Linear production game

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 $$a^j_k$$ amount of the kth resource. The products can be sold at a given market price $$\vec{c}$$ while the resources themselves can not. Each of the N players is given a vector $$\vec{b^i}=(b^i_1,...,b^i_m)$$ 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 $$P(S)$$ as follows.

Core
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 $$P(N)$$. Let $$\alpha$$ be the optimal dual solution of $$P(N)$$. The payoff to player i is $$x^i=\sum_{k=1}^m\alpha_k b^i_k$$. It can be proved by the duality theorems that $$\vec{x}$$ is in the core of v.

An important interpretation of the imputation $$\vec{x}$$ is that under the current market, the value of each resource j is exactly $$\alpha_j$$, 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.