Suppose an agent maintains a loss function and a convex set of probability distributions. Because an act is a function from states to consequences, the expected loss of an act can be calculated with respect to any distribution in the set of distributions. And now the agent can compare acts by comparing expected loss.
If an act a1 has smaller expected loss than another act a2, no matter which distribution the agent picks from his beliefs, then a1 has to be better than a2.
But suppose that the agent picks the distributions and notices that a1 has expected loss that is sometimes smaller, sometimes larger than the expected loss of a2. The agent concludes that a1 and a2 are not comparable with respect to his beliefs: for all the agent knows, a1 is not better than, worse than or equal to a2. The agent is indeterminate with respect to a1 and a2.
A Bayesian agent can always say that one option is better than, worse than, or equal to another option. Our agent here may be in a different situation, in an indeterminate state with respect to some acts.
When the agent is represented by a partial order of preferences (and therefore a set of probability distributions), it is not clear how the agent will choose between alternatives that are incomparable. An example clarifies the problem.
Suppose the agent has three alternatives, a1 (go to the park, a2 (go to the movies), and a3 (stay home. There are two states of nature, (sunny) and (cloudy).
The agent has a convex set of probability distributions:
Consider a utility function defined like this:
|a1 ( park)||10||-10|
|a2 ( market)||-5||4|
|a3 ( home)||0||0|
For a fixed value of , we have:
Figure 3 shows a picture of the expected utilities of the acts as varies:
If the agent has no preference among the possible values of , then there is no clear ranking of decisions. There is an interval of possible expected loss for a1, a2 and a3. As varies, these intervals overlap! To see this, consider:
The rational agent has freedom to choose among decisions that are incomparable by expected utility. As far as the agent is concerned, a1, a2 and a3 are incomparable; the agent needs some advice in order to choose a definite action.
There is considerable debate about how an agent makes a decision using sets of probabilities. Some representative examples:
Take the example above. For Levi, the agent excludes dominated alternatives (like a3) and then looks at the worst situation in each decision, and picks the decision that presents the best worst case. Decision a1 can lead to an expected utility of -4; decision a2 can lead to an expected utility of -1.3; therefore a2 is better. This approach is non-Bayesian, as can be seen by dropping a1 from consideration; if this happens, a3 is the recommended decision. The exclusion of a1 leads to reversal of preferences (a2 was preferred to a3, but now a3 is preferred to a2), something inconceivable in Bayesian theory. Now, for the -minimax approach, a3 is the best option since it has the best worst case.