Onset of limit cycles in population games with attractiveness driven strategy choice

We consider population games in which players are motivated in their choice of strategies by payoffs, popularity and transcendent factors. The dynamics of players is governed by balance conservation equations, with attractiveness driven strategy choice. We demonstrate that the increase of the attractiveness of strategy does not necessarily lead to the increase of its frequency in the population in the long run. In particular we discover the existence of limit cycles in the Iterated Prisoner's Dilemma and in the Rock-Paper-Scissors games in such populations.