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 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.
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Core
Every LP game v is a
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
- OWEN, Guillermo (1975), "