Gal Kaminka ^1,2* Yinon Douchan ¹

• ¹ Department of Computer Science, Faculty of Exact Sciences, Bar-Ilan University, Ramat Gan, Israel
• ² Bar-Ilan University, Ramat Gan, Tel Aviv District, Israel

Inspired by natural phenomena, generations of researchers have been investigating how a swarm of robots can act coherently and purposefully, when individual robots can only sense and communicate with nearby peers, with no means for global communications and coordination.In this paper, we will show how swarms can perform better, when they self-adapt to admit heterogeneous behavior roles.We begin by modeling a foraging swarm task as an extensive-form cooperative game, in which the swarm goal is an additive function of individual contributions. If robots could predict future collisions, they might make optimal collision-avoidance decisions. In practice, swarm robots cannot make such predictions, as they can only sense locally. We derive a reward function from a game-theoretic view of swarm foraging as a fully-cooperative repeating game with an unknown horizon. We demonstrate analytically that the total coordination overhead of the swarm (total time spent on collision-avoidance, rather than foraging per-se) is directly tied to the total utility of the swarm: less overhead, more items collected. Treating every collision as a stage in the repeating game, the overhead is bounded by the total EI of all robots. We then use a marginal-contribution (difference-reward) formulation to derive individual rewards from the total EI. The resulting Aligned Effective Index (AEI) reward has the property that each individual can estimate the impact of its decisions on the swarm: individual improvements translate to swarm improvements. We show that AEI provably generalizes previous work, adding a component that computes the effect of counterfactual robot absence. Different assumptions on this counterfactual lead to bounds on AEI from above and below. While the theoretical analysis clarifies both assumptions and gaps with respect to the reality of robots, experiments with real and simulated robots empirically demonstrate the efficacy of the approach in practice, and the importance of behavioral (decision-making) diversity in optimizing swarm goals.