Final Report Summary - NEWDIRMD (New directions in market design)
Executive summary
The first sub-project has focused on the many-to-one matching with contracts framework of Hatfield and Milgrom (2005). This framework considers a two-sided market, consisting of firms and workers, in which each worker can sign at most one employment contract and firms can hire multiple workers. The main objective was to discover the largest class of preferences for the firms under which a centralised mechanism exists that (a) is guaranteed to produce a stable outcome, i.e. no group of workers and firms has an incentives to veto the outcome in order to sign new, more preferred, contracts among themselves, and (b) is strategy-proof for workers, i.e. each worker always finds it in her best interest to reveal her true ranking of employment contracts to the mechanism. In order to achieve this objective, I have worked in collaboration with John Hatfield and Scott Kominers. We have been able to identify three conditions that characterise the largest class of preferences for the firms for which a stable and strategy-proof mechanism exists. Furthermore, our results imply that whenever a stable and strategy-proof mechanism exists, it has to coincide with the well known cumulative offer mechanism. This last result provides us with an important foundation for the use of this mechanism. Our results answer important long-standing open questions in the literature and have the potential to open the door for further high-profile practical applications of matching theory.
For the second sub-project, the plan was to focus on decentralised matching markets in which participants try to learn from information about past outcomes. One key example is a decentralised market for admissions to universities. Here, universities typically receive applications from interested students and then have to decide on how many offers to make. Due to the shorter than initially expected project duration, I focused on an extension of a simpler model that was developed in an earlier working paper of mine. Here, in each period, students apply to all universities and universities simultaneously decide on how many offers to make. In my earlier paper on this model I had identified the conditions under which “better reply”-learning by universities is guaranteed to converge to market clearing, that is, a situation in which all universities exactly reach their target capacities. I am close to establishing the following result:
Consider a better reply dynamic for which there is an integer K such that, from any given period to the next, no university changes the number of offers it makes by more than K. Such a better reply dynamic must eventually reach a point after which no university ever misses its target enrolment by more than K students.
This is an important approximation result for better reply learning dynamics in the context of decentralised university admissions. In particular, if K is small, then a better reply dynamic in which per-period adjustments are limited by K is guaranteed to come very close to market-clearing. I expect this result to be very useful for further work on learning in decentralised matching markets.
As a further extension, I have engaged in joint research (with John Hatfield, Scott Kominers, Alexandru Nichifor, and Michael Ostrovsky) on generalised matching models. Among other things, these models are helpful for understanding the formation of supply chains in complex production processes. This line of research is a natural extension of my research on two-sided matching problems that was summarised in the first paragraph. Our research on generalised matching problems has (1) unified and extended various approaches to defining what “substitutability of preferences” means, (2) uncovered a new class of substitutable preferences that allows us to model intermediaries with production capacity, (3) showed that substitutability is preserved under economically important transformations such as trade endowments, mergers, and limited liability, and (4) showed that a suitably adapted chain stability concept is equivalent to the most widely used notion of stability.
Summary description
The aim of the first sub-project has been to characterise the conditions under which “good” centralised mechanisms exist for matching problems with contracts. One possible application is an entry-level labor market in which a matching of workers to firms and contractual terms, such as career specialisations or compensations, for each matched pair have to be determined simultaneously. In such settings, a mechanism should ideally satisfy two conditions:
The chosen outcome should be stable, i.e. there should not be a group of workers and firms that would like to veto the outcome in order to sign more preferred contracts among themselves.
The mechanism should be strategy-proof, i.e. each participant should always be best advised to reveal privately held information about her preferences over possible match partners and contractual terms truthfully.
In the more realistic case where firms can hire more than one worker, it has been known since Roth (1982) that no mechanism can satisfy properties (A) and (B). However, Hatfield and Milgrom (2005) have established that as long as each worker demands at most one employment contract and firms’ preferences over contracts are substitutable and size monotonic, there exist mechanisms that are stable and strategy-proof for the workers. Roughly speaking, substitutability requires that whenever the set of available contracts expands (in the superset sense), the set of rejected contracts also expands, and size monotonicity requires that whenever the set of available contracts expands (in the superset sense), the number of chosen contracts weakly increases. Interestingly. several recent studies have shown that the two key properties of stability and strategy-proofness can be achieved even when substitutability or size monotonicity are violated. See, among others, Sönmez and Switzer (2013) for a study of branch selection at the United States Military Academy, Kamada and Kojima (2013) for an investigation of regional caps in entry level labor markets, and Dimakopoulos and Heller (2015) for a study of the matching process for new lawyers in Germany. This poses important open questions for the literature on matching with contracts:
Question 1: What is the largest class of preferences for which a strategy-proof and stable centralised mechanism exists?
Question 2: What can we say about the class of stable and strategy-proof (for workers) mechanisms?
For the second sub-project, the aim was to analyse decentralised matching markets. So far, the literature on matching theory had almost exclusively focused on the case where matching can be organised via a fully centralised mechanism. However, full centralisation is often very difficult to achieve, if not impossible, and it is therefore important to develop a better understanding of decentralised markets. One highly relevant example is admission to public and private universities, which is decentralised in many countries, such as Germany and the United States. A key difference to a market organised via a centralised clearinghouse is that prospective students directly apply to universities they are interested in (instead of submitting a preference list to a clearinghouse) and admission offices enjoy complete autonomy in their decisions (instead of following the recommendations of a centralised algorithm to coordinate admission policies). Since aspiring students typically send more than one application, a university can never be completely sure that a given offer of admission will be accepted. As a response, universities typically overbook, that is, make more offers than they’d ultimately like to be accepted. There is ample anecdotal evidence that it is not uncommon for universities to regret admission decisions ex-post. One extreme example dates back to 2009, when the University of Bonn estimated that out of its admission offers for legal studies, only one tenth would end up accepted. Consequently it made 3500 admission offers to fill the 350 available places. Unfortunately, studying law at Bonn proved to be more popular than the administration thought that year and 590 students accepted their offers, leading to many administrative problems. In the other direction, universities in the United States often fill a significant fraction of their places from waitlists indicating that too few of their initial offers were accepted, while each year up to 18 000 places at German universities remain unfilled despite significant over-demand (see Hüber and Kübler, 2011). It is reasonable to assume that admission offices try to learn from such “mistakes”. Due to the limited time I had to work on the second sb-project, I focused on extending a model of decentralised matching market in which only universities learn and students always apply to all universities they are interested in. The main question that I tried to answer was the following.
Question 3: Under which conditions on universities’ learning behaviours will a decentralised admission market eventually reach a point after which each university is guaranteed to be very close to its enrolment target?
Finally, a natural extension of my work on the first subproject was to try to gain a better understanding of the conditions under which “good” mechanisms exist in generalised matching problems. This class of problems can be used to model, among other things, the formation of supply chains in complex production processes. An earlier research project, Hatfield et al. (2014), had already shown that a specific generalisation of the substitutability-condition for two-sided markets was crucial for ensuring that stable outcomes and competitive equilibria exist. The key questions that my co-authors and I have tried to answer during my work on the Marie-Curie project were the following.
Question 4: How are various approaches to defining “substitutability” that have been used in the literature on generalised matching problems related?
Question 5: Which changes in market structures “preserve” substitutability?
Question 6: What is the “right” stability concept for generalised matching problems?
Description of the main S & T results/foregrounds
For the first sub-project, I have worked together with John Hatfield and Scott Kominers. We have been able to identify identify three conditions that characterise the largest class of preferences for the firms for which a stable and strategy-proof mechanism exists.
Key to our characterisation is a partition of violations of substitutes and size monotonicity into those violations that are “relevant” and those that are not. This partition is based on the following notion of observability: A sequence of contracts with a given firm f is observable, if, at each point of the sequence, the worker who is supposed to propose the next contract in the sequence to f is not currently employed. Intuitively, the observability of a sequence of contracts is equivalent to requiring that there is some way of defining workers’ preferences so that a cumulative offer process actually produces that sequence of contracts. For example, suppose that some firm f always wants to employ some worker w, no matter which other contracts f has available, or which contract with f worker w has proposed. It might well be that the specific type of contract that f would like to sign with w depends on what other contracts f has available, so that in particular the preferences of f would not be substitutable. However, if we consider the worker-proposing cumulative offer process, this violation will never be observed since w will essentially dictate the terms of his employment with f by his first proposal to f.
Hence, we develop a weakening of the substitutability condition, observable substitutability: a firm’s preferences are observably substitutable if that firm never chooses any previously-rejected contract during any observable offer process. Similarly, observable size monotonicity requires that a firm, upon receiving a new offer, never newly rejects two (or more) contracts during an observable offer process. We show that, in contrast to the usual definitions of substitutability and size monotonicity, our concepts of observable substitutability and observable size monotonicity are necessary for the guaranteed existence of a stable and strategy-proof mechanism.
Unfortunately, and somewhat surprisingly, observable substitutability and observable size monotonicity are not sufficient for the existence of a stable and strategy-proof mechanism. In particular, we present an example in which the cumulative offer mechanism always proceeds as if all firms had substitutable and size monotonic choice functions but is still manipulable by workers. To complete our characterisation, our third condition requires that the choice function of every firm f is non-manipulatable, that is, if f is the only firm, the cumulative offer mechanism is strategy-proof for workers. While the necessity of such a condition is straightforward, it is far from obvious that such a condition will help us close the gap in our characterisation. Our final main result shows that the combination of observable substitutability, observable size monotonicity, and non-manipulatability is sufficient for the existence of a stable and strategy-proof mechanism. Put differently, observable substitutability and observable size monotonicity imply that whenever the cumulative offer mechanism can be manipulated, there is a firm f such that the cumulative offer mechanism can also be manipulated in an economy in which f is the only firm. Combining our results, we see that a stable and strategy-proof mechanism is guaranteed to exist if and only if firms’ preferences are observably substitutable, observably size monotonic, and non-manipulatable. This answers Question 1 that was posed above and closes a major gap in the theoretical literature on matching with contracts. We are currently working on finding a reinterpretation of the non-manipulatability condition as an extension of the observable substitutability condition.
Apart from characterising the conditions under which a stable and strategy-proof mechanism exists, it is also relevant, especially for practical purposes, to know more about the class of mechanisms that satisfy these properties. We show that when the preferences of every hospital are observably substitutable, any stable and strategy-proof mechanism is equivalent to a cumulative offer mechanism. Furthermore, the cumulative offer mechanism outcome is order-independent, i.e. the outcome does not depend on the order in which proposals are made. Hence, when a stable and strategy-proof mechanism is guaranteed to exist, it must be equivalent to the cumulative offer mechanism. This is an important extension of earlier uniqueness results in the literature and answers Question 2 that was posed above. We have also shown that our sufficient conditions for the existence of a stable and strategy-proof mechanism are strictly weaker than all previously known sufficient conditions. Currently, we are working on finding further practical applications of our results.
For the second sub-project, I focused on a model of “better reply”-learning by universities in which, in each period, each university chooses a number of offers that is guaranteed to be a weakly better reply to the choices that other universities made in the most recent period. Here, I am close to establishing the following result:
Consider a better reply dynamic for which there is an integer K such that, from any given period to the next, no university changes the number of offers it makes by more than K. Such a better reply dynamic must eventually reach a point after which no university ever misses its target enrolment by more than K students.
This is an important approximation result for better reply learning dynamics in the context of decentralised university admissions. In particular, if K is small, then a better reply dynamic in which per-period adjustments are limited by K is guaranteed to come very close to market-clearing. Hence, the result above provides a satisfactory answer to Question 3. I expect this result to be very useful for further work on learning in decentralised matching markets. A working paper summarising my findings will be made available to the general public soon.
Finally, I engaged in a joint research project with John Hatfield, Scott Kominers, Alexandru Nichifor, and Michael Ostrovsky. In the first part of this project, we have shown how the different definitions of substitutability are related to each other, while dispensing with some of the restrictions in the preceding literature. We consider agents who can simultaneously be buyers in some transactions and sellers in others, which allows us to embed the key substitutability concepts from the matching, auctions, and exchange economy literatures. Our main result shows that all the substitutability concepts are equivalent, thereby answering Question 4. We call preferences satisfying these conditions fully substitutable. We have also introduced a new class of fully substitutable preferences that models the preferences of intermediaries with production capacity. Finally, we proved that full substitutability is preserved under several economically important transformations: trade endowments and obligations, mergers, and limited liability. This answers Question 5. In the second part of our work on generalised matching models, we have established an equivalence between two key notions of stability. This result generalises classical results on the equivalence of the core and pairwise stability for two-sided matching markets. Our results hold for a very rich setting—trading networks with bilateral contracts. We allow agents to be buyers in some contracts and sellers in other contracts, and do not impose any restrictions on the network of possible trades. Our model subsumes settings with discrete and continuous prices, with quasilinear and non-quasilinear utility functions, and with and without indifferences in agents’ preferences. Our main result shows that if all agents’ preferences are fully substitutable and satisfy the Laws of Aggregate Supply and Demand, the notion of stability (under which all possible deviations by groups of agents need to be considered) is equivalent to chain stability, under which only deviations by agents along a chain of contracts need to be considered. This provides us with an answer to Question 6: Chain stability is the right stability concept for generalised matching problems since it can be checked much more easily than stability (since fewer possible deviations have to be checked) but is as robust as stability.
Potential impact, dissemination, and exploitation of results
The first sub-project opens the door for further practical applications of the matching with contracts framework. Furthermore, it provides an important foundation for the well known cumulative offer mechanism. The reason is that it is often hard to ensure that the preference restrictions developed in the theoretical literature are actually satisfied in real-life. However, if the only goal is to satisfy stability and strategy-proofness whenever these two properties can actually be achieved, nothing is lost by restricting attention to cumulative offer mechanisms: Whenever the cumulative offer mechanism fails to be stable or strategy-proof, we do not have to worry that there is another mechanism, which we have not discovered yet, that does a better job. Numerous dissemination activities of my research on this topic have been conducted and several more are planned for the near future. First, I have already presented my research at two highly regarded conferences: Match-Up 2015 in Glasgow, Scotland; The Society for Economic Design Conference 2015 in Istanbul, Turkey. The first of the just mentioned conferences is particularly interesting since it has a strong interdisciplinary focus and brings together researchers from the areas of Economics and Computer Science. Furthermore, I have already presented results of the first sub-project in a faculty seminar at the Université Paris Dauphine in March 2015. In 2015, I will also present my findings at the SAET Conference on Current Trends in Economics, which will take place in Cambridge, UK at the end of July. Apart from the dissemination of the research on the first sub-project to the scientific community, I see a huge opportunity to use our results to increase awareness for research on matching theory among practitioners.
The approximation result that I have discovered during my work on the second sub-project can be interpreted as saying that if the admissions environment, i.e. the distribution of students’ talents and preferences as well as universities’ evaluation criteria and target enrolments, is sufficiently stable, then a decentralised admission process may actually work quite well (in the sense that all universities will eventually be guaranteed to come “very close” to reaching their target enrolments). Of course, so far such an approximation result is in reach only for the case where universities use a specific form of learning rule (better-reply) and students always apply to all universities. The task of future research on this topic is to uncover similar approximation results for more general models of learning in decentralised matching markets. One possible avenue for the exploitation of the results would be to see whether such results can be helpful in identifying feasible interventions (apart from full centralisation) that can help improve the performance of real-life admission systems. A presentation of my research on the second sub-project is planned for the World Congress of the Econometric Society 2015 in Montreal, Canada.
The results on generalised matching models has at least two potential avenues of impact. Our first paper unifies and extends various approaches to defining “substitutability” that have been used in the areas of Economics, Computer Science, and Discrete Mathematics. My co-authors and I believe that one benefit of our results is that they have the potential to bring together researchers from various fields who work on topics related to generalised matching models and thereby potentially also enable interdisciplinary research. Incidentally, my co-author Scott Kominers has already presented the results of our first paper at the Match-Up 2015 conference in Glasgow, Scotland, and at the ACM-EC 2015 conference in Portland, Oregon, USA. The latter is one of the most highly regarded conferences at the intersection of Economics and Computer Science. Our research on “Full Substitutability in Trading Networks” will also be presented at the North American Winter Meeting of the Econometric Society 2016 in San Francisco, California, USA. Our second paper on generalised matching models established an equivalence between a general notion of stability and a version of the chain stability concept that was introduced in Ostrovsky (2008). Our result has the potential to be very useful for econometric work on generalised matching problems since it allows econometricians, who are interested in using cooperative solution concepts to estimate parameters of interest, to restrict attention to the easier-to-handle concept of chain stability. One reason to expect that our equivalence result will be useful for applied work is that similar equivalence results for much less general two-sided matching models are by now heavily used in econometric work. My co-author Alexandru Nichifor has already presented the results of our second paper at the “Workshop on Social and Information Networks” that was held in conjunction with the ACM-EC conference 2015. Our research on “Chain Stability in Trading Networks” will also be presented at the SAET Conference on Current Trends in Economics 2015.
Additional References
Dimakopoulos, P. D. and C.-P. Heller, 2014. Matching with waiting times: The German entry-level labour market for lawyers. Mimeo, Humboldt University of Berlin
Hatfield, J. and P. Milgrom, 2005. Matching with Contracts. American Economic Review 95, 913 - 935
Kamada, Y. and F. Kojima, 2015. Efficient matching unde distributional constraints: Theory and applications. American Economic Review 105, 67 - 99
Sönmez, T. and T. B. Switzer, 2013. Matching with (branch-of-choice) contracts at the United States Military Academy. Econometrica 81, 451 - 488
Ostrovsky, M., 2008. Stability in supply chain networks. American Economic Review 98, 897 - 923
The first sub-project has focused on the many-to-one matching with contracts framework of Hatfield and Milgrom (2005). This framework considers a two-sided market, consisting of firms and workers, in which each worker can sign at most one employment contract and firms can hire multiple workers. The main objective was to discover the largest class of preferences for the firms under which a centralised mechanism exists that (a) is guaranteed to produce a stable outcome, i.e. no group of workers and firms has an incentives to veto the outcome in order to sign new, more preferred, contracts among themselves, and (b) is strategy-proof for workers, i.e. each worker always finds it in her best interest to reveal her true ranking of employment contracts to the mechanism. In order to achieve this objective, I have worked in collaboration with John Hatfield and Scott Kominers. We have been able to identify three conditions that characterise the largest class of preferences for the firms for which a stable and strategy-proof mechanism exists. Furthermore, our results imply that whenever a stable and strategy-proof mechanism exists, it has to coincide with the well known cumulative offer mechanism. This last result provides us with an important foundation for the use of this mechanism. Our results answer important long-standing open questions in the literature and have the potential to open the door for further high-profile practical applications of matching theory.
For the second sub-project, the plan was to focus on decentralised matching markets in which participants try to learn from information about past outcomes. One key example is a decentralised market for admissions to universities. Here, universities typically receive applications from interested students and then have to decide on how many offers to make. Due to the shorter than initially expected project duration, I focused on an extension of a simpler model that was developed in an earlier working paper of mine. Here, in each period, students apply to all universities and universities simultaneously decide on how many offers to make. In my earlier paper on this model I had identified the conditions under which “better reply”-learning by universities is guaranteed to converge to market clearing, that is, a situation in which all universities exactly reach their target capacities. I am close to establishing the following result:
Consider a better reply dynamic for which there is an integer K such that, from any given period to the next, no university changes the number of offers it makes by more than K. Such a better reply dynamic must eventually reach a point after which no university ever misses its target enrolment by more than K students.
This is an important approximation result for better reply learning dynamics in the context of decentralised university admissions. In particular, if K is small, then a better reply dynamic in which per-period adjustments are limited by K is guaranteed to come very close to market-clearing. I expect this result to be very useful for further work on learning in decentralised matching markets.
As a further extension, I have engaged in joint research (with John Hatfield, Scott Kominers, Alexandru Nichifor, and Michael Ostrovsky) on generalised matching models. Among other things, these models are helpful for understanding the formation of supply chains in complex production processes. This line of research is a natural extension of my research on two-sided matching problems that was summarised in the first paragraph. Our research on generalised matching problems has (1) unified and extended various approaches to defining what “substitutability of preferences” means, (2) uncovered a new class of substitutable preferences that allows us to model intermediaries with production capacity, (3) showed that substitutability is preserved under economically important transformations such as trade endowments, mergers, and limited liability, and (4) showed that a suitably adapted chain stability concept is equivalent to the most widely used notion of stability.
Summary description
The aim of the first sub-project has been to characterise the conditions under which “good” centralised mechanisms exist for matching problems with contracts. One possible application is an entry-level labor market in which a matching of workers to firms and contractual terms, such as career specialisations or compensations, for each matched pair have to be determined simultaneously. In such settings, a mechanism should ideally satisfy two conditions:
The chosen outcome should be stable, i.e. there should not be a group of workers and firms that would like to veto the outcome in order to sign more preferred contracts among themselves.
The mechanism should be strategy-proof, i.e. each participant should always be best advised to reveal privately held information about her preferences over possible match partners and contractual terms truthfully.
In the more realistic case where firms can hire more than one worker, it has been known since Roth (1982) that no mechanism can satisfy properties (A) and (B). However, Hatfield and Milgrom (2005) have established that as long as each worker demands at most one employment contract and firms’ preferences over contracts are substitutable and size monotonic, there exist mechanisms that are stable and strategy-proof for the workers. Roughly speaking, substitutability requires that whenever the set of available contracts expands (in the superset sense), the set of rejected contracts also expands, and size monotonicity requires that whenever the set of available contracts expands (in the superset sense), the number of chosen contracts weakly increases. Interestingly. several recent studies have shown that the two key properties of stability and strategy-proofness can be achieved even when substitutability or size monotonicity are violated. See, among others, Sönmez and Switzer (2013) for a study of branch selection at the United States Military Academy, Kamada and Kojima (2013) for an investigation of regional caps in entry level labor markets, and Dimakopoulos and Heller (2015) for a study of the matching process for new lawyers in Germany. This poses important open questions for the literature on matching with contracts:
Question 1: What is the largest class of preferences for which a strategy-proof and stable centralised mechanism exists?
Question 2: What can we say about the class of stable and strategy-proof (for workers) mechanisms?
For the second sub-project, the aim was to analyse decentralised matching markets. So far, the literature on matching theory had almost exclusively focused on the case where matching can be organised via a fully centralised mechanism. However, full centralisation is often very difficult to achieve, if not impossible, and it is therefore important to develop a better understanding of decentralised markets. One highly relevant example is admission to public and private universities, which is decentralised in many countries, such as Germany and the United States. A key difference to a market organised via a centralised clearinghouse is that prospective students directly apply to universities they are interested in (instead of submitting a preference list to a clearinghouse) and admission offices enjoy complete autonomy in their decisions (instead of following the recommendations of a centralised algorithm to coordinate admission policies). Since aspiring students typically send more than one application, a university can never be completely sure that a given offer of admission will be accepted. As a response, universities typically overbook, that is, make more offers than they’d ultimately like to be accepted. There is ample anecdotal evidence that it is not uncommon for universities to regret admission decisions ex-post. One extreme example dates back to 2009, when the University of Bonn estimated that out of its admission offers for legal studies, only one tenth would end up accepted. Consequently it made 3500 admission offers to fill the 350 available places. Unfortunately, studying law at Bonn proved to be more popular than the administration thought that year and 590 students accepted their offers, leading to many administrative problems. In the other direction, universities in the United States often fill a significant fraction of their places from waitlists indicating that too few of their initial offers were accepted, while each year up to 18 000 places at German universities remain unfilled despite significant over-demand (see Hüber and Kübler, 2011). It is reasonable to assume that admission offices try to learn from such “mistakes”. Due to the limited time I had to work on the second sb-project, I focused on extending a model of decentralised matching market in which only universities learn and students always apply to all universities they are interested in. The main question that I tried to answer was the following.
Question 3: Under which conditions on universities’ learning behaviours will a decentralised admission market eventually reach a point after which each university is guaranteed to be very close to its enrolment target?
Finally, a natural extension of my work on the first subproject was to try to gain a better understanding of the conditions under which “good” mechanisms exist in generalised matching problems. This class of problems can be used to model, among other things, the formation of supply chains in complex production processes. An earlier research project, Hatfield et al. (2014), had already shown that a specific generalisation of the substitutability-condition for two-sided markets was crucial for ensuring that stable outcomes and competitive equilibria exist. The key questions that my co-authors and I have tried to answer during my work on the Marie-Curie project were the following.
Question 4: How are various approaches to defining “substitutability” that have been used in the literature on generalised matching problems related?
Question 5: Which changes in market structures “preserve” substitutability?
Question 6: What is the “right” stability concept for generalised matching problems?
Description of the main S & T results/foregrounds
For the first sub-project, I have worked together with John Hatfield and Scott Kominers. We have been able to identify identify three conditions that characterise the largest class of preferences for the firms for which a stable and strategy-proof mechanism exists.
Key to our characterisation is a partition of violations of substitutes and size monotonicity into those violations that are “relevant” and those that are not. This partition is based on the following notion of observability: A sequence of contracts with a given firm f is observable, if, at each point of the sequence, the worker who is supposed to propose the next contract in the sequence to f is not currently employed. Intuitively, the observability of a sequence of contracts is equivalent to requiring that there is some way of defining workers’ preferences so that a cumulative offer process actually produces that sequence of contracts. For example, suppose that some firm f always wants to employ some worker w, no matter which other contracts f has available, or which contract with f worker w has proposed. It might well be that the specific type of contract that f would like to sign with w depends on what other contracts f has available, so that in particular the preferences of f would not be substitutable. However, if we consider the worker-proposing cumulative offer process, this violation will never be observed since w will essentially dictate the terms of his employment with f by his first proposal to f.
Hence, we develop a weakening of the substitutability condition, observable substitutability: a firm’s preferences are observably substitutable if that firm never chooses any previously-rejected contract during any observable offer process. Similarly, observable size monotonicity requires that a firm, upon receiving a new offer, never newly rejects two (or more) contracts during an observable offer process. We show that, in contrast to the usual definitions of substitutability and size monotonicity, our concepts of observable substitutability and observable size monotonicity are necessary for the guaranteed existence of a stable and strategy-proof mechanism.
Unfortunately, and somewhat surprisingly, observable substitutability and observable size monotonicity are not sufficient for the existence of a stable and strategy-proof mechanism. In particular, we present an example in which the cumulative offer mechanism always proceeds as if all firms had substitutable and size monotonic choice functions but is still manipulable by workers. To complete our characterisation, our third condition requires that the choice function of every firm f is non-manipulatable, that is, if f is the only firm, the cumulative offer mechanism is strategy-proof for workers. While the necessity of such a condition is straightforward, it is far from obvious that such a condition will help us close the gap in our characterisation. Our final main result shows that the combination of observable substitutability, observable size monotonicity, and non-manipulatability is sufficient for the existence of a stable and strategy-proof mechanism. Put differently, observable substitutability and observable size monotonicity imply that whenever the cumulative offer mechanism can be manipulated, there is a firm f such that the cumulative offer mechanism can also be manipulated in an economy in which f is the only firm. Combining our results, we see that a stable and strategy-proof mechanism is guaranteed to exist if and only if firms’ preferences are observably substitutable, observably size monotonic, and non-manipulatable. This answers Question 1 that was posed above and closes a major gap in the theoretical literature on matching with contracts. We are currently working on finding a reinterpretation of the non-manipulatability condition as an extension of the observable substitutability condition.
Apart from characterising the conditions under which a stable and strategy-proof mechanism exists, it is also relevant, especially for practical purposes, to know more about the class of mechanisms that satisfy these properties. We show that when the preferences of every hospital are observably substitutable, any stable and strategy-proof mechanism is equivalent to a cumulative offer mechanism. Furthermore, the cumulative offer mechanism outcome is order-independent, i.e. the outcome does not depend on the order in which proposals are made. Hence, when a stable and strategy-proof mechanism is guaranteed to exist, it must be equivalent to the cumulative offer mechanism. This is an important extension of earlier uniqueness results in the literature and answers Question 2 that was posed above. We have also shown that our sufficient conditions for the existence of a stable and strategy-proof mechanism are strictly weaker than all previously known sufficient conditions. Currently, we are working on finding further practical applications of our results.
For the second sub-project, I focused on a model of “better reply”-learning by universities in which, in each period, each university chooses a number of offers that is guaranteed to be a weakly better reply to the choices that other universities made in the most recent period. Here, I am close to establishing the following result:
Consider a better reply dynamic for which there is an integer K such that, from any given period to the next, no university changes the number of offers it makes by more than K. Such a better reply dynamic must eventually reach a point after which no university ever misses its target enrolment by more than K students.
This is an important approximation result for better reply learning dynamics in the context of decentralised university admissions. In particular, if K is small, then a better reply dynamic in which per-period adjustments are limited by K is guaranteed to come very close to market-clearing. Hence, the result above provides a satisfactory answer to Question 3. I expect this result to be very useful for further work on learning in decentralised matching markets. A working paper summarising my findings will be made available to the general public soon.
Finally, I engaged in a joint research project with John Hatfield, Scott Kominers, Alexandru Nichifor, and Michael Ostrovsky. In the first part of this project, we have shown how the different definitions of substitutability are related to each other, while dispensing with some of the restrictions in the preceding literature. We consider agents who can simultaneously be buyers in some transactions and sellers in others, which allows us to embed the key substitutability concepts from the matching, auctions, and exchange economy literatures. Our main result shows that all the substitutability concepts are equivalent, thereby answering Question 4. We call preferences satisfying these conditions fully substitutable. We have also introduced a new class of fully substitutable preferences that models the preferences of intermediaries with production capacity. Finally, we proved that full substitutability is preserved under several economically important transformations: trade endowments and obligations, mergers, and limited liability. This answers Question 5. In the second part of our work on generalised matching models, we have established an equivalence between two key notions of stability. This result generalises classical results on the equivalence of the core and pairwise stability for two-sided matching markets. Our results hold for a very rich setting—trading networks with bilateral contracts. We allow agents to be buyers in some contracts and sellers in other contracts, and do not impose any restrictions on the network of possible trades. Our model subsumes settings with discrete and continuous prices, with quasilinear and non-quasilinear utility functions, and with and without indifferences in agents’ preferences. Our main result shows that if all agents’ preferences are fully substitutable and satisfy the Laws of Aggregate Supply and Demand, the notion of stability (under which all possible deviations by groups of agents need to be considered) is equivalent to chain stability, under which only deviations by agents along a chain of contracts need to be considered. This provides us with an answer to Question 6: Chain stability is the right stability concept for generalised matching problems since it can be checked much more easily than stability (since fewer possible deviations have to be checked) but is as robust as stability.
Potential impact, dissemination, and exploitation of results
The first sub-project opens the door for further practical applications of the matching with contracts framework. Furthermore, it provides an important foundation for the well known cumulative offer mechanism. The reason is that it is often hard to ensure that the preference restrictions developed in the theoretical literature are actually satisfied in real-life. However, if the only goal is to satisfy stability and strategy-proofness whenever these two properties can actually be achieved, nothing is lost by restricting attention to cumulative offer mechanisms: Whenever the cumulative offer mechanism fails to be stable or strategy-proof, we do not have to worry that there is another mechanism, which we have not discovered yet, that does a better job. Numerous dissemination activities of my research on this topic have been conducted and several more are planned for the near future. First, I have already presented my research at two highly regarded conferences: Match-Up 2015 in Glasgow, Scotland; The Society for Economic Design Conference 2015 in Istanbul, Turkey. The first of the just mentioned conferences is particularly interesting since it has a strong interdisciplinary focus and brings together researchers from the areas of Economics and Computer Science. Furthermore, I have already presented results of the first sub-project in a faculty seminar at the Université Paris Dauphine in March 2015. In 2015, I will also present my findings at the SAET Conference on Current Trends in Economics, which will take place in Cambridge, UK at the end of July. Apart from the dissemination of the research on the first sub-project to the scientific community, I see a huge opportunity to use our results to increase awareness for research on matching theory among practitioners.
The approximation result that I have discovered during my work on the second sub-project can be interpreted as saying that if the admissions environment, i.e. the distribution of students’ talents and preferences as well as universities’ evaluation criteria and target enrolments, is sufficiently stable, then a decentralised admission process may actually work quite well (in the sense that all universities will eventually be guaranteed to come “very close” to reaching their target enrolments). Of course, so far such an approximation result is in reach only for the case where universities use a specific form of learning rule (better-reply) and students always apply to all universities. The task of future research on this topic is to uncover similar approximation results for more general models of learning in decentralised matching markets. One possible avenue for the exploitation of the results would be to see whether such results can be helpful in identifying feasible interventions (apart from full centralisation) that can help improve the performance of real-life admission systems. A presentation of my research on the second sub-project is planned for the World Congress of the Econometric Society 2015 in Montreal, Canada.
The results on generalised matching models has at least two potential avenues of impact. Our first paper unifies and extends various approaches to defining “substitutability” that have been used in the areas of Economics, Computer Science, and Discrete Mathematics. My co-authors and I believe that one benefit of our results is that they have the potential to bring together researchers from various fields who work on topics related to generalised matching models and thereby potentially also enable interdisciplinary research. Incidentally, my co-author Scott Kominers has already presented the results of our first paper at the Match-Up 2015 conference in Glasgow, Scotland, and at the ACM-EC 2015 conference in Portland, Oregon, USA. The latter is one of the most highly regarded conferences at the intersection of Economics and Computer Science. Our research on “Full Substitutability in Trading Networks” will also be presented at the North American Winter Meeting of the Econometric Society 2016 in San Francisco, California, USA. Our second paper on generalised matching models established an equivalence between a general notion of stability and a version of the chain stability concept that was introduced in Ostrovsky (2008). Our result has the potential to be very useful for econometric work on generalised matching problems since it allows econometricians, who are interested in using cooperative solution concepts to estimate parameters of interest, to restrict attention to the easier-to-handle concept of chain stability. One reason to expect that our equivalence result will be useful for applied work is that similar equivalence results for much less general two-sided matching models are by now heavily used in econometric work. My co-author Alexandru Nichifor has already presented the results of our second paper at the “Workshop on Social and Information Networks” that was held in conjunction with the ACM-EC conference 2015. Our research on “Chain Stability in Trading Networks” will also be presented at the SAET Conference on Current Trends in Economics 2015.
Additional References
Dimakopoulos, P. D. and C.-P. Heller, 2014. Matching with waiting times: The German entry-level labour market for lawyers. Mimeo, Humboldt University of Berlin
Hatfield, J. and P. Milgrom, 2005. Matching with Contracts. American Economic Review 95, 913 - 935
Kamada, Y. and F. Kojima, 2015. Efficient matching unde distributional constraints: Theory and applications. American Economic Review 105, 67 - 99
Sönmez, T. and T. B. Switzer, 2013. Matching with (branch-of-choice) contracts at the United States Military Academy. Econometrica 81, 451 - 488
Ostrovsky, M., 2008. Stability in supply chain networks. American Economic Review 98, 897 - 923