Multi-Choices Minority Game

– The generalization of the problem of adaptive competition, known as the minority game, to the case of K possible choices for each player is addressed, and mapped onto a system of interacting perceptrons with input and output units of the type of K-states Potts-spins. An optimal solution of the minority game as well as the dynamical evolution of the adaptive strategies of the players are solved analytically for a general K and compared with numerical simulations. Considerable progress in the theoretical understanding of market phenomena has been achieved by the study of the minority game. This prototypical model describes a system of agents interacting through a market mechanism [1–6]. The game is based on the idea that the behavior of the agents is determined by the economic rule of supply and demand. According to this rule, given the available options (such as buy/sell), an agent wins if he chooses the minority action. The research of this game has been focused on cases in which each agent can choose between two options using its most efficient strategy, where the strategies remain unchanged throughout the game. However, in real market, many situations of interest involve more than two decision options as well as agents with dynamical strategies. Therefore, it is tempting to investigate cases with more than two possible choices provided to agents with dynamical strategies. In a recent study of the later extension [7] it was shown, that a certain updating rule of the strategies of the agents improves the efficiency of the market, which is measured by the global profit of the agents. In this letter we generalize the aforementioned work to a multi-choices minority game, namely a game with general K decision states. The multi-choices minority game consists of N players (agents) and K possible decisions. In each step, each one of the players chooses one of the K states, aiming to choose the state which is chosen by the smallest number of agents. The minority decisions generate a time series, whose term at time t, xt, has an integer value between 1 and K according to the minority decision. The information provided to each player is the history vector of size M , whose components are the last M minority states. The multi-choices minority game can be modeled by neural networks in the following way: each one of the N players is represented by a perceptron of a size M . The weights belonging to the ith player are {wij} where j = 1, ...,M . All N perceptrons have a common input which