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Definable strategies for Games and Equilibria

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.

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Definable strategies for Games and Equilibria

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  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

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