Three Essays on Matching with Contracts
This dissertation consists of three theoretical essays. In all essays matching with contracts is a key factor. The first essay tries to explain effects of choosing primitives of the model and irrelevance of rejected contracts condition on some key existence theorems and results in matching with contracts literature. The second essay analyzes the properties of cumulative offer algorithm and presents an application of matching with contracts. It studies the achievability of responsive choices under a constrained setup. The last essay presents a new market design application of program-student matching where affirmative action policies are effective. The first essay develops a hospital-doctor many-to-one matching with contracts model. Doctor preferences over contracts are part of primitive of the model. The other primitive of the model, our first essay suggests, hospital choice functions on sets of contracts. The first essay shows that if choice functions of hospitals are primitives of the model, then existence theorems used in many papers do not hold even when they satisfy strongest conditions. As a remedy, we introduced Irrelevance of Rejected Contracts (IRC) which guarantees stability if it is satisfied along with one substitutes condition. Next, we show the relation between IRC and law of aggregate demand (LAD) conditions. Since LAD is satisfied by many application naturally, many models satisfying LAD and the strongest substitutes conditions are immune to our criticism. On the other hand, many of the new and exiting applications satisfy only weakened substitutes condition. Therefore, assuming IRC explicitly does not only make their proofs accurate and also close the gap between theory and application. The second chapter studies properties of cumulative offer algorithm under weakened substitutes condition. In this part we showed that in many-to-one matching with contracts problems order of proposals of COA does not change the outcome, under bilateral substitutes and IRC conditions. Also, bilateral substitutes and IRC conditions make COA equivalent to generalized deferred acceptance algorithm which produces the outcome in fewer steps. This chapter also presents a new application area of matching with contracts. We used cadet-branch matching problem in USMA. In this application our main objective is, for a given branch, increasing cadet quality without giving up useful properties of allocation mechanism, such as stability and strategy-proofness. The third essay studies a college admission with affirmative action problem. With this application, for the first time in the literature, we presented an affirmative action problem where students need to claim privilege if they want to be subject to affirmative action. We analyzed the current system and showed that current guideline is unfair and causes incentive compatibility issues. Also we showed that it fails to satisfy affirmative action requirements described in affirmative action law. To solve these problems with the current system, we introduced a new choice function which is fair, respects affirmative action requirements and makes student optimal stable allocation stable and incentive compatible when used in conjunction with generalized deferred acceptance algorithm.