# Imputation (game theory)

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. 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 do they 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 a distribution of this sum is an imputation provided that 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 generalized. Suppose Mrs. Carlson and Mrs. Delacroix are also part of the club where each lady has made 3 gloves. Now the total is 12 gloves (six pairs) which nets €30. At the same time, one of the ladies on her own can still only make €5. Thus, imputations share €30 such that no-one gets less than €5. The following are possible 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.

## Time consistency in dynamic games

An important problem in the theory of cooperative dynamic games is the time-consistency of a given imputation function (in Russian literature it is termed dynamic stability of optimality principle). Let say that a number of players has made a cooperative agreement at the start of the game. Obviously, a rational player will leave the agreement if he/she can achieve a better outcome by abandoning, no matter what was announced before. The condition, which guarantees the sustaining of the cooperative agreement is known as time consistency. A number of regularization methods (integral and differential) based upon the IDP (imputation distribution procedures) was proposed.

## References

• Myerson Roger B.: Game Theory: Analysis of Conflict, Harvard University Press, Cambridge, 1991, ISBN 0-674-34116-3
• Petrosjan, Leon A. Differential games of pursuit, World Scientific, Singapore, London, 1993, pp. 340.
• Yeung, David W.K. and Petrosyan, Leon A. Cooperative Stochastic Differential Games (Springer Series in Operations Research and Financial Engineering), 2006, Springer pp. 242. ISBN 978-1441920942.
• Zaccour, Georges. Time Consistency in Cooperative Differential Games: A Tutorial. INFOR: Information Systems and Operational Research, Volume 46(1), 2008. ISSN 0315-5986.