Iterative voting and acyclic games

Multi-agent decision problems, in which independent agents have to agree on a joint plan of action or allocation of resources, are central to artificial intelligence. In such situations, agents' individual preferences over available alternatives may vary, and they may try to reconcile these differences by voting. We consider scenarios where voters cannot coordinate their actions, but are allowed to change their vote after observing the current outcome, as is often the case both in offline committees and in online voting. Specifically, we are interested in identifying conditions under which such iterative voting processes are guaranteed to converge to a Nash equilibrium state—that is, under which this process is acyclic. We classify convergence results based on the underlying assumptions about the agent scheduler (the order in which the agents take their actions) and the action scheduler (the actions available to the agents at each step). By so doing, we position iterative voting models within the general framework of acyclic games and game forms. In more detail, our main technical results provide a complete picture of conditions for acyclicity in several variations of Plurality voting. In particular, we show that (a) under the traditional lexicographic tie-breaking, the game converges from any state and for any order of agents, under a weak restriction on voters' actions; and that (b) Plurality with randomized tie-breaking is not guaranteed to converge under arbitrary agent schedulers, but there is always some path of better replies from any initial state of the game to a Nash equilibrium. We thus show a first separation between order-free acyclicity and weak acyclicity of game forms, thereby settling an open question from [Kukushkin 2011]. In addition, we refute another conjecture of Kukushkin regarding strongly acyclic voting rules, by proving the existence of strongly acyclic separable game forms