Expected utility

There are at least two senses in which one can be rational or irrational. One has to do with belief and is truth oriented. In this sense, a person is rational if her/his beliefs are held to the degree of confidence justified by truth-oriented norms of rational belief. Logic is the science that seeks to discover truth-oriented norms of rational belief.

The other sense of rationality has to do with action and is ends oriented. In this sense, we regard rational a person as rational acting in a way that maximizes this person's ends (these may be self interested or altruistic).

Actions are motivated by truth oriented beliefs and ends oriented desires. We act in the belief that our action will further our ends. But more often than not, we must act in the face of significant uncertainty. This is especially apparent when we recognize that doing nothing often amounts to maintaining the status quo which is itself an action.

What should one do when the outcome of a possible action is uncertain? Should one perform the action or not? You might be inclined to say that in the face of sufficient uncertainty, one should not act at all. But this strategy can easily turn out to be irrational. Consider the rabbit who opts not to run for the safety of his hole because of the uncertainty over whether or not the noise he just heard in the bushes was a hunter.

Some degree of uncertainty attends all of our actions. In the realm of business, the most profitable actions are typically taken with a higher degree of uncertainty. Of course, the greater the degree of uncertainty attending the profitable outcome of an action, the greater the risk of an unprofitable outcome. So how does one weigh the potential risks of an action against its possible costs or benefits in the face of significant uncertainties. The notion of the expected value of an action provides us with a method for deciding how to act in the face of uncertainty.

The expected value of an action equals the sum of the products of the probabilities and values for each possible outcome.

Let O1, O2, O3, . . . stand for possible outcomes of an action.

Let P(On) represent the probability assigned to outcome On.

And let V(On) represent the value of outcome On. Note that the value of the outcome of an action can be positive or negative.

The expected value of an action A can then be represented as follows

Expected value of A = (V(O1)*P(O1)) + (V(O2)*P(O2)) + (V(O3)*P(O3)) . . . + (V(On)*P(On))

In order to calculate the expected value of an action we must first distinguish all of its possible outcomes and assign probabilities and values to each. Often we can't assign numerical values to the probabilities or values of the outcomes of actions. But we can illustrate the notion of expected utility with a case where we can assign simple numerical values to the probabilities and values of the outcomes of a particular action. Consider the following scenario:

The action you are considering is betting one dollar on getting a six on a single roll of a fair six sided die. The payoff is $100 if six comes up. Otherwise, you lose your dollar.

There are only two possible outcomes for this bet. One is that you lose. That outcome has a negative value of one dollar and a probability of 5/6. The other possible outcome is that you win. This outcome has a positive utility of $99 ($100 minus the dollar you bet) and a probability of 1/6. Plugging these values into our formula for calculating expected value yields the following:

(-1)(5/6) + (99)(1/6) = 94/6

So the expected value of taking the bet is 94/6 or about $15.66 expressed in dollars.

We don't yet know whether or not you should accept the bet. The practically rational agent will perform the action that has the highest expected value of all available alternative actions. Whether or not you should accept this bet depends on what other actions you have the option of taking with your dollar. A bet with a bigger payoff and or better odds might yield a higher expected value yet. If such a bet is available, a rational agent (with only one dollar) would pass on the bet we've describe and take the one with the highest expected utility.