quick question

[Joe Halpern]

I think it's well known (and certainly easy to prove) that if
uncertainty is represented by a single probability distribution, then
the ordering on acts induced by minimizing expected regret is the same
as that induced by maximizing expected utility. Does someone have a
reference for this result? Thanks. -- Joe

[Patrick O'Callaghan]

Hi Joe,

they are only equivalent when the regret function is separable across across acts of course (that is in the absence of regret aversion), and there are papers such as Loomes and Sugden (1982) ``Regret theory: ..." which build on the case where regret aversion is present. The textbook Mas Colell, Whinston and Green called ``Microeconomic theory" has an exercise that is related to this in the 6th chapter. Fishburn's ``Skew-symmetric bi-linear" model can also be used to model regret. Finally, Savage's ``Foundations of Statistics" has some chapters (starting with chapter 9) on minimax which considers loss functions and may be what you're looking for.

Best wishes,

Patrick

[g charles-cadogan]

Hi Joe:

 

You might want to look at Bell, David E. "Risk premiums for decision regret." Management Science 29.10 (1983): 1156-1166. Section 4--particularly page 1164. There, the author presents analytics that may answer aspects of your question.

Regards,

G

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[Joe Halpern]

Thanks to all of you for the pointers. To be clear, I am assuming
separability, and defining the regret of an act a in state s as the
difference between the utility of the outcome of the best act in state s
and the utility of a(s), just as Savage did. In that setting, it is
easy to show that the ordering on acts induced by expected regret is
identical to the order on acts induced by expected utility.

With regard to the comments in the emails below, Bell's framework is
somewhat different from the standard regret setting; he has a relative
utility function u(x,y), where x and y are outcomes, which is taken to
have the form v(x) - f(v(x) - v(y)), where v(x) is the more traditional
utility on outcomes. I had forgotten that Savage had six chapters on
regret (he calls it minimax theory); his focus is on non-probabilistic
regret (where there is no probability on states). As near as I can
tell, he does not talk about expected regret. I haven't checked
Mas-Colell, Green, and Whinston yet.

Uzi Segal pointed me to his paper "Transitive Regret" (joint with
Bikhchandari) that refers to the result that I'm interested in as well
known, as do Renou and Schlag in "Minimax regret and strategic
uncertainty". Hayashi gives a 3-line proof in "Regret aversion and
opportunity dependence" (which appeared in JET in 2008) of a slightly
weaker result: that the act that minimizes expected regret is the same
as the act that maximizes expected utility (although the proof of the
more general result is essentially the same). Since the result is
clearly well known, I suspect it appears in the literature much earlier.

-- Joe

On 4/3/2013 5:34 AM, g charles-cadogan wrote:
> Hi Joe:
> You might want to look at Bell, David E. "Risk premiums for decision

> regret <http://www.jstor.org/stable/10.2307/2631346>." /Management
> Science/ 29.10 (1983): 1156-1166. Section 4--particularly page 1164.

> <mailto:decision_theory_forum%2Bunsu...@googlegroups.com>.

> To post to this group, send email to
> decision_t...@googlegroups.com

> <mailto:decision_t...@googlegroups.com>.