Definable strategies for Games and Equilibria

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Definable strategies for Games and Equilibria. Université Paris II Michel de Rougemont [email protected] http://www.lri.fr/~mdr. Games as models for defining the value of information Definable Strategies There are games with inherent difficult equilibria.

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### Definable strategies for Games and Equilibria

Université Paris II

Michel de Rougemont

http://www.lri.fr/~mdr

• Games as models for defining the value of information
• Definable Strategies
• There are games with inherent difficult equilibria

ICEB, December 2002

Game theory, Complexity Economy of information
• Classical Game theory
• Equibria, Mechanisms
• Complexity and Computer Science
• Agents are limited in power
• Permissive computing
• Economy of information
• Models for the Value of information
• Feasible mechanisms
• Our contribution:
• Definable strategies,
• Definable equilibria

ICEB, December 2002

Classical games

Prisoners Dilemma

II

• Normal forms (2 players)
• Extensive forms

I

I

C

D

II

I

II

(2,2)

(6,6)

ICEB, December 2002

Symbolic Games

Nord-East of size (n,m)

I plays II plays I plays …. loses!

State of the game :

Equilibrium = winning strategy

Unknown for n,m>7 distinct!

How to restrict players to strategies

computable in polynomial time?

FACT: exponential blow in the transformation

Symbolic ->Extensive ->Normal Form

ICEB, December 2002

Nash Equilibria

Prisoners Dilemma

(C,D) gives 0 for I and 4 for II

Def: (x,y) is a Nash Equilibrium if none of the players can make a better decision.

Theorem. There always exists a pair (p,q) which is a Nash equilibrium.

Complexity of Nash ?

If the players are LIMITED, the equilibria change

ICEB, December 2002

Definable strategies

Finite structure for a game. North-East:

({1,…k},{1,….n}, <, +,h1, h2)

h1, h2 : {1,…k}->{1,….n}. {1,….n}

Positions I : h1(t)=(i,j)

Positions II : h2(t)=(i,j)

Strategy : -> {1,….n}. {1,….n}

Inductive definition

Definition: A strategy is First-order if

is a first-order.

Theorem : A strategy is P-computable iff it is inductive.

Theorem. There are symbolic games where the complexity of the equilibria is at any level of PH.

ICEB, December 2002

Value of a formKleinberg,Papadimitriou, Raghavan 2001

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Value based on the Shapley value of a cooperative game

ICEB, December 2002

Conclusion
• Games and Mechanisms
• Model of permissive computing
• Value of information
• Definable strategies
• Restrict the power of agents
• Approximation
• Complexity and Logic help to understand the Economy of information

ICEB, December 2002