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Show Results For
- All HBS Web
(1,940)
- People (2)
- News (200)
- Research (1,361)
- Events (14)
- Multimedia (14)
- Faculty Publications (899)
- January 2018
- Background Note
Math Tools for Strategists
By: Tarun Khanna and Jan W. Rivkin
Great strategists rely heavily on numbers as they go about their work. This note offers an overview of the highbrow and lowbrow quantitative tools that individuals commonly encounter during strategy courses and in actual strategy work. The note focuses especially on... View Details
Khanna, Tarun, and Jan W. Rivkin. "Math Tools for Strategists." Harvard Business School Background Note 718-477, January 2018.
- December 1998
- Background Note
Note on Low-Tech Marketing Math
By: Robert J. Dolan
Describes basic calculations useful in marketing analysis, break-even analysis, and price-volume relationships. View Details
Dolan, Robert J. "Note on Low-Tech Marketing Math." Harvard Business School Background Note 599-011, December 1998.
- June 2008
- Article
Minimally Acceptable Altruism and the Ultimatum Game
By: Julio J. Rotemberg
I suppose that people react with anger when others show themselves not to be minimally altruistic. With heterogeneous agents, this can account for the experimental results of ultimatum and dictator games. Moreover, it can account for the surprisingly large fraction of... View Details
Rotemberg, Julio J. "Minimally Acceptable Altruism and the Ultimatum Game." Journal of Economic Behavior & Organization 66, nos. 3-4 (June 2008).
- November 1977
- Article
On the Pricing of Contingent Claims and the Modigliani-Miller Theorem
By: Robert C. Merton
Merton, Robert C. "On the Pricing of Contingent Claims and the Modigliani-Miller Theorem." Journal of Financial Economics 5 (November 1977): 241–249. (Chapter 13 in Continuous-Time Finance.)
- August 1970
- Case
Hawthorne Plastics
An "imperfect tester" problem involving the decision of how to produce batches of plastic strapping, given uncertainty about the length of the molecular chain in the raw material. A decision on whether to test the raw material and a choice of production process must be... View Details
Hammond, John S. "Hawthorne Plastics." Harvard Business School Case 171-004, August 1970.
- 2021
- Article
Fair Algorithms for Infinite and Contextual Bandits
By: Matthew Joseph, Michael J Kearns, Jamie Morgenstern, Seth Neel and Aaron Leon Roth
We study fairness in linear bandit problems. Starting from the notion of meritocratic fairness introduced in Joseph et al. [2016], we carry out a more refined analysis of a more general problem, achieving better performance guarantees with fewer modelling assumptions... View Details
Joseph, Matthew, Michael J Kearns, Jamie Morgenstern, Seth Neel, and Aaron Leon Roth. "Fair Algorithms for Infinite and Contextual Bandits." Proceedings of the AAAI/ACM Conference on Artificial Intelligence, Ethics, and Society 4th (2021).
- Other Unpublished Work
The Role of Inventory in Empowered Work Settings: Model and Empirical Analysis
By: S. Datar, M. Alles and R. Sarkar
- 1987
- Chapter
Money in the Utility Function: An Empirical Implementation
By: Julio J. Rotemberg and James Poterba
Rotemberg, Julio J., and James Poterba. "Money in the Utility Function: An Empirical Implementation." In New Approaches to Monetary Economics, edited by W. Barnett and K. Singleton, 219–240. Cambridge University Press, 1987.
- Article
Scale Economies in Statistical Analyses of Market Power
By: R. E. Caves, J. Khalizadeh-Shirazi and M. E. Porter
Caves, R. E., J. Khalizadeh-Shirazi, and M. E. Porter. "Scale Economies in Statistical Analyses of Market Power." Review of Economics and Statistics 57, no. 2 (May 1975): 133–140.
- 1976
- Other Unpublished Work
Simulation of the Market Effects of Housing Allowance
By: Dutch Leonard, Gregory K. Ingram and Robert Schafer
- 2023
- Article
Which Models Have Perceptually-Aligned Gradients? An Explanation via Off-Manifold Robustness
By: Suraj Srinivas, Sebastian Bordt and Himabindu Lakkaraju
One of the remarkable properties of robust computer vision models is that their input-gradients are often aligned with human perception, referred to in the literature as perceptually-aligned gradients (PAGs). Despite only being trained for classification, PAGs cause... View Details
Srinivas, Suraj, Sebastian Bordt, and Himabindu Lakkaraju. "Which Models Have Perceptually-Aligned Gradients? An Explanation via Off-Manifold Robustness." Advances in Neural Information Processing Systems (NeurIPS) (2023).
- March 2022 (Revised July 2022)
- Technical Note
Statistical Inference
This note provides an overview of statistical inference for an introductory data science course. First, the note discusses samples and populations. Next the note describes how to calculate confidence intervals for means and proportions. Then it walks through the logic... View Details
Keywords: Data Science; Statistics; Mathematical Modeling; Mathematical Methods; Analytics and Data Science
Bojinov, Iavor I., Michael Parzen, and Paul Hamilton. "Statistical Inference." Harvard Business School Technical Note 622-099, March 2022. (Revised July 2022.)
- March 2020
- Article
Diagnosing Missing Always at Random in Multivariate Data
By: Iavor I. Bojinov, Natesh S. Pillai and Donald B. Rubin
Models for analyzing multivariate data sets with missing values require strong, often assessable, assumptions. The most common of these is that the mechanism that created the missing data is ignorable—a twofold assumption dependent on the mode of inference. The first... View Details
Keywords: Missing Data; Diagnostic Tools; Sensitivity Analysis; Hypothesis Testing; Missing At Random; Row Exchangeability; Analytics and Data Science; Mathematical Methods
Bojinov, Iavor I., Natesh S. Pillai, and Donald B. Rubin. "Diagnosing Missing Always at Random in Multivariate Data." Biometrika 107, no. 1 (March 2020): 246–253.
- June 2000
- Article
On the Regulatory Application of Efficiency Measures
The last decade has witnessed a change to more powerful incentive schemes and the adoption by a large number of regulators of some form of price cap regimes. The efficiency frontiers literature tackles the problem of measuring the X factor in a price cap regime... View Details
Ruzzier, Christian Alejandro. "On the Regulatory Application of Efficiency Measures." Utilities Policy 9, no. 2 (June 2000): 81–92. (with M. Rossi.)
- August 2001 (Revised July 2008)
- Technical Note
A Technical Note and Discussion on Real Estate Valuation (IBET): Back of the Envelope (BOE) on Bonhomme Place: A Case within a Case
By: Arthur I Segel
Discusses real estate valuation. Reviews "back of the envelope" valuation; real estate appraisal methods, including the income method; market comparables and replacement costs; and more complex computer modeling. Also discusses other variables that could influence... View Details
Segel, Arthur I. "A Technical Note and Discussion on Real Estate Valuation (IBET): Back of the Envelope (BOE) on Bonhomme Place: A Case within a Case." Harvard Business School Technical Note 802-025, August 2001. (Revised July 2008.)
- 1981
- Chapter
Risk Aversion and Solutions to Nash's Bargaining Problem
By: R. Kihlstrom, A. E. Roth and D. Schmeidler
- May 2020
- Article
Inventory Auditing and Replenishment Using Point-of-Sales Data
By: Achal Bassamboo, Antonio Moreno and Ioannis Stamatopoulos
Spoilage, expiration, damage due to employee/customer handling, employee theft, and customer shoplifting usually are not reflected in inventory records. As a result, records often report phantom inventory, i.e., units of good not available for sale. We derive an... View Details
Keywords: Shelf Availability; Inventory Record Inaccuracy; Optimal Replenishment; Retail Analytics; Performance Effectiveness; Analysis; Mathematical Methods
Bassamboo, Achal, Antonio Moreno, and Ioannis Stamatopoulos. "Inventory Auditing and Replenishment Using Point-of-Sales Data." Production and Operations Management 29, no. 5 (May 2020): 1219–1231.
- April 1979
- Article
Statistical Models of Bond Ratings: A Methodological Inquiry
By: Robert S. Kaplan and Gabriel Urwitz
Kaplan, Robert S., and Gabriel Urwitz. "Statistical Models of Bond Ratings: A Methodological Inquiry." Journal of Business (April 1979): 231–261.
- 1980
- Chapter
Tests of Capital Market Theory and Implications of the Evidence
By: Michael Jensen
Jensen, Michael. "Tests of Capital Market Theory and Implications of the Evidence." In Handbook of Financial Economics, edited by J. L. Bicksler. North-Holland Publishing Company, 1980. (Originally published in Is Financial Analysis Useless? Proceedings of a Seminar on the Efficient Market and Random Walk Hypotheses (The Financial Analysts Research Foundation, 1975).)