Quantifying a certain ''advantage law'': minority game with above the rules agents

Abstract

In this work the properties of minority games containing agents which try to winning all the time are studied by means of computational simulations. We have considered several ways of introducing above the rules clever players using "strategies" which try to outdo the others endowed with statistically equivalent strategies and compared the resulting behaviors of the ensemble. It is shown that by introducing such agents the overall performance of the system gets significantly poorer. While the introduction of a very small fraction of these never-loosing-players may not destroys the unordered / ordered phase transition of the standard minority game we find that even for a low concentration of their presence only a state "worse" than random coin toss choices sets in. These special agents/players have the role of impurities or vacancies in spin systems and their presence may lead to a critical concentration where the usual phases are washed out.The minority game (MG) is a very simple model of interacting agents. Nevertheless it shows fascinating properties and poses novel challenging questions to statistical mechanics and its methods and ideas. It is a very simple toy model for complex systems and may come to play the same role in science as the Ising model, applicable in many interdisciplinary subjects from physics to economy and biology to name a few (see [1] for comprehensive and actual information). In short, its main idea is the following: an odd number of selfish players repeatedly has to choose one out of two possible actions. These could be for example buying or selling at the stock market, taking the bus or the metro, which internet router is the fastest (ie. less crowded). As it will become clearer in the examples, the players that are in the minority side will win the game. If there are more sellers than buyers, the price will fall and the buyers will be in advantage. If there are lots of people in the metro it would have been much more comfortable to take the bus and so on. This kind of game was inspired by Arthur's "El Farol" problem [2]. There it was an overcrowded bar that rose his interest of how a population of heterogeneous players interacts.In this section we give a short and necessarily incomplete summary of the MG and their well-known properties. Then we shall present our modified models and compare them with the standard model. In the MG that was introduced in [3,4], N agents (or players) choose at each time step i one out of two possible options, say 0 and 1. Agents that belong to the smaller of both groups win. They act completely independent, without any possibility of communication or interaction. Their only available information about the other player's actions is a "history" (or "memory") of the last m games' winners. There are two possible values for each of the m entries of the history, so there are 2 m possible different histories. Every player is equipped with a set of S strategies to determine their next choice. As the decision depends on the current history of the last games, these...

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