A model of influence with a continuum of actions

We generalize a two-action (yes–no) model of influence to a framework in which every player has a continuum of actions, among which he has to choose one. We assume the set of actions to be an interval. Each player has an inclination to choose one of the actions. Due to the influence among players, the final decision of a player, i.e., his choice of one action, may be different from his original inclination. In particular, a coalition of players with the same inclination may influence another player with different inclination, and as a result of this influence, the decision of the player is closer to the inclination of the influencing coalition than his inclination was. We introduce a measure of such a positive influence of a coalition on a player. Several unanimous influence functions in this generalized framework are considered. Also the set of fixed points under a given influence function is analyzed. Furthermore, we study linear influence functions and discuss their convergence. For a linear unanimous function, we find necessary and sufficient conditions for the existence of the positive influence of a coalition on a player, and we calculate the value of the influence index. We also introduce a measure of a negative influence of a coalition on a player.