Graphical Games Kjartan A. Jónsson

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

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

Nash equilibrium: Proposal Ex: 2 action game Ex: 2 action game Tree graph Tree graph Consider only actions from neighbors Consider only actions from neighbors n players n players k neighbors k neighbors Then propagate result upwards Then propagate result upwards Less expensive Less expensive CEO Root Manager A K=1 Employee A k=1 Employee B k=2 Manager B K=2 Employee C k=1 Employee D k=2 Employee E k=3

Abstract Tree Algorithm Downstream Pass: Downstream Pass: Each node V receives T(v,ui) from each Ui Each node V receives T(v,ui) from each Ui V computes T(w,v) and witness lists for each T(w,v) = 1 V computes T(w,v) and witness lists for each T(w,v) = 1 Upstream Pass: Upstream Pass: V receives values (w,v) from W, T(w,v) = 1 V receives values (w,v) from W, T(w,v) = 1 V picks witness u for T(w,v), passes (v,ui) to Ui V picks witness u for T(w,v), passes (v,ui) to Ui U1U2U3 W V 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 Borrowed from Michael Kearns

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” “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 Solutions “Approximate” “Approximate” “Exact” “Exact”

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

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

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

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

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

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