Adaptive Competition , Market Efficiency , Phase Transitions and Spin - Glasses

In this paper we analyze a simple model of adaptive competition which captures essential features of a variety of adaptive competitive systems in the social and biological sciences. In this model, each of N agents, at each time step of a game, joins one of two groups. The agents in the minority group are awarded a point, while the agents in the majority group get nothing. Each agent has a fixed set of strategies drawn at the beginning of the game from a common pool, and chooses his current best-performing strategy to determine which group to join. We find that for a fixed N, the system exhibits a phase change as a function of the size of the common strategy pool from which the agents initially draw their strategies. For small pool sizes, the system is in an efficient market phase in which all information that can be used by the agents' strategies is traded away, and no agent can accumulate more points than would an agent making random guesses. In addition, in this phase the commons suffer, and relatively few points are awarded to the agents in total. For large initial strategy pool sizes, the system is in an inefficient market phase, in which there is predictive information available to the agents' strategies, and some agents can do better than random at accumulating points. In addition, in this phase, the total number of points awarded to the agents is greater than in a game in which all agents guess randomly, and so the commons do relatively well. At a critical size of the strategy pool marking the cross-over from the efficient market to the inefficient market phases, the commons do best. This critical size of the pool grows monotonically, and in a very simple way with N. The behavior of this system has features reminiscent of a spin-glass in statistical physics, with the small pool size phase being, in a certain sense, more glassy than the large pool phase. We argue that the structure we have elucidated has important implications for a range of phenomena in the social and biological sciences, as well as for the general study of adaptive, competitive systems.