Bisemivalues for bicooperative games

We introduce bisemivalues for bicooperative games and we also provide an interesting characterization of this kind of values by means of weighting coefficients in a similar way as it was given for semivalues in the context of cooperative games. Moreover, the notion of induced bisemivalues on lower cardinalities also makes sense and an adaptation of Dragan’s recurrence formula is obtained. For the particular case of (p, q)-bisemivalues, a computational procedure in terms of the multilinear extension of the game is given.Peer ReviewedPostprint (author's final draft

A Review on Liao’s Dissertation Entitled “The Solutions on Multi-choice Games” and Related Publications

In 2007, Liao finished his Ph.d. dissertation[18](Liao 2007) entitled “The Solutions on Multi-choice Games”. Chapter 1 of the dissertation mainly worked on two special cases of the H&R multi-choice Shapley value. One assumes that the weight function w(j) is a positive constant function for all j 6= 0 with w(0) = 0 and the other one assumes that the weight function w(j) = j for all j. If w(j) ’s are equal for all j > 0 then the formula of H&R multi-choice Shapley value can be significantly simplified to the original formula of the traditional Shapley value for the traditional games. Therefore, as a matter of fact, Definitions 1 and 2 in Chapter 1 of the dissertation [18] are simply the traditional Shapley value. Hence, in most part of Chapter 1, Liao was just writing “new results” of traditional games in terms of the notations of multi-choice games. Furthermore, the dissertation [18] did not cited [7](1994), [8](1995a) and [10](1996) which held the original ideas of its main part of chapter 1.Multi-choice TU games, Shapley value, potential, w-consistency

Decomposition of the space of TU-games, Strong Transfer Invariance and the Banzhaf value

We provide a new and concise characterization of the Banzhaf value on the (linear) space of all TU-games on a fixed player set by means of two transparent axioms. The first one is the well-known Dummy player axiom. The second axiom, called Strong transfer invariance, indicates that a player's payoff is invariant to a transfer of worth between two coalitions he or she belongs to. To prove this result we derive direct-sum decompositions of the space of all TU-games. We show that, for each player, the space of all TU-games is the direct sum of the subspace of TU-games where this player is dummy and the subspace spanned by the TU-games used to construct the transfers of worth. This decomposition method has several advantages listed as concluding remarks

Parallel characterizations of a generalized shapley value and a generalized banzhaf value for cooperative games with level structure of cooperation

We present parallel characterizations of two different values in the framework of restricted cooperation games. The restrictions are introduced as a finite sequence of partitions defined on the player set, each of them being coarser than the previous one, hence forming a structure of different levels of a priori unions. On the one hand, we consider a value first introduced in Ref. [18], which extends the Shapley value to games with different levels of a priori unions. On the other hand, we introduce another solution for the same type of games, which extends the Banzhaf value in the same manner. We characterize these two values using logically comparable properties

New axiomatizations of the Shapley interaction index for bi-capacities

International audienceBi-capacities are a natural generalization of capacities (or fuzzy measures) in a context of decision making where underlying scales are bipolar. They are able to capture a wide variety of decision behaviours. After a short presentation of the basis structure, we introduce the Shapley value and the interaction index for capacities. Afterwards, the case of bi-capacities is studied with new axiomatizations of the interaction index

Appraising Diversity with an Ordinal Notion of Similarity: An Axiomatic Approach

This paper provides an axiomatic characterization of two rules for comparing alternative sets of objects on the basis of the diversity that they offer. The framework considered assumes a finite universe of objects and an a priori given ordinal quadernary relation that compares alternative pairs of objects on the basis of their ordinal dissimilarity. Very few properties of this quadernary relation are assumed (beside completeness, transitivity and a very natural form of symmetry). The two rules that we characterize are the maxi-max criterion and the lexi-max criterion. The maxi-max criterion considers that a set is more diverse than another if and only if the two objects that are the most dissimilar in the former are weakly as dissimilar as the two most dissimilar objects in the later. The lexi-max criterion is defined as usual as the lexicographic extension of the maxi-max criterion. Some connections with the broader issue of measuring freedom of choice are also provided.Diversity, Measurement, Axioms, Freedom of choice

Games with Graph Restricted Communication and Levels Structure of Cooperation

We analyze surplus allocation problems where cooperation between agents is restricted both by a communication graph and by a sequence of embedded partitions of the agent set. For this type of problem, we define and characterize two new vàlues extending the Shapley value and the Banzhaf value respectively. Our results enable the axiomatic comparison between the two values and provide some basic insights for the analysis of fair resource allocation in nowadays fully integrated societies

A characterization of the Logarithmic Least Squares Method

We provide an axiomatic characterization of the Logarithmic Least Squares Method (sometimes called row geometric mean), used for deriving a preference vector from a pairwise comparison matrix. This procedure is shown to be the only one satisfying two properties, correctness in the consistent case, which requires the reproduction of the inducing vector for any consistent matrix, and invariance to a specific transformation on a triad, that is, the weight vector is not influenced by an arbitrary multiplication of matrix elements along a 3-cycle by a positive scalar.Comment: 11 page

Strategic marketing, production, and distribution planning of an integrated manufacturing system

Production Scheduling;Distribution;CIM;production