The proportional Shapley value and applications

Axiomatizations of the Proportional Division Value

“…Thus, we call it the proportional division value, shortly denoted by PD value, which also in order to distinguish it from the proportional rule in claim problems, bargaining problems, insurance, law and so on. For other proportional solutions, we refer to the proportional value (Ortmann 2000;Khmelnitskaya and Driessen 2003;Kamijo and Kongo 2015), the proper Shapley values (Vorob'ev and Liapunov 1998;van den Brink et al 2015), and the proportional Shapley value (Béal et al 2018;Besner 2019). 2 The PD value depends only on the worths of one-person coalitions and the grand coalition, but ignores the worths of any other intermediate coalitions.…”

Sharing the Surplus and Proportional Values

“…For any non-empty coalition S, let s or |S| be the cardinality of S. We denote by G N the set of all games with player set N . Béal et al (2018). We restrict our discussion to the class of all individually positive games and all individually negative games, and denote this class by Carreras and Owen (2013)…”

“…For two-player games, this can be formalized in axioms such as standardness (assigning each player its stand-alone worth and allocating the surplus equally over all players), egalitarian standardness (ignoring individual entitlements and allocating the full worth equally over the players), and proportional standardness (allocating the full surplus proportional to the stand-alone worths of the players). For example, the Shapley value (Shapley, 1953) and the equal surplus division value (Driessen and Funaki, 1991) satisfy standardness, the equal division value (axiomatized in van den Brink (2007)) satisfies egalitarian standardness, and various proportional values, such as the proportional Email addresses: z.zou@vu.nl (Zhengxing Zou), j.r.vanden.brink@vu.nl (René van den Brink), funaki@waseda.jp (Yukihiko Funaki) value (Ortmann, 2000), the proportional Shapley value (Béal et al, 2018;Besner, 2019), and the proper Shapley value (Vorob'ev and Liapunov, 1998;van den Brink et al, 2015) satisfy proportional standardness. The values can be extended to games with more than two players by, for example reduced game consistency or balanced contributions type of axioms that relate payoffs of players in a game with their payoffs in a game on a reduced player set.…”

“…Separatorization 1 refers to a player's obstruction of cooperation in the sense that the worth of any coalition containing him equals the sum of the stand-alone worths of the players in this coalition, while the worth of any coalition without this player remains unchanged. This is not to be confused with dummification as introduced in Béal et al (2018) (strengthening nullification studied in, e.g., Gómez-Rúa and Vidal-Puga (2010); Béal et al (2016b); Ferrières (2017)), where a player becomes a dummy player. The first axiom, called proportional loss under separatorization, requires that if a player becomes a separator, then all other player's payoff change in proportion to their stand-alone worths.…”

“…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.…”