Foraging motion of swarms as nash equilibria of differential games

Abstract

The question of whether foraging swarms can form as a result of a non-cooperative game played by individuals is shown here to have an affirmative answer. A dynamic (or, differential) game played by N agents in one-dimensional motion is introduced and models, for instance, a foraging ant colony. Each agent controls its velocity to minimize its total work done in a finite time interval. The agents in the game start from a set of initial positions and migrate towards a target foraging location. Such swarm games are shown to have unique Nash equilibra under two different foraging location specifications and both equilibria display many features of a foraging swarm behavior observed in biological swarms. Explicit expressions are derived for pairwise distances between individuals of the swarm, swarm size, and swarm center location during foraging. Foraging swarms in one-dimensional motion with four different information structures are studied. These are complete and partial information structures, hierarchical leadership and one leader structures. In the complete information structure, every agent observes its distance to every other agent and makes use of this information in its effort optimization. In partial information structure, the agents know the position of only its neighboring agents. In the hierarchical leadership structure, the agents look only forward and measures its distance to the agents ahead. In single leader structure, the agents know the position of only leader. In all cases, a Nash equilibrium exists under some realistic assumptions on the sizes of the weighing parameters in the cost functions. The consequences of having a “passive” leader in a swarm are also investigated. We model foraging swarms with leader and followers again as non-cooperative, multi-agent differential games. We consider two types of leadership structures, namely, hierarchical leadership and a single leader structure. In both games, the type of leadership is assumed to be passive since a leader is singled out only due to its rank in the initial queue. We identify the realistic assumptions under which a unique Nash equilibrium exists in each game and derive the properties of the Nash equilibriums in detail. It is shown that having a passive leader economizes in the total information exchange at the expense of aggregation stability in a swarm.