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Efficient Algorithm for Nash Equilibrium in Graphical Games

Explore computational strategies for determining Nash equilibrium in graphical games, considering various approaches and complexities. Discuss approximate and exact solutions, benefits, and future research directions.

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Efficient Algorithm for Nash Equilibrium in 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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