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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.