Modelling%20Large%20Games

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Modelling Large Games

• by
• Ehud Kalai
• Northwestern University

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Full paper to come
Related past papers Kalai, E., Large Robust
Games, Econometrica, 72, No. 6,
November 2004, pp 1631-1666. 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).
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In semi anonymous gamesmany players
structural robustness

• Lecture plan
• Overview and motivating examples (3 slides).
• Definition of structural robustness (4 slides).
• Implications of structural robustness (4 slides).
• Sufficient conditions for structural robustness
  (3 slides).
• More formally (4 slides)
• 6. Future work (1 slide)

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Message and Motivating Examples
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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.
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Example Ex-post Nash in Match Pennies
Players k males and k females.
Strategies H or T.
Males payoff The proportion of females his
choice matches.
Females payoff The proportion of males her
choice mismatches.
The mixed strategy equilibrium becomes ex-post
Nash as k increases.
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Example Computer choice game
Players 1,2,,n
Players types I-type or M-type, iid w.p. .50-.50
Strategies I or M
Individuals payoff .1 if he chooses
his computer type (0 otherwise) .9 x
(the proportion of opponents he matches).
identical payoffs and priors are not needed in
the general model
The favorite computer equilibrium survives
sequential play as n becomes large.
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Definitions
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Want A general definition that accommodates both
previous robustness notions and more.
Idea of definition An equilibrium s 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.
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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
Examples (1) A game with revision (or one dry
run), (2) sequential play
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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 s
Example mixed strategies in match pennies.
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It is (e,r) structurally robust if in every
alteration and adaptation as above
Pr(every G-player is e-optimizing at all
his positive probability information sets) gt 1-r.
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Implications of structural robustness
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• 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.
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3. Revelation and delegation.
Ex Computer Choice game with delegation.
Players the original n computer choosers one
outsider, Pl. n1.
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. n1.
Next simultaneously, self-players choose own
computers, Pl.
n1 chooses computers for the delegators.
Payoffs of original computer choosers as in
CC. Payoff of Pl n1 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.
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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 s 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
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5. Market games Nash prices are competitive.
Ex Shapley-Shubik market game.
Players n traders.
Types .50-.50 iid probs, 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(1991)/(991)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
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Partial invariance to institutions Markets in
two island economy
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Sufficient conditions for structural robustness
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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).
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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.
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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.
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Formal statement
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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 xTi, p, A xAi, u
(u1,,un)).
N µ N, a set of n-players. Ti µ T , possible
types of player i. Independent priors, p(t)Pi
pi(ti). Ai µ A , possible actions of player
i. ui, utility of player i, is a fn of his
type-actn and the empirical dist over opponents
type-actns to 0,1, i.e., semi anonymous payoff
functions.
The uis are uniformly equicontinuous.
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Structural Robustness Theorem
Given the family F and an e gt 0, there exist
positive constants a and b, b lt1, s.t. for
n1,2, all the n-player equilibria of
games in F are (e, abn)
structurally robust.
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Method of proof

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

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A bit more precisely
Step 1. For an eqm of the simultaneous move
game Prob(outcome not being weakly e-ex-post
Nash) lt abn, with a,b gt 0, b lt 1.

• Step 2. For any strategy profile of the simult.
  move game
• If Prob(outcome not being weakly e-ex-post
  Nash) lt abn,
• then in any alteration and every adaptation
• Prob( some original player not being
• 2e optimal at some information set) lt n
  abn/e.

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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

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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?
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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.