Toward a theory of local resource competition: the minority game with private information

In many social and biological systems, agents compete for limited resources over an extended, connected network, but at any moment compete for resources semi-locally using incomplete, private information. To understand such systems, it is important to understand the nature of competition for scarce resources with private information. In this paper we study this question by examining the behavior of minority games in which agents make their decisions based on private information. We first introduce a framework, based on bi-graphs, for discussing local resource competition on a network. We then consider the special case in which the bi-graph model reduces to a minority game. We study several variations of the minority game with private information, including games with a mixture of public and private information. We find that the games with private information share a general structural similarity to the standard minority game with public information in that there are, typically, two phases, a maladaptive phase and an adaptive phase, the latter encompassing a region of emergent coordination in which the scarce resource is well utilized. There are, however, very significant differences between the games with private information and the standard minority game. In both public and private games, the maladaptive phase is characterized by dynamics with a periodicity, but the nature of the periodicity is markedly different in the two cases. Scaling behavior is strongly affected by private information, and, in general, coordination among agent's choices, while possible, is more subtle and often more difficult to achieve.