Contract design and stability in many-to-many matching

Abstract

We develop a model of many-to-many matching with contracts that subsumes as special cases many-tomany matching markets and buyer-seller markets with heterogeneous and indivisible goods. In our setting, substitutable preferences are sufficient to guarantee the existence of stable outcomes; moreover, in contrast to results for the setting of many-to-one matching with contracts, if any agent's preferences are not substitutable, then the existence of a stable outcome cannot be guaranteed. In many-to-many matching with contracts, a new market design issue arises: The design of the contract language can impact the set of stable outcomes. Bundling contractual primitives encourages substitutability of agents' preferences over contracts and makes stable outcomes more likely to exist; however, bundling also makes the contractual language less expressive. Consequently, in choosing contract language, market designers face a tradeoff between expressiveness and stability.

Binary operations and lattice structure for a model of matching with contracts

“…Such set of fixed points were closely related to the set of stable allocations but, as we show through an example, there was not necessarily a bijection between both sets. Hatfield and Kominers (2012) obtained analogous results, with a similar drawback, for a model of many-to-many matching with contracts. Accordingly, the issue about the structure of the set of stable allocations was not fully clarified in Hatfield and Milgrom (2005) or Hatfield and Kominers (2012).…”

“…Hatfield and Kominers (2012) obtained analogous results, with a similar drawback, for a model of many-to-many matching with contracts. Accordingly, the issue about the structure of the set of stable allocations was not fully clarified in Hatfield and Milgrom (2005) or Hatfield and Kominers (2012). Our paper offsets this deficiency for the restricted model that we consider by proving constructively that the set of stable allocations has lattice structure concerning two partial orderings well-known in matching theory.…”

Probabilistic stable rules and Nash equilibrium in two-sided matching problems

“…Each agent v has a linear preference ordering P v over 2 Sv . 8 We write P f : {w 1 , w 2 }, w 2 , ∅, {w 2 , w 3 }, ..., w 3 , for example, to indicate that f 's first choice is being matched to w 1 and w 2 , its second choice is being matched to w 2 only, and it prefers remaining unmatched to being matched to any other subset of workers.…”

“…This, together with f ∈ µ (w)\µ(w) implies that µ(f ) ∪ {w} ⊆ µ (f ). By (8) and substitutability, w / ∈ Ch(µ (f ), Q f ), contradicting µ ∈ S(Q ). Thus, µ(f )\µ (f ) = ∅.…”

“…Noting that w ∈ µ(f ), we have (µ (f ) ∪ {w}) ⊆ (µ(f ) ∪ {w}). By (8) and substitutability, w ∈ Ch(µ (f ) ∪ {w}, Q f ).…”