On the Core and Dual Set of Linear Programming Games

We study the relation between the core of a given LP-game and the set of payoff vectors generated by optimal dual solutions to the corresponding linear program. It is well known that the set of dual payoffs is contained in the core, and that cores of games in which players are replicated converge to the set of dual payoffs when the number of replications tends to infinity. We give a necessary and sufficient condition for finite convergence. As corollaries we strengthen a sufficient condition due to Owen and obtain new conditions as well. We also study conditions in which the core and the set of dual payoffs coincide even without replication. We give a necessary and sufficient condition for this phenomenon and present two classes of LP-games with this property which properly subsumes all examples of this type discussed in the literature.