We provide new axiomatic characterizations of the proportional Shapley value, a weighted value with the worths of the singletons as weights. This value satisfies anonymity and therefore symmetry just as the Shapley value and has characterizations which are proportional counterparts to the famous characterizations of the Shapley value in Shapley (1953b), Myerson (1980) and Young (1985a). If the stand alone worths are plausible weights the proportional Shapley value is a convincing alternative to the Shapley value for example in cost allocation. We introduce two new axioms, called proportionality and player splitting respectively. Each of which gives a main difference between the proportional Shapley value and the Shapley value.
A new concept for TU-values, called value dividends, is introduced. Similar to Harsanyi dividends, value dividends are defined recursively and provide new characterizations of values from the Harsanyi set. In addition, we generalize the Harsanyi set where each of the TU-values from this set is defined by the distribution of the Harsanyi dividends via sharing function systems and give an axiomatic characterization. As a TU value from the generalized Harsanyi set, we present the proportional Harsanyi solution, a new proportional solution concept. A new characterization of the Shapley value is proposed as a side effect. None of our characterizations uses additivity.
In this paper we present a new class of values for cooperative games with level structure. We use a multi-step proceeding, suggested first in Owen (1977), applied to the weighted Shapley values. Our first axiomatization is an generalisation of the axiomatization given in Gómez-Rúa and Vidal-Puga (2011), itselves an extension of a special case of an axiomatization given in Myerson (1980) and Hart and Mas-Colell (1989) respectively by efficiency and weighted balanced contributions. The second axiomatization is completely new and extends the axiomatization of the weighted Shapley values introduced in Hart and Mas-Colell (1989) by weighted standardness for two player games and consistency. As a corollary we obtain a new axiomatization of the Shapley levels value.
We present new axiomatic characterizations of five classes of TU-values, the classes of the weighted, positively weighted, and multiweighted Shapley values, random order values, and the Harsanyi set. The axiomatizations are given in parallel, i. e., they differ only in one axiom. In conjunction with marginality, a new property, called coalitional differential dependence, is the key that allows us to dispense with additivity. In addition, we propose new axiomatizations of the above five classes, in which, in part new, different versions of monotonicity, associated with the strong monotonicity in Young (1985), are decisive.
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