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 ${\displaystyle a_{k}^{j}}$ amount of the kth resource. The products can be sold at a given market price ${\displaystyle {\vec {c}}}$ while the resources themselves can not. Each of the N players is given a vector ${\displaystyle {\vec {b^{i}}}=(b_{1}^{i},...,b_{m}^{i})}$ 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 ${\displaystyle P(S)}$ as follows.

${\displaystyle v(S)=\max _{x\geq 0}(c_{1}x_{1}+...+c_{n}x_{n})}$ ${\displaystyle s.t.\quad a_{j}^{1}x_{1}+a_{j}^{2}x_{2}+...+a_{j}^{n}x_{n}\leq \sum _{i\in S}b_{j}^{i}\quad \forall j=1,2,...,m}$

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

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

• OWEN, Guillermo (1975), "On the Core of Linear Production Games", Mathematical Programming, Mathematical Programming , 9: 358–370, doi:10.1007/BF01681356