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From Preference to Utility
This article is for those who are willing to explore the mathematics involved in computing the utility function u. It is still quite intuitive though. High school math is sufficient.
Recall that a preference ordering is a list of pairs (x,y) such that x is more preferred than y [1]. We are going to try to find a corresponding utility function u. However, we first need several assumptions about the preference ordering and about the person which created the ordering:
- The person must be able to judge, for every two options, which one is preferable, or whether they are equally preferred.
- The preference ordering must be transitive.
- The person must be able to trade off values against each other.
The first two conditions were explained in [1].
A relatively intuitive way to explain the third condition is that it is equivalent to the following: If x is more preferable than y, and y is more preferable than z, then there is some gamble involving x and z (i.e. outcome x is likely with probability p and outcome z with probability 1 - p) which is equally preferred as y.
We are going to use this last explanation extensively in order to compute u.
Intuition
Before we continue lets explore an intuitive example. Consider Jerry, who faces a decision of buying a lottery ticket. If he does not buy a ticket then the outcome he gains is zero dollars, if he buys a ticket (which costs 10 dollars) then there is a small probability, p, that he will win a million dollars. So there are three outcomes
- x = 1000,000$ - 10$ = 999990$
- y = 0
- z = -10$
Clearly, for Jerry, x is more preferable than y, and y is more preferable than z. Is there a probability p such that that Jerry finds the possibilities of buying and not buying the ticket equally preferable?
Obviously, if the probability of winning is very high, then Jerry would buy a ticket, whereas if the probability is extremely low then Jerry would not buy a ticket. In an extreme case, if p = 0, then there is no chance of winning, and no person would throw away 10$ for no reason.
Intuitively, since there is a p such that Jerry would buy a ticket, and a different p such that Jerry would not buy a ticket, there is some value of p in between the values of the two previous p, for which Jerry finds it difficult to decide, since both options are equally preferable.
Utility
Lets translate this into the language of utility. Suppose we have outcomes x,y,z such that x is more preferable than y, and y is more preferable than z. So we know that the utility function would be such that:
u(x) > u(y) > u(z)
Furthermore, we assume that there is some probability p, such that y is equally preferred to a gamble between x and z ( where outcome x is of probability p). Using the utility function, this could be expressed as:
u(y) = p * u(x) + (1 - p) * u(z)
Note that our intention is for u to represent "preference" in a numeric form. So saying that two options are of equal preference means that their utilities are equal. On the left hand side we have the utility of y, and on the right hand side, the utility of the gamble.
This is an important equation since it will allow us to determine the value of the utility function for new outcomes.
Calculating Utility
Suppose that for the outcomes of x ,y, z, we already know the value u for two of the three outcomes. If, in addition, the person (e.g. Jerry) provides us with p, we can use the equation
u(y) = p * u(x) + (1 - p) * u(z)
to calculate the utility of the remaining outcome.
Suppose we want to find the utility of a new outcome, w. All we need is:
- The utility of two outcomes w1,w2.
- The person's preference ordering of w1,w2,w and an additional probability p, also provided by the person
Finding Two Utility Values
We have to start from somewhere. How will we be able to determine a utility for two first outcomes. Note that we only need two to start, and from there on, we could calculate the rest of the utilities using the first two.
One candidate to consider is an outcome in which nothing happens. So we will set the utility of nothing as:
u(Nothing) = 0
One might argue that an outcome of Nothing may have different values depending on the context. For example, if you expected to have a surprise birthday, and Nothing happened, then you would be disappointed and it seems that this should have negative utility. But another way to look at it is that it is not Nothing which caused negative utility, but the fact that you developed an expectation which wasn't fulfilled.
We need another utility. Lets just randomly choose an outcome. For example, the outcome of eating an Ice Cream Portion. How are we going to determine the utility of that? What we are going to do is to set the utility of an ice cream portion (ICP) as:
u(ICP) = 1
Essentially, we are using u(ICP) as the standard measuring unit for preference, in the same way that the standard meter is used for measuring distance. The size of such units are determined arbitrarily. Their power is in that they are used consistently, and thus allow comparison between different outcomes or different distances.
Now that we have determined two utilities, we demonstrate how to determine utilities of other outcomes.
u(Pizza)
Suppose Jerry likes Pizza slightly more than an Ice Cream Portion. So:
u(Pizza) > u(ICP) > U(Nothing)
Now lets use the equation:
u(ICP) = p * u(Pizza) + (1 - p) * u(Nothing)
We ask Jerry for the value of p. Since Jerry only slightly prefers pizza, he is probably not willing to make a gamble where the chances for him getting Nothing are high. So Jerry would provide a high value of p, for example 4/5 .
So we get
u(ICP) = p * u(Pizza) + (1 - p) * u(Nothing)
1 = p * u(Pizza) + (1 - p) * 0
1 = p * u(Pizza)
u(Pizza) = 1/p = 1 / (4/5) = 5/4 = 1.25
So u(Pizza) = 1.25 which is indeed slightly more than the utility of an Ice Cream Portion.
u(Vacation)
Like most of us, Jerry prefers a vacation much much more than an Ice Cream Portion. But we still have
u(Vacation) > u(ICP) > U(Nothing)
Using the equation we get:
u(ICP) = p * u(Vacation) + (1 - p) * u(Nothing)
Now one might complain that it is unreasonable to ask for Jerry to provide a value for p. One might claim that an ice cream portion is so insignificant in value when compared to a vacation, that no matter what the value of p is, it is always better to take the gamble.
Supposedly, we could solve this by using p = 0, but this will not allow us to compute u(Vacation) (try to assign p = 0 in the equation and see what you get).
Even though we cannot use p=0, we could use a p which is very small. We could use a p which is the probability that somebody wins a national lottery week after week, for an entire year. Even though this is not zero, we assume there is such a p for which Jerry would prefer the Ice Cream Portion, and therefore there is a different p, for which Jerry would be indifferent as to whether or not to take the gamble. Lets assume p = 1/100,000 . So:
u(ICP) = p * u(Vacation) + (1 - p) * u(Nothing)
1 = p * u(Vacation)
u(Vacation) = 1/p = 1/(1/100,000) = 100,000
So u(Vacation) = 100,000 , which is significantly larger than u(ICP).
u(Peanut)
Jerry likes a (single) Peanut less than an Ice Cream Portion, so
u(ICP) > u(Peanut) > u(Nothing)
Using the equation, we get:
u(Peanut) = p * u(ICP) + (1 - p) * u(Nothing) = p * 1 + (1 - p) * 0 = p
Suppose that Jerry considers a single Peanut of little value. So he would likely prefer the gamble between the Ice Cream Portion and Nothing. It would take a small value of p for Jerry to consider the Peanut and the gamble of equal preference. So if p = 0.1 then u(Peanut) = p = 0.1 which corresponds to our assumption that although a single Peanut is better than Nothing, it is much less than an Ice Cream Portion.
u(Tack)
Here we consider an outcome of negative utility: Jerry accidentally stepping on a small nail (a Tack). In this case:
u(ICP) > u(Nothing) > u(Tack)
Using the equation, we get:
u(Nothing) = p * u(ICP) + (1 - p) * u(Tack)
0 = p + (1 - p) * u(Tack)
u(Tack) = -(p/(1-p))
Assume that Jerry considers Nothing as preferable as a gamble between an Ice Cream Portion and a Tack only when the probability of Ice Cream is quite high, for example, p = 9/10. Then we get:
u(Tack) = -( 9/10 /(1-(9/10))) = - ((9/10)/(1/10)) = -9
So the utility of the Tack outcome is negative.
Conclusion
We have shown how, using a preference order, and some additional information from the person, we can construct a utility function, which can later be used in making decisions involving the preferences of the person.
Note that although the explanation was quite detailed, as we used concrete examples and actual numbers, it is probably unrealistic to employ this method in practice, since it requires a tremendous amount of assessments from a person.
What we are actually trying to show is that it is theoretically possible to build a utility function. Without this, it would be difficult to justify our presentation of rational choice using utility.
REFERENCES
[1] EverDawn, "Rational Choice", 2b_rational.html
