Generating functional analysis of batch minority games with arbitrary strategy numbers

TL;DR

This paper provides an analytical solution for the strategy frequency statistics in batch minority games with any number of strategies, revealing detailed phase behavior and matching numerical simulations.

Contribution

It introduces a generating functional analysis for MGs with arbitrary strategies, enabling exact calculation of strategy selection statistics and phase diagrams.

Findings

Exact expressions for strategy selection statistics

Closed equations for order parameters in ergodic regime

Phase diagrams confirming theoretical predictions

Abstract

Both the phenomenology and the theory of minority games (MG) with more than two strategies per agent are different from those of the conventional and extensively studied case S=2. MGs with $S>2$ exhibit nontrivial statistics of the frequencies with which the agents select from their available decision making strategies, with far-reaching implications. In the few theoretical MG studies with $S>2$ published so far, these statistics could not be calculated analytically. This prevented solution even in ergodic stationary states; equations for order parameters could only be closed approximately, using simulation data. Here we carry out a generating functional analysis of fake history batch MGs with arbitary values of $S$, and give an analytical solution of the strategy frequency problem. This leads to closed equations for order parameters in the ergodic regime, exact expressions for strategy selection statistics, and phase diagrams. Our results find perfect confirmation in numerical simulations.