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Stable Matchings, Optimal Assignments, and Linear Programming

Alvin E. Roth, Uriel G. Rothblum and John H. Vande Vate
Mathematics of Operations Research
Vol. 18, No. 4 (Nov., 1993), pp. 803-828
Published by: INFORMS
Stable URL: http://www.jstor.org/stable/3690124
Page Count: 26
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Stable Matchings, Optimal Assignments, and Linear Programming
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Abstract

Vande Vate (1989) described the polytope whose extreme points are the stable (core) matchings in the Marriage Problem. Rothblum (1989) simplified and extended this result. This paper explores a corresponding linear program, its dual and consequences of the fact that the dual solutions have an unusually direct relation to the primal solutions. This close relationship allows us to provide simple proofs both of Vande Vate and Rothblum's results and of other important results about the core of marriage markets. These proofs help explain the structure shared by the marriage problem (without sidepayments) and the assignment game (with sidepayments). The paper further explores "fractional" matchings, which may be interpreted as lotteries over possible matches or as time-sharing arrangements. We show that those fractional matchings in the Stable Marriage Polytope form a lattice with respect to a partial ordering that involves stochastic dominance. Thus, all expected utility functions corresponding to the same ordinal preferences will agree on the relevant comparisons. Finally, we provide linear programming proofs of slightly stronger versions of known incentive compatibility results.

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