The Intermediate Price of Anarchy ([formula omitted]) in bin packing games

Abstract

In the case of bin packing items with positive sizes are packed into a minimum number of bins, such that the total size of items in any bin is at most 1. We consider the bin packing game where the cost of a bin (i.e., 1) is shared among the items in the bin proportionally to their sizes. Any item can move to another bin if it fits and after the move the item will pay less for being in the new bin. A packing is called a Nash Equilibrium (NE) if no item can benefit from moving in this way. The price of anarchy (PoA) is the asymptotic worst-case ratio between the number of bins in an NE and of bins in a (socially) optimal solution. A much stronger concept of a stable packing is the Strong NE (SNE), where no coalition of items can move so that each member of the coalition reduces its own cost by the move. If we compare the SNE packings with the optimal packings, we get the measure called SPoA (strong price of anarchy).We define a new kind of equilibrium as follows. An item i in bin B with size si has an improving step to move into a target bin B′ if there is an empty or not empty set S of items in the target bin, so that (i) si is strictly bigger than the total size of items of S, (ii) after removing the items in S from the target bin (and putting them into B) item i fits into the target bin, and (iii) the level of the target bin after the move is bigger than the level of the bin where item i was before the move. It is clear that if such move is possible, then moving is advantageous for item i, as it will have to pay less. If no such move is possible, the packing is called Intermediate NE (INE), and the related measure is called IPoA. By definition, the value of IPoA is betweenSPoA and PoA.It is known (Epstein and Kleiman, 2011) that the PoA lies between 1.6413 and 1.6428; moreover the exact value of the SPoA was found recently Epstein et al. (2016) (this value is approximately 1.6067). We will prove that the IPoA of the bin packing game is very close to the value of SPoA, namely it is no bigger than 1.6095.