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Expected Utility Theory

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Expected utility theory is a tool aimed to help make decisions amongst various possible choices. It is a way to balance risk versus reward using a formal, mathematical function.

When faced with a number of different choices, expected utility theory recommends that you calculate the expected utility of each choice and then choose the one with highest expected utility.

Utility

Utility is simply a measure of a person's preferences amongst different things. From these preferences (if they are rational!) we can deduce a utility function which represents preferences by order relations between numbers.

This only works if a person's preferences are, in a certain sense, rational. If someone prefers the Angels to the Dodgers, then they shouldn't also prefer the Dodgers to the Angels. And, if they prefer the Angels to the Dodgers, and the Dodgers to the Giants, then they shouldn't prefer the Giants to the Angels. We also require that, for any two things, a person prefers one to the other, or is indifferent between the two.

If a person's preferences are rational in the above sense, then we can define a utility function as follows:

$u\left(\cdot \right)$ is a function that assigns numbers to things (represented by variables $x,y,z,\dots$)

$u\left(x\right)>u\left(y\right)$ if and only if this person prefers $x$ to $y$

$u\left(x\right)=u\left(y\right)$ if and only if this person is indifferent between $x$ and $y$

The numbers assigned by $u\left(\cdot \right)$ should also match how much one thing is preferred to another. If someone assigns $u\left(A\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}g\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}s\phantom{\rule{0.333em}{0ex}}w\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}n\right)=100$ and $u\left(D\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}d\phantom{\rule{0}{0ex}}g\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}s\phantom{\rule{0.333em}{0ex}}w\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}n\right)=1$, then they'd prefer to see an Angels win 100 times more than a Dodgers win.

Example of Utility Functions

Suppose we want to create a utility function for a fan of the "Planet of the Apes" movies. There were five movies in the series: "Planet of the Apes", "Beneath the Planet of the Apes", "Escape From the Planet of the Apes", "Conquest of the Planet of the Apes" and "Battle for the Planet of the Apes".

Our fan likes "Escape" the best out of the five, prefers "Escape" to "Beneath," is indifferent between "Beneath" and "Conquest," prefers "Beneath" to "Planet," and prefers "Planet" to "Battle." Here is a utility function that could represent our fan's preferences:

$u\left(E\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}p\phantom{\rule{0}{0ex}}e\right)=10,u\left(B\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}h\right)=u\left(C\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}q\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}t\right)=8,u\left(P\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}t\right)=5,u\left(B\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}e\right)=1$

One way to think of utility is in terms of how much you would pay for each of these things, or how much these things are worth to you.

Expected Utility Calculations

How appealing a certain choice is depends not only on the payoffs of that choice, but how likely those payoffs are. The multi-million dollar payoff of a lottery is certainly appealing, but it is so unlikely that buying a lottery ticket is virtually a waste of money. Expected utility calculations are meant to balance risk versus reward.

We think of an act (like buying a lottery ticket) as having a number of possible outcomes (in this case, winning or losing). Given a person's utility function (see above) and their degrees of belief in each of the possible outcomes, we can figure out the expected utility of any act. This is done as follows:

Let the act in question be labelled $A$. Let ${o}_{1},{o}_{2},\dots ,{o}_{n}$ be the various possible outcomes of $A$ (there needs to be at least one outcome, but there could be many).

To each outcome ${o}_{i}$ is an associated probability $P\phantom{\rule{0}{0ex}}r\left({o}_{i}\right)$ which measures how likely that outcome is, and a utility $u\left({o}_{i}\right)$ which measures that outcome's spot in this person's preference relation.

The expected utility of $A$ is:

 $E\left(A\right)=u\left({o}_{1}\right)\cdot P\phantom{\rule{0}{0ex}}r\left({o}_{1}\right)+u\left({o}_{2}\right)\cdot P\phantom{\rule{0}{0ex}}r\left({o}_{2}\right)+\dots u\left({o}_{n}\right)\cdot P\phantom{\rule{0}{0ex}}r\left({o}_{n}\right)$

Now, when faced with a choice between multiple acts ${A}_{1},{A}_{2},\dots {A}_{n}$, expected utility theory says that a person should choose the act with the highest expected utility. That is, calculate $E\left({A}_{1}\right),E\left({A}_{2}\right),\dots E\left({A}_{n}\right)$ and then choose the act with the highest associated utility.

Example of Expected Utility Calculations

You are trying to decide what to do tonight, and you have three options. There is a party in your housing complex that you could go to, or you could go to an Anaheim Ducks game, or you could just stay home and watch one of your favorite DVDs.

If you stay home and watch the movie, you know for sure you will receive a utility of 50.

You assign a utility of 200 to a Ducks win, but you also know that they only have a 25% chance of winning tonight. A Ducks loss has a utility of 10 to you.

If you go to the party, there's a 90% chance it will be lame, and the utility of a lame party for you is only 25. There's a 9% chance that the party will have decent music, an outcome that you assign utility 50 to. Finally, there's a 1% chance that you will randomly meet the person of your dreams at the party, which would yield utility 1000.

What should you do? Figure out the expected utility of each of the three acts:

$E\left(m\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}v\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}e\right)=u\left(m\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}v\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}e\right)\cdot P\phantom{\rule{0}{0ex}}r\left(m\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}v\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}e\right)=50\cdot 1.00=50$

$E\left(D\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}k\phantom{\rule{0}{0ex}}s\phantom{\rule{0.333em}{0ex}}g\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}e\right)=u\left(D\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}k\phantom{\rule{0}{0ex}}s\phantom{\rule{0.333em}{0ex}}w\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}n\right)\cdot P\phantom{\rule{0}{0ex}}r\left(D\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}k\phantom{\rule{0}{0ex}}s\phantom{\rule{0.333em}{0ex}}w\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}n\right)+u\left(D\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}k\phantom{\rule{0}{0ex}}s\phantom{\rule{0.333em}{0ex}}l\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}e\right)\cdot P\phantom{\rule{0}{0ex}}r\left(D\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}k\phantom{\rule{0}{0ex}}s\phantom{\rule{0.333em}{0ex}}l\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}e\right)$

$=200\cdot 0.25+10\cdot 0.75=57.5$

$E\left(p\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}y\right)=u\left(l\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}e\right)\cdot P\phantom{\rule{0}{0ex}}r\left(l\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}e\right)+u\left(g\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}d\phantom{\rule{0.333em}{0ex}}m\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}c\right)\cdot P\phantom{\rule{0}{0ex}}r\left(g\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}d\phantom{\rule{0.333em}{0ex}}m\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}c\right)+u\left(d\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}s\right)\cdot P\phantom{\rule{0}{0ex}}r\left(d\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}s\right)$

$=0.90\cdot 25+0.09\cdot 50+0.01\cdot 1000=37$

Based on these calculations, the best choice is to go to the Ducks game.