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Using Minigames to Explain Imperfect Outcomes in the Ultimatum Game

In evolutionary game theory, “minigames” with reduced strategy sets are sometimes analysed in lieu of more complex models with many strategies. Are these simplified versions up to the task of explaining pertinent dynamic features of the larger models? This paper looks at the ultimatum game, in which it is known that a noisy evolutionary model leads to stable dynamic equilibriums that are far away from the game’s unique subgame perfect solution. It is argued that a naive approach is unsatisfactory and that the minigame analysis is more useful when related to the full game explicitly. A constellation of embedded minigames is identified in the full game, one for each imperfect equilibrium of the full game, with each playing out on its own conditional frequency space. It is shown that the conditional frequency dynamics applicable to these minigames have the same form as a full game’s dynamics with a reduced strategy set. While the minigames thus identified are still not two-dimensional, it is shown that two critical variables in each can be treated separately from the others, and these indeed behave like the variables in a two-dimensional standalone minigame. A graphical analysis based on selection-mutation equilibrium loci allows a clear understanding of why stable imperfect equilibriums exist and which factors tend to stabilize particular equilibriums. For example, lower-offer equilibriums are easier to stabilize, because a) proposers have more to lose by deviating from them and b) responder mutation aims at a higher target for the relevant conditional frequency.
Working paper 657
1 January 2017
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