Allocation of Bilateral Surplus in Networks

We study a new class of network models in which players occupy physical positions on a tree and can only generate bilateral surplus with adjacent players. We study the allocation problem when players are positioned such that the total amount of realized surplus is maximal, which we call a bilateral surplus problem. We focus on bilateral allocations, in which each pair of adjacent players shares their own bilateral surplus among themselves. Taking a game-theoretic approach, we assume that coalitions that are connected in the network can freely rearrange their locations and define the worth of a coalition in the associated bilateral surplus game to be the maximum amount of internally realized bilateral surplus it can obtain. We show that the core of a bilateral surplus game is non-empty, and every core element is a bilateral allocation. The converse need not hold. We provide a closed-form expression for the allocations of random tree solutions based on the following principle: along each edge, each cone first receives its worth, and the additional worth created by connecting the cones is split according to the solution’s weights. We consider the average tree solution as a special case, and show that, on the class of bilateral surplus problems, it is the unique solution satisfying the properties of efficiency and equal per-capita cone excesses. It is seen that this characterization can also be extended to the domain of acyclic graph games, resulting in a new characterization of the average tree solution on this class.