Motivated by the problem of utility allocation in a portfolio under a Markowitz meanvariance choice paradigm, we propose an allocation criterion for the variance of the sum of n possibly dependent random variables. This criterion, the Shapley value, requires to translate the problem into a cooperative game. The Shapley value has nice properties, but, in general, is computationally demanding. Th...

This paper introduces a measure of uncertainty in the determination of the Shapley value, illustrates it with examples, and studies some of its properties. The introduced measure of uncertainty quantifies random variations in a player’s marginal contribution during the bargaining process. The measure is symmetric with respect to exchangeable substitutions in the players, equal to zero for dummy...

We consider the problem of the axiomatization of the Shapley value on the class of assignment games. We show that Shapley’s original [21], Young’s [24], Chun’s [7], van den Brink’s [2], (5-6) Hart and Mas-Colell’s [12] potential function and consistency approaches and Roth’s [19] characterization do not work on the class of assignment games. We also consider Myerson’s [15] axiomatization of the...

For f a weighted voting scheme used by n voters to choose between two candidates, the n Shapley-Shubik Indices (or Shapley values) of f provide a measure of how much control each voter can exert over the overall outcome of the vote. Shapley-Shubik indices were introduced by Lloyd Shapley and Martin Shubik in 1954 [SS54] and are widely studied in social choice theory as a measure of the “influen...

We study the asymptotic properties of the Shapley value of patent licensing games with the Cournot competition, shedding light on its relations to the nucleolus, core and bargaining set. The Shapley value of the outside patentee of a non-drastic cost-reducing innovation converges to 2(a − c), which coincides with the patentee’s profit through non-cooperative licensing by means of upfront fee (o...

The main aim of this article is to compare the results of classical Shapley value concept with results of the Shapley value extended to the cooperative games with fuzzy coalitions applied on the real data of the cooperative simple game – in this case the data from the voting in the Lower House of the Czech Parliament 20022012. One of the most intriguing tasks is to describe real system problems...

The paper starts by proposing an explicit and intuitive bargaining process to analyze coalitional externalities, which we then use to generate a value allocation extending the Shapley value to partition function from games, the Rational Belief Shapley value (RBS). We investigate the strategic foundation of the value by constructing a mechanism implementing it. It extends existing models of mult...

We will discuss some relations among values, i.e., global/overall importance of each single element of the object to be considered, interaction indices, i.e., simultaneous interaction among the elements in a coalition, and inclusion-exclusion coverings, i.e., decomposability of non-additive measures and/or of the Choquet integral. So, we show that inclusion-exclusion coverings are characterized...

Let f be a finite variance function of d independent input variables. Sobol’ indices are used to measure the importance of input variables and subsets of them. They are based on a variance decomposition. A similar problem arises in economics, when the value produced through the joint efforts of a team is to be attributed to individual members of that team. The Shapley value is widely used to so...