Rational Choice

Most of classical decision theory was developed over the first sixty years of the twentieth century. The theory focuses on instrumental rationality, that is, on reasoning about how agents can best achieve their desires in light of their beliefs. Decisions take place under three conditions: certainty (outcomes of actions are certain), risk (outcomes are not certain but their probabilities are known, as in some games of chance), and uncertainty (probabilities of outcomes are unknown). There are various ways of handling decision-making under uncertainty, but it is usually reduced to decision-making under risk by using the agent's subjective probabilities, and it will be the focus here.

Figure 1.
SOURCE: Courtesy of the author

Building blocks of the classical theory.

The following example is illustrative: Tom must decide whether to get a flu shot. Allergic to the vaccine, he realizes that getting the shot means he will be sick for a day. He also believes there may be an outbreak of flu that would make him very sick for a week.

Actions, conditions, and outcomes.

Decision theory starts with three fundamental concepts:

1. Actions. These are the options an agent ponders (here getting a flu shot or not). Actions are often represented by rows in a table.
2. Conditions. These are how things turn out independently of actions (e.g., whether there is a flu outbreak or not). Conditions are represented by columns.
3. Outcomes. These are the states that result from actions under various conditions (e.g., getting sick for a day in the absence of a flu outbreak).

Desires and beliefs.

Classical decision theory represents agents as having preferences over outcomes that capture their desires. For example, Tom prefers health to illness, and less illness to more. Preferences are mathematically represented by subjective utility functions, subject to certain constraints (e.g., the expected utility of one outcome is greater than that of a second only if the agent prefers the first to the second). U(A, S) is the utility of the outcome of action A in condition S. Many sets of numbers can reflect the same preferences, as long as the intervals among them reflect analogous relationships among intensities of desires.

Classical decision theory also represents agents as having degrees of belief about conditions. For example, Tom might believe that a flu outbreak is less likely than not. Degrees of belief are mathematically represented by subjective probability functions

Figure 2.
SOURCE: Courtesy of the author

that specify how likely the agent thinks various outcomes would be. P(S|A) represents an agent's degree of belief that condition S will come about given that he performs action A. For example, Tom might surmise that P (flu outbreak | shot) is 0.4. Probabilities are represented by the first member of a pair of numbers in a cell, and utilities by the second number in Figure 2 (which has a hypothetical set of numerical values).

Expected utility.

The subjective expected utility of each action is the sum of products in each cell in the action's row.

• Shot (row one): (.4)(1) + (.6)(−1) = (.4) + (−.6) = −.2
• No shot (row two): (.4)(−6) + (.6)(3) = (−2.4) + (1.8) = −.6

Here the first action has a higher expected utility than the second (in symbols): EU (shot) > EU (not shot), and decision theory reveals that getting a shot is the rational thing to do. There is often a compromise between beliefs and desires; for example, it is frequently often more rational to do something likely to lead to a moderate payoff than to pursue a higher payoff with less chance of success.

There can be more than two actions or situations, and the general formula is:
EU(A) = Σ_i P(S|A) X U(A, S_i)
The fundamental claim of decision theory is that a rational decision is one with the highest subjective expected utility (there may be more than one due to ties). And the centerpiece is a representation theorem proving that any agent whose beliefs and desires conform to certain plausible constraints (e.g., whose preference ranking is transitive) behaves as if she were maximizing expected utility.

Interpretations of classical decision theory.

Some see utility maximization as a descriptive claim; people in fact behave pretty much as the theory says they should. Others see it as a normative claim: a rational person should choose a utility maximizing action.