Consistency, population solidarity, and egalitarian solutions for TU-games

Abstract

A (point-valued) solution for cooperative games with transferable utility, or simply TU-games, assigns a payoff vector to every TU-game. In this paper we discuss two classes of equal surplus sharing solutions. The first class consists of all convex combinations of the equal division solution (which allocates the worth of the 'grand coalition' consisting of all players equally over all players) and the center-of-gravity of the imputation-set value (which first assigns every player its singleton worth and then allocates the remainder of the worth of the grand coalition, N , equally over all players). The second class is the dual class consisting of all convex combinations of the equal division solution and the egalitarian non-separable contribution value (which first assigns every player its contribution to the 'grand coalition' and then allocates the remainder equally over all players). We provide characterizations of the two classes of solutions using either population solidarity or a reduced game consistency in addition to other standard properties.

Coalitional desirability and the equal division value

“…This desirability relation among the players originates from Isbell (1958) and has been studied extensively in order to evaluate the influence of voters on the class of simple games (see also Courtin and Tchantcho, 2015;Molinero et al, 2015, among others). The axiom of Desirability is often invoked in the characterization of classes of values such as the two classes of equal sharing values (van den Brink and Funaki, 2009;van den Brink et al, 2016), the procedural values (Malawski, 2013), the egalitarian Shapley values (Casajus and Huettner, 2013), a class of solidarity values (Béal et al, 2017) or to delimit subclasses of the linear, efficient and symmetric values (Levínský and Silársky, 2004;Radzik and Driessen, 2013).…”

Sharing the Surplus and Proportional Values

“…Specifically, it requires that each player receives at least a fraction α ∈ [0, 1] of the average stand-alone worth if it is feasible to do so. We compare this axiom with a known lower bound axiom for α-essential games which requires that in such games, every player earns at least a fraction α of its stand-alone worth, see van den Brink et al (2016). 4…”

“…Besides these two extreme values, our family consists of all convex combinations of the PD value and the EPSD value, and thus can be viewed as making a trade-off between egocentrism and egalitarianism. This family of values is in line with a recent and growing literature that combine different allocation principles by considering convex combinations of two extreme values, such as the egalitarian Shapley values (being convex combinations of the Shapley value and equal division value, see Joosten (1996) and van den Brink et al (2013)), the consensus values (being convex combinations of the Shapley value and equal surplus division value, see Ju et al (2007)) and the family of convex combinations of the equal division value and the equal surplus division value (axiomatized in, e.g., van den Brink and Funaki (2009);van den Brink et al (2016); Xu et al (2015); Ferrières (2017)). Also, our family of values is in line with a recent and growing literature on non-symmetric surplus sharing values, such as the weighted division value (Béal et al, 2015(Béal et al, , 2016a, the weighted surplus division value Llerena, 2017, 2019), the weighted equal allocation of non-separable contributions value (Hou et al, 2019), and the PD value (Zou et al, 2019).…”

“…The difference between the PD value and the proportional Shapley value 5 is pinpointed to one axiom. With Theorem 6, it immediately follows that the PD value is 4 Recall that α-individual rationality is used to characterize the convex combinations of the ED and ESD values in van denBrink et al (2016) andXu et al (2015).5 The proportional Shapley value is defined as:ψ P Sh i (N, v) = S⊆N,i∈S v({i}) j∈S v({j}) ∆ v (S) for all (N, v) ∈ G N nz and i ∈ N , where ∆ v (S) = v(S) − T ⊆S,T =∅ ∆ v (T ) is the Harsanyi dividend of S.characterized by efficiency, proportional balanced contributions under separatorization, and the inessential game property Béal et al (2018). offer a characterization of the proportional Shapley value on G N nz by employing efficiency, proportional balanced contributions under dummification 6 , and the inessential game property.…”

A new axiomatization of a class of equal surplus division values for TU games

“…Other axiomatizations of the convex combinations can also be found in van den Brink et al [13]. First, they characterized it by simplifying the β-consistency of [12] into projection reduced game consistency.…”