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Graphical Games

Graphical Games. Kjartan A. Jónsson. Nash equilibrium. Nash equilibrium N players playing a dominant strategy is a Nash equilibrium When one has a dominant strategy and the other chooses accordingly is also Nash equilibrium Computationally expensive for n players. Computing Nash equilibrium.

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Graphical Games

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  1. Graphical Games Kjartan A. Jónsson

  2. Nash equilibrium • Nash equilibrium • N players playing a dominant strategy is a Nash equilibrium • When one has a dominant strategy and the other chooses accordingly is also Nash equilibrium • Computationally expensive for n players

  3. Computing Nash equilibrium • Ex: 2 action game • Tabular representation • Consider all possible actions from all players • n players • Expensive

  4. Nash equilibrium: Proposal • Ex: 2 action game • Tree graph • Consider only actions from neighbors • n players • k neighbors • Then propagate result upwards • Less expensive

  5. Abstract Tree Algorithm Downstream Pass: Each node V receives T(v,ui) from each Ui V computes T(w,v) and witness lists for each T(w,v) = 1 Upstream Pass: V receives values (w,v) from W, T(w,v) = 1 V picks witness u for T(w,v), passes (v,ui) to Ui U1 U2 U3 T(w,v) = 1 <-->  an “upstream” Nash where V = v given W = w <--> u: T(v,ui) = 1 for all i, and v is a best response to u,w V W Borrowed from Michael Kearns

  6. Problem • “Since v and ui are continues variables, it is not obvious that the table T(v,ui) can be represented compactly, or finitely, for arbitrary vertices in a tree” • Solutions • “Approximate” • “Exact”

  7. Approximation • Approximation algorithm • Run time: polynomial in 2^k • Represent an approx. to every Nash • Generates random Nash or specific Nash

  8. Exact Extension to exact algorithm • Run time: exponential • Each table is a finite union of rectangles • Exponential in depth

  9. Benefits • We can represent a multiplayer game using a graph • Natural relationship between graphical games and modern probabilistic modeling more tools • Local Markov Networks language to express correlated equilibria

  10. Future research • Efficient algorithm for Exact Nash Computation • Strategy-proof • Loose now to win later • Cooperative and behavioral actions • Cooperation between a set of players

  11. Conclusion • Theoretically: works fine • Practically? • An employee in division A can influence division B (email correspondence) • Circled graph • Considered in both divisions • Ignored

  12. References • Book: Algorithmic Game Theory, chapter on Graphical Games • Paper: Graphical Models for Game Theory – Michael Kearns, Michael L. Littman, Satinder Singh • Presentation: by Michael Kearns (NIPS-gg.ppt)

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