Envy, Multi Envy, and Revenue Maximization

We study the envy free pricing problem faced by a seller who wishes to maximize revenue by setting prices for bundles of items. Consistent with standard usage [9] [10], we define an allocation/pricing to be envy free if no agent wishes to replace her allocation (and pricing) with those of another agent. If there is an unlimited supply of items and agents are single minded then we show that finding the revenue maximizing envy free allocation/pricing can be solved in polynomial time by reducing it to an instance of weighted independent set on a perfect graph. We also consider a generalization of envy freeness. We define an allocation/pricing as multi envy free if no agent wishes to replace her allocation with the union of the allocations of some set of other agents and her price with the sum of their prices. We show that even though such allocation/pricing can be approximated by O(log m+ log n) factor [3], it is coNP -hard to decide if a given allocation/pricing is multi envy free. We also show that revenue maximization multi envy free allocation/pricing is APX hard. An interesting restricted version of the subset pricing problem is when all items are intervals of a line segment and all requests are a contiguous set of items along the line. The motivation here is that one can think of the agents as drivers on a highway when each product is highway segment(or guests in a hotel — items are translated to dates). In this setting, determining if a given allocation/pricing is multi envy free is polynomial time. If the highway has bounded capacities then a revenue maximizing envy free allocation/pricing can be computed in polynomial time and we also give an FPTAS for the revenue maximizing multi envy free allocation/pricing.