Moore interval subtraction and interval solutions for TU-games

Standard solutions for cooperative transferable utility (TU-) games assign to every player in a TU-game a real number representing the player's payoff. In this paper, we introduce interval solutions for TU-games which assign to every player in a game a payoff interval. Even when the worths of coalitions are known, it might be that the individual payoff of a player is not known. According to an interval solution, every player knows at least a lower- and upper bound for its individual payoff. Therefore, interval solutions are useful when there is uncertainty about the payoff allocation even when the worths that can be earned by coalitions are known. Specifically, we consider two interval generalizations of the famous Shapley value that are based on marginal contributions in terms of intervals. To determine these marginal interval contributions, we apply the subtraction operator of Moore. We provide axiomatizations for the class of totally positive TU-games. We also show how these axiomatizations can be used to extend any linear TU-game solution to an interval solution. Finally, we illustrate these interval solutions by applying them to sequencing games.