# Linear production game

Template:Orphan 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 $a_{k}^{j}$ 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 ${\vec {b^{i}}}=(b_{1}^{i},...,b_{m}^{i})$ 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.
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.