Imputation (game theory)
Encyclopedia
In fully cooperative game
s 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.
The example can be generalised. If Mrs Carlson and Mrs Delacroix 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).
. In general the core is a selection from the set of imputations.
Cooperative game
In game theory, a cooperative game is a game where groups of players may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players...
s players act efficiently when they form a single coalition, the grand coalition
Grand coalition
A grand coalition is an arrangement in a multi-party parliamentary system in which the two largest political parties of opposing political ideologies unite in a coalition government...
. 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 Arnold and Mrs Bauer 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 Carlson and Mrs Delacroix 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 coreCore (economics)
The core is the set of feasible allocations that cannot be improved upon by a subset of the economy's consumers. A coalition is said to improve upon or block a feasible allocation if the members of that coalition are better off under another feasible allocation that is identical to the first...
. In general the core is a selection from the set of imputations.