Recall from the beginning of this chapter that epistemic rationality is reasonableness of belief, whereas practical (or pragmatic) rationality is reasonableness of choices or actions. Decision theory is the branch of philosophy that deals with practical rationality: it investigates what kinds of choices and actions are reasonable. A popular account of practical rationality is Bayesian decision theory, which will be introduced in what follows.
In Bayesian decision theory, a choice between several alternative actions can be represented in a decision matrix—a table in which the rows represent possible choices or actions, and the columns represent a relevant set of mutually exclusive and exhaustive propositions about the world, like this:
W_{1}  W_{2}  W_{3}  

A_{1}  U_{11}  U_{12}  U_{13} 
A_{2}  U_{21}  U_{22}  U_{23} 
A_{3}  U_{31}  U_{32}  U_{33} 
A_{1}, A_{2}, A_{3} (etc.) represent the possible acts you are considering. W_{1}, W_{2}, W_{3} (etc.) are mutually exclusive and exhaustive propositions about the world that are relevant to your deliberations. U_{11}, U_{12}, (etc.) represent the utilities you assign to each possible outcome. That is, they represent how much you value or disvalue each act + proposition pair.
The expected utility of each possible act (A_{1}, A_{2}, A_{3}, …) is calculated by multiplying the utility of each possible outcome times your credence in the corresponding proposition, then adding these numbers. For example, if pr(W_{1}), pr(W_{2}), and pr(W_{3}) are your credences in the propositions W_{1}, W_{2}, and W_{3}, then the expected utility of act A_{1}, given the above decision matrix, is:
U_{11} pr(W_{1}) + U_{12} pr(W_{2}) + U_{13} pr(W_{3})
According to Bayesian decision theory, the rational thing to do is to choose whichever act has the highest expected utility. If there’s a tie, and two options both have the highest expected utility, either option is rational.
Suppose you don’t know whether one of your professors will give a pop quiz tomorrow, but you think it’s 30% likely. In other words, your credence in the proposition that there will be a quiz tomorrow is:
pr(Q) = .3
You are considering whether to study, just in case. The choice is between two possible actions: to study or not to study, that is the question! The propositions Q and ~Q are mutually exclusive and exhaustive propositions relevant to your deliberation. So, we’ll need two rows and two columns in the decision matrix, and there are four possible outcomes to consider:
The following decision matrix represents your situation:
Quiz pr(Q) = .3 
No quiz pr(~Q) = .7 


Study  40  10 
Don’t study  50  20 
What should you do? According to Bayesian decision theory, the practically rational choice is whichever option has the highest expected utility. In this example, the expected utility of studying is:
(40 × .3) + (10 × .7) = 5
The expected utility of not studying is:
(50 × .3) + (20 × .7) = 1
Studying has the highest expected utility. So, according to Bayesian decision theory, studying is the rational thing to do in this situation.