Philosophers disagree about what probability is. Somewhat roughly put, there are two camps in this debate, objectivists and subjectivists. Objectivists maintain that statements about probability refer to facts in the external world. If you claim that the probability is fifty per cent that the coin I hold in my hand will land heads up when tossed, then you are – according to the objectivist – referring to a property of the external world, such as the physical propensity of the coin to land heads up about every second time it is tossed. From an intuitive point of view, this seems to be a very plausible idea. However, subjectivists disagree with this picture. Subjectivists deny that statements about probability can be understood as claims about the external world. What is, for instance, the probability that your suitor will ask you to marry him? It seems rather pointless to talk about some very complex propensity of that person, or count the number of marriage proposals that other people make, because this does not tell us anything about the probability that you will be faced with a marriage proposal. If it is true that the probability is, say, 1/2 then it is true because of someone's mental state.

According to the subjective view, statements about probability refer to the degree to which the speaker believes something.

Jane is having a romantic dinner with her fiancé in a newly opened French bistro in Santa Barbara, California. After having enjoyed 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 as long as it is cooked by a first-class chef. However, she has some vague suspicions that this may not be the case in this particular restaurant. The starter was rather poor and cooking monkfish is difficult. Virtually any restaurant can serve edible hamburgers, however.

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) she 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.

Before you make a decision you have somehow to determine what to decide about. Or, to put it differently, you have to specify what the relevant acts, states and outcomes are. Suppose, for instance, that you are thinking about taking out fire insurance on your home. Perhaps it costs $100 to take out insurance on a house worth $100,000, and you ask: Is it worth it? Before you decide, you have to get the formalisation of the decision problem right. In this case, it seems that you face a decision problem with two acts, two states, and four outcomes. It is helpful to visualise this information in a decision matrix; see Table 2.1.

To model one's decision problem in a formal representation is essential in decision theory, since decision rules are only defined relative to such formalisations. For example, it makes no sense to say that the principle of maximising expected value recommends one act rather than another unless there is a formal listing of the available acts, the possible states of the world and the corresponding outcomes. However, instead of visualising information in a decision matrix it is sometimes more convenient to use a decision tree. The decision tree in Figure 2.1 is equivalent to the matrix in Table 2.1.

The square represents a choice node, and the circles represent chance nodes. At the choice node the decision maker decides whether to go up or down in the tree.

The following decision problem appeared some years ago in Marilyn vos Savant's Ask Marilyn column in the Parade Magazine.

The Monty Hall problem has a simple and undisputable solution, which all decision theorists agree on. If you prefer to win a brand new car rather than a goat, then you ought to switch. It is irrational not to switch. Yet, a considerable number of Marilyn vos Savant's readers argued that it makes no difference whether you switch or not, because the probability of winning the car must be 1/3 no matter how you behave. Although intuitively plausible, this conclusion is false.

The point is that Monty Hall's behaviour actually reveals some extra information that makes it easier for you to locate the car.