Measuring Influence Among Players with an Ordered Set of Possible Actions

In the paper, we introduce and study generalized weighted influence indices of a coalition on a player, where players have an ordered set of possible actions. Each player has an inclination to choose one of the actions. Due to influence of a coalition of other players, a final decision of the player may be different from his original inclination. An influence in such situations is measured by the general weighted influence index. In a particular case, the decision of the player may be closer to the inclination of the influencing coalition than his inclination was. The weighted influence index which captures such a case is called the positive weighted influence index. We also consider the negative weighted influence index, where a final decision of the player goes farther away from the inclination of the influencing coalition. Some special cases of the weighted influence indices, called a possibility influence index and an equidistributed influence index, are also defined. We consider different influence functions and study their properties. A set of followers and a set of a conditional followers of a given coalition are defined, and their properties are analyzed. We define the concepts of success, decisiveness, luck, and failure for the multi-choice model of influence.