Evolutionary analysis of a simple Minority Game: Coexistence, dominance, and paradoxical outcomes

The Minority Game models the El Farol Bar problem, where individuals decide whether to attend the bar based on expected crowd size. Despite extensive study, the interactions between specific strategies remain underexplored due to the model’s complexity. In this work, we analyze a simplified version of the Minority Game where each player follows a fixed strategy based solely on the outcome of the last winning action. We demonstrate that the long-term population dynamics is determined by two key inequalities that are defined in terms of the number of strategies in the population. Using an evolutionary framework with a death-and-birth process, we perform an invasion analysis to study how a single mutant strategy interacts with a resident population. Our results reveal two possible evolutionary outcomes: stable coexistence of competing strategies or dominance of one strategy, which paradoxically leads to its own disadvantage by becoming the majority. However, by extending our evolutionary analysis to the full set of strategies, we demonstrate that evolutionary dynamics consistently drive the system toward the coexistence of two strategies, preventing the paradoxical outcome.