Phenomenology of minority games in efficient regime

TL;DR

This paper analyzes the utility functions and state dynamics of the minority game in its efficient regime, revealing bounded utilities, finite states, and explaining demand fluctuations and periodicity through Markov processes and graph analysis.

Contribution

It provides an explicit Markov process representation for the minority game with specific payoff functions and analyzes state finiteness, utility boundedness, and demand fluctuations.

Findings

Finite number of states for g(x)=sgn(x)

Bounded utility functions in the game

Explanation of demand fluctuations and periodicity

Abstract

We present a comprehensive study of utility function of the minority game in its efficient regime. We develop an effective description of state of the game. For the payoff function $g(x)=\sgn (x)$ we explicitly represent the game as the Markov process and prove the finitness of number of states. We also demonstrate boundedness of the utility function. Using these facts we can explain all interesting observable features of the aggregated demand: appearance of strong fluctuations, their periodicity and existence of prefered levels. For another payoff, $g(x)=x$, the number of states is still finite and utility remains bounded but the number of states cannot be reduced and probabilities of states are not calculated. However, using properties of the utility and analysing the game in terms of de Bruijn graphs, we can also explain distinct peaks of demand and their frequencies.