Equilibria of Large Games: Information-Proofness

1. The Equilibria of Large Games are Information Proof by Ehud Kalai Main Message Nash equilibrium is often a bad modeling tool Problems disappear in large games, as the equilibria become “information proof.” Information-proof equilibria are extremely robust

2. Lecture plan:  Twoexamples (one Nash one Bayesian) Convergence to information-proofness (uniform for many games at an exponential rate)  Propertiesof information proof equilibria (both stability and structure, relationship to Nash enterchangeability) Quick Definition:A Bayesian eq. is information proof if knowledge of its outcome (realized types and selected pure actions of all the players) gives no player incentive to change his own selected action (ex-post Nash).

3. Complete information example (uncertainty due only to mixed strategies) Mis/Matching at the pool Very difficult to model The only (mixed) equilibrium is not good because it is not information proof.

4. Comment: no problem with both Matching (coordination game) Pool home pool home Both pure strategy equilibria are info proof and highly stable

5. With many players, mis/matching does work nmalesand mfemales choices:cityorbeach payoffs at either location for a malefor a female % males selecting same location % females selecting same location Every equilibrium is ()info proof and highly robust

6. Incomplete information example Computer choice game  n players: 80% matchers, 1% poets, 19% poets lovers  choices:IBMor Mac  types:like IBMorlike Mac  priors: independent, equally likely types  Matcher’s payoff: .10(if he chooses the computer he likes) +.90(the proportion of others he matches)  Poet’s payoff:.10(if he chooses the computer he likes) - .90(the proportion of others he matches)  Poets lover’s:.10(if he chooses the computer he likes) +.90(the proportion of poets that like his choice) Very difficult to model for n =100, but players buy what they like is () info proof for large n

7. But more generally:  Players from different locations, professions, genders, etc.  More computer choices  More types  Players with different priors and different utility functions Still, if types are independent and payoff functions are continuous and anonymous, all the equilibria become information proof and robust as the number of players increases

8. General Asymptotic Result Г is a family of Bayesian game satisfying: 1. Universal finite set of types T and of actions A 2. Finitely many anonymous continuous payoff functions U : T×A × dist (T×A) [0,1] 3. Independent priors over types Thm: All the equilibria of games in Г with m or more players are ε information proof.

9. Local continuity suffices Majority voting payoff for a Sharon supporter An equilibrium with 80% expected for Sharon (20% for Barak) is highly information proof 50 100 % for Sharon payoff for a Bush supporter An equilibrium with 50.01% expected for Bush (49.99% for Gore) is not 50 100 % for Bush

10. Stability Properties of Information- Proof Equilibrium Invariance to: sequential games with revision (Nash, without subgame perfection) prior type probabilities (full info-proofness only) mixed strategyprobabilities (full info-proofness only)

11. A sequential revisional version of a given Bayesian game is a finite extensive perfect recall game with: Initial node is nature’s. Arcs identify player type profiles. Prob’s are the given priors Every other node belongs to a player, arcs are the actions of this player.  At everyinformation set a player knows at least his type. Everyplay pathvisits every player at least once.

12. The outcomeof a play path is the initial type profile and the last action selected by each player. Payoffsdefined by the outcome. A player’s induced strategy: randomize as in the given Bayesian gamestrategy and never revise Stability Characterization Thm: A Bayesian eq. is info-proof  it induces Nash equilibrium in every sequential revisional version of the game.

13. Subgame perfection in large games: example A million men and a million women each chooses IBM or Mac Man’s payoff = .10 (if he chooses IBM) +.90(the proportion of others he matches) Woman’s payoff = .10 (if she chooses Mac) +.90(the proportion of others she matches) All choosing IBMis information proof Not subgame perfect, if the women move first, BUT The number of deviation from the play path, to a get to a non credible subgame, is huge

14. Structure of Info-Proof Equilibria normal form example: .60 .40 0 0 .25 8 , 67 , 6 9 , 1 5 , 2 .50 8 , 47 , 4 0 , 2 3 , 1 .25 8 , 97 , 9 3 , 6 5 , 7 0 2 , 9 1 , 8 9 , 9 8 , 8

15. Structure characterization Thm: A Bayesian eq is information proof  the outcomes in its support are interchangeable in a generalized sense of Nash. NE NE? The NE’s are interchangeable if the NE?’s are also Nash eq. NE? NE A player does not care which equilibrium his opponents play (restricted local dominance).

16. Complete info anonymous games: Purification and Schmeidler’s Results with a continuum of players Schmeidler shows: •Existence of a “mixed” strategy equilibrium •Purification: every “mixed” strategy equilibrium has an equivalent pure strategy equilibrium. the finite asymptotic results here: •Existence is automatic by Nash’s Theorem. •Purification: for every mixed strategy eq. every realization is an equivalent pure strategy eq’m

17. Related concepts and Properties Ex post Nash implementation. (Green and Laffont, Wilson criticism) No regret equilibrium. (Minehart&Scotchmer) Rational expectationsproperties. (Grossman..., Radner, Jordan, … Minelli& Polemarchakis, Forges&Minelli)

18. Related Large Games Continuum of players (Schmeidler, Kahn, Rath, Al-Najjar) Large markets and resource allocation (Groves&Hart, Hart, Hildenbrand&Kohlberg) Large auctions (Rustichini, Satterthwaite &Williams, Pesendorfer&Swinkels, Dekel&Wolinsky, Chung&Ely) Large voting games (Feddersen&Pesendorfer)

19.  Large repeated games (Green, Sabourian, Al-Najjar&Smorodinsky) Player smallness (Fudenberg, Levine & Pesendorfer, Gul&Postlewaite, Mailath& Postlewaite, McLean&Postlewaite) Recurring Games, learning unknown priors (Jackson&Kalai)