A Dynamic Game Model of Collective Choice in Multi-Agent Systems

TL;DR

This paper develops a mean field game model for collective decision-making in multi-agent systems, inspired by biological behaviors, analyzing equilibrium conditions and demonstrating how large populations simplify the computation of equilibria.

Contribution

It introduces a novel mean field game framework for collective choice, including existence conditions and the transition to epsilon Nash equilibria as population size grows.

Findings

Multiple equilibria can exist in the model.

Large populations allow agents to compute equilibria using probability distributions.

Fixed point equilibria become epsilon Nash equilibria with increasing agents.

Abstract

Inspired by successful biological collective decision mechanisms such as honey bees searching for a new colony or the collective navigation of fish schools, we consider a mean field games (MFG)-like scenario where a large number of agents have to make a choice among a set of different potential target destinations. Each individual both influences and is influenced by the group's decision, as well as the mean trajectory of all the agents. The model can be interpreted as a stylized version of opinion crystallization in an election for example. The agents' biases are dictated first by their initial spatial position and, in a subsequent generalization of the model, by a combination of initial position and a priori individual preference. The agents have linear dynamics and are coupled through a modified form of quadratic cost. Fixed point based finite population equilibrium conditions are identified and associated existence conditions are established. In general multiple equilibria may exist and the agents need to know all initial conditions to compute them precisely. However, as the number of agents increases sufficiently, we show that 1) the computed fixed point equilibria qualify as epsilon Nash equilibria, 2) agents no longer require all initial conditions to compute the equilibria but rather can do so based on a representative probability distribution of these conditions now viewed as random variables. Numerical results are reported.