DADSS Lecture 11: Decision Analysis with Utility Elicitation and Use

Administrative Details Project proposal feed back soon. Homework 6 due March 13 after Spring break. Exam 1 – results

Exam 1 Results

An Opportunity Toss a fair coin If it comes up heads, you win $2 If it comes up tails, you toss again If it comes up heads on the 2 nd toss, you win $4, otherwise you toss gain Heads on the third toss: $8 Heads on the fourth toss: $16 … Heads on the n th toss: $2 n What would you be willing to pay to play this game?

The Concept of Utility Expected value is not entirely satisfying Risk preference Non-monetary values Satiation Utility is measure of: happiness, well-being, satisfaction, welfare… In practical terms: a subjective index of personal value

The St. Petersburg Paradox Daniel Bernoulli, 1738 What is the expected value of the coin toss gamble? Probability of H: 0.50 Probability of TH: 0.25 Probability of TTH: Probability of TTT … H: (½) n

The St. Petersburg Paradox The expected value of the game is infinite, but virtually no one is willing to pay more than $10 for it! Risk Aversion

Utility’s Conceptual Origins A quantitative measure of personal value Preferences Risk Attitudes/Preferences/Tolerance Most interesting decisions have consequences that are either non-monetary or not entirely monetary All-monetary consequences still require considering satiation and risk aversion

Satiation Your net worth is $100,000 and you inherit another $100,000 You’re ecstatic! You’re Bill Gates and you inherit $100,000 You don’t even notice Decreasing marginal utility from consumption/income/wealth

Risk Preference Why do some people buy insurance? Lottery tickets? …both? Risk Seeking: Willing to pay > expected value (accept <) Risk Neutral: Only willing to transact at ( = ) expected value Risk Averse: Willing to pay )

Utility as an Expression of Value For now, we will deal only with utility functions over final wealth Later we’ll consider utility functions that reflect preferences for a variety of factors Utility is a function that converts monetary values into a unique, subjective “desirability index” Probabilities are outside of the utility function!

Utility Function Requirements Utility is unique to the decision maker Mary’s utility for Lottery A is 100; Bill’s utility for Lottery A is 50. Mary likes Lottery A twice as much as Bill? NO! Comparing utilities opens philosophical black holes into other universes. Avoid it.

Utility Function Properties Utility functions are unique to an affine (positive linear) transformation (it is unit-free) U(x) = 4x 1/2 + 7 U(x) = 8x 1/ Index Neutrality: The end points of your indexing system don’t matter For simplicity, we’ll scale all utility functions from 0 to 1 U(min) = 0, U(max) = 1 What matters is the curvature of the line between those two points reflects your risk tolerance These are equivalent

Utility Function Properties Separable Utility Functions Comparing U(A+B) with U(A)+U(B) What about 50% chance of A and 50% chance of B? EU = 0.5 U(A) U(B) Why not U(0.5 A B)?

Where Does Utility Come From? Elicitation Consider the following utility function: Does that accurately express your preferences?

Elicitation Payoffs now as expected utilities instead of expected values But…we will still face problems that involve monetary values How do we transform monetary values into utilities?

Common Utility Functions x = Final Wealth u(x) = Utility of Final Wealth Quadratic (0 < k < 1/2w) Logarithmic Negative Exponential General Exponential

Elicitation of Utility The variables other than x in the previous functional forms: Scaling factors Coefficients of risk aversion Elicitation of utility functions: Selecting an appropriate functional form Estimating the scaling factors and/or risk aversion coefficients These are many ways of eliciting a person’s utility function

The Curve-Fitting Approach: Assessing Outcomes 1. Pick two end-point values and assign arbitrary utility values to them, giving the “better” value the higher utility $0 and $1,000 U(0) = 0, U(1000) = Present the decision maker with a choice using an arbitrary intermediate value Alternative 1: $1,000 with probability.5 and $0 with probability 1 –.5 Alternative 2: $x for certain 3. Find the value of $x that makes the decision maker indifferent between Alt 1 and Alt 2. This implies: U(x) =.5 × U(1000) + (1 –.5) × U(0) 4. For example, suppose $x = $350 U(350) = 0.5 × × 0 = Repeat as necessary for other intermediate values

The Curve-Fitting Approach: Assessing Probabilities 1. Again, pick two arbitrary values and assign arbitrary utility values 1. $0 and $1, U(0) = 0, U(1000) = Present the decision maker with a choice using an arbitrary intermediate value 1. Alternative 1: $500 for certain 2. Alternative 2: $1,000 with probability p and $0 with probability 1 – p 3. Find the probability that makes the decision maker indifferent 1. U(500) = p × U(1000) + (1 – p) × U(0) 4. Suppose p = U(500) = 0.6 × × 0 = Repeat as necessary for other intermediate values

The Curve-Fitting Approach Using the original assumptions U(0) = 0 U(1000) = 100 …and our derived information U(500) = 60 …and continuing the process… We can trace out a curve and then use Excel’s curve-fitting function (right-click on the graphed points) to determine an appropriate functional form

The Curve-Fitting Approach 1. Excel will produce a best-fit curve Sometimes gives insight, but often doesn’t. 2. Assume a curve with a single (risk-aversion) parameter, then estimate that parameter. Use Excel’s Solver to minimize the sum of the squared errors between your fitted curve and your elicited curve by changing the risk aversion parameter

The Curve-Fitting Approach Candidate Function #1: Exponential Utility a = Candidate Function #2: Negative Exponential R = -351

Response to Risk Aversion

Working with Utility At this point, we will assume in the future that we have used some method to elicit the decision maker’s utility function In most problems, you will now be given a utility function that you are to assume reflects the decision maker’s preferences E.g., u(x) = 1 - e -ax with a = 0.03

The Party Problem with EV Outdoors Indoors Sun Rain Sun Rain EV = 120 Wealth Basing on EV, select Outdoors for $120

The Party Problem with Utility Outdoors Indoors Sun Rain Sun Rain EU = 0.96 Wealthu(Wealth) u(x) = 1 - e -  x  = 0.03 EV selects Outdoors, but EU suggests Indoors

Consequences of Utility With utility, because it recognizes risk preference, you can often get very different answers than with expected value In this version of the party problem, risk aversion makes it very difficult to take the risk of going outside and getting $0

Certainty Equivalence Suppose we face a lottery (a risky choice): How much would we have to receive (or be willing to pay) in order to give up/avoid the risky choice? What amount, for certain, is equivalent to the utility of the risky option? This amount is known as a certainty equivalent, Plays a very important role in decision making with utility

Certainty Equivalence Our utility function elicitation procedure used certainty equivalence CE will also be used to determine the value of information with utility Take the “Outdoors” choice resulting in the following chance node: EU = 0.6 WealthU(Wealth)

Certainty Equivalence 1. What amount, received for certain, would I have to receive in order to be indifferent between the certain amount (payoff with probability 1) and the lottery? 2. Invert the utility function and solve for x

Certainty Equivalence This decision maker is indifferent between receiving $30.54 for certain and a 60% chance of receiving $200. This gives us another way of conceptualizing risk aversion Risk Averse: CE < EV Risk Neutral: CE = EV Risk Seeking: CE > EV

Certainty Equivalence A quick, useful summary statistic With exponential utility, there are several “short-cuts” possible through the use of certainty equivalents

Problems with Utility Don’t be fooled into thinking that utility, as wonderful as it is, solves everyone’s problems Measurement WTP/WTA Comparability/Pseudocertainty Prospect Theory (gains/losses & framing) Utility function “loading up”

The Allais Paradox Which of the following do you prefer? Alternative 1: Receive $1 million with certainty Alternative 2: Receive $2.5 million with probability 0.10, $1 million with probability 0.89, and $0 with probability 0.01 Most people prefer Alternative 1 (either is OK – remember, these are expressions of personal preference)

The Allais Paradox A preference for 1 over 2 in utility terms implies that Set U($2.5 million) = 1 and U($0) = 0 Then, 0.11 U($1 million) > 0.10 OK… U($1 million) > 0.10 U($2.5 million) U($1 million) U($0)

The Allais Paradox Which of the following do you prefer? Alternative A: $1 million with probability 0.11, otherwise $0 Alternative B: $2.5 million with probability 0.10, otherwise $0 Most people would prefer B here (again, either is OK – it’s personal preference…) But wait – these alternatives are equivalent (1 = A, 2 = B)! People changed their preferences! Unless you were consistent in your choices, you have a problem (or utility theory has a problem)

The Allais Paradox What’s going on? In the first comparison, why accept a 1-in-100 chance of getting $0 when you can get $1 million for sure? In the second comparison, the most likely outcome is nothing, so why not take an additional 1-in-100 risk of getting nothing in exchange for increasing your potential payout 2 ½ times?