STABLE MATCHINGS, OPTIMAL ASSIGNMENTS, AND LINEAR-PROGRAMMING

Vande Vate (1989) described the polytope whose extreme points are the stable (core) matchings in the Marriage Problem. Rothblum (1989) simpl ified and extended this result. This paper explores a corresponding li near program, its dual and consequences of the fact that the dual solu tions have an unusually direct relation to the primal solutions. This close relationship allows us to provide simple proofs both of Vande Va te and Rothblum's results and of other important results about the cor e 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'' matchi ngs, 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 utili ty functions corresponding to the same ordinal preferences will agree on the relevant comparisons. Finally, we provide linear programming pr oofs of slightly stronger versions of known incentive compatibility re sults.