Effects of payoff functions and preference distributions in an adaptive population

TL;DR

This paper investigates how initial preferences and payoff functions influence the dynamics and volatility in adaptive populations modeled by the Minority Game, revealing phase transitions and bifurcations depending on these factors.

Contribution

It introduces a detailed analysis of the effects of payoff functions and preference distributions on population dynamics, highlighting phase transitions in linear payoffs and basin boundaries in quadratic payoffs.

Findings

Diversity in initial preferences affects volatility and collective behavior.

Linear payoffs exhibit dynamical phase transitions with decreasing diversity.

Quadratic payoffs show basin boundaries separating different dynamical regimes.

Abstract

Adaptive populations such as those in financial markets and distributed control can be modeled by the Minority Game. We consider how their dynamics depends on the agents' initial preferences of strategies, when the agents use linear or quadratic payoff functions to evaluate their strategies. We find that the fluctuations of the population making certain decisions (the volatility) depends on the diversity of the distribution of the initial preferences of strategies. When the diversity decreases, more agents tend to adapt their strategies together. In systems with linear payoffs, this results in dynamical transitions from vanishing volatility to a non-vanishing one. For low signal dimensions, the dynamical transitions for the different signals do not take place at the same critical diversity. Rather, a cascade of dynamical transitions takes place when the diversity is reduced. In contrast, no phase transitions are found in systems with the quadratic payoffs. Instead, a basin boundary of attraction separates two groups of samples in the space of the agents' decisions. Initial states inside this boundary converge to small volatility, while those outside diverge to a large one. Furthermore, when the preference distribution becomes more polarized, the dynamics becomes more erratic. All the above results are supported by good agreement between simulations and theory.