Minimax and the value of information

In his discussion of minimax decision rules, Savage (The foundations of statistics, Dover Publications Inc., Mineola 1954, p. 170) presents an example purporting to show that minimax applied to negative expected utility (referred to by Savage as “negative income”) is an inadequate decision criterion for statistics; he suggests the application of a minimax regret rule instead. The crux of Savage’s objection is the possibility that a decision maker would choose to ignore even “extensive” information. More recently, Parmigiani (Theor Decis 33:241–252, 1992) has suggested that minimax regret suffers from the same flaw. He demonstrates the existence of “relevant” experiments that a minimax regret agent would never pay a positive cost to observe. On closer inspection, I find that minimax regret is more resilient to this critique than would first appear. In particular, there are cases in which no experiment has any value to an agent employing the minimax negative income rule, while we may always devise a hypothetical experiment for which a minimax regret agent would pay. The force of Parmigiani’s critique is further blunted by the observation that “relevant” experiments exist for which a Bayesian agent would never pay. I conclude with a discussion of pessimism in the context of minimax decision rules.