Decision Trees and Utility Theory Chapter 4 Decision Trees and Utility Theory

I am going to focus on the Utility Theory component of this chapter I am going to focus on the Utility Theory component of this chapter. Before we do so let’s consider an example. Say one option for you is to take a bet that pays $5,000,000 if a coin flipped comes up tails and you get $0 if the coin comes up heads. The other option is that you will get $2,000,000 with certainty. (Say your grandmother will give you $2,000,000 if you do not bet.) EMV of the bet = .5(5,000,000) + .5(0) = 2,500,000 EMV sure deal = 1(2,000,000) = 2,000,000 Choosing the option with the highest EMV has been our decision rule. But, now with a sure bet we may decide to avoid the risky alternative. Would you take a sure $2,000,000 over a risky $5,000,000? Is that your final answer?

Utility Theory is a methodology that incorporates our attitude toward risk into the decision making process. It is useful to employ a graph like this in our analysis. In the graph we will consider a rule or function that translates monetary values into utility values. The utility values are our subject views of preference for monetary values. Typically we assume higher money values have higher utility. Utility value Monetary value

Say we observe a person always buying chocolate ice cream over vanilla ice cream when both are available and both cost basically the same, or even when chocolate is more expensive and always when chocolate is the same price or cheaper. So by observing what people do we can get a feel for what is preferred over other options. When we assign utility numbers to options the only real rule we follow is that higher numbers mean more preference or utility. Even when we have financial options we can study or observe the past to get a feel for our preferences. The book we use goes through an elaborate story for assigning utility values. It is just one story and is valid, but other ways have validity as well. Our point is to become aware of the method and see how the method works, assuming the values assigned are realistic to the problem at hand.

In general we say people have one of three attitudes toward risk In general we say people have one of three attitudes toward risk. People can be risk avoiders, risk seekers , or indifferent toward risk. Utility value Utility values are assigned to monetary values and the general shape for each type of person is shown at the left. Note that for equal increments in dollar value the utility either rises at a decreasing rate(avoider), constant rate or increasing rate. Risk avoider Risk indifferent Risk seeker Monetary value

Here we show a generic example with a risk avoider Here we show a generic example with a risk avoider. Two monetary values of interest are, say, X1 and X2 and those values have utility U(X1) and U(X2), respectively Utility U(X2) U(X1) $ X1 X2

Say the outcome of a risky decision is to have X1 occur q% of the time and X2 occur (1 – q)% . Then the EMV is q(X1) + (1 – q)(X2). The expected utility of the risky decision is found in a similar way and without proof I tell the expected utility is Utility U(X2) U(X1) EU $ EMV X1 X2 along the straight line connecting the points on the curve directly above the EMV for the decision. We have the expected utility as EU = qU(X1) + (1 – q)U(X2)

The decision maker may have an option that is certain The decision maker may have an option that is certain. If so, the EU is simply the utility along the utility curve. So in this diagram we see that any sure bet greater than Y has an expected utility greater than the expected utility of the risky option. Utility U(X2) U(X1) EU $ Y EMV X1 X2

Utility theory then suggests that the alternative that is chosen is the one that has the highest expected utility. Example: Say a risky alternative has 45% chance of getting $10,000 and a 55% chance of getting -$10,000. Say U(10,000) = .3 and U(-10,000) = .05 and the U(0) = .15 and say a certain alternative has a value of 0. EU of risky deal = .45(.3) + .55(.05) = .1625 EU of the certain deal = 1(.15) = .15 The person will choose the risky deal.

Another Example Say Utility U = square root of X, where X is a dollar amount received by a person. Then U(4) = 2 and U(16) = 4, for example. Say a risky option will pay 4 50% of the time and 16 50% of the time. The expected value is 10 because .5(4) + .5(16) = 10 and the expected utility is 3 because .5U(4) + .5U(16) = .5(2) + .5(4) = 3. Now, if there is an option that will pay more than 9 with certainty, than the certain option is better. Let’s see this on the next slide.

U(x) U(16)=4 U(x) EU = 3 U(4)=2 4 9 10 16 x Any certain option above 9 gives a utility value greater than the expected utility of the uncertain option.

Assignment 4 – 10 points Say utility for a person is represented by the function U = 5 times square root x What is u for x from 1 to 25 (give me a list that has the following – if x = 1, u =‘s ?, if x = 2 u =? and so on through x = 25) ? Say an individual has an investment opportunity that will pay 4 50% of the time and 16 50% of the time. What is the expected monetary value of the option? What is the expected utility of the option? Explain why an alternative investment opportunity that would pay 9.50 with certainty would be better than the risky opportunity in b) for the person.