Graphical Games Kjartan A Jnsson Nash equilibrium n
Graphical Games Kjartan A. Jónsson
Nash equilibrium n Nash equilibrium N players playing a dominant strategy is a Nash equilibrium n When one has a dominant strategy and the other chooses accordingly is also Nash equilibrium n n Computationally expensive for n players
Computing Nash equilibrium n n Ex: 2 action game Tabular representation n Consider all possible actions from all players n n n players Expensive
Nash equilibrium: Proposal n n Ex: 2 action game Tree graph n Consider only actions from neighbors n n n players k neighbors Then propagate result upwards CEO Root Less expensive Manager A K=1 Employee A k=1 Manager B K=2 Employee B k=2 Employee C k=1 Employee D k=2 Employee E k=3
Abstract Tree Algorithm U 1 U 2 V U 3 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 W n Downstream Pass: n Each node V receives T(v, ui) from each Ui n V computes T(w, v) and witness lists for each T(w, v) = 1 Borrowed from Michael Kearns n Upstream Pass: n V receives values (w, v) from W, T(w, v) = 1 n V picks witness u for T(w, v), passes (v, ui) to Ui
Problem n n “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” n “Exact” n
Approximation n Approximation algorithm Run time: polynomial in 2^k n Represent an approx. to every Nash n Generates random Nash or specific Nash n
Exact Extension to exact algorithm n Run time: exponential n Each table is a finite union of rectangles n Exponential in depth
Benefits n We can represent a multiplayer game using a graph n n Natural relationship between graphical games and modern probabilistic modeling more tools Local Markov Networks language to express correlated equilibria
Future research n n Efficient algorithm for Exact Nash Computation Strategy-proof n n Loose now to win later Cooperative and behavioral actions n Cooperation between a set of players
Conclusion n n Theoretically: works fine Practically? n An employee in division A can influence division B (email correspondence) Circled graph n Considered in both divisions n Ignored n
References n n n 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 (NIPSgg. ppt)
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