# Linear production game

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*Linear Production Game* (*LP Game*) is a N-person game in which the value of a coalition can be obtained by solving a Linear Programming problem. It is widely used in the context of resource allocation and payoff distribution. Mathematically, there are *m* types of resources and *n* products can be produced out of them. Product *j* requires amount of the *kth* resource. The products can be sold at a given market price while the resources themselves can not. Each of the *N* players is given a vector of resources. The value of a coalition *S* is the maximum profit it can achieve with all the resources possessed by its members. It can be obtained by solving a corresponding Linear Programming problem as follows.

## The core of the LP game

Every LP game *v* is a totally balanced game. So every subgame of *v* has a non-empty core. One imputation can be computed by solving the dual problem of . Let be the optimal dual solution of . The payoff to player* i* is . It can be proved by the duality theorems that is in the core of *v*.

An important interpretation of the imputation is that under the current market, the value of each resource *j* is exactly , although it is not valued in themselves. So the payoff one player i should receive is the total value of the resources he possesses.

However, not all the imputations in the core can be obtained from the optimal dual solutions. There are a lot of discussions on this problem. One of the mostly widely used method is to consider the r-fold replication of the original problem. It can be shown that if an imputation *u* is in the core of the r-fold replicated game for all r, then *u* can be obtained from the optimal dual solution.

## References

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