June 17, 2011 The Effect of Exchange Rates on Statistical Decisions Summary: June 17, 2011 The Effect of Exchange Rates on Statistical Decisions Mark J. Schervish, Teddy Seidenfeld, and Joseph B. Kadane Statistical decision theory, whether based on Bayesian principles or other concepts such as minimax or admissibility, relies on the idea of minimizing expected loss or maximizing ex- pected utility. Loss and utility functions are generally treated as unitless numerical measures of how costly or valuable are the various consequences of potential decisions. In this paper, we address directly the issue of the units in which loss and utility are settled and the implications that those units have on the rankings of potential decisions. The simplest example is to imagine that the loss will be paid in units of some currency. If there are multiple currencies available for paying the loss, one must take explicit account of which currency is used as well as the exchange rates between the various available currencies. KEY WORDS: Currency; numeraire; scoring rule; state-dependence; utility. 1. INTRODUCTION Statistical decision theory is generally based on minimizing a loss function or maximizing a utility function, whether that basis stems from an axiomatic foundation or is merely posited as a principle. The corresponding loss function or the utility function is generally assumed to be unitless. In the various axiomatic derivations of expected utility theory (see, e.g., Anscombe and Aumann 1963; Savage 1954) a unitless utility is derived, but its argument list includes explicit prizes or consequences of decision making which have value to the decision maker. For exam- Collections: Mathematics