An Introduction to Decision Theory

Jane is having a romantic dinner with her fiancé in a newly opened French bistro in Santa Barbara, California. After enjoying a vegetarian starter, Jane has to choose a main course. There are only two options on the menu, Hamburger and Lotte de mer. Jane recalls that Lotte de mer means monkfish, and she feels that this would be a nice option so long as it is cooked by a first-class chef. However, she has some vague suspicions that this may not be the case at this particular restaurant. The starter was rather poor and cooking monkfish is difficult. She knows that almost every chef can make a decent hamburger.

Jane feels that she cannot assign any probability to the prospect of getting good monkfish. She simply knows too little about this newly opened restaurant. Because of this, she is in effect facing a decision under ignorance. In decision theory ignorance is a technical term with a very precise meaning. It refers to cases in which the decision maker (i) knows what her alternatives are and what outcomes they may result in, but (ii) is unable to assign any probabilities to the states corresponding to the outcomes (see Table 3.1). Sometimes the term “decision under uncertainty” is used synonymously.

Jane feels that ordering a hamburger would be a safe option, and a hamburger would also be better than having no main course at all. Furthermore, she feels that good monkfish is better than an edible hamburger, but terrible monkfish is worse than having no main course. This ranking can be represented on an ordinal scale, as explained in Chapter 2. Since ordinal scales only preserve the order between objects, any set of numbers will do as long as better outcomes are represented by higher numbers. An example is given in Table 3.2.

Which alternative should Jane choose? To answer this question the decision maker has either to apply some decision rule or provide some other kind of structured reasoning for choosing one alternative over another. This chapter surveys some of the most influential rules for making decisions under ignorance.