Plateaux formation, abrupt transitions, and fractional states in a competitive population with limited resources

We study, both numerically and analytically, a Binary-Agent-Resource (B-A-R) model consisting of N agents who compete for a limited resource 1/2<L/N <1, where L is the maximum available resource per turn for all N agents. As L increases, the system exhibits well-defined plateaux regions in the success rate which are separated from each other by abrupt transitions. Both the maximum and the mean success rates over each plateau are 'quantized' -- for example, the maximum success rate forms a well-defined sequence of simple fractions as L increases. We present an analytic theory which explains these surprising phenomena both qualitatively and quantitatively. The underlying cause of this complex behavior is an interesting self-organized phenomenon in which the system, in response to the global resource level, effectively avoids particular patterns of historical outcomes.