On the Connection between Individual Scaled Vickrey Payments and the Egalitarian Allocation

TL;DR

This paper reveals a fundamental connection between individual scaled Vickrey payments and the egalitarian allocation, showing they are equivalent to a weighted egalitarian allocation and providing efficient computation methods.

Contribution

It establishes that individual scaled Vickrey payments are identical to a weighted egalitarian allocation, demonstrating their uniqueness and offering a polynomial-time algorithm for special cases.

Findings

ISV payments are equivalent to a weighted EA

The uniqueness of ISV payments is proven

A polynomial-time algorithm for computing a weighted EA is provided

Abstract

The Egalitarian Allocation (EA) is a well-known profit sharing method for cooperative games which attempts to distribute profit among participants in a most equal way while respecting the individual contributions to the obtained profit. Despite having desirable properties from the viewpoint of game theory like being contained in the core, the EA is in general hard to compute. Another well-known method is given by Vickrey Payments (VP). Again, the VP have desirable properties like coalitional rationality, the VP do not fulfill budget balance in general and, thus, are not contained in the core in general. One attempt to overcome this shortcoming is to scale down the VP. This can be done by a unique scaling factor, or, by individual scaling factors. Now, the individual scaled Vickrey Payments (ISV) are computed by maximizing the scaling factors lexicographically. In this paper we show that the ISV payments are in fact identical to a weighted EA, thus exhibiting an interesting connection between EA and VP. With this, we conclude the uniqueness of the ISV payments and provide a polynomial time algorithm for computing a special weighted EA.