6 research outputs found
On the expected number of equilibria in a multi-player multi-strategy evolutionary game
In this paper, we analyze the mean number of internal equilibria in
a general -player -strategy evolutionary game where the agents' payoffs
are normally distributed. First, we give a computationally implementable
formula for the general case. Next we characterize the asymptotic behavior of
, estimating its lower and upper bounds as increases. Two important
consequences are obtained from this analysis. On the one hand, we show that in
both cases the probability of seeing the maximal possible number of equilibria
tends to zero when or respectively goes to infinity. On the other hand,
we demonstrate that the expected number of stable equilibria is bounded within
a certain interval. Finally, for larger and , numerical results are
provided and discussed.Comment: 26 pages, 1 figure, 1 table. revised versio
On Tractable Convex Relaxations of Standard Quadratic Optimization Problems under Sparsity Constraints
Standard quadratic optimization problems (StQPs) provide a versatile
modelling tool in various applications. In this paper, we consider StQPs with a
hard sparsity constraint, referred to as sparse StQPs. We focus on various
tractable convex relaxations of sparse StQPs arising from a mixed-binary
quadratic formulation, namely, the linear optimization relaxation given by the
reformulation-linearization technique, the Shor relaxation, and the relaxation
resulting from their combination. We establish several structural properties of
these relaxations in relation to the corresponding relaxations of StQPs without
any sparsity constraints, and pay particular attention to the rank-one feasible
solutions retained by these relaxations. We then utilize these relations to
establish several results about the quality of the lower bounds arising from
different relaxations. We also present several conditions that ensure the
exactness of each relaxation.Comment: Technical Report, School of Mathematics, The University of Edinburgh,
Edinburgh, EH9 3FD, Scotland, United Kingdo
On the Support Size of Stable Strategies in Random Games
Abstract. In this paper we study the support sizes of evolutionary stable strategies (ESS) in random evolutionary games. We prove that, when the elements of the payoff matrix behave either as uniform, or normally distributed independent random variables, almost all ESS have support sizes o(n), where n is the number of possible types for a player. Our arguments are based exclusively on the severity of a stability property that the payoff submatrix indicated by the support of an ESS must satisfy. We then combine our normal–random result with a recent result of McLennan and Berg (2005), concerning the expected number of Nash Equilibria in normal–random bimatrix games, to show that the expected number of ESS is significantly smaller than the expected number of symmetric Nash equilibria of the underlying symmetric bimatrix game