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  • All HBS Web  (854)
    • News  (79)
    • Research  (639)
    • Events  (14)
    • Multimedia  (4)
  • Faculty Publications  (632)
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  • 2009
  • Working Paper

Smith and Rawls Share a Room: Stability and Medians

By: Bettina-Elisabeth Klaus and Flip Klijn
We consider one-to-one, one-sided matching (roommate) problems in which agents can either be matched as pairs or remain single. We introduce a so-called bi-choice graph for each pair of stable matchings and characterize its structure. Exploiting this structure we... View Details
Keywords: Fairness; One-Sided Platforms; Two-Sided Platforms; Marketplace Matching; Mathematical Methods; Balance and Stability
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Klaus, Bettina-Elisabeth, and Flip Klijn. "Smith and Rawls Share a Room: Stability and Medians." Harvard Business School Working Paper, No. 09-111, March 2009.
  • 2008
  • Chapter

Matching and Market Design

By: Muriel Niederle, Alvin E. Roth and Tayfun Sonmez
Matching is the part of economics concerned with who transacts with whom and how. Models of matching, starting with the Gale-Shapley deferred acceptance algorithm, have been particularly useful in studying labour markets and in helping design clearinghouses to fix... View Details
Keywords: Market Design; Marketplace Matching; Outcome or Result; Mathematical Methods
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Niederle, Muriel, Alvin E. Roth, and Tayfun Sonmez. "Matching and Market Design." In The New Palgrave Dictionary of Economics. 2nd ed. Edited by Steven Derlauf and Larry Blume. Hampshire, U.K.: Palgrave Macmillan, 2008.
  • January 2003 (Revised February 2011)
  • Tool

Business Analysis and Valuation Model (Version 5)

By: Paul M. Healy, Krishna G. Palepu and Jonathan Barnett
Once you enter company financial statements, this software enables you to standardize them to a common format, make any needed adjustments to the company's accounting, and make assumptions about the company's future performance. The model then provides financial ratios... View Details
Keywords: Financial Statements; Standards; Mathematical Methods; Valuation
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"Business Analysis and Valuation Model (Version 5)." Harvard Business School Tool 103-701, January 2003. (Revised February 2011.)
  • 2025
  • Working Paper

Incentive-Compatible Recovery from Manipulated Signals, with Applications to Decentralized Physical Infrastructure

By: Jason Milionis, Jens Ernstberger, Joseph Bonneau, Scott Duke Kominers and Tim Roughgarden
We introduce the first formal model capturing the elicitation of unverifiable information from a party (the "source") with implicit signals derived by other players (the "observers"). Our model is motivated in part by applications in decentralized physical... View Details
Keywords: Mathematical Methods; Infrastructure; Information Infrastructure
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Milionis, Jason, Jens Ernstberger, Joseph Bonneau, Scott Duke Kominers, and Tim Roughgarden. "Incentive-Compatible Recovery from Manipulated Signals, with Applications to Decentralized Physical Infrastructure." Working Paper, March 2025.
  • May–June 2025
  • Article

Slowly Varying Regression Under Sparsity

By: Dimitris Bertsimas, Vassilis Digalakis Jr, Michael Lingzhi Li and Omar Skali Lami
We consider the problem of parameter estimation in slowly varying regression models with sparsity constraints. We formulate the problem as a mixed integer optimization problem and demonstrate that it can be reformulated exactly as a binary convex optimization problem... View Details
Keywords: Mathematical Methods; Analytics and Data Science
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Bertsimas, Dimitris, Vassilis Digalakis Jr, Michael Lingzhi Li, and Omar Skali Lami. "Slowly Varying Regression Under Sparsity." Operations Research 73, no. 3 (May–June 2025): 1581–1597.
  • 2024
  • Working Paper

Empirical Guidance: Data Processing and Analysis with Applications in Stata, R, and Python

By: Melissa Ouellet and Michael W. Toffel
This paper describes a range of best practices to compile and analyze datasets, and includes some examples in Stata, R, and Python. It is meant to serve as a reference for those getting started in econometrics, and especially those seeking to conduct data analyses in... View Details
Keywords: Empirical Methods; Empirical Operations; Statistical Methods And Machine Learning; Statistical Interferences; Research Analysts; Analytics and Data Science; Mathematical Methods
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Ouellet, Melissa, and Michael W. Toffel. "Empirical Guidance: Data Processing and Analysis with Applications in Stata, R, and Python." Harvard Business School Working Paper, No. 25-010, August 2024.
  • 2021
  • Working Paper

Impact Investing: A Theory of Financing Social Enterprises

By: Benjamin N. Roth
I present a model of financing social enterprises to delineate the role of impact investors relative to “pure” philanthropists. I characterize the optimal scale and structure of a social enterprise when financed by grants, and when financed by investments. Impact... View Details
Keywords: Impact Investing; Social Entrepreneurship; Finance; Investment; Mathematical Methods
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Roth, Benjamin N. "Impact Investing: A Theory of Financing Social Enterprises." Harvard Business School Working Paper, No. 20-078, February 2020. (Revised June 2021.)
  • Article

Quantile Evaluation, Sensitivity to Bracketing, and Sharing Business Payoffs

By: Y. Grushka-Cockayne, K. C. Lichtendahl, V.R.R. Jose and R.L. Winkler
From forecasting competitions to conditional value-at-risk requirements, the use of multiple quantile assessments is growing in practice. To evaluate them, we use a rule from the general class of proper scoring rules for a forecaster’s multiple quantiles of a single... View Details
Keywords: Forecast; Aggregation; Operations; Forecasting and Prediction; Mathematical Methods
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Grushka-Cockayne, Y., K. C. Lichtendahl, V.R.R. Jose, and R.L. Winkler. "Quantile Evaluation, Sensitivity to Bracketing, and Sharing Business Payoffs." Operations Research 65, no. 3 (May–June 2017): 712–728.
  • January 2000
  • Article

Maxmin Expected Utility through Statewise Combinations

By: Ramon Casadesus-Masanell, Peter Klibanoff and Emre Ozdenoren
This paper provides an axiomatic foundation for a maxmin expected utility over a set of priors (MMEU) decision rule in an environment where the elements of choice are Savage acts. The key axioms are stated using statewise combinations as in Gul (1992). View Details
Keywords: Decision Choices and Conditions; Mathematical Methods
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Casadesus-Masanell, Ramon, Peter Klibanoff, and Emre Ozdenoren. "Maxmin Expected Utility through Statewise Combinations." Economics Letters 66, no. 1 (January 2000): 49–54.
  • March 2016 (Revised January 2020)
  • Teaching Note

Behavioural Insights Team (A) and (B)

By: Michael Luca and Patrick Rooney
The Behavioural Insights Team case introduces students to the concept of choice architecture and the value of experimental methods (sometimes called A/B testing) within organizational contexts. The exercise provides an opportunity for students to apply these principles... View Details
Keywords: Behavioral Economics; Experiments; Choice Architecture; Public Entrepreneurship; Decision Choices and Conditions; Mathematical Methods; United Kingdom
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Luca, Michael, and Patrick Rooney. "Behavioural Insights Team (A) and (B)." Harvard Business School Teaching Note 916-050, March 2016. (Revised January 2020.)
  • 1977
  • Chapter

A Reexamination of the Capital Asset Pricing Model

By: Robert C. Merton
Keywords: Capital; Asset Pricing; Mathematical Methods
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Merton, Robert C. "A Reexamination of the Capital Asset Pricing Model." In Studies in Risk and Return, edited by J. Bicksler and I. Friend. Cambridge, MA: Ballinger Publishing Company, 1977.
  • March 2011 (Revised April 2021)
  • Case

The Whiz Kids

By: Tom Nicholas and David Chen
In October 1945, Henry Ford II received a telegram in his office at the Ford Motor Company in Dearborn, Michigan written by Charles "Tex" Thornton, a U.S. Air Force colonel. The telegram presented an opportunity for Ford to deploy a system of statistical control which... View Details
Keywords: Ford Motor Company; Statistical Control; Management Systems; Accounting; Operations; Strategy; Mathematical Methods; Auto Industry; United States
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Nicholas, Tom, and David Chen. "The Whiz Kids." Harvard Business School Case 811-042, March 2011. (Revised April 2021.)
  • 2023
  • Article

Balancing Risk and Reward: An Automated Phased Release Strategy

By: Yufan Li, Jialiang Mao and Iavor Bojinov
Phased releases are a common strategy in the technology industry for gradually releasing new products or updates through a sequence of A/B tests in which the number of treated units gradually grows until full deployment or deprecation. Performing phased releases in a... View Details
Keywords: Product Launch; Mathematical Methods; Product Development
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Li, Yufan, Jialiang Mao, and Iavor Bojinov. "Balancing Risk and Reward: An Automated Phased Release Strategy." Advances in Neural Information Processing Systems (NeurIPS) (2023).
  • January 2020
  • Article

The Long-Run Dynamics of Electricity Demand: Evidence from Municipal Aggregation

By: Tatyana Deryugina, Alexander MacKay and Julian Reif
We study the dynamics of residential electricity demand by exploiting a natural experiment that produced large and long-lasting price changes in over 250 Illinois communities. Using a flexible difference-in-differences matching approach, we estimate that the price... View Details
Keywords: Electricity Demand; Consumption Dynamics; Energy; Policy; Demand and Consumers; Price; Mathematical Methods
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Deryugina, Tatyana, Alexander MacKay, and Julian Reif. "The Long-Run Dynamics of Electricity Demand: Evidence from Municipal Aggregation." American Economic Journal: Applied Economics 12, no. 1 (January 2020): 86–114.
  • July 2019
  • Article

Optimal Capital Structure and Bankruptcy Choice: Dynamic Bargaining vs Liquidation

By: Samuel Antill and Steven R. Grenadier
We model a firm’s optimal capital structure decision in a framework in which it may later choose to enter either Chapter 11 reorganization or Chapter 7 liquidation. Creditors anticipate equityholders’ ex-post reorganization incentives and price them into the ex-ante... View Details
Keywords: Default; Dynamic Bargaining; Capital Structure; Insolvency and Bankruptcy; Mathematical Methods
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Antill, Samuel, and Steven R. Grenadier. "Optimal Capital Structure and Bankruptcy Choice: Dynamic Bargaining vs Liquidation." Journal of Financial Economics 133, no. 1 (July 2019): 198–224.
  • 2018
  • Working Paper

Algorithm Appreciation: People Prefer Algorithmic to Human Judgment

By: Jennifer M. Logg, Julia A. Minson and Don A. Moore
Even though computational algorithms often outperform human judgment, received wisdom suggests that people may be skeptical of relying on them (Dawes, 1979). Counter to this notion, results from six experiments show that lay people adhere more to advice when they think... View Details
Keywords: Algorithms; Accuracy; Advice Taking; Forecasting; Theory Of Machine; Mathematical Methods; Decision Making; Forecasting and Prediction; Trust
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Logg, Jennifer M., Julia A. Minson, and Don A. Moore. "Algorithm Appreciation: People Prefer Algorithmic to Human Judgment." Harvard Business School Working Paper, No. 17-086, March 2017. (Revised April 2018.)
  • 2011
  • Working Paper

Price Competition under Multinomial Logit Demand Functions with Random Coefficients

In this paper, we postulate a general class of price competition models with Mixed Multinomial Logit demand functions under affine cost functions. We first characterize the equilibrium behavior of this class of models in the case where each product in the market is... View Details
Keywords: Customers; Income Characteristics; Price; Product Marketing; Mathematical Methods; Competition; Segmentation
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Allon, Gad, Awi Federgruen, and Margaret Pierson. "Price Competition under Multinomial Logit Demand Functions with Random Coefficients." Harvard Business School Working Paper, No. 12-030, October 2011.
  • May 2004
  • Article

The Case for International Coordination of Electricity Regulation: Evidence from the Measurement of Efficiency in South America

A decade of experience has shown that monitoring the performance of public and private monopolies is the hardest part of electricity sector reform in South America—because operators control most of the information needed for effective regulation. South American... View Details
Keywords: Information; Mathematical Methods; Monopoly; Globalization; Energy Sources; Energy Industry; South America
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Ruzzier, Christian Alejandro, A. Estache, and M. Rossi. "The Case for International Coordination of Electricity Regulation: Evidence from the Measurement of Efficiency in South America." Journal of Regulatory Economics 25, no. 3 (May 2004): 271–295.
  • April 1990
  • Case

Clark Material Handling Group-Overseas: Brazilian Product Strategy (A&B) (Condensed)

By: Robert J. Dolan
Assumes some knowledge of conjoint analysis. Permits analysis of basic results and dynamic market simulations in one class session. View Details
Keywords: Mathematical Methods; Marketing
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Dolan, Robert J. "Clark Material Handling Group-Overseas: Brazilian Product Strategy (A&B) (Condensed)." Harvard Business School Case 590-081, April 1990.
  • February 2021
  • Article

A Dynamic Theory of Multiple Borrowing

By: Daniel Green and Ernest Liu
Multiple borrowing—a borrower obtains overlapping loans from multiple lenders—is a common phenomenon in many credit markets. We build a highly tractable, dynamic model of multiple borrowing and show that, because overlapping creditors may impose default externalities... View Details
Keywords: Commitment; Multiple Borrowing; Common Agency; Misallocation; Microfinance; Investment; Mathematical Methods
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Green, Daniel, and Ernest Liu. "A Dynamic Theory of Multiple Borrowing." Journal of Financial Economics 139, no. 2 (February 2021): 389–404.
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