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...

Economists have long argued that the existence of increasing returns to scale or unique complementary inputs may lead to indeterminacy in how the gains to team production are shared among team members. Cooperative games provide a framework in which to formalize and explore this intuition. In games with side payments, the notion of a convex game introduced by Shapley (1971) provides a natural wa...

The paper studies the vector space of set functions on a finite set X, which can be alternatively seen as pseudo-Boolean functions, and including as a special cases games. We present several bases (unanimity games, Walsh and parity functions) and make an emphasis on the Fourier transform. Then we establish the basic duality between bases and invertible linear transform (e.g., the Möbius transfo...

Designing fair compensation mechanisms for demand response (DR) is challenging. This paper models the problem in a game theoretic setting and designs a payment distribution mechanism based on the Shapley Value. As exact computation of the Shapley Value is in general intractable, we propose estimating it using a reinforcement learning algorithm that approximates optimal stratified sampling. We a...

We suggest a new game-theory-based ranking method for wines, in which the Shapley Value of each wine is computed, and wines are ranked according to their Shapley Values. Judges should find it simpler to use, since they are not required to rank order or grade all the wines, but merely to choose the group of those that they find meritorious. Our ranking method is based on the set of reasonable ax...

Hamiache’s recent axiomatization of the well-known Shapley value for TU games states that the Shapley value is the unique solution verifying the following three axioms: the inessential game property, continuity and associated consistency. Driessen extended Hamiache’s axiomatization to the enlarged class of efficient, symmetric, and linear values, of which the Shapley value is the most important...

Shapley values, also known as Shapley–Shubik indices, measure the power that agents have in a weighted voting game. Suppose that agent weights are chosen randomly according to some distribution. We show that the expected Shapley values for the smallest and largest agent are independent of the quota for a large range of quotas, and converge exponentially fast to an explicit value depending only ...

The core and the Shapley value are important solution concepts in cooperative game theory. While the core is designed for the stability of the game, the Shapley value aims for fairness among the players. However, the Shapley value might not lie within the core and a core solution might not be ‘fair’. We introduce a new solution concept called the ‘fairest core’, one that aims for both stability...

This paper makes the case for using Shapley value to quantify the importance of random input variables to a function. Alternatives based on the ANOVA decomposition can run into conceptual and computational problems when the input variables are dependent. Our main goal here is to show that Shapley value removes the conceptual problems. We do this with some simple examples where Shapley value lea...

Lotteries. A lottery L = (p, x) (or gamble) is a vector of payoffs (x1, x2, . . . , xn) with probabilities (p1, p2, . . . , pn). The payoffs might be consumption bundles or monetary payoffs. St Petersburg Paradox. The payoff is 2, where n is the first time the coin comes up heads. The expected value is infinite. If the game lasts just two rounds, it is worth 1 + 1. If it lasts 3 rounds, it is w...