We consider the following class of problems. The value of an outcome to a society is measured via a submodular utility function (submodularity has a natural economic interpretation: decreasing marginal utility). Decisions, however are controlled by non-cooperative agents who seek to maximise their own private utility. We present, under some basic assumptions, guarantees on the social performance of Nash equilibria. For submodular utility functions, any Nash equilibrium gives an expected social utility within a factor ¾ of optimal, subject to a function-dependent additive term. For non-decreasing, submodular utility functions, any Nash equilibrium gives an expected social utility within a factor ½ · AE of optimal, where ¼ AE ½ is a number based upon the discrete curvature of the function. A condition under which all sets of social and private utility functions induce pure strategy Nash equilibria is presented. The case in which agents, themselves, make use of approximation algorithms in decision making is discussed and performance guarantees given. Finally we present some specific problems that fall into our framework. These include the competitive versions of the facility location problem andmedian problem, a maximisation version of the traffic routing problem studied by Roughgarden and Tardos [9], and multiple-item auctions.

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Abstract. We present an approximation algorithm for the problem of finding a minimum-cost k-vertex connected spanning subgraph, assuming that the number of vertices is at least 6k 2 . The approximation guarantee is six times the kth harmonic number (which is O(log k)), and this is also an upper bound on the integrality ratio for a standard linear programming relaxation.Key words. approximation algorithm, -critically k-connected graph, linear programming relaxation, network design, k-outconnected graph, vertex connectivity AMS subject classifications. Primary, 68W25, 05C40; Secondary, 68R10, 90C27 DOI. 10.1137/S00975397013922871. Introduction. Let G = (V, E) be an undirected graph, let each edge e ∈ E have a nonnegative cost c e , and let k be a positive integer. The minimum-cost kvertex connected spanning subgraph (mincost k-VCSS) problem is to find a spanning subgraph H of minimum cost such that H is k-vertex connected. (A graph is called k-vertex connected if it has at least k +1 vertices, and the removal of any k −1 vertices leaves a connected graph.) The problem is NP-hard for k ≥ 2, and for k = 1 it is the minimum spanning tree problem. Our paper addresses the "special case" of the problem where the graph has order |V | ≥ 6k 2 ; this too is NP-hard for k ≥ 2. (So for a fixed k, our method handles all graphs except a finite set of "small" graphs, and our method fails on each of the "small" graphs.) Our approximation guarantee is six times the kth harmonic number, which is O(log k). Also, this is an upper bound on the integrality ratio for a standard linear programming relaxation. Several previous papers have attacked the mincost k-VCSS problem (without restrictions on |V |), with the goal of improving on the approximation guarantee (see the references). An approximation guarantee of more than k/2 has been presented in [11]; also, an upper bound of O(k) on the integrality ratio was known [4,5]. Better results were not known for our "special case," but we mention that our results may not improve on previous results for small k (k = 2, 3, 4, . . .). (An O(log k) approximation guarantee was claimed earlier in [15], but subsequently an error has been found, and that claim has been withdrawn; see [16].) For more discussion on related problems and results, see the introduction of [3].Our algorithm is based on two results: (1) a polynomial-time algorithm of Frank and Tardos [5] for finding a minimum-cost k-outconnected subdigraph of a digraph

Two-player win-lose games have a simple directed graph representation. Exploiting this, we develop graph theoretic techniques for finding Nash equilibria in such games. In particular, we give a polynomial time algorithm for finding a Nash equilibrium in a two-player win-lose game whose graph representation is planar.