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We continue our study of evolution in minority games by examining games in which agents with poorly performing strategies can trade in their strategies for new ones from a different strategy space. In the context of the games discussed in this paper, this means allowing for strategies that use information from different numbers of time lags, m. We find, in all the games we study, that after evolution, wealth per agent is high for agents with strategies drawn from small strategy spaces (small m), and low for agents with strategies drawn from large strategy spaces (large m). In the game played with N agents, wealth per agent as a function of m is very nearly a step function. The transition is at m=mt, where mt~mc-1. Here mc is the critical value of m at which N agents playing the game with a fixed strategy space (fixed m) have the best emergent coordination and the best utilization of resources. We also find that overall system-wide utilization of resources is independent of N. Furthermore, although overall system-wide utilization of resources after evolution varies somewhat depending on some other aspects of the evolutionary dynamics, in the best cases, utilization of resources is on the order of the best results achieved in evolutionary games with fixed strategy spaces. Simple explanations are presented for some of our main results., Comment: 18 pages text, 8 figures. Postscript from msword for text, from matlab for figures

In this paper we study the minority game in the presence of evolution. In particular, we examine the behavior in games in which the dimension of the strategy space, m, is the same for all agents and fixed for all time. We find that for all values of m, not too large, evolution results in a substantial improvement in overall system performance. We also show that after evolution, results obey a scaling relation among games played with different values of m and different numbers of agents, analogous to that found in the non-evolutionary, adaptive games. Best system performance still occurs, for a given number of agents, at m_c, the same value of the dimension of the strategy space as in the non-evolutionary case, but system performance is now nearly an order of magnitude better than the non- evolutionary result. For m

In this paper we present results and analyses of a class of games in which heterogeneous agents are rewarded for being in a minority group. Each agent possesses a number of fixed strategies each of which are predictors of the next minority group. The strategies use a set of aggregate, publicly available information (reflecting the agents' collective previous decisions) to make their predictions. An agent chooses which group to join at a given moment by using one of his strategies. These games are adaptive in that agents can choose, at different points of the game, to exercise different strategies in making their choice of which group to join. We find, rather generally, that such systems evidence a phase change from a maladaptive, informationally efficient phase in which the system performs poorly at generating resources, to an inefficient phase in which there is an emergent cooperation among the agents, and the system more effectively generates resources. The best emergent coordination is achieved in a transition region between these two phases. This transition occurs when the dimension of the strategy space is of the order of the number of agents playing the game. We present explanations for this general behavior, based in part on an information theoretic analysis of the system and its publicly available information. We also propose a mean-field like model of the game which is most accurate in the maladaptive, efficient phase. In addition, we show that the best individual agent performance in the two different phases is achieved by sets of strategies with markedly different characteristics. We discuss implications of our results for various aspects of the study of complex adaptive systems., Comment: 41 pages text, 26 pages of figures. Related papers at http://www.pscs.umich.edu/RESEARCH/pscs-tr.html

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. 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. 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. All information that can be used by the agents' strategies is traded away, no agent can accumulate more points than would an agent making random guesses, and thus the commons suffer,since 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 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 with N. The behavior of this system has some features reminiscent of a spin-glass., Comment: A tar.gz file with three postscript files: 16 pages of text, figure 1, and figures 2-6