A dynamic game model of collective choice in multi-agent systems

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 represented by the mean trajectory of all 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 (i) the computed fixed point equilibria qualify as epsilon Nash equilibria, (ii) 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.

Bio-inspired evolutionary dynamics on complex networks under uncertain cross-inhibitory signals

“…Evolutionary dynamics in structured environment is discussed in Tan et al (2014), Piccardi and Casagrandi (2008), Ranjbar-Sahraei et al (2014). A game perspective in collective decision making is provided in Salhab et al (2015). Consensus and games are studied in Yin et al (2012).…”

Dynamic Collective Choice: Social Optima

“…Theorem 8] , where Assumption 1 is used to construct a bounded set that is mapped by G into itself. When the parameters are the same for all the agents, by similar techniques than those used in [18,Theorem 6], one can show that a fixed point of G is the optimal state of an LQR problem of running cost q 2 x T (L + I n )x + r 2 u 2 . The existence and uniqueness of an optimal solution of this LQR problem is a consequence of Assumption 2.…”

“…In [18], we studied the non-cooperative version of our model and developed via the MFG methodology approximate Nash strategy profiles that converge to exact Nash equilibria as the number of players increases to infinity. Since the person-by-person solutions are Nash-like solutions, we rely in this note on some results established in [18] to establish the existence of the person-by-person solutions and compute them.…”

“…In [18], we studied the non-cooperative version of our model and developed via the MFG methodology approximate Nash strategy profiles that converge to exact Nash equilibria as the number of players increases to infinity. Since the person-by-person solutions are Nash-like solutions, we rely in this note on some results established in [18] to establish the existence of the person-by-person solutions and compute them. A static discrete choice model with social interactions was also studied by Brock and Durlauf in [19], where the authors develop a noncooperative and a cooperative game involving a large number of players.…”

Mean field production output control with sticky prices: Nash and social solutions

“…Email addresses: bcwang@sdu.edu.cn (Bingchang Wang), mhuang@math.carleton.ca (Minyi Huang). nance, economics [46], [15], [6], [23], operation research [28], [1], [33], [36], and social sciences [29], [21], [2], etc. This paper aims to present an application of mean field control to production output adjustment in a large market with many firms and sticky prices.…”