Dynamic Collective Choice: Social Optima

TL;DR

This paper addresses dynamic collective choice problems with many agents, proposing a scalable decentralized strategy based on Mean Field Games that approximates social optima and analyzing the impact of social effects on group behavior.

Contribution

It introduces a mean field game approach to find decentralized strategies for large populations in dynamic collective choice problems, reducing computational complexity.

Findings

Decentralized strategies approximate social optima in large populations.

The number of linear quadratic regulator problems grows exponentially with population size.

Social effects influence majority size differently in cooperative versus noncooperative scenarios.

Abstract

We consider a dynamic collective choice problem where a large number of players are cooperatively choosing between multiple destinations while being influenced by the behavior of the group. For example, in a robotic swarm exploring a new environment, a robot might have to choose between multiple sites to visit, but at the same time it should remain close to the group to achieve some coordinated tasks. We show that to find a social optimum for our problem, one needs to solve a set of Linear Quadratic Regulator problems, whose number increases exponentially with the size of the population. Alternatively, we develop via the Mean Field Games methodology a set of decentralized strategies that are independent of the size of the population. When the number of agents is sufficiently large, these strategies qualify as approximately socially optimal. To compute the approximate social optimum, each player needs to know its own state and the statistical distributions of the players' initial states and problem parameters. Finally, we give a numerical example where the cooperative and noncooperative cases have opposite behaviors. Whereas in the former the size of the majority increases with the social effect, in the latter, the existence of a majority is disadvantaged.