On the minority game: Analytical and numerical studies

Abstract

We investigate further several properties of the minority game we have recently introduced. We explain the origin of the phase transition and give an analytical expression of σ 2 /N in the N ≪ 2 M region. The ability of the players to learn a given payoff is also analyzed, and we show that the Darwinian evolution process tends to a self-organized state, in particular, the life-time distribution is a power-law with exponent -2. Furthermore, we study the influence of identical players on their gain and on the system's performance. Finally, we show that large brains always take advantage of small brains.Recently we have studied a simple model of a minority game [1] that captures the essential features of the "barproblem" of Arthur ( [2] and [3]). N players compete with each other and act by induction and adaptation. They must choose one side between the two at each time step and those who happen to be in the minority side win. They receive a reward when making the right choice, keep in memory the M last sides which were the right ones and use this knowledge to act at the next time step. Each player possesses a finite set of S strategies and uses the one which would have been the most rewarding if it had been used since the beginning. A strategy is a behavior rule that stipulates an action for every information possible (see [1] or [4] for more details). Recently, Savit, Manuca and Riolo [5] have studied the step payoff case where each player has the same memory M and only two strategies. They have found a phase transition with parameter ρ = 2 M /N .Here, we continue the study of the payoff learning and the Darwinism's process initiated in [1] and introduce other interesting sights of the model. We also give a geometrical explanation of the phase transition and find an analytical expression of σ 2 /N in the N ≪ 2 M region.

Dynamical spin-glass-like behavior in an evolutionary game

“…An impotrant approach seems to be modelling the collective effect found in economic systems by assemblies of individually acting heterogeneous agents. One of the basic models is the Minority game [3,4], an evolutionary game which mimics the adaptive behavior of agents with bounded rationality, studied by W. B. Arthur in his El Farol bar problem [5].…”

“…Such a sort of an on-line learning and agaptation leads to betterthan-random performance of the system as a whole. It was found, that the properties of the game depend on the memory length M and number of players N through the scaling variable α = 2 M /N [4,6,7]. The Minoriry game was thoroughly studied both numerically and analytically [8][9][10][11][12][13][14][15][16][17][18][19] along with the study of the original bar attendance problem [20,21] and various modifications of the Minority game [22][23][24][25][26][27][28][29] An important role is attributed to the observation that the dynamics of the memorized pattern and the strategies' scores are in certain regimes decoupled [30][31][32] and that the thermal noise can ge introduced in the players' decisions [33][34][35].…”

Proposal of a Network-Based Minority Game Model with Observation and Modification Processes

“…Social efficiency (volatility) has been studied in MG related research [10], [11]. It is defined as how many agents acquire the reward by selecting a minority choice, which expresses global efficiency of resource allocation and provides a measure of cooperation among agents.…”

Diversity of scales makes an advantage: The case of the Minority Game

“…The Minority Game (MG), introduced slightly more than a decade ago [1,2,3,4,5], gained the reputation of an "Ising model" in the field of Econophysics [6,7]. It is surely an exaggeration, as the complexity of the Ising model is many orders of magnitude higher, but nevertheless, the MG is a widely accepted framework for testing a variety of ideas in modelling socio-economic phenomena.…”