Matching Market Equilibrium Algorithms

Years and Authors of Summarized Original Work

• 1971; Shapley, Shubik

• 1982; Kelso, Crawford

• 1986; Demange, Gale and Sotomayor

Problem Definition

The study of matching market equilibrium was initiated by Shapley and Shubik [13] in an assignment model. A classical instance of the matching market involves a set B of n unit-demand buyers and a set Q of m indivisible items, where each buyer wants to buy at most one item and each item can be sold to at most one buyer. Each buyer i has a valuation v _ij  ≥ 0 for each item j, representing the maximum amount that i is willing to pay for item j. Each item j has a reserve price r _j  ≥ 0, below which it won’t be sold. Without loss of generality, one can assume there is a null item whose value is zero to all buyers and whose price is always zero.

An output of the matching market is a tuple (p, x), where p = (p₁, …, p _m ) ≥ 0 is a price vector with p _j denoting the price charged for item j and x = (x₁, …, x _n ) ≥ 0 is an allocation vector with x_...