1. Pure Nash Equilibria: Complete Characterization of Hard and Easy Graphical Games Albert Xin Jiang U. of British Columbia MohammadAli Safari Sharif U. of Technology

2. Computational Game Theory Computing solution concepts, given description of game  Normal form: size grows exponentially  Compact game representations Graphical Games [Kearns, Littman & Singh 2001]  A player’s utility only depends on a subset of other players.  Model as directed graph: Vertices: players Edge from u to v: utility of v depends on u  Size: exponential in max in-degree of graph

3. Pure-strategy Nash equilibrium Pure-strategy Nash equilibrium (PSNE)  Each player is playing a pure-strategy best response Computing PSNE for graphical games  NP-complete [Gottlob et al 2003]  Tractable for certain classes of graphs: Bounded treewidth [Daskalakis & Papadimitriou 2006] Bounded hypertree-width [Gottlob et al 2003]  Are there other tractable classes of graphs?

4. Our Results for Graphical Games PUREGG(C, -): problem of deciding the existence of PSNE in graphical games, whose graphs are restricted to the class C. Assumption: FPT  W[1] We give a complete characterization of tractable classes of bounded-indegree graphs:  PUREGG(C,-) in P iff graphs in C have bounded treewidth (after iterated removal of sinks)

5. Colored Hypergraphical Games Certain players have identical utility functions Colored Hypergraph (V,E,C): each edge e is an ordered tuple of vertices, labeled with a color Colored Hypergraphical Games (CHG)  Vertices: players  Edge with starting i: players affecting i’s utility  Players with same-colored edges have the same utility function.

6. Graphical Games as CHGs A graphical game can be encoded as a CHG, with a different color for each player’s edge. Given a directed graph G=(V,E), the induced colored hypergraph H(G) is as follows:  Same set of vertices as G  For each v  V, there is a hyperedge e consisting of vertices in the neighborhood of v  This hyperedge is labeled with color v

7. Results for CHGs Computational problem: PURECHG(C, -)  Our result: tractable if and only if bounded treewidth modulo homomorphic equivalence (after iterated removal of sinks)  This is a wider family of tractable games compared to PUREGG(C, -)

8. Background: Homomorphism Given colored hypergraphs G and H, a homomorphism from G to H is a mapping h: V(G)  V(H) that preserves both adjacency and color  For every edge e=(a 1, …,a k )  E(G), h(e)  (h(a 1 ),…,h(a k ))  E(H) and h(e) has same color as e. G has treewidth w modulo homomorphic equivalence if there is a graph H with treewidth w such that there is a homomorphism from G to H and vice versa.  Treewidth mod. hom. equivalence can be smaller than treewidth  e.g. bipartite graphs are hom. equivalent to an edge

9. Homomorphism Problem Homomorphism problem: given G, H, decide whether there is a homomorphism from G to H HOM(C, D): homomorphism problem where left hand graph is restricted to class C and right hand graph to class D.  Use ‘-’ where graph unrestricted: e.g. HOM(C, -) Many well-known problems can be formulated as homomorphism problems  k-colorability  CSP

10. Grohe’s result Theorem [Grohe 2003] : Assume FPT  W[1]. Then for every recursively enumerable class C of colored hypergraphs with bounded arity, the following statements are equivalent: 1.HOM(C,-) is in polynomial time 2.C has bounded treewidth modulo homomorphic equivalence

11. Graphs with Sinks Is there equivalence between PURECHG(C,-) and HOM(C,-)?  Not directly. Example: graphical games on DAGs  Pure NE always exist; can be computed efficiently  Class of all DAGs have unbounded treewidth (mod hom equivalence) More generally, whenever there is a sink (vertex with out-degree zero), that vertex doesn’t affect the existence of pure NE PUREGG and HOM have slightly different structure:  In a graphical game, whatever actions others chose, each player has at least one best response.

12. Main theorems Definition: a directed graph G is irreducible if it does not have a sink Given digraph G, define the reduced graph red(G) to be the result of repeatedly removing vertices with out- degree zero. Theorem: PURECHG(C,-) in P iff red(C) has bounded treewidth mod hom equivalence Corollary for PUREGG(C,-)  the treewidth mod hom equivalence of H(C) is equal to the treewidth of H(C)  PUREGG(C, -) in P iff red(C) has bounded treewidth Key step: prove the irreducible case

13. Tractability If H(C) has bounded treewidth mod homomorphism equivalence, then PUREGG(C,-) is in P Proof sketch: given a graphical game, construct an instance (G’,H’) of homomorphism problem  G’: same as H(G)  H’: a vertex for every action of every player a hyperedge of color i for each tuple of actions of i and N(i) corresponding to a best response of i against her neighbors’ actions There is a PSNE in the game iff there exists a homomorphism By Grohe’s theorem, such homomorphism problem can be solved in poly time

14. Hardness for graphical games Given irreducible C, if PUREGG(C,-) is in P, then H(C) must have bounded treewidth mod homomorphic equivalence First try: given arbitrary hom problem (G,H), construct equivalent graphical game  Treewidth preserved  Problem: resulting graphical game has specific structure Instead: we want PUREGG(C,-) for arbitrary class C of irreducible digraphs

15. Hardness (cont’d) Proof sketch: given class C of digraphs, consider H(C), the class of the induced colored hypergraphs For colored hypergraphs (G’, H’) where G’  H(C), construct graphical game on digraph G where G’=H(G)  Each player i’s action set is V(H’), plus “failure actions” T and B.  Utilities are constructed so that the graphical game has a pure NE iff there is a homomorphism from G’ to H’ High rewards for strategy profiles corresponding to homomorphism Otherwise, players are forced to play the failure actions

16. Conclusion PUREGG(C, -): complete characterization of tractable classes of bounded-indegree graphs PURECHG(C,-): complete characterization of tractable classes of bounded-indegree colored hypergraphs Future work: use similar techniques for related problems, other representations  action-graph games [Jiang et al. 2008]