Cost Sharing under Uncertainty: An Algorithmic Approach to Cooperative Interval-Valued Games

Abstract

Recently, Branzei, Dimitrov, and Tijs (2003) introduced cooperative interval-valued games. Among other insights, the notion of an interval core has been coined and proposed as a solution concept for interval-valued games. In this paper we will present a general mathematical programming algorithm which can be applied to find an element in the interval core. As an example, we discuss lot sizing with uncertain demand to provide an application for interval-valued games and to demonstrate how interval core elements can be computed. Also, we reveal that pitfalls exist if interval core elements are computed in a straightforward manner by considering the interval borders separately.

Interval-Valued Fuzzy Cooperative Games Based on the Least Square Excess and Its Application to the Profit Allocation of the Road Freight Coalition

“…In this section, on the basis of the square excess, we focus on developing one kind of mathematical programming model for solving interval-valued cooperative games. Obviously, we can obtain the interval-valued least square excess solution through solving the mathematical programming model of Equation (9). Equation (9) can be rewritten in the form of the Lagrange function, as follows:…”

“…Alparslan [2] analysed the properties of one kind of interval-valued Shapley value and showed its axiomatic characterization. Kimms et al [9] proposed one kind of mathematical programming method, which was used to determine the interval-valued core. Alparslan et al [10] generalized several kinds of interval-valued cooperative game's set-value solutions, such as interval stable sets and the interval Shapley value.…”

On the interval Shapley value

“…Also, there exist several problems, which can be solved using interval games in the additive cone under consideration. These are connection problems where the cost of the connections are affected by interval uncertainty [17], lot-sizing problems with uncertain demands [15], sequencing problems where parameters are compact intervals of real numbers [3] and airport situations with interval data [5].…”

How to Handle Interval Solutions for Cooperative Interval Games

“…Recently, many papers appeared on modeling economic and Operational Research situations with interval data by using game theory, in particular cooperative interval games, as a tool. [1][2][3][4][5][6][7][8][9] A cooperative interval game in coalitional form is an ordered pair <N, w> where N = {1, 2, . .…”