In fully cooperative games players act efficiently when they form a single coalition, the grand coalition. The focus of the game is to find acceptable distributions of the payoff of the grand coalition. Distributions where a player receives less than it could obtain on its own, without cooperating with anyone else, are unacceptable - a condition known as individual rationality. Imputations are distributions that are efficient and are individually rational.
Example
Mrs A and Mrs B are knitting gloves. The gloves are one-size-fits-all, and two gloves make a pair that they sell for €5. They have each made 3 gloves. How to share the proceeds from the sale? The problem can be described by a characteristic function form game with the following characteristic function: Each lady has 3 gloves, that is 1 pair with a market value of €5. Together, they have 6 gloves or 3 pair, having a market value of €15. Then all possible distributions of this sum are imputations, where none of the ladies gets less than €5, the amount they can achieve on their own. For instance (7.5, 7.5) is an imputation, but so is (5, 10) or (9, 6).
The example can be generalised. If Mrs C and Mrs D are also part of the club and still each lady has made 3 gloves, now the total to distribute is 12 gloves, six pairs, that is, €30. At the same time one of the ladies, on her own can still get only €5. Thus imputations share €30, such that no-one gets less than €5. The following are all imputations: (7.5, 7.5, 7.5, 7.5), (10, 5, 10, 5), (5, 15, 5, 5) or (7, 5, 9, 9).
Properties
For 2-player games the set of imputations coincides with the core. In general the core is a selection from the set of imputations.
References
- Myerson Roger B.: Game Theory: Analysis of Conflict, Harvard University Press, Cambridge, 1991, ISBN 0-674-34116-3
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