The Weighted Surplus Division Value for Cooperative Games

Abstract

The weighted surplus division value is defined in this paper, which allocates to each player his individual worth and then divides the surplus payoff with respect to the weight coefficients. This value can be characterized from three different angles. First, it can be obtained analogously to the scenario of getting the procedural value whereby the surplus is distributed among all players instead of among the predecessors. Second, endowing the exogenous weight to the surplus brings about the asymmetry of the distribution. We define the disweighted variance of complaints to remove the effect of the weight and prove the weighted surplus division value is the unique solution of an optimization model. Lastly, the paper offers axiomatic characterizations of the weighted surplus division value through proposing new properties, including the ω -symmetry for zero-normalized game and individual equity.

Cooperative Multi-Path Routing Algorithm for Integrated Satellite-Maritime Networks

“…In addition, the weighted surplus division value (WSdV) is presented as a new distribution rule for marginal contributions. e WSdV is similar to the philosophical idea of the Shapley value, but it focuses on how the marginal contributions are divided among the predecessors and considers the exogenous weights when distributing the surplus portion after every player takes his individual worth [11,12].…”

“…For each player i ∈ N, (a) real number c i (N, v) represents an assessment made by the player i; it is his gain from participating in the game. For all S⊆N and all i ∈ S, the marginal contribution of player i to coalition S is denoted as Δ i (v, S), and let Γ(v, S) be the average marginal contribution of all players in the coalition S; they are given by [10][11][12]…”

“…According to (1), the player i's Shapley value(Sh i (•)) and Solidarity value(So i (•)) are defined as follows [10][11][12]:…”

“…e equal division (ED) solution is the solution that distributes the worth v(N) of the grand coalition equally among all players, and thus assigns to every (N, v); the payoff is obtained as ED i (N, v) � v(N)/n for all i ∈ N; it averages the worth of the grand coalition to all players. e player i's equal surplus division value(ES i (•)) allocates the surplus evenly after giving every player his individual worth [11].…”

Procedural Implementation and Axiomatization of the Weighted Nonseparable Cost Value

“…e weighted Shapley value is defined in [11] and be portrayed axiomatically by Kalai and Samet [12]. Yang et al [13] researched the weighted surplus division value in cooperative games. In this paper, the weighted-NSC value is defined applying the exogenous determined positive weight.…”