Definable strategies for Games and Equilibria

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Definable strategies for Games and Equilibria

• Université Paris II
• Michel de Rougemont
• mdr_at_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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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

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Classical games

• Normal forms (2 players)
• Extensive forms

Prisoners Dilemma
II
I
I
C
D
II
I
II
(2,2)
(6,6)
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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,mgt7
distinct! How to restrict players to strategies
computable in polynomial time? FACT
exponential blow in the transformation Symbolic
-gtExtensive -gtNormal Form
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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
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Definable strategies

• Finite structure for a game. North-East
• (1,k,1,.n, lt, ,h1, h2)
• h1, h2 1,k-gt1,.n. 1,.n
• Positions I h1(t)(i,j)
• Positions II h2(t)(i,j)
• Strategy -gt 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.

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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

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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