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: 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 .

3. 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.

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
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

6. 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

7. 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

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
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.

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
% 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
% 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 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.

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

12. 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.

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 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

14. Structure of Info-Proof Equilibria
normal form example:
, , , , 2
, , , , 1
, , , , 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
, , , , 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
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).

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
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)

17. 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.

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)
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

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