In the classical approach to cooperative games, the worth of every coalition is assumed to be known. However, in real applications there may be situations in which the worth of some coalitions is unknown. The corresponding games are called partially defined games.

Partially defined cooperative games were first studied by Willson (1993). However, this author restricted attention to partially defined games in which if the worth of a particular coalition is known, then it is also known the worth of all the coalitions with the same cardinality. Moreover, Wilson (1993) proposed and characterized an extension of the Shapley value for partially defined games, which coincides with the ordinary Shapley value of a complete game. In this complete game coalitions whose worth were unknown are assigned a worth zero, that seems to be not well justified.

In this work we propose another extension of the Shapley value for general partially defined games by following the Harsanyi's approach by using dividends. That is, it is assumed that each coalition guarantees certain payments, called the Harsanyi dividends (Harsanyi, 1963), to its members. We assume that coalitions whose worth is not known assign a dividend equal to zero. The final payoff will be the sum of these dividends. We also study properties satisfied by this new value. And we characterize the proposed value using four axioms. Three of them are the well known axioms of carrier, additivity and positivity. The fourth one, called indispensable coalition axiom, is a weaker version of the anonymity axiom when referring to full defined games.

Further, we restrict attention to simple partially games in order to measure power with our new value when not all the possible coalitions are feasible. We study properties satisfied by this new power index.

M. Josune Albizuri

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The Shapley Value for Partially Defined Cooperative Games