Modifying the Hoede-Bakker index to the Shapley-Shubik and Holler-Packel indices

In the paper, we present some modifications of the Hoede-Bakker index defined in a social network in which players may influence each other. Due to influences of the other actors, the final decision of a player may be different from his original inclination. These modifications are defined for an arbitrary probability distribution over all inclination vectors. In particular, they concern a situation in which the inclination vectors may be not equally probable. Furthermore, byassuming special probability distributions over all inclination vectors, we construct modificationsof the Hoede-Bakker index that coincide with the Shapley-Shubik index and with the Holler-Packelindex, respectively.Hoede-Bakker index ; inclination vector ; probability distribution ; Shapley-Shubik index ; Holler-Packel index

On the not-preference-based Hoede-Bakker index

The paper concerns a certain modification of the generalized Hoede-Bakker index - a notion defined for a social network of players. In the original Hoede-Bakker set up, preferences of players are involved. It is assumed that a player has an inclination either to accept or to reject a proposal, but due to the influence of others, his final decision may be different from his original inclination. In this paper, we propose the not-preference-based (NPB) generalized Hoede-Bakker index, where feasible strategies instead of players' inclinations are considered. We show that if all feasible strategy profiles are equally probable, then the NPB generalized Hoede-Bakker index is a 'net' Success, i.e., 'Success - Failure', where Success and Failure of a player is defined as the probability that the player is successful and fails, respectively. Moreover, under the assumption of equal probabilities of all feasible strategy profiles, we show that the probability that a player is lucky (Luck) equals the probability that he fails (Failure). Since Success - Luck = Decisiveness, it follows that, under the same assumption, the NPB generalized Hoede-Bakker index is equal to the probability that a player is decisive.Hoede-Bakker index ; feasible strategy ; success ; failure ; decisiveness

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.decisiveness ; follower of a coalition ; influence function ; influence indices ; success

Measuring influence in command games

In the paper, we study a relation between command games proposed by Hu and Shapley and an influence model. We show that our framework of influence is more general than the framework of the command games. We define several influence functions which capture the command structure. These functions are compatible with the command games, in the sense that each commandable player for a coalition in the command game is a follower of the coalition under the command influence function. Some of the presented influence functions are equivalent to the command games, that is, they are compatible with the command games, and additionally each follower of a coalition under the command influence function is also a commandable player for that coalition in the command games. For some influence functions we define the equivalent command games. We show that not for all influence functions the compatible command games exist. Moreover, we propose a more general definition of the influence index and show that under some assumptions, some power indices, which can be used in the command games, coincide with some expressions of the weighted influence indices. Both the Shapley-Shubik index and the Banzhaf index are equal to a difference between the weighted influence indices under some influence functions, and the only difference between these two power indices lies in the weights for the influence indices. An example of the Confucian model of society is broadly examined.influence function; follower; influence index; command game; commandable player; Shapley-Shubik index; Banzhaf index

Modifying the Hoede-Bakker index to the Shapley-Shubik and Holler-Packel indices

Working Paper GATE 2007-29In the paper, we present some modifications of the Hoede-Bakker index defined in a social network in which players may influence each other. Due to influences of the other actors, the final decision of a player may be different from his original inclination. These modifications are defined for an arbitrary probability distribution over all inclination vectors. In particular, they concern a situation in which the inclination vectors may be not equally probable. Furthermore, byassuming special probability distributions over all inclination vectors, we construct modificationsof the Hoede-Bakker index that coincide with the Shapley-Shubik index and with the Holler-Packelindex, respectively

A model of influence in a social network

In the paper, we study a model of influence in a social network. It is assumed that each player has an inclination to say YES or NO which, due to influence of other players, may be different from the decision of the player. The point of departure here is the concept of the Hoede-Bakker index - the notion which computes the overall decisional 'power' of a player in a social network. The main drawback of the Hoede-Bakker index is that it hides the actual role of the influence function, analyzing only the final decision in terms of success and failure. In this paper, we separate the influence part from the group decision part, and focus on the description and analysis of the influence part. We propose among other descriptive tools a definition of a (weighted) influence index of a coalition upon an individual. Moreover, we consider different influence functions representative of commonly encountered situations. Finally, we propose a suitable definition of a modified decisional power.influence function, influence index, decisional power, social network

Wage bargaining with non-stationary preferences under strike decision

In this paper, we present a non-cooperative wage bargaining model in which preferences of both parties, a union and a firm, are expressed by the sequences of discount rates varying in time. For such a wage bargaining with non-stationary preferences, we determine subgame perfect equilibria between the union and the firm for the case when the union is supposed to go on strike in each period in which there is a disagreement. A certain generalization of the original Rubinstein bargaining model is applied to determine these equilibria.union - firm bargaining ; alternating offers ; varying discount rates ; subgame perfection

Different Approaches to Influence Based on Social Networks and Simple Games

We present an overview of research on a certain model of influence in a social network. Each agent has to make an acceptance/rejection decision, and he has an inclination to choose either the yes-action or the no-action. The agents are embedded in a social network which models influence between them. Due to the influence, a decision of an agent may differ from his preliminary inclination. Such a transformation between the agents' inclinations and their decisions are represented by an influence function. Follower functions encode the players who constantly follow the opinion of a given unanimous coalition. We examine properties of the influence and follower functions and study the relation between them. The model of influence is also compared to the framework of command games in which a simple game is built for each agent. We study the relation between command games and influence functions. We also define influence indices and determine the relations between these indices and some well-known power indices. Furthermore, we enlarge the set of possible yes/no actions to multi-choice games and investigate the analogous tools related to influence in the multi-choice model.influence ; social network ; influence function ; command game ; follower ; voting

A model of influence in a social network

In the paper, we study a model of influence in a social network. It is assumed that each player has an inclination to say YES or NO which, due to influence of other players, may be different from the decision of the player. The point of departure here is the concept of the Hoede-Bakker index - the notion which computes the overall decisional "power" of a player in a social network. The main drawback of the Hoede-Bakker index is that it hides the actual role of the influence function, analyzing only the final decision in terms of success and failure. In this paper, we separate the influence part from the group decision part, and focus on the description and analysis of the influence part. We propose among other descriptive tools a definition of a (weighted) influence index of a coalition upon an individual. Moreover, we consider different influence functions representative of commonly encountered situations. Finally, we propose a suitable definition of a modified decisional power.Influence function, influence index, decisional power, social network.

Iterating influence between players in a social network

We generalize a yes-no model of influence in a social network with a single step of mutual influence to a framework with iterated influence. Each agent makes an acceptance- rejection decision and has an inclination to say either ‘yes' or ‘no'. Due to influence by others, an agent's decision may be different from his original inclination. Such a transformation from the inclinations to the decisions is represented by an influence function. We analyze the decision process in which the mutual influence does not stop after one step but iterates. Any classical influence function can be coded by a stochastic matrix, and a generalization leads to stochastic influence functions. We apply Markov chains theory to the analysis of stochastic binary influence functions. We deliver a general analysis of the convergence of an influence function and then study the convergence of particular influence functions. This model is compared with the Asavathiratham model of influence. We also investigate models based on aggregation functions. In this context, we give a complete description of terminal classes, and show that the only terminal states are the consensus states if all players are weakly essential.Social network, influence, stochastic influence function, convergence, terminal class, Markov chains, aggregation functions.