Linear production game - Alchetron, The Free Social Encyclopedia

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 c → while the resources themselves can not. Each of the N players is given a vector 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.

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 P ( N ) . Let α be the optimal dual solution of P ( N ) . The payoff to player i is x i = ∑ k = 1 m α k b k i . It can be proved by the duality theorems that x → is in the core of v.

An important interpretation of the imputation x → is that under the current market, the value of each resource j is exactly α 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.

References

Linear production game Wikipedia

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