Main Message

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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
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Lecture plan ? 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).
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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.
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Comment no problem with both Matching (coordinati
on game)
Pool home
pool home
Both pure strategy equilibria are info proof and
highly stable
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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
males selecting same location
females selecting same location
Every equilibrium is (?) info proof and highly
robust
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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 ?
Matchers payoff .10(if he chooses the computer
he likes) .90?(the
proportion of others he matches) ? Poets
payoff .10(if he chooses the computer he likes)
- .90?(the proportion
of others he matches) ? Poets lovers
.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
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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
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General Asymptotic Result ? 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 TA dist (TA)
0,1
3. Independent priors over types
Thm All the equilibria of games in ? with m or
more players are e information proof.
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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
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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)
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A sequential revisional version of a given
Bayesian game is a finite extensive perfect
recall game with ? Initial node is natures.
Arcs identify player type profiles. Probs 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.
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? 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
players 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.
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Subgame perfection in large games example
A million men and a million women each chooses
IBM or Mac Mans payoff .10 (if he chooses
IBM) .90?(the
proportion of others he matches) Womans
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 number of deviation from the play path, to a
get to a non credible subgame, is huge
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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
0 2 , 9 1 , 8 9 , 9 8 , 8
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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 NEs 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).
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Complete info anonymous games Purification and
Schmeidlers 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 Nashs Theorem. Purification
for every mixed strategy eq. every realization
is an equivalent pure strategy eqm
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Related concepts and Properties ? Ex post Nash
implementation. (Green and Laffont, Wilson
criticism) ? No regret equilibrium.
(MinehartScotchmer) ? Rational expectations
properties. (Grossman..., Radner, Jordan,
Minelli Polemarchakis, ForgesMinelli)
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Related Large Games ? Continuum of players
(Schmeidler, Kahn, Rath, Al-Najjar) ? Large
markets and resource allocation (GrovesHart,
Hart, HildenbrandKohlberg) ? Large auctions
(Rustichini, Satterthwaite Williams,
PesendorferSwinkels, DekelWolinsky,
ChungEly) ? Large voting games
(FeddersenPesendorfer)
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? Large repeated games (Green, Sabourian,
Al-NajjarSmorodinsky) ? Player smallness
(Fudenberg, Levine Pesendorfer,
GulPostlewaite, Mailath Postlewaite,
McLeanPostlewaite) ? Recurring Games, learning
unknown priors (JacksonKalai)