Population Protocols that Correspond to Symmetric Games

Population protocols have been introduced by Angluin et al. as a model of networks consisting of very limited mobile agents that interact in pairs but with no control over their own movement. A collection of anonymous agents, modeled by finite automata, interact pairwise according to some rules that update their states. The model has been considered as a computational model in several papers. Input values are initially distributed among the agents, and the agents must eventually converge to the the correct output. Predicates on the initial configurations that can be computed by such protocols have been characterized under various hypotheses. The model has initially been motivated by sensor-networks, but it can be seen more generally as a model of networks of anonymous agents interacting pairwise. This includes sensor networks, ad-hoc networks, or models from chemistry. In an orthogonal way, several distributed systems have been termed in literature as being realizations of games in the sense of game theory. In this paper, we investigate under which conditions population protocols, or This work and all authors were partly supported by ANR Project SOGEA and by ANR Project SHAMAN, Xavier Koegler was partly supported by COST Action 295 DYNAMO and ANR Project ALADDIN Preprint submitted to Elsevier July 17, 2009 more generally pairwise interaction rules, can be considered as the result of a symmetric game. We prove that not all rules can be considered as symmetric games.We characterize the computational power of symmetric games. We prove that they have very limited power: they can count until 2, compute majority, but they can not even count until 3. As a side effect of our study, we also prove that any population protocol can be simulated by a symmetric one (but not necessarily a game).