We study a dichotomous decision model, where individuals can make the decision yes or no and can influence the decisions of others. We characterize all decisions that form Nash equilibria. Taking into account the way individuals influence the decisions of others, we construct the decision tilings where the axes reflect the personal preferences of the individuals for making the decision yes or no. These tilings characterize geometrically all the pure and mixed Nash equilibria. We show, in these

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... ilings, that Nash equilibria form degenerated hystereses with respect to the replicator dynamics, with the property that the pure Nash equilibria are asymptotically stable and the strict mixed equilibria are unstable. These hystereses can help to explain the sudden appearance of social, political and economic crises. We observe the existence of limit cycles for the replicator dynamics associated to situations where the individuals keep changing their decisions along time, but exhibiting a periodic repetition in their decisions. We introduce the notion of altruist and individualist leaders and study the way that the leader can affect the individuals to make the decision that the leader pretends. 2000 Mathematics Subject Classification. Primary: 91A35, 91A25; Secondary: 91A80. Key words and phrases. Dichotomous decision model, pure and mix Nash equilibria. 1 2 RENATO SOEIRO, ABDELRAHIM MOUSA, TÂNIA OLIVEIRA AND ALBERTO PINTO Received xxxx 20xx; revised xxxx 20xx.