1. 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

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 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

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 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

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 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

8. 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