Modelling Large Games by Ehud Kalai Northwestern University

Full paper to come Related past papers: Kalai, E., “Large Robust Games,” Econometrica, 72, No. 6, November 2004, pp 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).

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)

Message and Motivating Examples

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) robust in a strong sense.

Example: Ex-post Nash in Match Pennies Players: k males and k females. Strategies: H or T. Male’s payoff: The proportion of females his choice matches. Female’s payoff: The proportion of males her choice mismatches. The mixed strategy equilibrium becomes ex-post Nash as k increases.

Example: Computer choice game Players: 1,2,…,n Strategies: I or M Player’s types: I-type or M-type, iid w.p Individual’s 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

Definitions

Want: A general definition that accommodates both previous robustness notions and more. Idea of definition: An equilibrium  of a one-simultaneous-move Bayesian game G is structurally robust, if it “remains equilibrium” in all “alterations” of G. “Alterations” of G are described by extensive games, A’ s.  “remains equilibrium” in an alteration A, if every adaptation of  to A,, is equilibrium in A.

G: any n-person one-simultaneous-move Baysian game. An alteration of G is any finite extensive game A s.t. A includes the G players: { A Players} = {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 are the same as in G Preservation of G choices: Every pure strategy of a G-player i,, has at least one adaptation,, in A. Examples: (1) A game with revision (or one dry run), (2) sequential play Playing A means playing G: with every final node z of A there is an associated profile of G pure strategies,. That is: playing leads to final nodes z with (z) =, no matter what strategies are used by the opponents.

Given an alteration A and a G-pure-strategy. An adaptation of to A is a strategy of player i in A, that leads to a final nodes z with no matter what strategies are used by the opponents. Given a G-strategy-profile  An adaptation of  is an A -strategy-profile,, s.t. for every G player i, is an adaptation of. Example: mixed strategies in match pennies. Given a G-mixed-strategy of player i. An adaptation of is an A -strategy,, s.t for every G-pure-strategy :for some.

Definition: An equilibrium of G,  is structurally robust if in every alteration A and in every adaptation, every G- player i is best responding, i.e. is best response to. It is (  ) structurally robust if in every alteration and adaptation as above: Pr(every G-player is  -optimizing at all his positive probability information sets) > 1- .

Implications of structural robustness

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.

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.

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  of the one simultaneous move game can be adapted. If G is a reduced form of a game U with unknown structure, the equilibria of G may serve as equilibria of U

5. Market games: Nash prices are competitive. Ex: Shapley-Shubik market game. Players: n traders. Types: iid prob’s, 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 traded price=(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 aggregate data of opponent types and fruit ownership (externalities). Every Nash equilibrium prices is competitive, i.e., strong rational expectations properties

Partial invariance to institutions: Markets in two island economy

Sufficient conditions for structural robustness

Structural-Robustness Thm (rough statement): The equilibria of a finite, one-simultaneous-move Bayesian game 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 have different payoff functions and different prior type- probabilities (publicly known).

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.

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 fails structural robustness), especially when n becomes large.

Formal statement

The model T – vocabulary of types (finite). A – vocabulary of actions (finite). N – Names of players. A family F : for any number of players n =1,2,…, F contains infinitely many simul. move Bayesian games G = (N, T= xT i, , A = xA i, u = (u 1,…,u n )). The u i ’s are uniformly equicontinuous. N µ N, a set of n-players. T i µ T, possible types of player i. Independent priors,  (t)=  i  i  t i . A i µ A, possible actions of player i. u i, utility of player i, is a fn of his type-act’n and the empirical dist over opponents type-actn’s to [0,1], i.e., semi anonymous payoff functions.

Structural Robustness Theorem: Given the family F and an  > 0, there exist positive constants  and ,  <1, s.t. for n=1,2,… all the n-player equilibria of games in F are ( ,  n ) structurally robust.

Method of proof Two steps: 1.By Chernoff bounds: as the number of players increases all the equilibria become (weakly)  ex-post Nash at an exponential rate. 2. This implies that they become  structurally robust at an exponential rate.

A bit more precisely Step 1. For an eq’m of the simultaneous move game Prob(outcome not being weakly  -ex-post Nash) <  n, with  > 0,  < 1. Step 2. For any strategy profile of the simult. move game If Prob(outcome not being weakly  -ex-post Nash) <  n, then in any alteration and every adaptation  Prob( some original player not being  2  optimal at some information set) < n  n / .

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

Modeling large games Sampling Models 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 aggregate data about the players?). methods help the modelers and players identify the game and equilibria?

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.