In a role game, players can condition their strategies on their player position in the base game. If the base game is strategically equivalent to a zero-sum game, the set of Nash equilibria of the role game is globally asymptotically stable under the best response dynamics. If the base game is 2 ×2, then in the cyclic case the set of role game equilibria is a continuum. We identify a single equilibrium in this continuum that attracts all best response paths outside the continuum.
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