In classical game theory, the players are assumed to be rational and intelligent, which is often contradictory to the reality. We consider more realistic behavioral game dynamics where the players choose actions in a turn-by-turn manner and exhibit two prominent behavioral traits -α-fraction of them are myopic who strategically choose optimal actions against the empirical distribution of the previous plays, while the rest exhibit herding behavior by choosing the most popular action till then. The utilities are realised for all, at the end of the game, and each player gets to play only once. Our analysis focuses on scenarios when players encounter two possible choices, common in applications like participation games (e.g., crowd-sourcing) or minority games.

To begin with, we derive the almost sure mean-field limits of such dynamics. The proof is constructive and progressively narrows down the potential limit set and finally establishes the existence of a unique limit for almost all sample paths. We argue that the dynamics at the limit is captured by a differential inclusion (and not the usual ordinary differential equation) due to the discontinuities arising from the switching behavioral choices. It is noteworthy that our methodology can be easily modified to analyse the avoid-the-crowd behavior, in place of herding behavior.

We conclude with two interesting examples, named participation game and routing game, which encapsulates several real-life scenarios. Interestingly, for the first game, we observe that the game designer can induce a higher level of participation in an activity with smaller reward, by leveraging upon the presence of herding players.