A Tutorial on Matching Theory & Contract Theory

Transcript:A Tutorial on Matching Theory & Contract Theory:
A Tutorial on Matching Theory & Contract Theory Yanru Zhang 1 OUTLINE INTRODUCTION MATCHING THEORY Stable Matching with Complete Preference Lists Stable Matching with Relaxed Preference Lists CONTRACT THEORY Optimal Employment Contracts without Uncertainty, Hidden Information, or Hidden Actions Optimal Contracts under Uncertainty Information and Incentives Optimal Contracting with Multilateral Asymmetric Information 2 INTRODUCTION Economics a field that aims to understand the process by which scarce resources are allocated to their most efficient uses markets playing a central role in this allocation process 3 INTRODUCTION Examples Matching doctors and hospitals Matching students and high-schools Matching kidneys and patients Investment under risk Insurance Hiring employees Existing Theories of Matching Theory Contract Theory 4 MATCHING THEORY The Nobel Prize in Economic Sciences 2012 to Lloyd Shapley and Alvin Roth Lloyd Shapley Developed the theory in the 1960s Alvin Roth Generated further analytical developments practical design of market institutions 5 MATCHING THEORY Definition on Wikipedia a mathematical framework attempting to describe the formation of mutually beneficial relationships over time. Where is it used The U.S. National Resident Matching Program (NRMP) for medical school graduates Public schools in Boston and NYC Many medical and other labor markets across 6 PART I: Matching Problem with Complete Preference List Three types of matching problems One-to-one (stable marriage) One-to-many (college admissions) Many-to-many (complex scenarios, P2P) 7 Solution of a Matching Problem We seek to find a stable matching such that There does not exist any pair of players, i and j matches, respectively to players, a and b but.. j prefers a to b and a prefers j to i How can we find a stable matching? many approaches(minimizing sum/ max of ranks, minimizing diff of total ranks, Gale and Shapley algorithm) Most popular: Deferred acceptance or GS algorithm Illustrated via an example 8 Example 1: Matching partners 9 women and men be matched respecting their individual preferences Example 1: Matching partners The Gale-Shapley algorithm can be set up in two alternative ways: men propose to women women propose to men Each man proposing to the woman he likes the best Each woman looks at the different proposals she has received (if any) retains what she regards as the most attractive proposal (but defers from accepting it) and rejects the others The men who were rejected in the first round Propose to their second-best choices The women again keep their best offer and reject the rest Continues