On the rationality of cycling in the theory of moves framework

Theory of Moves (TOM) is a novel approach to game theory for determining rational strategies during the play of dynamic games [2]. While alternate models such as normal form games exist, players of these games are limited to single shot interactions with each other, but within TOM, sequences of moves and counter moves are allowed. As a consequence of this framework potential cyclic behavior may arise. Unfortunately, standard TOM framework suggests that agents do not move from the initial state if the possibility of cyclic behavior is detected. However, in a plethora of real life scenarios, cycling can benefit a player over time. We first extend the TOM framework by allowing agents to choose how much time to stay in each state while specifying time limits for moves. This generalization allows for cycling behavior in addition to normal, acyclic TOM play. We present additional rationality rules to handle the choice of move time and cyclic play and identify conditions for the existence of solutions that involve cycles. Moreover, if solutions do exist, equilibria are determined so a player can predict the rational outcome upon engaging a cycle. A variety of time constraints on move times are investigated and the effects of these contrasts on the solution space and equilibria are analyzed.