Linear production game

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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 akj 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 bi=(b1i,...,bmi) 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.

v(S)=maxx0(c1x1+...+cnxn)

s.t.aj1x1+aj2x2+...+ajnxniSbjij=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 α be the optimal dual solution of P(N). The payoff to player i is xi=k=1mαkbki. 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 αj, 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, 9, Mathematical Programming : 358–370, doi:10.1007/BF01681356