Value dividends, the Harsanyi set and extensions, and the proportional Harsanyi solution
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Value dividends, the Harsanyi set and extensions, and the proportional Harsanyi solution Manfred Besner1 Accepted: 23 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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. Keywords TU-game · Value dividends · (Generalized) Harsanyi set · Weighted Shapley values · (Proportional) Harsanyi solution · Sharing function systems
1 Introduction The concept of Harsanyi dividends was introduced by Harsanyi (1959). They can be defined inductively: the dividend of the empty set is zero and the dividend of any other possible coalition of a player set equals the worth of the coalition minus the sum of all dividends of proper subsets of that coalition. Hence, Harsanyi dividends can be interpreted as “the pure contribution of cooperation in a TU-game” (Billot and Thisse 2005). Harsanyi could show that if the Harsanyi dividends of all possible coalitions are spaced evenly among its members, each player’s payoff equals the Shapley value (Shapley 1953b). The weighted Shapley values (Shapley 1953a) distribute the Harsanyi dividends proportionally to players’ personal given weights. The ratio of the weights of two players is equal for all coalitions containing them. However,
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Manfred Besner [email protected] Department of Geomatics, Computer Science and Mathematics, HFT Stuttgart, University of Applied Sciences, Schellingstr. 24, 70174 Stuttgart, Germany
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M. Besner
sometimes this seems unrealistic. For example, in some coalitions, the influence of one of two players on the other players may be higher than in other coalitions with other players. Hammer et al. (1977) and Vasil’ev (1978) proposed the Harsanyi set, a class of TU-values called Harsanyi solutions, also known as sharing values (Derks et al. 2000) which take this into account. There the players’ weights, assigned to the Harsanyi dividends via a sharing system, can differ for all coalitions. Myerson (1980) introduced the balanced contributions axiom which allows, along with efficiency, an elegant axiomatization of the Shapley value. It states for two players i and j that j’s presence contributes as much to i’s payoff as i’s presence contributes to j’s payoff. The w-balanced contributions properties, the ratio of the winnings or losses of two players when the other player leaves the game is proportional to their weights, joint with efficiency, characterize the TU
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