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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

- 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

Prisoners Dilemma

II

- Normal forms (2 players)
- Extensive forms

I

I

C

D

II

I

II

(2,2)

(6,6)

ICEB, December 2002

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

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

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

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

- 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