Whose deletion does not affect your payoff? The difference between the Shapley value, the egalitarian value, the solidarity value, and the Banzhaf value

Axioms of invariance for TU-games

“…As such, our results are in the spirit of the comparable characterizations of the equal division rule and of the Shapley value studied in van den Brink [22] and Kamijo and Kongo [9]. Kamijo and Kongo [9] also use axioms of invariance.…”

“…As such, our results are in the spirit of the comparable characterizations of the equal division rule and of the Shapley value studied in van den Brink [22] and Kamijo and Kongo [9]. Kamijo and Kongo [9] also use axioms of invariance. The main difference is that the invariance axioms compare the payoffs in a TU-game before and after the deletion of a player from the player set instead of keeping the same player set and altering the TU-game under consideration as in the present article.…”

“…Among the rules satisfying these three axioms, the equal division rule is the only one that satisfies the nullifying property (Theorem 1) while the Shapley value is the only one that satisfies the dummy player property (Theorem 3). This comparison between the equal division rule and the Shapley value is in the spirit of the comparative axiomatic characterizations studied by van den Brink [22] and Kamijo and Kongo [9]. In particular, as in these articles, it is enough to replace the nullifying player property by the dummy player property in the set of axioms characterizing the equal surplus division rule to obtain a set of axioms that characterizes the Shapley value.…”

“…Besides Kamijo and Kongo [9], different principles of invariance appear in the literature. The closest article is maybe Young [24], in which the axiom of marginality requires that a player's payoff is the same in two TU-games if all his contributions are identical in the two TU-games.…”

“…The main difference is that the invariance axioms compare the payoffs in a TU-game before and after the deletion of a player from the player set instead of keeping the same player set and altering the TU-game under consideration as in the present article. As such Kamijo and Kongo [9] have to consider a class of TU-games with a variable player set whereas we deal with a fixed but arbitrary player set. Nonetheless, both articles share the use of axioms such as the dummy player property and the nullifying player property in order to distinguish between egalitarian and marginalist rules.…”