Reformulation of Public Help Index θ Using Null Player Free Winning Coalitions

This paper proposes a new representation for the Public Help Index θ (briefly, PHI θ). Based on winning coalitions, the PHI θ index was introduced by Bertini et al. in (2008). The goal of this article is to reformulate the PHI θ index using null player free winning coalitions. The set of these coalitions unequivocally defines a simple game. Expressing the PHI θ index by the winning coalitions that do not contain null players allows us in a transparent way to show the parts of the power assigned to null and non-null players in a simple game. Moreover, this new representation may imply a reduction of computational cost (in the sense of space complexity) in algorithms to compute the PHI θ index if at least one of the players is a null player. We also discuss some relationships among the Holler index, the PHI θ index, and the gnp index (based on null player free winning coalitions) proposed by Álvarez-Mozos et al. in (2015).

Public Good Indices for Games with Several Levels of Approval

This work focuses on (j, 2) games in which there are several levels of approval in the input, i. e. games with n players, j ordered qualitative alternatives in the input level and 2 possible ordered quantitative alternatives in the output. When considering (j, 2) games, we extend the Public Good index (PGI), the Null Player Free index (NPFI) and the Shift index (SI) and provide full characterizations of these extensions.

An Axiomatization for Two Power Indices for (3,2)-Simple Games

The aim of this work is to give a characterization of the Shapley–Shubik and the Banzhaf power indices for (3,2)-simple games. We generalize to the set of (3,2)-simple games the classical axioms for power indices on simple games: transfer, anonymity, null player property and efficiency. However, these four axioms are not enough to uniquely characterize the Shapley–Shubik index for (3,2)-simple games. Thus, we introduce a new axiom to prove the uniqueness of the extension of the Shapley–Shubik power index in this context. Moreover, we provide an analogous characterization for the Banzhaf index for (3,2)-simple games, generalizing the four axioms for simple games and adding another property.

Allocation Rules for Games with Optimistic Aspirations

A game with optimistic aspirations specifies two values for each coalition of players: the first value is the worth that the players in the coalition can guarantee for themselves in the event that they coordinate their actions, and the second value is the amount that the players in the coalition aspire to get under reasonable but very optimistic assumptions about the demands of the players who are not included in the coalition. In this paper, in addition to presenting this model and justifying its relevance, we introduce allocation rules and extend the properties of efficiency, additivity, symmetry, and null player property to this setting. We demonstrate that these four properties are insufficient to find a unique allocation rule and define three properties involving null players and nullifying players that allow the identification of unique allocation rules. The allocation rules we identify are the Midpoint Shapley Value and the Equal Division Rule.

A CONSISTENT IMPUTATION GENERATION METHOD FOR LINGUISTIC COOPERATIVE GAMES AND ITS APPLICATION TO RISK AVERSION

Cooperative game theory is very useful to risk aversion problems in economics and management systems. The existing methods only focus on the situation payoffs take the form of numerical values, ones take the form of linguistic labels are seldom discussed. The aim of this study is to propose the consistent imputation for cooperative games under a linguistic environment. To support risk aversion, a 2-tuple linguistic representation is employed to obtain the valid results and avoid the loss of linguistic information. This paper firstly defines some concepts for linguistic cooperative games, such as linguistic imputation, carrier, core and null player. A set of their desirable properties are also discussed. The linguistic Shapley value is then presented based on three axioms. Moreover, the existence and uniqueness of the linguistic Shapley value are discussed in detail. To adjust the linguistic imputation in accordance with the cardinality of a given original linguistic label set, an adjustment algorithm for generating consistent imputation is proposed. Finally, we give the application of linguistic imputation in solving risk aversion problems to illustrate the validity of the consistent imputation generation (CIG) method.