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

Lecture plan: Quick Definition: A Bayesian eq. is information proof  Two examples (one Nash one Bayesian)  Convergence to information-proofness (uniform for many games at an exponential rate)  Properties of 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). Example of a Bayesian game- Poker, where Bayesian equilibrium is not info pf The desirability of ex-post stability in Implementation goes back to Wilson and Green-Laffont Our type of large games is a generalization of Schmeidler’s games with a continuum of players and complete info But we do finite asymptotic results with incomplete information .

Complete information example (uncertainty due only to mixed strategies) Mis/Matching at the pool Ex-post sustainability of equilibrium in the actual game, beyond the decision phase In complete info games: Pure stgy eq are info-proof Mixed stgy eq are usually not But in incomplete info games even pure stgy eq are typically not info pf Very difficult to model The only (mixed) equilibrium is not good because it is not information proof.

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

With many players, mis/matching does work n males and m females choices: city or beach payoffs at either location for a male for a female In particular, the mixed stgy eq is almost info-proof Robustness: the order of moves, observations, revisions, the precise values of n and m. Note: the payoff functions are continuous The same conclusion will hold under any continuous functions % males selecting same location % females selecting same location Every equilibrium is () info proof and highly robust

Incomplete information example Computer choice game  n players: 80% matchers, 1% poets, 19% poets lovers  choices: IBM or Mac  types: like IBM or like 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) The problems of instability for a population of 100 people. The problem is less sever in a population with 1 million people (10,000 poets) Very difficult to model for n =100, but players buy what they like is () info proof for large n

Still, if types are independent and payoff functions are 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

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 Example discontinuous payoffs: match 1 poet First prove comparative static bounds imply uniform convergence at an exponential rate in the number of players Thm: All the equilibria of games in Г with m or more players are ε information proof.

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 Continuity may be weakened to local continuity near the expected play of the equilibrium. Example, voting games Local continuity may be weakended to low strategic interdependence. zero strategic An equilibrium with 50.01% expected for Bush (49.99% for Gore) is not 50 100 % for Bush

Stability Properties of Information- Proof Equilibrium Invariance to:  sequential games with revision (Nash, without subgame perfection)  prior type probabilities (full info-proofness only)  mixed strategy probabilities (full info-proofness only) Example: in the computer choice game. The poets in NY choose first. After observing their choices, the ordinary people all over the country choose. After observing them, each NY poet gets a 50% chance to revise. After observing a random sample of 10%, every west coast poet chooses. A random group of ordinary people revise, and the game ends. If the types were equally likely, every body choosing what they like with no revision is an equilibrium.

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 every information set a player knows at least his type.  Every play path visits every player at least once.

profile and the last action selected by each player.  The outcome of a play path is the initial type profile and the last action selected by each player. Payoffs defined by the outcome.  A player’s induced strategy: randomize as in the given Bayesian game strategy and never revise Stability Characterization Thm: A Bayesian eq. is info-proof  it induces Nash equilibrium in every sequential revisional version of the game. Stability Thm implies that info-pf eqm can serve the same function as rational expectations eqm. It overcomes much of the criticizm of game theoretic models, e.g. in IO, not knowing the order of moves.

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 IBM is information proof Not subgame perfect, if the women move first, BUT The only info proof equilibria are all choosing IBM and all choosing Mac, but .50,.50 is an approximate info pf equilibria The number of deviation from the play path, to a get to a non credible subgame, is huge

Structure of Info-Proof Equilibria normal form example: .60 .40 0 0 .25 8 , 6 7 , 6 9 , 1 5 , 2 .50 8 , 4 7 , 4 0 , 2 3 , 1 .25 8 , 9 7 , 9 3 , 6 5 , 7 A player does not care which equilibrium action his opponent ends up choosing Weakly dominant actions in the game restricted to the equilibrium actions see why the mixing probabilities do not matter 0 2 , 9 1 , 8 9 , 9 8 , 8

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 The same under generalized notions Generalizations to multi player. Two actions in my support are equally good, no matter the profile of realized (at the equilibrium) opponents’ choices Generalization to Bayesian. Given own type, two actions in my support are equally good no matter the realized profile of opponents equilibrium types and actions So in large game all equilibria have approximate versions of this property A player does not care which equilibrium his opponents play (restricted local dominance).

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 purification is automatic, you cannot get away from it. This should work even better in the limit, Schmeidler’s case, but is impossible to formulate (see Judd)

Related concepts and Properties  Ex post Nash implementation. (Green and Laffont, Wilson criticism)  No regret equilibrium. (Minehart&Scotchmer)  Rational expectations properties. (Grossman..., Radner, Jordan, … Minelli& Polemarchakis, Forges&Minelli) Ex-Post implementation in large society should be easier.

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) Schmeidler result should be substantially stronger. But difficulty with the law of large numbers with a continuum of players. More general sufficient conditions based on auctions Local continuity, or even small strategic dependence may suffice, as seen by voting

 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) Why player smallness condition may help Learning priors is important But may be hard to model here