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Show Results For

  • All HBS Web  (854)
    • News  (79)
    • Research  (639)
    • Events  (14)
    • Multimedia  (4)
  • Faculty Publications  (632)
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  • November 2006
  • Case

Introducing Frequent Flyer Programs

By: Dennis A. Yao
Allows students to explore the value to American Airlines of introducing a frequent flyer program in 1981. View Details
Keywords: Value; Management Analysis, Tools, and Techniques; Mathematical Methods; Game Theory; Competitive Advantage; Programs; Customers; Air Transportation Industry
Citation
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Yao, Dennis A. "Introducing Frequent Flyer Programs." Harvard Business School Case 707-479, November 2006.
  • May–June 2025
  • Article

Branch-and-Price for Prescriptive Contagion Analytics

By: Alexandre Jacquillat, Michael Lingzhi Li, Martin Ramé and Kai Wang
Contagion models are ubiquitous in epidemiology, social sciences, engineering, and management. This paper formulates a prescriptive contagion analytics model where a decision maker allocates shared resources across multiple segments of a population, each governed by... View Details
Keywords: COVID-19; Mathematical Methods
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Jacquillat, Alexandre, Michael Lingzhi Li, Martin Ramé, and Kai Wang. "Branch-and-Price for Prescriptive Contagion Analytics." Operations Research 73, no. 3 (May–June 2025): 1558–1580.
  • 2020
  • Article

Fast Exact Matrix Completion: A Unified Optimization Framework for Matrix Completion

By: Dimitris Bertsimas and Michael Lingzhi Li
We formulate the problem of matrix completion with and without side information as a non-convex optimization problem. We design fastImpute based on non-convex gradient descent and show it converges to a global minimum that is guaranteed to recover closely the... View Details
Keywords: Mathematical Methods
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Bertsimas, Dimitris, and Michael Lingzhi Li. "Fast Exact Matrix Completion: A Unified Optimization Framework for Matrix Completion." Journal of Machine Learning Research 21, no. 1 (2020).
  • Article

How Much Should We Trust Staggered Difference-In-Differences Estimates?

By: Andrew C. Baker, David F. Larcker and Charles C.Y. Wang
We explain when and how staggered difference-in-differences regression estimators, commonly applied to assess the impact of policy changes, are biased. These biases are likely to be relevant for a large portion of research settings in finance, accounting, and law that... View Details
Keywords: Difference In Differences; Staggered Difference-in-differences Designs; Generalized Difference-in-differences; Dynamic Treatment Effects; Mathematical Methods
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Baker, Andrew C., David F. Larcker, and Charles C.Y. Wang. "How Much Should We Trust Staggered Difference-In-Differences Estimates?" Journal of Financial Economics 144, no. 2 (May 2022): 370–395. (Editor's Choice, May 2022; Jensen Prize, First Place, June 2023.)
  • Mar 2021
  • Conference Presentation

Descent-to-Delete: Gradient-Based Methods for Machine Unlearning

By: Seth Neel, Aaron Leon Roth and Saeed Sharifi-Malvajerdi
We study the data deletion problem for convex models. By leveraging techniques from convex optimization and reservoir sampling, we give the first data deletion algorithms that are able to handle an arbitrarily long sequence of adversarial updates while promising both... View Details
Keywords: Machine Learning; Unlearning Algorithm; Mathematical Methods
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Neel, Seth, Aaron Leon Roth, and Saeed Sharifi-Malvajerdi. "Descent-to-Delete: Gradient-Based Methods for Machine Unlearning." Paper presented at the 32nd Algorithmic Learning Theory Conference, March 2021.
  • Article

Oracle Efficient Private Non-Convex Optimization

By: Seth Neel, Aaron Leon Roth, Giuseppe Vietri and Zhiwei Steven Wu
One of the most effective algorithms for differentially private learning and optimization is objective perturbation. This technique augments a given optimization problem (e.g. deriving from an ERM problem) with a random linear term, and then exactly solves it.... View Details
Keywords: Machine Learning; Algorithms; Objective Perturbation; Mathematical Methods
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Neel, Seth, Aaron Leon Roth, Giuseppe Vietri, and Zhiwei Steven Wu. "Oracle Efficient Private Non-Convex Optimization." Proceedings of the International Conference on Machine Learning (ICML) 37th (2020).
  • Article

Aztec Castles and the dP3 Quiver

By: Megan Leoni, Gregg Musiker, Seth Neel and Paxton Turner
Bipartite, periodic, planar graphs known as brane tilings can be associated to a large class of quivers. This paper will explore new algebraic properties of the well-studied del Pezzo 3 (dP3) quiver and geometric properties of its corresponding brane tiling. In... View Details
Keywords: Brane Tiling; Del Pezzo 3 Lattice; Dimer Model; Mathematical Methods
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Leoni, Megan, Gregg Musiker, Seth Neel, and Paxton Turner. "Aztec Castles and the dP3 Quiver." Journal of Physics A: Mathematical and Theoretical 47, no. 47 (November 28, 2014).
  • 2021
  • Working Paper

How Much Should We Trust Staggered Difference-In-Differences Estimates?

By: Andrew C. Baker, David F. Larcker and Charles C.Y. Wang
Difference-in-differences analysis with staggered treatment timing is frequently used to assess the impact of policy changes on corporate outcomes in academic research. However, recent advances in econometric theory show that such designs are likely to be biased in the... View Details
Keywords: Difference In Differences; Staggered Difference-in-differences Designs; Generalized Difference-in-differences; Dynamic Treatment Effects; Mathematical Methods
Citation
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Baker, Andrew C., David F. Larcker, and Charles C.Y. Wang. "How Much Should We Trust Staggered Difference-In-Differences Estimates?" European Corporate Governance Institute Finance Working Paper, No. 736/2021, February 2021. (Harvard Business School Working Paper, No. 21-112, April 2021.)
  • 1990
  • Chapter

Refinement of Nash Equilibrium: The Main Ideas

By: E. Kohlberg
Keywords: Mathematical Methods
Citation
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Kohlberg, E. "Refinement of Nash Equilibrium: The Main Ideas." In Game Theory and Applications, edited by T. Ichiishi, A. Neyman, and Y. Tauman. San Diego: Academic Press, 1990.
  • Article

Refined Configuration Results for Extremal Type II Lattices of Ranks 40 and 80

By: 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... View Details
Keywords: Mathematical Methods
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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.
  • June 2006 (Revised March 2007)
  • Teaching Note

Polyphonic HMI: Mixing Music and Math

By: Anita Elberse
Keywords: Mathematical Methods; Music Industry
Citation
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Elberse, Anita. "Polyphonic HMI: Mixing Music and Math." Harvard Business School Teaching Note 506-081, June 2006. (Revised March 2007.)
  • February 2006
  • Teaching Note

Cost-Volume-Profit Models (TN)

By: David F. Hawkins, V.G. Narayanan, Michele Jurgens and Jacob Cohen
Keywords: Mathematical Methods
Citation
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Hawkins, David F., V.G. Narayanan, Michele Jurgens, and Jacob Cohen. "Cost-Volume-Profit Models (TN)." Harvard Business School Teaching Note 106-046, February 2006.
  • August 2005 (Revised April 2008)
  • Teaching Note

Store24 (A): Statistics Review with Data Desk (TN)

By: Frances X. Frei
Presents an overview of the statistical analysis covered in the case discussion. View Details
Keywords: Mathematical Methods; Retail Industry
Citation
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Frei, Frances X. "Store24 (A): Statistics Review with Data Desk (TN)." Harvard Business School Teaching Note 606-036, August 2005. (Revised April 2008.)
  • 2006
  • Chapter

Advanced Regression Models

By: Raghuram Iyengar and Sunil Gupta
Keywords: Mathematical Methods
Citation
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Iyengar, Raghuram, and Sunil Gupta. "Advanced Regression Models." In Handbook of Marketing Research, edited by Rajiv Grover. Sage Publications, 2006.
  • 1978
  • Chapter

Matrix-Weighted Averages: Computation and Presentation

By: Dutch Leonard
Keywords: Mathematical Methods
Citation
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Leonard, Dutch. "Matrix-Weighted Averages: Computation and Presentation." In Proceedings of the Eleventh Symposium on the Interface of Computers and Statistics, edited by Ronald A. Gallant and Thomas Michael Gerig. Raleigh, NC: North Carolina State University, Institute of Statistics, 1978.
  • July 1982 (Revised March 1984)
  • Background Note

Worked Examples in Dynamic Programming

By: David E. Bell
Keywords: Mathematical Methods
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Bell, David E. "Worked Examples in Dynamic Programming." Harvard Business School Background Note 183-028, July 1982. (Revised March 1984.)
  • 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
Citation
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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.
  • 2024
  • Working Paper

Fiscal Policy under Convex Supply Curves

By: Shlok Goyal, Avi Lipton and Borui Niklas Zhu
Recent empirical evidence suggests that supply curves are convex. Supply curve convexity is at odds with conventional Phillips curves, which rely on an infinitely elastic underlying supply curve. This paper explores the effect of supply curve convexity on the... View Details
Keywords: Fiscal Stimulus; Fiscal Policy; Inflation; Inflation and Deflation; Macroeconomics; Policy; Mathematical Methods; United States
Citation
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Goyal, Shlok, Avi Lipton, and Borui Niklas Zhu. "Fiscal Policy under Convex Supply Curves." Working Paper, August 2024.
  • 2009
  • Working Paper

Patent Policy, Patent Pools, and the Accumulation of Claims in Sequential Innovation

By: Gaston Llanes and Stefano Trento
We present a dynamic model where the accumulation of patents generates an increasing number of claims on sequential innovation. We study the equilibrium innovation activity under three regimes: patents, no-patents and patent pools. Patent pools increase the probability... View Details
Keywords: Innovation and Invention; Patents; Rights; Mathematical Methods
Citation
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Llanes, Gaston, and Stefano Trento. "Patent Policy, Patent Pools, and the Accumulation of Claims in Sequential Innovation." Harvard Business School Working Paper, No. 10-005, July 2009.
  • Article

The Shapley Value as a von Neumann-Morgenstern Utility

By: A. E. Roth
Keywords: Value; Mathematical Methods
Citation
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Roth, A. E. "The Shapley Value as a von Neumann-Morgenstern Utility." Econometrica 45, no. 3 (April 1977): 657–664.
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