Abstract
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 tilings, that Nash equilibria form degenerated hysteresis with respect to the dynamics, with the property that the pure Nash equilibria are asymptotically stable and the strict mixed equilibria are unstable. These hysteresis can help to explain the sudden appearance of social, political and economic crises.We observe the existence of limit cycles for the 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.
Highlights
The main goal in Planned Behavior or Reasoned Action theories, as developed in the works of Ajzen and Baker, is to understand and predict the way individuals turn intentions into behaviors
We characterize all the pure Nash equilibria and we show that the pure Nash equilibria are, in general, asymptotically stable with respect to the replicator dynamics
We characterize all the strict mixed Nash equilibria and we prove that the strict mixed Nash equilibria are, in general, unstable
Summary
The main goal in Planned Behavior or Reasoned Action theories, as developed in the works of Ajzen (see [1]) and Baker (see [4]), is to understand and predict the way individuals turn intentions into behaviors. We will show that the relative decision preferences and the influence matrix together with the total number of individuals of each type encode all the relevant information for characterizing the Nash equilibria.
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