Combinatorial optimization games

We introduce a general integer programming formulation for a class of combinatorial optimization games, immediately allows us to improve the algorithmic result for finding imputations in the core (an important solution concept in cooperative game theory) of the network Bow game on simple networks by Kalai and Zemel [16]. An interesting result is a general theorem that the core for this class of games is nonempty if and only if a related Linear program has an integer optimal solution. We study the properties for this mathematical condition to hold for several interesting problems, and apply them to resolve algorithmic and complexity issues for their cores along the Line as put forward in [2,19]: decide whether the core is empty; if the core is empty, find an imputation in the core; given an imputation 2, test whether x is in the core. We also explore the properties of totally balanced games in this succinct formulation of cooperative games.