Minority game with arbitrary cutoffs

Abstract

We study a model of a competing population of N adaptive agents, with similar capabilities, repeatedly deciding whether to attend a bar with an arbitrary cutoff L. Decisions are based upon past outcomes. The agents are only told whether the actual attendance is above or below L. For L∼N/2, the game reproduces the main features of Challet and Zhang's minority game. As L is lowered, however, the mean attendances in different runs tend to divide into two groups. The corresponding standard deviations for these two groups are very different. This grouping effect results from the dynamical feedback governing the game's time-evolution, and is not reproduced if the agents are fed a random history.

Introduction

Complex adaptive systems [1] have been the subject of much recent attention. These systems typically exhibit rich global behaviour which cannot be straightforwardly deduced from the microscopic details of the constituent objects (agents). Fascinating self-organized phenomena [2] have been observed or discussed for a wide range of systems such as sandpiles, traffic flow, financial markets and other social phenomena. Within the physics community, moreover, many ‘microscopic’ economics-based models have been proposed [3], [4], [5], [6], [7] in the growing subfield of ‘econophysics’.

Challet and Zhang [8], [9] recently introduced a simple minority game in which agents repeatedly compete to be in the minority group. These agents have similar capabilities and each of them makes decisions based on the past history of outcomes. Further work on the basic minority game is presented in [10], [11], [12], [13], [14], [15], [16]. While the minority game is stated in terms of agents choosing between two rooms, the model can be stated in different ways to fit different situations; for example, the agents may as well be deciding whether to buy or sell a certain stock in a simple market model. A more general version of the minority game, namely the bar-attendance model [17] in which agents decide whether to attend a bar with a certain seating capacity, has also been studied recently [18].

In this paper, we generalize the minority game to the case of arbitrary cutoff. There are N agents, each of whom possesses s strategies, deciding whether to attend a bar with a seating capacity, i.e. cutoff, equal to L. Good (bad) decisions correspond to attending an undercrowded (overcrowded) bar, or not attending an overcrowded (undercrowded) bar. The only information given to the agents after each turn is whether the actual attendance was above or below the cutoff. We present extensive numerical results for the mean attendances and corresponding standard deviations as a function of the agents’ capabilities and the cutoff L. For L=N/2, the game reduces to the minority game [8], [9], [10], [11], [12], [13], [14], [15], [16]. The present model for L≠N/2 thus represents an intermediate model between the minority game [8], [9], [10], [11], [12], [13], [14], [15], [16] and the bar-attendance model [17], [18]: it allows for arbitrary cutoff as in the bar-attendance model, but the actual attendances are not announced as in the minority game. As L is reduced below N/2, the mean attendances and standard deviations tend to be distributed into two groups of values for different runs of the same game. One group comprises a mean attendance which is insensitive to the agents’ capabilities, and is accompanied by a small standard deviation of attendance. The other group contains a spread in mean attendances, each with a larger corresponding standard deviation. An explanation of these features is given.

The plan of the paper is as follows. In Section 2, the game with arbitrary cutoff is defined. Results for the mean attendances and standard deviations for different cutoff L, and for different number of strategies s per agent, are presented and discussed in Section 3. The relationship between games with cutoff L and L′=N−L is also discussed. Section 4 summarizes our main findings.