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A NOTE ON MINIMAX REGRET RULES WITH MULTIPLE TREATMENTS IN FINITE SAMPLES

Published online by Cambridge University Press:  27 February 2025

Haoning Chen  and
Patrik Guggenberger Open the ORCID record for Patrik Guggenberger [Opens in a new window]
Haoning Chen
Affiliation:
Pennsylvania State University
Patrik Guggenberger*
Affiliation:
Pennsylvania State University
*
Address correspondence to Patrik Guggenberger, Department of Economics, Pennsylvania State University, State College, PA, USA; e-mail: pxg27@psu.edu
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Abstract

We study minimax regret treatment rules under matched treatment assignment in a setup where a policymaker, informed by a sample of size N, needs to decide between T different treatments for a $T\geq 2$. Randomized rules are allowed for. We show that the generalization of the minimax regret rule derived in Schlag (2006, ELEVEN—Tests needed for a recommendation, EUI working paper) and Stoye (2009, Journal of Econometrics 151, 70–81) for the case $T=2$ is minimax regret for general finite $T>2$ and also that the proof structure via the Nash equilibrium and the “coarsening” approaches generalizes as well. We also show by example, that in the case of random assignment the generalization of the minimax rule in Stoye (2009, Journal of Econometrics 151, 70–81) to the case $T>2$ is not necessarily minimax regret and derive minimax regret rules for a few small sample cases, e.g., for $N=2$ when $T=3.$

In the case where a covariate x is included, it is shown that a minimax regret rule is obtained by using minimax regret rules in the “conditional-on-x” problem if the latter are obtained as Nash equilibria.

Type
ARTICLES
Information
Econometric Theory , First View , pp. 1 - 27
Copyright
© The Author(s), 2025. Published by Cambridge University Press

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Footnotes

We would like to thank the Editor Peter Phillips, the Co-Editor Matias Cattaneo, and three referees for very helpful comments and suggestions. We thank Kei Hirano, Jiaqi Huang, Matt Masten, Chen Qiu, Karl Schlag, Joerg Stoye, and Aleksey Tetenov for comments and helpful information about this research agenda. We thank seminar participants at Brisbane, Cambridge, Colorado Boulder, Exeter, Frankfurt, Heidelberg, Konstanz, LMU, Macquarie, Manchester, NTU, NUS, McGill, Oxford, Regensburg, SMU, SUNY Stony Brook, UC Davis, UCL, UC Riverside, University of Sydney, and UNSW, and participants at the ‘Advances in Econometrics Conference and Festschrift in Honor of Joon Y. Park’ in 2023, the ‘New York Camp Econometrics conference’, and the Rochester conference in Econometrics in 2024.

References

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