game played on the web.

Instructions: Go to web site xyz before Friday and click in your choice.

Structural Questions: who are the players?the order of play?monitoring? communications? commitments? delegations? revisions?...

Equilibrium: any equilibrium s of the one simultaneous move gamecan be adapted.

If G is a reduced form of a game U with unknownstructure, the equilibria of G may serve as equilibria of U


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5. Market games: Nash prices are competitive.

Ex: Shapley-Shubik market game.

Players: n traders.

Types: .50-.50 iid probs, a banana owner or an apple owner.

Strategies: keep your fruit or trade it (for the other kind).

Proportionate Price: e.g., with 199 bananas and 99 apples tradedprice=(199+1)/(99+1)=2. (2 bananas for an apple, 0.5 apples for a banana).

Payoff: depends on your type and your final fruit, and on the aggregatedata of opponent types and fruit ownership (externalities).

Every Nash equilibrium prices is competitive, i.e., strong

rational expectations properties


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Partial invariance to institutions: Markets in two island

economy


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Sufficient conditions for

structural robustness


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Structural-Robustness Thm (rough statement):

The equilibria of a finite, one-simultaneous-move Bayesiangame are (approximately) structurally-robust provided that:

1. The number of players is large.

2. The players types are drawn independently.

3. The payoff functions are anonymous and continuous.

The players are only semi anonymous. They may havedifferent payoff functions and different prior type-

probabilities (publicly known).


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A discontinuous counter example.

Ex: Match the Expert.

Players: 1,2,,n

P1 Types: I expert (informed that I is better) or with equal prob.

M expert (informed that M is better).

Players 2,..,n Types: all non expert wp 1.

Payoffs: 1 if you choose the better computer, 0 otherwise.

Equilibrium: Pl. 1 chooses the better computer,Pl. 2,3,,n randomize.

The equilibrium fails to be ex-post Nash (hence, it fails

structural robustness), especially when n becomes large.


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Counter example with dependent types.

Ex: Computer choice game with noisy dependent information.

Players: 1,2,,n

Types: wp .50 I is better and (independently of each other)each chooser is told I better wp .90 and M better wp .10.

wp .50 M is better and .

Payoffs: 1 if you choose the better computer, 0 otherwise.

Equilibrium: Everybody chooses what he is told.

The equilibrium fails to be ex-post Nash (hence, it failsstructural robustness), especially when n becomes large.


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Formal statement


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The modelT vocabulary of types (finite).

A vocabulary of actions (finite).NNames of players.

A family F: for any number of players n =1,2,,Fcontains infinitely many simul. move Bayesian gamesG = (N, T= xTi, p, A = xAi, u = (u1,,un)).

The uis are uniformly equicontinuous.

N N, a set of n-players.

TiT, possible types of player i.

Independent priors, p(t)=Pipi(ti).AiA , possible actions of player i.ui, utility of player i, is a fn of his type-actn and the empirical dist overopponents type-actns to [0,1], i.e., semianonymous payoff functions.


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Structural Robustness Theorem:

Given the family F and an e > 0, there existpositive constants a and b, b


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Method of proof

Two steps:

1. By Chernoff bounds: as the number of

players increases all the equilibria become(weakly) e ex-post Nash at an exponentialrate.

2. This implies that they become e structurallyrobust at an exponential rate.


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A bit more precisely

Step 1. For an eqm of the simultaneous move game

Prob(outcome not being weakly e-ex-post Nash) 0,b< 1.

Step 2. For any strategy profile of the simult. move game

If Prob(outcome not being weakly e-ex-post Nash)


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Areas for future work:

Relaxing the independence condition

What are the weaker conditions we get

under reasonable weaker independence

assumptions.

Computing equilibria of large games


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Modeling large games

SamplingModels of large games. What are the best

parameters to include (e.g., do we really need the

prior and utility of every player, or is it better to have

the modeler and every player have some aggregatedata about the players?).

methods help the modelers and players identify the

game and equilibria?


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Broader issues

Bounded rationality and computational ability in games.

Modified equilibrium notions that incorporate complexity

limitations.

Explicit presentations of family of games, and complexity

restricted solutions in the data of the game, given the

language of the game.This has been done to some degree in cooperative game

theory, less so in non cooperative.

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