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I am a Junior Fellow at the Harvard Society of Fellows, an Affiliate of the Harvard Center of Mathematical Sciences and Applications, an Associate of the Harvard Center for Research on Computation and Society, and a Research Economist at the National Bureau of Economic Research. In Spring 2016, I was also a Visiting Fellow at the Oxford Martin School. From 2011-2013, I was the inaugural Research Scholar at the Becker Friedman Institute for Research in Economics at the University of Chicago.

I completed my PhD in Business Economics at Harvard University in Spring 2011, with the dissertation "Matching Models of Markets." Before that, I graduated from Harvard summa cum laude and Phi Beta Kappa inMathematics (with a minor in Ethnomusicology) in Spring 2009, and completed my Masters of Arts at Harvard in Spring 2010.

### Journal Articles

1. Contract Design and Stability in Many-to-Many Matching

John William Hatfield and Scott Duke Kominers

We develop a model of many-to-many matching with contracts that subsumes as special cases many-to-many 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, we see that, in choosing contract language, market designers face a tradeoff between expressiveness and stability.

Citation:

Hatfield, John William, and Scott Duke Kominers. "Contract Design and Stability in Many-to-Many Matching." Games and Economic Behavior (forthcoming). View Details
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2. Matching with Slot-Specific Priorities: Theory

Scott Duke Kominers and Tayfun Sönmez

We introduce a two-sided, many-to-one matching with contracts model in which agents with unit demand match to branches that may have multiple slots available to accept contracts. Each slot has its own linear priority order over contracts; a branch chooses contracts by filling its slots sequentially, according to an order of precedence. We demonstrate that in these matching markets with slot-specific priorities, branches’ choice functions may not satisfy the substitutability conditions typically crucial for matching with contracts. Despite this complication, we are able to show that stable outcomes exist in the slot-specific priorities framework and can be found by a cumulative offer mechanism that is strategy proof and respects unambiguous improvements in priority.

Citation:

Kominers, Scott Duke, and Tayfun Sönmez. "Matching with Slot-Specific Priorities: Theory." Theoretical Economics 11, no. 2 (May 2016): 683–710. View Details
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3. Big Data and Big Cities: The Promises and Limitations of Improved Measures of Urban Life

Edward L. Glaeser, Scott Duke Kominers, Michael Luca and Nikhil Naik

New, "big" data sources allow measurement of city characteristics and outcome variables at higher frequencies and finer geographic scales than ever before. However, big data will not solve large urban social science questions on its own. Big data has the most value for the study of cities when it allows measurement of the previously opaque, or when it can be coupled with exogenous shocks to people or place. We describe a number of new urban data sources and illustrate how they can be used to improve the study and function of cities. We first show how Google Street View images can be used to predict income in New York City, suggesting that similar image data can be used to map wealth and poverty in previously unmeasured areas of the developing world. We then discuss how survey techniques can be improved to better measure willingness to pay for urban amenities. Finally, we explain how Internet data is being used to improve the quality of city services.

Keywords: Data and Data Sets; Urban Scope; City;

Citation:

Glaeser, Edward L., Scott Duke Kominers, Michael Luca, and Nikhil Naik. "Big Data and Big Cities: The Promises and Limitations of Improved Measures of Urban Life." Economic Inquiry (forthcoming). View Details
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4. Crowdsourcing City Government: Using Tournaments to Improve Inspection Accuracy

Edward Glaeser, Andrew Hillis, Scott Duke Kominers and Michael Luca

The proliferation of big data makes it possible to better target city services like hygiene inspections, but city governments rarely have the in-house talent needed for developing prediction algorithms. Cities could hire consultants, but a cheaper alternative is to crowdsource competence by making data public and offering a reward for the best algorithm. A simple model suggests that open tournaments dominate consulting contracts when cities can tolerate risk and when there is enough labor with low opportunity costs. We also report on an inexpensive Boston-based restaurant tournament, which yielded algorithms that proved reasonably accurate when tested "out-of-sample" on hygiene inspections.

Citation:

Glaeser, Edward, Andrew Hillis, Scott Duke Kominers, and Michael Luca. "Crowdsourcing City Government: Using Tournaments to Improve Inspection Accuracy." American Economic Review: Papers and Proceedings 106, no. 5 (May 2016): 114–118. View Details
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5. The Growing Problem of Patent Trolling

Lauren Cohen, Umit Gurun and Scott Duke Kominers

The last decade has seen a sharp rise in patent litigation in the U.S., with 2015 having one of the highest patent lawsuit counts on record. In theory, this could be a consequence of growth in the commercialization of technology and innovation—patent lawsuits increase as more firms turn to intellectual property (IP) protection to safeguard their competitive advantages. However, the majority of recent patent litigation is driven by nonpracticing entities (NPEs), firms that generate no products but instead amass patent portfolios for the sake of enforcing IP rights afforded therein. We discuss new, large-sample evidence adding to a growing literature suggesting that NPEs—in particular, large patent aggregators—on average act as "patent trolls," suing cash-rich firms seemingly irrespective of actual patent infringement. This has a negative impact on future innovation activity at targeted firms. These results suggest a need to check the power of NPEs through changes in U.S. IP policy, in particular to screen out trolling early in the litigation process.

Citation:

Cohen, Lauren, Umit Gurun, and Scott Duke Kominers. "The Growing Problem of Patent Trolling." Science 352, no. 6285 (April 29, 2016): 521–522. (Explanatory Video.) View Details
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6. To Groupon or Not to Groupon: The Profitability of Deep Discounts

Benjamin Edelman, Sonia Jaffe and Scott Duke Kominers

We examine the profitability and implications of online discount vouchers, a relatively new marketing tool that offers consumers large discounts when they prepay for participating firms' goods and services. Within a model of repeat experience good purchase, we examine two mechanisms by which a discount voucher service can benefit affiliated firms: price discrimination and advertising. For vouchers to provide successful price discrimination, the valuations of consumers who have access to vouchers must generally be lower than those of consumers who do not have access to vouchers. Offering vouchers tends to be more profitable for firms which are patient or relatively unknown, and for firms with low marginal costs. Extensions to our model accommodate the possibilities of multiple voucher purchases and firm price re-optimization. Despite the potential benefits of online discount vouchers to certain firms in certain circumstances, our analysis reveals the narrow conditions in which vouchers are likely to increase firm profits.

Citation:

Edelman, Benjamin, Sonia Jaffe, and Scott Duke Kominers. "To Groupon or Not to Groupon: The Profitability of Deep Discounts." Marketing Letters 27, no. 1 (March 2016): 39–53. (First circulated in June 2011. Featured in Working Knowledge: Is Groupon Good for Retailers? Excerpted in HBR Blogs: To Groupon or Not To Groupon: New Research on Voucher Profitability.) View Details
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7. Agglomerative Forces and Cluster Shapes

William R. Kerr and Scott Duke Kominers

We model spatial clusters of similar firms. Our model highlights how agglomerative forces lead to localized, individual connections among firms, while interaction costs generate a defined distance over which attraction forces operate. Overlapping firm interactions yield agglomeration clusters that are much larger than the underlying agglomerative forces themselves. Empirically, we demonstrate that our model's assumptions are present in the structure of technology and labor flows within Silicon Valley. Our model further identifies how the lengths over which agglomerative forces operate influence the shapes and sizes of industrial clusters; we confirm these predictions using variations across patent technology clusters.

Citation:

Kerr, William R., and Scott Duke Kominers. "Agglomerative Forces and Cluster Shapes." Review of Economics and Statistics 97, no. 4 (October 2015): 877–899. View Details
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8. Design and Implementation of a Privacy Preserving Electronic Health Record Linkage Tool in Chicago

Abel Kho, John Cashy, Kathryn Jackson, Adam Pah, Satyender Goel, Jorn Boehnke, John Eric Humphries, Scott Duke Kominers and et al.

Objective
To design and implement a tool that creates a secure, privacy preserving linkage of electronic health record (EHR) data across multiple sites in a large metropolitan area in the United States (Chicago, IL), for use in clinical research.
Methods
The authors developed and distributed a software application that performs standardized data cleaning, preprocessing, and hashing of patient identifiers to remove all protected health information. The application creates seeded hash code combinations of patient identifiers using a Health Insurance Portability and Accountability Act compliant SHA-512 algorithm that minimizes re-identification risk. The authors subsequently linked individual records using a central honest broker with an algorithm that assigns weights to hash combinations in order to generate high specificity matches.
Results
The software application successfully linked and de-duplicated 7 million records across 6 institutions, resulting in a cohort of 5 million unique records. Using a manually reconciled set of 11 292 patients as a gold standard, the software achieved a sensitivity of 96% and a specificity of 100%, with a majority of the missed matches accounted for by patients with both a missing social security number and last name change. Using 3 disease examples, it is demonstrated that the software can reduce duplication of patient records across sites by as much as 28%.
Conclusions
Software that standardizes the assignment of a unique seeded hash identifier merged through an agreed upon third-party honest broker can enable large-scale secure linkage of EHR data for epidemiologic and public health research. The software algorithm can improve future epidemiologic research by providing more comprehensive data given that patients may make use of multiple healthcare systems.

Citation:

Kho, Abel, John Cashy, Kathryn Jackson, Adam Pah, Satyender Goel, Jorn Boehnke, John Eric Humphries, Scott Duke Kominers, and et al. "Design and Implementation of a Privacy Preserving Electronic Health Record Linkage Tool in Chicago." Journal of the American Medical Informatics Association 22, no. 5 (September 2015): 1072–1080. View Details
9. Multilateral Matching

John William Hatfield and Scott Duke Kominers

We introduce a matching model in which agents engage in joint ventures via multilateral contracts. This approach allows us to consider production complementarities previously outside the scope of matching theory. We show analogues of the first and second welfare theorems and, when agents' utilities are concave in venture participation, show that competitive equilibria exist, correspond to stable outcomes, and yield core outcomes. Competitive equilibria exist in our setting even when externalities are present.

Citation:

Hatfield, John William, and Scott Duke Kominers. "Multilateral Matching." Journal of Economic Theory 156 (March 2015): 175–206. View Details
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10. Convergence of Position Auctions under Myopic Best-Response Dynamics

Matthew Cary, Aparna Das, Benjamin Edelman, Ioannis Giotis, Kurtis Heimerl, Anna Karlin, Scott Duke Kominers, Claire Mathieu and Michael Schwarz

We study the dynamics of multi-round position auctions, considering both the case of exogenous click-through rates and the case in which click-through rates are determined by an endogenous consumer search process. In both contexts, we demonstrate that the dynamic auctions converge to their associated static, envy-free equilibria. Furthermore, convergence is efficient, and the entry of low-quality advertisers does not slow convergence. Because our approach predominantly relies on assumptions common in the sponsored search literature, our results suggest that dynamic position auctions converge more generally.

Citation:

Cary, Matthew, Aparna Das, Benjamin Edelman, Ioannis Giotis, Kurtis Heimerl, Anna Karlin, Scott Duke Kominers, Claire Mathieu, and Michael Schwarz. "Convergence of Position Auctions under Myopic Best-Response Dynamics." ACM Transactions on Economics and Computation 2, no. 3 (July 2014): 1–20. View Details
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11. Investment Incentives in Labor Market Matching

John William Hatfield, Fuhito Kojima and Scott Kominers

We provide an illustration of how the design of labor market clearing mechanisms can affect incentives for human capital acquisition. Specifically, we extend the labor market matching model (with discrete transfers) of Kelso and Crawford (1982) to incorporate the possibility that agents may invest in human capital before matching. We show that in this setting, the worker-optimal stable matching mechanism incentivizes workers to make (nearly) efficient human capital investments. En route to our main result, we show that so long as the space of salaries is sufficiently rich, every stable outcome in the Kelso and Crawford (1982) setting is approximately efficient.

Keywords: Human Capital; Marketplace Matching;

Citation:

Hatfield, John William, Fuhito Kojima, and Scott Kominers. "Investment Incentives in Labor Market Matching." American Economic Review: Papers and Proceedings 104, no. 5 (May 2014): 436–441. View Details
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12. Delayed-Response Strategies in Repeated Games with Observation Lags

Drew Fudenberg, Yuhta Ishii and Scott Kominers

We extend the folk theorem of repeated games to two settings in which players' information about others' play arrives with stochastic lags. In our first model, signals are almost-perfect if and when they do arrive, that is, each player either observes an almost-perfect signal of period-t play with some lag or else never sees a signal of period-t play. The second model has the same lag structure, but the information structure corresponds to a lagged form of imperfect public monitoring, and players are allowed to communicate via cheap-talk messages at the end of each period. In each case, we construct equilibria in “delayed-response strategies,” which ensure that players wait long enough to respond to signals that with high probability all relevant signals are received before players respond. To do so, we extend past work on private monitoring to obtain folk theorems despite the small residual amount of private information.

Citation:

Fudenberg, Drew, Yuhta Ishii, and Scott Kominers. "Delayed-Response Strategies in Repeated Games with Observation Lags." Journal of Economic Theory 150 (March 2014): 487–514. View Details
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13. Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

Daniel M. Kane and Scott Duke Kominers

For relatively prime positive integers and r, we consider the least common multiple L_n:=\mathop{\textrm{lcm}}(u_0,u_1,\dots, u_n) of the finite arithmetic progression \{u_k:=u_0+kr\}_{k=0}^n. We derive new lower bounds on L_n that improve upon those obtained previously when either u_0 or n is large. When r is prime, our best bound is sharp up to a factor of n+1 for u_0 properly chosen, and is also nearly sharp as n\to\infty.

Keywords: Mathematical Methods;

Citation:

Kane, Daniel M., and Scott Duke Kominers. "Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions." Canadian Mathematical Bulletin 57, no. 3 (September 2014): 551–561. View Details
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14. Stability and Competitive Equilibrium in Trading Networks

John William Hatfield, Scott Kominers, Alexandru Nichifor, Michael Ostrovsky and Alexander Westkamp

We introduce a model in which agents in a network can trade via bilateral contracts. We find that when continuous transfers are allowed and utilities are quasi-linear, the full substitutability of preferences is sufficient to guarantee the existence of stable outcomes for any underlying network structure. Furthermore, the set of stable outcomes is essentially equivalent to the set of competitive equilibria, and all stable outcomes are in the core and are efficient. By contrast, for any domain of preferences strictly larger than that of full substitutability, the existence of stable outcomes and competitive equilibria cannot be guaranteed.

Keywords: Balance and Stability; Markets;

Citation:

Hatfield, John William, Scott Kominers, Alexandru Nichifor, Michael Ostrovsky, and Alexander Westkamp. "Stability and Competitive Equilibrium in Trading Networks." Journal of Political Economy 121, no. 5 (October 2013): 966–1001. View Details
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15. On Derivatives Markets and Social Welfare: A Theory of Empty Voting and Hidden Ownership

Jordan M. Barry, John William Hatfield and Scott Duke Kominers

In the past twenty-five years, derivatives markets have grown exponentially. Large, modern derivatives markets increasingly enable investors to hold economic interests in corporations without owning voting rights, and vice versa. This leads to both empty voters—investors whose voting rights in a corporation exceed their economic interests— and hidden owners—investors whose economic interests exceed their voting rights.
We present formal analysis that shows how, when financial markets are opaque, empty voting and hidden ownership can render financial markets unpredictable, unstable, and inefficient. By contrast, we show that when financial markets are transparent, empty voting and hidden ownership have dramatically different effects: they follow predictable patterns, encourage stable outcomes, and promote efficiency. Our analysis lends insight into the operation of securities markets in general and derivatives markets in particular. It also provides a new justification for a robust mandatory disclosure regime and facilitates analysis of proposed substantive securities regulations.

Keywords: Voting; Corporate Disclosure; Financial Markets; Ownership;

Citation:

Barry, Jordan M., John William Hatfield, and Scott Duke Kominers. "On Derivatives Markets and Social Welfare: A Theory of Empty Voting and Hidden Ownership." Virginia Law Review 99, no. 6 (October 2013): 1103–1168. View Details
16. Vacancies in Supply Chain Networks

John William Hatfield and Scott Duke Kominers

We use the supply chain matching framework to study the effects of firm exit. We show that the exit of an initial supplier or end consumer has monotonic effects on the welfare of initial suppliers and end consumers but may simultaneously have positive and negative effects on intermediaries. Furthermore, we demonstrate that there are no clear comparative statics for the effects of intermediary exit on the welfare of other firms; most surprisingly, intermediary exit may diminish the welfare of other firms at the same level of the supply chain.

Citation:

Hatfield, John William, and Scott Duke Kominers. "Vacancies in Supply Chain Networks." Economics Letters 119, no. 3 (June 2013): 354–357. View Details
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17. Minimal S-Universality Criteria May Vary in Size

Noam D. Elkies, Daniel M. Kane and Scott Duke Kominers

In this note, we give simple examples of sets $\mathcal{S}$ of quadratic forms that have minimal $\mathcal{S}$-universality criteria of multiple cardinalities. This answers a question of Kim, Kim, and Oh [KKO05] in the negative.

Citation:

Elkies, Noam D., Daniel M. Kane, and Scott Duke Kominers. "Minimal S-Universality Criteria May Vary in Size." Journal de Théorie des Nombres de Bordeaux 25, no. 3 (2013): 557–563. View Details
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18. Holdout in the Assembly of Complements: A Problem for Market Design

Scott Duke Kominers and E. Glen Weyl

Holdout problems prevent private (voluntary and self-financing) assembly of complementary goods—such as land or dispersed spectrum—from many self-interested sellers. While mechanisms that fully respect sellers' property rights cannot alleviate these holdout problems, traditional solutions, such as the use of coercive government powers of "eminent domain" to expropriate property, can encourage wasteful and unfair assemblies. We discuss the problems holdout creates for the efficient operation of markets and how previous approaches have used regulated coercion to address these challenges. We then investigate when encouraging competition can partially or fully substitute for coercion, focusing particularly on questions of spectrum allocation.

Keywords: Governance; Market Design; Property;

Citation:

Kominers, Scott Duke, and E. Glen Weyl. "Holdout in the Assembly of Complements: A Problem for Market Design." American Economic Review: Papers and Proceedings 102, no. 3 (May 2012): 360–365. View Details
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19. On the Correspondence of Contracts to Salaries in (Many-to-Many) Matching

Scott Duke Kominers

In this note, I extend the work of Echenique (2012) to show that a model of many-to-many matching with contracts may be embedded into a model of many-to-many matching with wage bargaining whenever (1) all agentsʼ preferences are substitutable and (2) the matching with contracts model is unitary, in the sense that every contractual relationship between a given firm–worker pair is specified in a single contract. Conversely, I show that unitarity is essentially necessary for the embedding result.

Citation:

Kominers, Scott Duke. "On the Correspondence of Contracts to Salaries in (Many-to-Many) Matching." Games and Economic Behavior 75, no. 2 (July 2012): 984–989. View Details
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20. Testing Substitutability

John William Hatfield, Nicole Immorlica and Scott Duke Kominers

We provide an algorithm for testing the substitutability of a length-N preference relation over a set of contracts X in time O(|X|3⋅N3). Access to the preference relation is essential for this result: We show that a substitutability-testing algorithm with access only to an agentʼs choice function must make an expected number of queries exponential in |X|. An analogous result obtains when the agentʼs preferences are quasilinear in a numeraire and the algorithm only has access to the agentʼs underlying valuation function.

Citation:

Hatfield, John William, Nicole Immorlica, and Scott Duke Kominers. "Testing Substitutability." Games and Economic Behavior 75, no. 2 (July 2012): 639–645. View Details
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21. Discrete Choice Cannot Generate Demand That Is Additively Separable in Own Price

Sonia Jaffe and Scott Duke Kominers

We show that in a unit demand discrete choice framework with at least three goods, demand cannot be additively separable in own price. This result sharpens the analogous result of Jaffe and Weyl (2010) in the case of linear demand and has implications for testing of the discrete choice assumption, out-of-sample prediction, and welfare analysis.

Citation:

Jaffe, Sonia, and Scott Duke Kominers. "Discrete Choice Cannot Generate Demand That Is Additively Separable in Own Price." Economics Letters 116, no. 1 (July 2012): 129–132. View Details
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22. Matching in Networks with Bilateral Contracts

John William Hatfield and Scott Duke Kominers

We introduce a model in which firms trade goods via bilateral contracts which specify a buyer, a seller, and the terms of the exchange. This setting subsumes (many-to-many) matching with contracts, as well as supply chain matching. When firms' relationships do not exhibit a supply chain structure, stable allocations need not exist. By contrast, in the presence of supply chain structure, a natural substitutability condition characterizes the maximal domain of firm preferences for which stable allocations are guaranteed to exist. Furthermore, the classical lattice structure, rural hospitals theorem, and one-sided strategy-proofness results all generalize to this setting. (JEL C78, D85, D86, L14)

Citation:

Hatfield, John William, and Scott Duke Kominers. "Matching in Networks with Bilateral Contracts." American Economic Journal: Microeconomics 4, no. 1 (February 2012): 176–208. View Details
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23. Matching with Preferences over Colleagues Solves Classical Matching

Scott Duke Kominers

In this note, we demonstrate that the problem of "many-to-one matching with (strict) preferences over colleagues" is actually more difficult than the classical many-to-one matching problem, "matching without preferences over colleagues." We give an explicit reduction of any problem of the latter type to a problem of the former type. This construction leads to the first algorithm which finds all stable matchings in the setting of "matching without preferences over colleagues," for any set of preferences. Our construction directly extends to generalized matching settings.

Citation:

Kominers, Scott Duke. "Matching with Preferences over Colleagues Solves Classical Matching." Games and Economic Behavior 68, no. 2 (March 2010): 773–780. View Details
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24. On the Classification of Type II Codes of Length 24

Noam D. Elkies and Scott Duke Kominers

We give a new, purely coding-theoretic proof of Koch's criterion on the tetrad systems of Type II codes of length 24 using the theory of harmonic weight enumerators. This approach is inspired by Venkov's approach to the classification of the root systems of Type II lattices in R^{24}, and gives a new instance of the analogy between lattices and codes.

Keywords: Mathematical Methods;

Citation:

Elkies, Noam D., and Scott Duke Kominers. "On the Classification of Type II Codes of Length 24." SIAM Journal on Discrete Mathematics 23, no. 4 (2010). View Details
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25. Further Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

Shaofang Hong and Scott Duke Kominers

For relatively prime positive integers u_0 and r, we consider the arithmetic progression {u_k := u_0+k*r} (0 <= k <= n). Define L_n := lcm{u_0,u_1,...,u_n} and let a >= 2 be any integer. In this paper, we show that, for integers alpha,r >= a and n >= 2*alpha*r, we have L_n >= u_0*r^{alpha+a-2}*(r+1)^n. In particular, letting a = 2 yields an improvement to the best previous lower bound on L_n (obtained by Hong and Yang) for all but three choices of alpha,r >= 2.

Keywords: Mathematical Methods;

Citation:

Hong, Shaofang, and Scott Duke Kominers. "Further Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions." Proceedings of the American Mathematical Society 138, no. 3 (March 2010): 809–813. View Details
26. Improved Bounds on the Sizes of S.P Numbers

Paul Myer Kominers and Scott Duke Kominers

A number which is S.P in base r is a positive integer which is equal to the sum of its base-r digits multiplied by the product of its base-r digits. These numbers have been studied extensively in The Mathematical Gazette. Recently, Shah Ali obtained the first effective bound on the sizes of S.P numbers. Modifying Shah Ali's method, we obtain an improved bound on the number of digits in a base-r S.P number. Our bound is the first sharp bound found for the case r=2.

Keywords: Mathematical Methods;

Citation:

Kominers, Paul Myer, and Scott Duke Kominers. "Improved Bounds on the Sizes of S.P Numbers." Mathematical Gazette 94, no. 529 (March 2010): 127–129. View Details
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27. Refined Configuration Results for Extremal Type II Lattices of Ranks 40 and 80

Noam D. Elkies and Scott Duke Kominers

We show that, if L is an extremal Type II lattice of rank 40 or 80, then L is generated by its vectors of norm min(L)+2. This sharpens earlier results of Ozeki, and the second author and Abel, which showed that such lattices L are generated by their vectors of norms min(L) and min(L)+2.

Keywords: Mathematical Methods;

Citation:

Elkies, Noam D., and Scott Duke Kominers. "Refined Configuration Results for Extremal Type II Lattices of Ranks 40 and 80." Proceedings of the American Mathematical Society 138, no. 1 (January 2010): 105–108. View Details
28. Configurations of Extremal Even Unimodular Lattices

Scott Duke Kominers

We extend the results of Ozeki on the configurations of extremal even unimodular lattices. Specifically, we show that if L is such a lattice of rank 56, 72, or 96, then L is generated by its minimal-norm vectors.

Keywords: Mathematical Methods;

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29. On Universal Binary Hermitian Forms

Scott Duke Kominers

Earnest and Khosravani, Iwabuchi, and Kim and Park recently gave a complete classification of the universal binary Hermitian forms. We give a unified proof of the universalities of these Hermitian forms, relying upon Ramanujan's list of universal quadratic forms and the Bhargava-Hanke 290-Theorem. Our methods bypass the ad hoc arguments required in the original classification.

Keywords: Mathematical Methods;

Citation:

Kominers, Scott Duke. "On Universal Binary Hermitian Forms." A02. INTEGERS: Electronic Journal of Combinatorial Number Theory 9 (2009): 9–15. View Details
30. Leonard Bernstein's Doodles: Reading Outside the Lines at the Library of Congress

Scott Duke Kominers

Citation:

Kominers, Scott Duke. "Leonard Bernstein's Doodles: Reading Outside the Lines at the Library of Congress." Special Issue on Leonard Bernstein in Boston. Journal of the Society for American Music 3, no. 1 (February 2009): 26–33. (As an appendix to "Leonard Bernstein's Jewish Boston: Cross-Disciplinary Research in the Classroom" by Carol J. Oja and Kay Kaufman Shelemay.) View Details
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31. Configurations of Rank-40r Extremal Even Unimodular Lattices (r=1,2,3)

Scott Duke Kominers and Zachary Abel

We show that if L is an extremal even unimodular lattice of rank 40r with r=1,2,3 then L is generated by its vectors of norms 4r and 4r+2. Our result is an extension of Ozeki's result for the case r=1.

Keywords: Mathematical Methods;

Citation:

Kominers, Scott Duke, and Zachary Abel. "Configurations of Rank-40r Extremal Even Unimodular Lattices (r=1,2,3)." Journal de Théorie des Nombres de Bordeaux 20, no. 2 (2008): 365–371. View Details

### Book Chapters

1. Empirical Evidence on the Behavior and Impact of Patent Trolls: A Survey

Lauren Cohen, Umit Gurun and Scott Duke Kominers

We survey the empirical literature on non-practicing entity (NPE) litigation behavior and its consequences. We document both aggregate trends and cross-sectional differences amongst various types of NPEs. Survey evidence illustrates a number of ways in which NPEs can potentially act opportunistically and indicates at least some instances and consequences of observed NPE opportunism. Large-sample empirical work has recently begun corroborating and amplifying the findings from survey evidence. NPEs on average behave as "patent trolls." Indeed, NPEs hold and frequently litigate patents that are likely to be at least partially invalidated; moreover, NPEs target cash irrespective of its relation to alleged infringement. Cash-targeting is neither the main driver of practicing entity (PE) intellectual property (IP) litigation, nor of non-IP litigation against publicly traded firms. The empirical evidence suggests, however, that not all NPEs exhibit trolling behavior—the cash-targeting observed in the data is primarily the behavior of large patent aggregators, rather than small inventors. NPE patent trolling has a real negative impact on targeted firms, without any increase in innovation, technology transfer, or other counterbalancing benefits measured thus far.

Citation:

Cohen, Lauren, Umit Gurun, and Scott Duke Kominers. "Empirical Evidence on the Behavior and Impact of Patent Trolls: A Survey." In Patent Assertion Entities and Competition Policy, edited by D. Daniel Sokol. Cambridge University Press, forthcoming. View Details
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2. Weighted Generating Functions for Type II Lattices and Codes

Noam D. Elkies and Scott Duke Kominers

We give a new structural development of harmonic polynomials on Hamming space, and harmonic weight enumerators of binary linear codes, that parallels one approach to harmonic polynomials on Euclidean space and weighted theta functions of Euclidean lattices. Namely, we use the finite-dimensional representation theory of sl2sl2 to derive a decomposition theorem for the spaces of discrete homogeneous polynomials in terms of the spaces of discrete harmonic polynomials, and prove a generalized MacWilliams identity for harmonic weight enumerators. We then present several applications of harmonic weight enumerators, corresponding to some uses of weighted theta functions: an equivalent characterization of t-designs, the Assmus–Mattson Theorem in the case of extremal Type II codes, and configuration results for extremal Type II codes of lengths 8, 24, 32, 48, 56, 72, and 96.

Keywords: Mathematical Methods;

Citation:

Elkies, Noam D., and Scott Duke Kominers. "Weighted Generating Functions for Type II Lattices and Codes." In Quadratic and Higher Degree Forms. Vol. 31, edited by Alladi Krishnaswami, Manjul Bhargava, David Savitt, and Pham Huu Tiep, 63–108. Developments in Mathematics. Springer, 2013. View Details

### Working Papers

1. Strategy-Proofness, Investment Efficiency, and Marginal Returns: An Equivalence

John William Hatfield, Fuhito Kojima and Scott Duke Kominers

We show that a mechanism induces an agent to make efficient ex ante investment choices if and only if it rewards that agent with his marginal surplus; additionally, for an ex post efficient mechanism, these properties are equivalent to strategy-proofness for the agent. Our results extend to settings with uncertainty; moreover, they have analogs for mechanisms that are only approximately efficient and approximately incentive compatible. Among other applications, our results imply both that under the worker-optimal stable mechanism, workers are incentivized to make efficient human capital investments before entering the labor market, and that second-price auctions induce efficient bidder participation.

Citation:

Hatfield, John William, Fuhito Kojima, and Scott Duke Kominers. "Strategy-Proofness, Investment Efficiency, and Marginal Returns: An Equivalence." Working Paper, January 2015. View Details
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2. Hidden Substitutes

John William Hatfield and Scott Duke Kominers

In this paper, we show that preferences exhibiting some forms of complementarity in fact have an underlying substitutable structure. Specifically, we show that some preferences that are not substitutable in the setting of many-to-one matching with contracts become substitutable when an employer is allowed to sign multiple contracts with an individual worker. These substitutably completable preferences guarantee the existence of stable outcomes, even though stable outcomes are not guaranteed, in general, when complementarities are present. Our results imply the existence of a stable, strategy-proof mechanism for allocating workers with specialized skillsets; moreover, our results give new insight into the existing applications of matching with contracts to cadet–branch matching and the design of affirmative action programs.

Citation:

Hatfield, John William, and Scott Duke Kominers. "Hidden Substitutes." Working Paper, September 2014. View Details
3. Shielded Innovation

Lauren Cohen, Umit Gurun and Scott Duke Kominers

We show that increased litigation risk has driven innovators to shield themselves by shifting innovation out of industry and into universities. We show both theoretically and empirically that litigation by Non-Practicing Entities (NPEs) pushes innovation to spaces with reduced litigation threat. Innovation has shifted into universities (and away from public and private firms) in exactly those industries with the most aggressive NPE litigation, precisely following extensive NPE litigation. The extent of innovation shielding is large and significant. An increase of 100 NPE lawsuits in an industry shifts up the university share of innovation by roughly 70% in subsequent years (t=5.34).

Keywords: Patent trolls; patents; NPEs; PAEs; innovation; Patents;

Citation:

Cohen, Lauren, Umit Gurun, and Scott Duke Kominers. "Shielded Innovation." Working Paper, June 2015. View Details
4. Patent Trolls: Evidence from Targeted Firms

Lauren Cohen, Umit G. Gurun and Scott Duke Kominers

We develop a theoretical model of, and provide the first large-sample evidence on, the behavior and impact of non-practicing entities (NPEs) in the intellectual property space. Our model shows that NPE litigation can reduce infringement and support small inventors. However, the model also shows that as NPEs become effective at bringing frivolous lawsuits, the resulting defense costs inefficiently crowd out firms that, absent NPEs, would produce welfare-enhancing innovations without engaging in infringement. Our empirical analysis shows that on average, NPEs behave as opportunistic patent trolls. NPEs sue cash-rich firms—a one standard deviation increase in cash holdings roughly doubles a firm's chance of being targeted by NPE litigation. We find moreover that NPEs target cash unrelated to the alleged infringement at essentially the same frequency as they target cash related to the alleged infringement. By contrast, cash is neither a key driver of intellectual property lawsuits by practicing entities (e.g., IBM and Intel), nor of any other type of litigation against firms. We find further suggestive evidence of NPE opportunism, such as forum shopping and targeting of firms, that has reduced the ability of firms to defend themselves against litigation. We find that NPE litigation has a real negative impact on innovation at targeted firms: firms substantially reduce their innovative activity after settling with NPEs (or losing to them in court). Moreover, we neither find any markers of significant NPE pass-through to end innovators, nor of a positive impact of NPEs on innovation in the industries in which they are most prevalent.

Citation:

Cohen, Lauren, Umit G. Gurun, and Scott Duke Kominers. "Patent Trolls: Evidence from Targeted Firms." Harvard Business School Working Paper, No. 15-002, July 2014. (Revised August 2016.) View Details
5. Stability, Strategy-Proofness, and Cumulative Offer Mechanisms

John William Hatfield, Scott Duke Kominers and Alexander Westkamp

We consider the setting of many-to-one matching with contracts, where firms may demand multiple contracts but each worker desires at most one contract. We introduce three novel conditions—observable substitutability, observable size monotonicity, and non-manipulatability—and show that when these conditions are satisfied, the cumulative offer mechanism is the unique mechanism that is stable and strategy-proof (for workers). Moreover, when the choice function of some firm fails any of our three conditions, one can construct unit-demand choice functions for the other firms such that no stable and strategy-proof mechanism exists. In the final part of the paper, we characterize the class of choice functions for which the cumulative offer mechanism is guaranteed to yield a stable outcome.

Citation:

Hatfield, John William, Scott Duke Kominers, and Alexander Westkamp. "Stability, Strategy-Proofness, and Cumulative Offer Mechanisms." Working Paper, July 2015. View Details
6. Full Substitutability

John William Hatfield, Scott Duke Kominers, Alexandru Nichifor, Michael Ostrovsky and Alexander Westkamp

Various forms of substitutability are essential for establishing the existence of equilibria and other useful properties in diverse settings such as matching, auctions, and exchange economies with indivisible goods. We extend earlier models' canonical definitions of substitutability to settings in which an agent can be both a buyer in some transactions and a seller in others, and show that all these definitions are equivalent. We introduce a new class of substitutable preferences that allows us to model intermediaries with production capacity. We then prove that substitutability is preserved under economically important transformations such as trade endowments, mergers, and limited liability. We also show that substitutability corresponds to submodularity of the indirect utility function, the single improvement property, and a no complementarities condition. Finally, we show that substitutability implies the monotonicity conditions known as the Laws of Aggregate Supply and Demand.

Keywords: Market Design; Balance and Stability;

Citation:

Hatfield, John William, Scott Duke Kominers, Alexandru Nichifor, Michael Ostrovsky, and Alexander Westkamp. "Full Substitutability." Working Paper, May 2015. View Details
7. Chain Stability in Trading Networks

John William Hatfield, Scott Duke Kominers, Alexandru Nichifor, Michael Ostrovsky and Alexander Westkamp

We show that in general trading networks with bilateral contracts, a suitably adapted chain stability concept (Ostrovsky, 2008) is equivalent to stability (Hatfield and Kominers, 2012; Hatfield et al., 2013) if all agents' preferences are fully substitutable and satisfy the Laws of Aggregate Supply and Demand. Furthermore, in the special case of trading networks with transferable utility, an outcome is consistent with competitive equilibrium if and only if it is not blocked by any chain of contracts.

Keywords: Contracts; Market Design; Balance and Stability;

Citation:

Hatfield, John William, Scott Duke Kominers, Alexandru Nichifor, Michael Ostrovsky, and Alexander Westkamp. "Chain Stability in Trading Networks." Working Paper, April 2015. View Details
8. Costly Concessions: An Empirical Framework for Matching with Imperfectly Transferable Utility

Alfred Galichon, Scott Duke Kominers and Simon Weber

We introduce an empirical framework for models of matching with imperfectly transferable utility and unobserved heterogeneity in tastes. Our framework allows us to characterize matching equilibrium in a flexible way that includes as special cases the classic fully- and non-transferable utility models, collective models, and settings with taxes on transfers, deadweight losses, and risk aversion. We allow for the introduction of a very general class of unobserved heterogeneity on agents' preferences. Under minimal assumptions, we show existence and uniqueness of equilibrium. We provide two algorithms to compute the equilibria in our model. The first algorithm operates under any structure of heterogeneity in preferences. The second algorithm is more efficient, but applies only in the case when random utilities are logit. We show that the log-likelihood of the model has a particularly simple expression and we compute its derivatives. As an application, we build a model of marriage with two-sided preferences over the partner type and private consumption. We estimate our model using the 2013 "Living Costs and Food Survey" database.

Citation:

Galichon, Alfred, Scott Duke Kominers, and Simon Weber. "Costly Concessions: An Empirical Framework for Matching with Imperfectly Transferable Utility." Working Paper, January 2016. View Details
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9. Paying (for) Attention: The Impact of Information Processing Costs on Bayesian Inference

Scott Duke Kominers, Xiaosheng Mu and Alexander Peysakhovich

Human information processing is often modeled as costless Bayesian inference. However, research in psychology shows that attention is a computationally costly and potentially limited resource. We study a Bayesian individual for whom computing posterior beliefs is costly. Such an agent faces a tradeoff between economizing on attention costs and having more accurate beliefs. We show that even small processing costs can lead to significant departures from the standard costless processing model. There exist situations where beliefs can cycle persistently and never converge. In addition, when updating is costly, agents are more sensitive to signals about rare events than to signals about common events. Thus, these individuals can permanently overestimate the likelihood of rare events (e.g., the probability of a plane crash). There is a commonly held assumption in economics that individuals will converge to correct beliefs/optimal behavior given sufficient experience. Our results contribute to a growing literature in psychology, neuroscience, and behavioral economics suggesting that this assumption is both theoretically and empirically fragile.

Keywords: Behavior; Cognition and Thinking; Economics;

Citation:

Kominers, Scott Duke, Xiaosheng Mu, and Alexander Peysakhovich. "Paying (for) Attention: The Impact of Information Processing Costs on Bayesian Inference." Working Paper, February 2016. View Details
10. A Theory of Intergenerational Mobility

Gary Becker, Scott Duke Kominers, Kevin Murphy and Jorg L. Spenkuch

We develop a model of intergenerational resource transmission that emphasizes the link between cross-sectional inequality and intergenerational mobility. By drawing on first principles of human capital theory, we derive several novel results. In particular, we show that, even in a world with perfect capital markets and without differences in innate ability, wealthy parents invest, on average, more in their offspring than poorer ones. As a result, persistence of economic status is higher at the top of the income distribution than in the middle. Successive generations of the same family may even cease to regress towards the mean. Moreover, we demonstrate that government interventions intended to ameliorate inequality may in fact lower intergenerational mobility—even when they do not directly favor the rich. Lastly, we consider how mobility is affected by changes in the marketplace.

Citation:

Becker, Gary, Scott Duke Kominers, Kevin Murphy, and Jorg L. Spenkuch. "A Theory of Intergenerational Mobility." Working Paper, August 2015. View Details
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11. Do People Shape Cities, or Do Cities Shape People? The Co-evolution of Physical, Social, and Economic Change in Five Major U.S. Cities

Nikhil Naik, Scott Duke Kominers, Ramesh Raskar, Edward L. Glaeser and Cesar Hidalgo

Urban change involves transformations in the physical appearance and the social composition of neighborhoods. Yet, the relationship between the physical and social components of urban change is not well understood due to the lack of comprehensive measures of neighborhood appearance. Here, we introduce a computer vision method to quantify change in physical appearance of streetscapes and generate a dataset of physical change for five large American cities. We combine this dataset with socioeconomic indicators to explore whether demographic and economic changes precede, follow, or co-occur with changes in physical appearance. We find that the strongest predictors of improvement in a neighborhood’s physical appearance are population density and share of college-educated adults. Other socioeconomic characteristics, like median income, share of vacant homes, and monthly rent, do not predict improvement in physical appearance. We also find that neighborhood appearances converge to the initial appearances of bordering areas, supporting the Burgess “invasion” theory. In addition, physical appearance is more likely to improve in neighborhoods proximal to the central business district. Finally, we find modest support for “tipping” and “filtering” theories of urban change.

Keywords: Demographics; Urban Development; Change; Economics;

Citation:

Naik, Nikhil, Scott Duke Kominers, Ramesh Raskar, Edward L. Glaeser, and Cesar Hidalgo. "Do People Shape Cities, or Do Cities Shape People? The Co-evolution of Physical, Social, and Economic Change in Five Major U.S. Cities." Working Paper. View Details
12. Configurations of Extremal Type II Codes

Noam D. Elkies and Scott Duke Kominers

We prove configuration results for extremal Type II codes, analogous to the configuration results of Ozeki and of the second author for extremal Type II lattices. Specifically, we show that for n∈{8,24,32,48,56,72,96} every extremal Type II code of length n is generated by its codewords of minimal weight. Where Ozeki and Kominers used spherical harmonics and weighted theta functions, we use discrete harmonic polynomials and harmonic weight enumerators. Along the way we introduce "t12-designs" as a discrete analog of Venkov's spherical designs of the same name.

Keywords: Mathematical Methods;

Citation:

Elkies, Noam D., and Scott Duke Kominers. "Configurations of Extremal Type II Codes." Working Paper, March 2015. View Details

1. How U.S. Laws Protecting America's Best Ideas Are Killing Innovation

Lauren Cohen, Umit G. Gurun and Scott Duke Kominers

### Cases and Teaching Materials

1. Online Marketing at Big Skinny

Benjamin Edelman and Scott Duke Kominers

Describes a wallet maker's application of seven Internet marketing technologies: display ads, algorithmic search, sponsored search, social media, interactive content, online distributors, and A/B testing. Provides concise introductions to the key features of each technology, and asks which forms of online marketing the company should prioritize in the future. Discusses similarities and differences between online and off-line marketing, as well as issues of marketing campaign evaluation.

Citation:

Edelman, Benjamin, and Scott Duke Kominers. "Online Marketing at Big Skinny." Harvard Business School Case 911-033, February 2011. (Revised February 2012.) (request a courtesy copy.) View Details
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2. Online Marketing at Big Skinny

Benjamin Edelman and Scott Kominers

Teaching Note for 911033.

Citation:

Edelman, Benjamin, and Scott Kominers. "Online Marketing at Big Skinny." Harvard Business School Teaching Note 911-034, February 2011. (Revised June 2014.) View Details
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