We generalize the null player property (satisfied by the Shapley value) and nullifying player property (satisfied by the equal division solution) to the so-called delta-reducing player property, stating that a delta-reducing player (being a player such that any coalition containing this player earns a fraction delta in [0,1] of the worth of that coalition without that player) earns a zero payoff. This property yields the null player property for delta = 1 and the nullifying player property for delta = 0. We show that efficiency, symmetry, linearity and this delta-reducing player property characterizes the corresponding delta-discounted Shapley value. Moreover, we provide a strategic implementation of these solutions where delta is a discount factor that determines the decrease in value to be distributed in the next round after the proposal is rejected and the remaining players (without the proposer) play a new round of bidding.
Cooperative TU-game Shapley value equal division solution delta-discounted Shapley value Axiomatization Implementation Discounting