A tractable evolutionary model for the Minority Game with asymmetric payoffs

Abstract

We set up a simple behavioral model for a large population of agents who are repeatedly playing the Minority Game and whose interaction is modeled by means of the so-called replicator dynamics. This allows us to specify the dynamics of the aggregate variables, the number of agents choosing each side, in terms of a low-dimensional dynamical system that gives qualitatively the same results of the existing computational approaches. As an extension we introduce asymmetric payoffs, i.e., we analyze the case where the minority and majority payoffs are side dependent. In this case the fluctuations out of the equilibrium are qualitatively different. In particular, contrary to the previous case, they are associated with a difference in the average payoff gained by each side. Furthermore, a parameter region exists where the dynamics does not converge to any isolated periodic attractor

Evolutionary minority games with memory

“…Dindo in [14], in order to obtain a tractable model, constructs a dynamical system for the minority game and concludes that, in this case, infinite discount memory stabilizes the dynamics.…”

“…Following [14], we introduce in the evolutionary model (2) a way to represent players' memory by considering a fitness measure that represents a discounted sum of the payoffs gained along the whole story of the repeated minority game, obtained by taking, at each time step, a convex combination of the current payoff and the fitness measure observed in the previous time period:…”

“…Again, the parameter ω ∈ [0, 1] gives a measure of the memory, as U i (t) = R(t) for ω = 0, whereas the arithmetic mean of all the payoffs observed in the past is obtained in the other limiting case, ω = 1. If the recursive scheme (13) is plugged into the evolutionary model (2), then we get (14) and, subtracting the third equation from the second, we get the equivalent 2-dimensional model…”

“…The motivation of this paper is to modify the evolutionary model proposed by [14] in order to consider how past payoffs can affect current choices in two different ways. The two ways we consider are both based on a weighted average of the past outcomes and lead to a two-dimensional discrete dynamical system characterized by analytical tractability.…”

“…The first proposed method assumes that the players, in order to decide the next period strategy choice, consider a weighted average, i.e., a convex combination of the current payoffs and those observed in the previous time period (memory of length one). The second method is the same considered in [14], and considers a weighted sum of all previously observed payoffs with exponentially (or geometrically) distributed weights that discount past outcomes. Both these generalization of the evolutionary model without memory are performed in a way that does not modify the equilibrium points of the original model, and only affects their stability properties.…”