The Full Strategy Minority Game

TL;DR

This paper analytically solves the Full Strategy Minority Game (FSMG), demonstrating its symmetries, period-two dynamics, and providing approximations for key variables, thereby advancing understanding of the MG's behavior.

Contribution

The paper explicitly solves the FSMG using symmetries, proves its strict period-two dynamics, and extends results to biased strategies, offering new analytical insights into the MG.

Findings

FSMG can be explicitly solved thanks to its symmetries.

FSMG exhibits strict period-two dynamics with probability 1.

Analytical methods for approximating key variables in the MG are developed.

Abstract

The Full Strategy Minority Game (FSMG) is an instance of the Minority Game (MG) which includes a single copy of every potential agent. In this work, we explicitly solve the FSMG thanks to certain symmetries of this game. Furthermore, by considering the MG as a statistical sample of the FSMG, we compute approximated values of the key variable {\sigma}2/N in the symmetric phase for different versions of the MG. As another application we prove that our results can be easily modified in order to handle certain kind of initial biased strategies scores, in particular when the bias is introduced at the agents' level. We also show that the FSMG verifies a strict period two dynamics (i.e., period two dynamics satisfied with probability 1) giving, to the best of our knowledge, the first example of an instance of the MG for which this feature can be analytically proved. Thanks to this property, it is possible to compute in a simple way the probability that a general instance of the MG breaks the period two dynamics for the first time in a given simulation.