A Partially Defined Game with Costs

The present study explores a problem that can be resolved by employing the notion of a partially defined cooperative game, yet cannot by using a restricted game. The following situation is considered: First, it is assumed that the worth of the grand and singleton coalitions are known. It takes some amount of costs to obtain worth of unknown coalitions. If it is performed, then the worth of the grand coalition is decreased by the value of a cost function. With the view point of fairness of a payoff allocation, we should examine coalitional worth as much as possible. However, we should stop examining coalitional worth at some point since total payoff is reduced by continuing the examinations. We name the new decision making problem a partially defined cooperative game with costs. The problem of a partially defined cooperative game with costs is finding the solution of partially defined cooperative games at each point and the best exiting rule of examinations of coalitional worth.