# Equivalent Choice Functions and Stable Mechanisms

## TL;DR

This paper establishes a necessary and sufficient condition, called equivalence, for the existence of stable, group-strategy-proof matching mechanisms in models with contracts and monotone preferences, linking choice functions to observable substitutes.

## Contribution

It introduces the concept of equivalence for choice functions, providing a weak embedding result that characterizes when stable, strategy-proof mechanisms can exist in contract-based matching models.

## Key findings

- Equivalence to a choice profile with substitutes is necessary and sufficient.
- Stable, group-strategy-proof mechanisms exist under the equivalence condition.
- The result links observable substitutes to the existence of stable mechanisms.

## Abstract

We study conditions for the existence of stable and group-strategy-proof mechanisms in a many-to-one matching model with contracts if students' preferences are monotone in contract terms. We show that "equivalence", properly defined, to a choice profile under which contracts are substitutes and the law of aggregate holds is a necessary and sufficient condition for the existence of a stable and group-strategy-proof mechanism. Our result can be interpreted as a (weak) embedding result for choice functions under which contracts are observable substitutes and the observable law of aggregate demand holds.