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Definable strategies for Games and Equilibria
Université Paris II Michel de Rougemont Definable strategies for Games and Equilibria Games as models for defining the value of information Definable Strategies There are games with inherent difficult equilibria ICEB, December 2002
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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
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Classical games Normal forms (2 players) Extensive forms C D 3,3 0,4
Prisoners Dilemma II C D 3,3 0,4 4,0 1,1 I I C D II I II (2,2) (6,6) ICEB, December 2002
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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
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Nash Equilibria C D 3,3 0,4 4,0 1,1 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
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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
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Value of a form Kleinberg,Papadimitriou, Raghavan 2001
Amazon. Do you prefer: Books CDs DVDs Books CDs Value based on the Shapley value of a cooperative game ICEB, December 2002
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Conclusion Games and Mechanisms Definable strategies
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
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