Strategic games may exhibit symmetries in a variety of ways. A common aspect of symmetry, enabling the compact representation of games even when the number of players is unbounded, is that players cannot (or need not) distinguish between the other players. We define four classes of symmetric games by considering two additional properties: identical payoff functions for all players and the ability to distinguish oneself from the other players. Based on these varying notions of symmetry, we investigate the computational complexity of pure Nash equilibria. It turns out that in all four classes of games equilibria can be found efficiently when only a constant number of actions is available to each player, a problem that has been shown intractable for other succinct representations of multi-player games. We further show that identical payoff functions simplify the search for equilibria, while a growing number of actions renders it intractable. Finally, we show that our results extend to wider classes of threshold symmetric games where players are unable to determine the exact number of players playing a certain action.

doi 10.1287/moor.1080.0322

Abstract not found

We define a generalized strategy eliminability criterion for bimatrix games that considers whether a given strategy is eliminable relative to given dominator & eliminee subsets of the players ’ strategies. We show that this definition spans a spectrum of eliminability criteria from strict dominance (when the sets are as small as possible) to Nash equilibrium (when the sets are as large as possible). We show that checking whether a strategy is eliminable according to this criterion is coNP-complete (both when all the sets are as large as possible and when the dominator sets each have size 1). We then give an alternative definition of the eliminability criterion and show that it is equivalent using the Minimax Theorem. We show how this alternative definition can be translated into a mixed integer program of polynomial size with a number of (binary) integer variables equal to the sum of the sizes of the eliminee sets, implying that checking whether a strategy is eliminable according to the criterion can be done in polynomial time, given that the eliminee sets are small. Finally, we study using the criterion for iterated elimination of strategies. Categories and Subject Descriptors

We present a Garey/Johnson-style list of problems known to be complete for the second and higher levels of the polynomial-time Hierarchy (polynomial hierarchy, or PH for short). We also include the best-known hardness of approximation results. The list will be updated as necessary. Updates The compendium currently lists more than 80 problems. Latest changes include: • added [GT26] SUCCINCT k-KING, • added [GT25] SUCCINCT k-DIAMETER, • added [GT4] SUCCINCT k-RADIUS at third level, • added [GT24] MINIMUM VERTEX COLORING DEFINING SET, • added [GT23] GRAPH SANDWICH PROBLEM FOR Π, • added [L24] MINIMUM 3SAT DEFINING SET,

Abstract We review the concepts of hypertree decomposition and hypertree width from a graph theoretical perspective and report on a number of recent results related to these concepts. We also show – as a new result – that computing hypertree decompositions is fixed-parameter intractable. 1 Hypertree Decompositions: Definition and Basics This paper reports about the recently introduced concept of hypertree decomposition and the associated notion of hypertree-width. The latter is a cyclicity measure for hypergraphs, and constitutes a hypergraph invariant as it is preserved under hypergraph isomorphisms. Many interesting NP-hard problems are polynomially solvable for classes of instances associated with hypergraphs of bounded width. This is also true for other hypergraph invariants such as treewidth, cutset-width, and so on. However, the advantage of hypertree-width with respect to other known hypergraph invariants is that it is more general and covers larger classes of instances of bounded width. The main concepts of hypertree decomposition and hypertree-width are introduced in the present section. A normal form for hypertree decompositions is described in Section 2. Section

We analyze the problem of computing pure Nash equilibria in action graph games (AGGs), which are a compact gametheoretic representation. While the problem is NP-complete in general, for certain classes of AGGs there exist polynomial time algorithms. We propose a dynamic-programming approach that constructs equilibria of the game from equilibria of restricted games played on subgraphs of the action graph. In particular, if the game is symmetric and the action graph has bounded treewidth, our algorithm determines the existence of pure Nash equilibrium in polynomial time.

We present a technique for reducing a normal-form (aka. (bi)matrix) game, O, to a smaller normal-form game, R, for the purpose of computing a Nash equilibrium. This is done by computing a Nash equilibrium for a subcomponent, G, of O for which a certain condition holds. We also show that such a subcomponent G on which to apply the technique can be found in polynomial time (if it exists), by showing that the condition that G needs to satisfy can be modeled as a Horn satisfiability problem. We show that the technique does not extend to computing Pareto-optimal or welfaremaximizing equilibria. We present a class of games, which we call ALAGIU (Any Lower Action Gives Identical Utility) games, for which recursive application of (special cases of) the technique is su#cient for finding a Nash equilibrium in linear time. Finally, we discuss using the technique to compute approximate Nash equilibria.

In this paper we present a novel generic mapping between Graphical Games and Markov Random Fields so that pure Nash equilibria in the former can be found by statistical inference on the latter. Thus, the problem of deciding whether a graphical game has a pure Nash equilibrium, a well-known intractable problem, can be attacked by well-established algorithms such as Belief Propagation, Junction Trees, Markov Chain Monte Carlo and Simulated Annealing. Large classes of graphical games become thus tractable, including all classes already known, but also new classes such as the games with O(log n) treewidth.

Pure Nash equilibria in games with a large