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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.

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| v(S)=\max_{x\geq 0} (c_1x_1+...+c_nx_n)

s.t. \quad a^1_jx_1+a^2_jx_2+...+a^n_jx_n\leq \sum_{i\in S}b^i_j \quad \forall j=1,2,...,m

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.

References

Category:Cooperative games