Unformatted text preview: EXPECTED UTILITY THEORY VERSUS PROSPECT THEORY
The this lecture will attempt to summarize the most important discoveries in behavioral economics. AEM 414 Lecture 7 EXPECTED UTILITY THEORY VERSUS PROSPECT THEORY The Nobel Prize in economics was awarded in 2002 to Daniel Kahneman (Psychologist) of Princeton and Vernon Smith (Economist) of Charles Mason. Kahneman, along with Amos Tversky (Stanford, deceased), ran hypothetical psychology experiments that tested the theory of rational choice which is called Expected Utility Theory. AEM 414 Lecture 7 EXPECTED UTILITY THEORY VERSUS PROSPECT THEORY They published their results in a now famous paper in 1979 in the journal Econometrica, and called their alternative theory, that is based on a number of anomalies that showed up in testing, Prospect Theory. Vernon Smith is responsible for introducing experimentation into economics. Prior to his work, economics was primarily viewed as observational, like astronomy. Acceptance of experimentation in economics paved the way for widespread acceptance of Prospect Theory. Lecture 7 AEM 414 Expected Utility Theory What would the perfectly rational person do if faced with an uncertain world? The initial attempt to define rational behavior actually looked at Expected Value (EV). Thus, if you are faced with a choice of accepting a coin toss which pays you $10,000 if you win (heads with p=.5 odds) or you pay $10,000 if you lose (tails with 1p=.5 odds), expected value theory says you should look at the probability weighted money outcomes to see if you should accept the bet. Lecture 7 AEM 414 Example Let’s say that you now have 30k in income this year to spend. If you accept the bet, the expected value of the outcome is: EV = .5*(30k + 10k) + .5*(30k – 10k) = 30k Thus, if you compare the expected value of the bet to your certain income of 30k, you should be indifferent! How many of you would really accept this bet? Not half? Lecture 7 AEM 414 Bernoulli’s Solution Daniel Bernoulli solved this problem when he solved a puzzle called the St. Petersburg Paradox. “The expected utility hypothesis stems from Daniel Bernoulli's (1738) solution to the famous St. Petersburg Paradox posed in 1713 by his cousin Nicholas Bernoulli (it is common to note that Gabriel Cramer, another Swiss mathematician, also provided effectively the same solution ten years before Bernoulli). The Paradox challenges the old idea that people value random ventures according to its expected return. The Paradox posed the following situation: a fair coin will be tossed until a head appears; if the first head appears on the nth toss, then the payoff is 2n ducats. How much should one pay to play this game? The paradox, of course, is that the expected return is infinite.” (http://cepa.newschool.edu/het/essays/uncert/bernoulhyp.htm). Lecture 7 AEM 414 Bernoulli’s Solution (cont.) Bernoulli argued that the expected value measure was not correct because each successive increase in the amount of money you have increases your happiness less than the previous increase. This is called the principle of diminishing marginal utility where utility is a numerical measure of happiness. Based on observed behavior (risk aversion) a reasonable utility function is : U = lnW where U=utility, W= wealth, and ln denotes the natural logarithm.
Lecture 7 AEM 414 Bernoulli’s Solution (cont.) If we go back to the “fair bet” example, we can show that most people will turn down even odds of 10k as follows. Expected utility is the probabilityweighted sum of utilities: EU = .5*ln(30k + 10k) + .5*ln(30k – 10k) = .5*(10.60) + .5*(9.90) = 10.25 expected utils. Which compares to a utility of refusing the bet of: EU = 1.0*ln(30k) = 10.31 certain utils. Clearly, a person who would like to be happier would refuse the bet since 10.31 utils is better than 10.25 utils.
Lecture 7 AEM 414 Importance of Differences Note also in the example that the gain in utility of winning an extra 10k is (10.6-10.31) = .29 but the loss in utility of losing 10k is -(10.31-9.90) = -.41. As we will see later from psychology, people are very sensitive to changes. This is why it is so easy to reject a fair bet since the loss in utility if you lose looms much larger than the gain in utility if you win. Note also, that the normal concave utility function implies that people are risk averse so they usually will refuse a fair bet for a large amount of money. However, some people are clearly risk seekers (for example, yours truly). The theory can handle lunatic risk seekers, they just have convex utility functions and will accept any fair bet they find and some that are not so fair!
AEM 414 Lecture 7 The natural log is a reasonable utility function for a risk averse individual. Note that it becomes nearly linear. QuickTime ™ and a TIFF (Uncompressed) decompressor are needed to see this picture. AEM 414 Lecture 7 Two Thought Experiments: Experiment 1 Imagine that your are faced with the following decision: A) You can have $100 or B) You can have a coin toss for $200 or $0. How many of you would choose A? ________ How many of you would choose B? ________ EXPLANATION: YOU ARE RISK AVERSE OR RISK NEUTRAL. Lecture 7 AEM 414 Two Thought Experiments: Experiment 2 Imagine that you are faced with the following decision: A) You must pay $100 or B) You can have a coin toss for paying $200 or $0 How many of you would choose A? ________ How many of you would choose B? ________ EXPLANATION: YOU MUST BE RISK SEEKERS! Lecture 7 AEM 414 Anomaly HOW CAN YOU BE BOTH RISK AVERSE/NEUTRAL IN GAINS AND RISK SEEKING IN LOSSES? WE HAVE AN ANOMALY! PROSPECT THEORY ATTEMPTS TO EXPLAIN THIS ANOMALY BY POSITING THAT THE UTILITY FUNCTION IS FUNDAMENTALLY WRONG FROM BIOLOGICAL PRINCIPLES AS FOLLOWS: AEM 414 Lecture 7 Prospect Theory
1) The nervous system is set up to primarily to detect differences, not absolute levels. 2) A gain is perceived as a pleasurable change from the status quo (reference point) and the nervous system shows a decreasing response both to the intensity and duration of pleasurable stimuli. 3) A loss is perceived as a painful change from the status quo (reference point) and the nervous system shows a decreasing response both to the intensity and duration of painful stimuli.
AEM 414 Lecture 7 Prospect Theory (cont.)
4) The reference point is usually where you are (the status quo), but can be where you would like to be, or think you should be. 5) We do not yet fully understand how reference points are determined, but they are subject to phenomena such as adaptation and social pressure (social norms), etc. These observations add up to the VALUE FUNCTION as a replacement for the utility function
Lecture 7 AEM 414 The value function may provide survival value by making humans risk seeking and loss averse when they fall below a reference point of subsistence! DOMAINS Pleasure
QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Neutral Reference Point (Slope above to the Neutral right is half the slope below to the left) right Pain AEM 414 Lecture 7 The Value Function of Prospect Theory There are many examples of phenomena that can be explained by the Value Function of Prospect Theory. These include: – – – – Risk Seeking Status Quo Bias The Sunk Cost Fallacy Loss Aversion AEM 414 Lecture 7 Risk Seeking Behavior motivated by an attempt to make up for perceived losses. Examples: – The Watts Riots of 1965 – Employees who become belligerent because they feel they have been wronged in salary adjustments (especially if a colleague got more!) – Customers who argue and ask for unreasonable compensation because their expectations were not fulfilled. Lecture 7 AEM 414 Status Quo Bias The tendency of people to remain at the status quo even when it is in their interest to change what they are doing. Examples: – Failure to sell stocks when the market tanks because people do not want to admit to losses. – Failure to adopt new technology and accept changes in production methods because the existing skills and knowledge become worthless. – Staying in a bad relationship too long (more to come). – The failure to react to price changes.
Lecture 7 AEM 414 The Sunk Cost Fallacy Most people, if they have put a lot of effort or money into something, are unwilling to give up. Examples: – Students stay with a research topic way too long before switching to another one. – Continued investment into engineering a new product that has become a black hole for money with no production in sight. – Continued marketing expenditures on a product that consumers hate. Lecture 7 AEM 414 Loss Aversion When losses are valued twice as much as gains. This has enormous importance for economics since when people sell things, they value them twice as much as when they buy things! The name for the value of selling something (minimum offer) is WILLINGNESS TO ACCEPT The name for the value of buying something (maximum bid) is WILLINGNESS TO PAY.
Lecture 7 AEM 414 Loss Aversion (cont.) Under normal circumstances Expected Utility Theory predicts that WTA = WTP. Prospect Theory predicts that WTA = 2WTP. We will look at WTA and WTP for insurance policies and lottery tickets from last weeks experiment. AEM 414 Lecture 7 The weighting function of prospect theory QuickTime ™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Edwards, a psychologist at USC in the 1960s, showed overweighting of low probabilities. His weighting function is incorporated in Prospect Theory. The distortion is exaggerated in the figure to the left. In terms of WTP little distortion at high probabilities but large over bidding can be shown at low probabilities called overweighting. However, this low probability response is made up of zero bids (risk dismissal) and very high bids which cannot be explained by risk aversion. AEM 414 Lecture 7 Bids for insurance in the lab are bimodal for low probabilities. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Pooled frequency distribution of bids over 50 rounds. Subjects received $1 cash on each round and 8 subjects bid for 4 insurance policies in a competitive auction. Subjects had an initial balance of $100 and faced 1% odds of a $40 loss. Subjects switched from overbidding to underbidding as experiment proceeded. AEM414 Lecture 7 The mathematical versions of Expected Utility and Prospect Theory
Define: Y0 = reference point income, Y = income, V(Y - Y0) = value function p= of event 1, (1 - p) = probability of event 2, and π (p) = weighting function. and EU= (1-p)U(Y2 ) + pU(Y1) pU(Y The value of a prospect = π (1-p)V(Y2 - Y0) + π (p)V(Y1 - Y0)
Lecture 7 AEM 414 Your Experiment on Insurance and Lottery Tickets What is the value of 2% odds of a $150 gain or loss? EV=$3.00. What is the value of 12% odds of a $25 gain or loss? EV=$3.00. What is the value of 96% odds of a $3.12 gain or loss? EV=$3.00. Given that even $150 is a small fraction of your lifetime wealth, your value should be EV. AEM 414 Lecture 7 Overall Results
WTA Gain Mean Median Standard Deviation WTP/EV or WTA/EV
2.50 1.57 3.25 2.02 2.04 1.39 2.52 1.89 1.40 1.09 1.45 1.23 WTA/WTP
WTAG = amount you would accept to sell a lottery ticket with a given probability of winning WTAL = amount you would accept to sell an insurance policy protecting you from a given probability of losing money WTPG = amount you are willing to pay to buy a lottery ticket with a given probability of winning WTPL = amount you are willing to pay to buy an insurance policy protecting you from a given probability of losing money Lecture 7 Lottery Ticket: 2% chance of winning $150.00
$ 3.00 $ 12.43 $ 7.50 WTP Gain $ 3.00 $ 5.88 $ 4.71 Insurance Policy: 2% chance of losing $150.00 $ 3.00 $ 16.81 $ 9.74 WTP Loss $ 3.00 $ 9.23 $ 6.05 Lottery Ticket: 12% chance of winning $25.00 WTA Loss $ 3.00 $ 5.65 $ 6.12 WTP Gain $ 3.00 $ 3.71 $ 4.18 Insurance Policy: 12% chance of losing $25.00 WTA Gain $ 3.00 $ 8.92 $ 7.55 WTP Loss $ 3.00 $ 5.55 $ 5.68 Lottery Ticket: 96% chance of winning $3.12 WTA Loss $ 3.00 $ 3.74 $ 4.19 WTP Gain $ 3.00 $ 2.16 $ 3.28 Insurance Policy: 96% chance of losing $3.12 WTA Gain WTA Loss WTP Loss 1.591 1.609 1.464 1.330 1.277 $ 4.34 $ 3.70 $ 3.00 $ 3.00 $ 4.71 $ 3.20 1.176 AEM 414 Discussion Loss aversion is supported: WTA/WTP = 1.6 for 2%, but WTA/WTP = 1.2 for 96% odds. Note for induced values odds are 100% and there is no loss aversion. If you are buying or selling $3.00 you would bid or offer $3.00. So loss aversion depends on uncertainty and the more uncertain, the larger is loss aversion! Overweighting is supported: Average Bid or Offer/EV=2.3 for 2% odds of gain or loss and 1.3 for 96% odds of gain or loss. Lecture 7 AEM 414 Losses (red) and Gains (green)
5.00 R atios : W T P/EV and W T A/EV f or Gains and Los s es by Probability (Pooled D ata)
WTA Loss WTP Loss WTA Gai n
3. 25 4.00 WTP Gai n Ratio 3.00
2. 02 2. 50 2. 52 1. 89 1. 57 2. 04 1. 39 1. 45 1. 23 1. 40 1. 09 2.00 1.00 0.00 2% of $150 12% of $25 Probability of Gain or Loss 96% of $3.12 AEM 414 Lecture 7 Losses
5.00 R atios : WT P/EV and W T A/EV f or Los s es by Probability (Pooled D ata)
WTA Loss 4.00 3.25 WTP Loss Ratio 3.00 2.02 2.52 1.89 1.45 1.23 2.00 1.00 0.00 2% of $150 12% of $25 Probability of Loss 96% of $3.12 AEM 414 Professor Schulze Lecture 7 Gains
3.50 3.00 2.50 2.50 2.04 R atios : W T P/EV and W T A/EV f or Gains by Probability (Pooled D ata)
WTA Gain WTP Gain Ratio 2.00 1.57 1.50 1.00 0.50 0.00 1.39 1.40 1.09 2% of $150 12% of $25 Probability of Gain 96% of $3.12 AEM 414 Lecture 7 ...
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- Fall '08