On the Sensitivity of Linear Resource Sharing Problems to the Arrival of New Agents

We consider a multi-agent optimal resource sharing problem that is represented by a linear program. The amount of resource to be shared is fixed, and agents belong to a population that is characterized probabilistically so as to allow heterogeneity among the agents. In this article, we provide a characterization of the probability that the arrival of a new agent affects the resource share of other agents, which means that accommodating the new agent request at the detriment of the other agents allocation provides some payoff. This probability represents a sensitivity index for the optimal solution of a linear programming resource sharing problem when a new agent shows up, and it is of fundamental importance for a correct and profitable operation of the multi-agent system. Our developments build on the equivalence between the resource sharing problem and certain dual reformulations that can be interpreted as scenario programs with the number of scenarios corresponding to the number of agents in the primal problem. The recent "wait-and-judge " scenario approach is then used to obtain the sought sensitivity index. Our theoretical findings are demonstrated through a numerical example on optimal cargo aircraft loading.