Approximate core allocations and integrality gap for the bin packing game

A cooperative (uniform) bin packing game is an N-person game, where the player set consists of k bins of capacity 1 each and $n$ items of sizes $a_1,...,a_n$. The value of a coalition of players is defined to be the maximum total size of items in the coalition that can be packed into the bins of the coalition. We aim at finding a multiplicative ϵ-core allocation with ϵ as small as possible, thus approximating the core as closely as possible. Our main result shows that the 1/4-core is nonempty for all instances of the uniform bin packing game