A NOTE ON MINIMAX REGRET RULES WITH MULTIPLE TREATMENTS IN FINITE SAMPLES
研究了在有限样本下，政策制定者从多个处理中做选择时，最小最大遗憾规则的推广问题，证明了当处理数大于2时，已有规则仍是最优的，并给出了随机分配下的新规则。
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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>T</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:math> $T\geq 2$ upper T greater than or equals 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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> $T=2$ upper T equals 2 is minimax regret for general finite <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>T</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:math> $T>2$ upper T greater than 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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>T</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:math> $T>2$ upper T greater than 2 is not necessarily minimax regret and derive minimax regret rules for a few small sample cases, e.g., for <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> $N=2$ upper N equals 2 when <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3.</mml:mn> </mml:math> $T=3.$ upper T equals 3 period 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.