Dynamic Collective Choice: Social Optima

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.

A unified approach to mean-field team: homogeneity, heterogeneity and quasi-exchangeability

“…Please refer [8,10,18,29,34] for the limit relation between mean-field games (MFG) and non-cooperate N -player games. The interested readers may refer e.g., [27,33,37], for detailed analysis comparison between MFG and MT, and [36,38] for some recent MT study from various perspectives with different modeling variants. In particular, see [23] for social optima in mean field control problems with volatility uncertainty; see [25] for linearquadratic-Gaussian (LQG) mean-field social optimization with a major player; and [39] for social optima in LQG models with Markov jump parameters.…”

Social optima in linear quadratic mean field control with unmodeled dynamics and multiplicative noise

“…In [22], authors studied the mean field LQ social control model with Markov jump parameters and gave an asymptotic team‐optimal solution. For further references, e.g., see [23] for team‐optimal control with finite population and partial information, [24] for social optima in mixed games, and [25] for the dynamic cooperative collective choice.…”

Linear quadratic mean field social optimization: Asymptotic solvability

“…Social optima are analyzed in [52] for a mean field jump LQ model governed by a common Markovian chain. An LQ social optimum model is studied in [47] for a large number of weakly coupled agents choosing cooperatively between multiple destinations. A nonlinear social optimization problem for an infinite horizon economy is analyzed in [41], where necessary conditions of the social optimum are derived by using Gateaux derivatives and Lagrangian multipliers due to market clearing equality constraints.…”