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In 1990, motivated by applications in the social sciences, Thomas Schwartz made a conjecture about tournaments which would have had numerous attractive consequences. In particular, it implied that there is no tournament with a partition A, B of its vertex set, such that every transitive subset of A is in the out-neighbour set of some vertex in B, and vice(More)
We prove that for any constant k and any < 1, there exist bimatrix win-lose games for which every-WSNE requires supports of cardinality greater than k. To do this, we provide a graph-theoretic characterization of win-lose games that possess-WSNE with constant cardinality supports. We then apply a result in additive number theory of Haight [8] to construct(More)
An important instance of the Caccetta-Häggkvist conjecture asserts that an n-vertex digraph with minimum outdegree at least n/3 contains a directed triangle. Improving on a previous bound of 0.3532n due to Hamburger, Haxell, and Kostochka we prove that a digraph with minimum outdegree at least 0.3465n contains a directed triangle. The proof is an(More)
In an-approximate Nash equilibrium, a player can gain at most in expectation by unilateral deviation. An-well-supported approximate Nash equilibrium has the stronger requirement that every pure strategy used with positive probability must have payoff within of the best response payoff. Daskalakis, Mehta and Papadimitriou [8] conjectured that every win-lose(More)
We examine strategy-proof elections to select a winner amongst a set of agents, each of whom cares only about winning. This impartial selection problem was introduced independently by Holzman and Moulin [5] and Alon et al. [1]. Fisher and Klimm [4] showed that the permutation mechanism is impartial and 1 2-optimal, that is, it selects an agent who gains, in(More)
Ramsey's theorem says that for every clique H 1 and for every graph H 2 with no edges, all graphs containing neither of H 1 , H 2 as induced subgraphs have bounded order. What if, instead, we exclude a graph H 1 with a vertex whose deletion gives a clique, and the complement H 2 of another such graph? This no longer implies bounded order, but it implies(More)