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We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recently-established equivalence between polynomial time solvability of normal form games and graphical games, establishing that(More)
Gaussian processes provide natural non-parametric prior distributions over regression functions. In this paper we consider regression problems where there is noise on the output, and the variance of the noise depends on the inputs. If we assume that the noise is a smooth function of the inputs, then it is natural to model the noise variance using a second(More)
The Vapnik-Chervonenkis (V-C) dimension is an important combinatorial tool in the analysis of learning problems in the PAC framework. For polynomial learnability, we seek upper bounds on the V-C dimension that are polynomial in the syntactic complexity of concepts. Such upper bounds are automatic for discrete concept classes, but hitherto little has been(More)
Weighted threshold games are coalitional games in which each player has a weight (intuitively corresponding to its voting power), and a coalition is successful if the sum of its weights exceeds a given threshold. Key questions in coali-tional games include finding coalitions that are stable (in the sense that no member of the coalition has any rational(More)
Suppose that a set of <i>m</i> tasks are to be shared as equally as possible amongst a set of <i>n</i> resources. A game-theoretic mechanism to find a suitable allocation is to associate each task with a "selfish agent", and require each agent to select a resource, with the cost of a resource being the number of agents to select it. Agents would then be(More)
This paper studies a load balancing game introduced by Koutsoupias and Papadimitriou, that is intended to model a set of users who share several internet-based resources. Some of the recent work on this topic has considered the problem of constructing <i>Nash equilibria</i>, which are choices of actions where each user has optimal utility given the actions(More)
Various economic interactions can be modeled as two-sided markets. A central solution concept to these markets are stable matchings, introduced by Gale and Shapley. It is well known that stable matchings can be computed in polynomial time, but many real-life markets lack a central authority to match agents. In those markets, matchings are formed by actions(More)
In this paper we consider the following scenario. A set of <i>n</i> jobs with different threads is being run concurrently. Each job has an associated weight, which gives the proportion of processor time that it should be allocated. In a single time quantum, <i>p</i> threads of (not necessarily distinct) jobs receive one unit of service, and we require a(More)
The j-State General Markov Model of evolution (due to Steel) is a stochastic model concerned with the evolution of strings over an alphabet of size j. In particular, the Two-State General Markov Model of evolution generalises the well-known Cavender-Farris-Neyman model of evolution by removing the symmetry restriction (which requires that the probability(More)