# Why can't subjective utilities take probabilities into account?

If my understanding of expected utility theory is correct,

• it is rational for a decision maker to have subjective utilities for objective consequences. For example, it can be rational for a decision maker to value 5 dollars twice as much as 4 dollars
• however, it is not rational for a decision maker to value a 100% chance of 5 dollars twice as much as an 80% chance of 5 dollars

That is, probabilities must be "outside" of- not taken into account by- subjective utilities.

Is this understanding correct, and if so, why is the theory structured this way?

Note: My understanding is based on Hastie & Dawes (2010), Rational Choice in an Uncertain World.

You certainly did not understand it correctly, because the statement that the second case is either rational or not cannot be stated based only on the information you provided.

The rationality of the second statement depends on the specific decision-maker's attitude to risk only. It is not possible to infer it based only on the information you are providing.

1) Let's consider the first case: "You value 5 dollars twice as much as 4 dollars"

A possible decision situation would be a day in which you are so starving that you might well eat one hamburguer as two. However, one hamburguer costs 4 dollars in the store where you are; but there is a promotion in which you can buy two hamburgueres for the price of 5 dollars.

You could represent this situation stating that you value the same: a) 4 additional dollars, in relation to your current status quo; b) as the 1 additional dollar, in relation to the moment after receiving the previous 4 dollars.

In this situation, there is no uncertainty involved about the consequences of your decision. If you pay 4 dollars, you eat one hamburguer. If you pay 5 dollars, you eat two hamburguers. In certain situations, you can model your preference by means of a value function (rather than an utility function).

A value function measures only how much you value (i.e. how much you consider attractive, desirable, preferable) distinct consequences, so that consequence levels with higher values mean that you prefer them in relation to consequence levels with lower values.

Utility not only measures this value, but it also measures your attitude in relation to uncertainty (risk). Since this is a certain situation, your utility function is exactly the same as your value function.

2) Now for the second case: "Why is not also rational to value 5 dollars twice as much as an 80% chance of 5 dollars"

This situation is distinct from the previous one, because now one of the alternatives is uncertain, there is only a 80% chance of occurring.

A possible decision situation would be the case in which the store has offering a promotion in which you could take a 20% chance of eating your first hamburguer for free, but if you lose you would have to pay 5 dollars for it.

The only possible way that you could "value 5 dollars twice as much as 80% chance of 5 dollars" is if you are indifferent between: a) paying 4 dollars for eating one hamburguer (as in the previous case); b) or accepting this new promotion.

If you are indifferent, then IT IS RATIONAL that you value 5 dollars twice as much as an 80% chance of 5 dollars! In this case, it is said that you are risk-neutral, and once more, your utility function is exactly the same as your value function.

If you prefer choice a) then it means you are not willing to risk having to pay 5 dollars for the hamburguer. Then, you no longer value 5 dollars twice as much as an 80% chance of 5 dollars! You now value 5 dollars MORE THAN TWICE AS an 80% chance of 5 dollars! In this case, it is said that you are risk-averse, and now your utility function is distinct from your value function.

Likewise, If you prefer choice b) then, once more, you no longer value 5 dollars twice as much as an 80% chance of 5 dollars! You now value 5 dollars LESS THAN TWICE AS an 80% chance of 5 dollars! In this case, it is said that you are risk-seeking, and again your utility function is distinct from your value function.

PS: Note that utilites (and values) are always subjective, so the term is redundant (that's why I did not mentioned it at all)