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# Definable strategies for Games and Equilibria PowerPoint PPT Presentation

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

Definable strategies for Games and Equilibria

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## Definable strategies for Games and Equilibria

Université Paris II

Michel de Rougemont

mdr@lri.fr

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

Amazon.

Do you prefer:

Books

CDs

DVDs

www.amazon.com

Books

CDs

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