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Decision Making Under Uncertainty

Decision Making Under Uncertainty. Econ 301. Bernoulli Game. How much would you pay for a gamble that paid you for the nth head in a sequence of coin tosses ending at the first tail? For example if the sequence is HHHT you get 1+2+4. If the sequence is HHHHT you get 1+2+4+8.

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Decision Making Under Uncertainty

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  1. Decision Making Under Uncertainty Econ 301

  2. Bernoulli Game • How much would you pay for a gamble that paid you for the nth head in a sequence of coin tosses ending at the first tail? • For example if the sequence is HHHT you get 1+2+4. If the sequence is HHHHT you get 1+2+4+8

  3. Bernoulli Game • What is the expected value of this bet? • How much would you pay to participate?

  4. Expected Utility Theory • Expected Utility Theory is a resolution of St. Petersburg Paradox • While EV is infinite, expected utility is finite (and possibly small because of diminishing marginal utility of money).

  5. Probability Review • A probability is a number between 0 and 1 that indicates that a particular outcome will occur. • How is probability estimated? • Frequencies: If we know a history with which a particular outcome occurred we can use frequency as an estimate of probability.

  6. Probability Review • n is the number of times a particular outcome occurred out of N total times an event occurred. • A house either burns down or doesn’t. If n=13 similar houses burned in your neighborhood of N=1000 homes, you might estimate the probability of your house burning down as 13/1000=1.3%

  7. Probability Review If we don’t have a history to estimate frequency, we could use available information to form subjective probability (our own best estimate that a particular outcome will occur). What kind of information will you use to form subjective probabilities?

  8. Probability Review • Estimates of subjective probabilities are often biased because people tend to • Overweigh probabilities of unlikely events • A typical American’s chance of dying from a shark attach is 1 in 350 million; a bee sting 1 in 6 million; falling into a hole 1 in 2.8 million; a handgun 1 in 1.9 million; lightning 1 in 600,000; homicide, 1 in 15,000; flu, 1 in 3025; cancer 1 in 514; heart disease, 1 in 384 • Overall people tend to overestimate likelihood of deaths from infrequent causes and underestimate likelihood of deaths from common causes. • Use law of “small numbers” (like in the coin toss on Friday) • Framing

  9. Probability Review • Probability distribution relates the probability of occurrence to each possible outcome.

  10. Probability Distribution

  11. Expected Value

  12. Expected value maximization is a good approximation for a lot of problems • Expected value and expected utility maximization are the same under risk neutrality.

  13. Expected Value: an example • Outdoor concert • V=$15 if doesn’t rain • V=$-5 if it rains • 50% chance of rain; needs to decide whether to schedule a concert before finding out the weather • EV= Pr(no rain)V(no rain)+Pr(rain)V(rain)= • .5*15+.5(-5)=$5

  14. Expected Value • What if you could get information before scheduling a concert • V(rain)=0 (because don’t schedule) • V(no rain)=$15 • EV=.5(15)+.5(0)=7.5 • Expected gain from perfect information is the difference between EV with information and EV without information: 7.5-5=2.5 or the savings from not hiring the band if it rains .5*5=2.5

  15. Variance • Variance measures the spread of a probability distribution. • Formally, the variance is the probability-weighted average of the squares of the differences between the observed outcome and the expected value.

  16. Variance

  17. Variance • Variance = .5(15-5)^2+.5(-5-5)^2=100 • Standard deviation=10 • Holding the expected value constant, the smaller the standard deviation (or variance), the smaller the risk. • Indoor concert: with no rain V=$10 and with rain V=0. • EV=.5*10+.5*0=5 (the same as before),

  18. Variance • but variance = .5(10-5)^2+.5(0-5)^2=25 • Variance is lower if holds concert indoors. • Where will he hold the concert? • To know an answer to this question, we need to know his attitude towards risk. • Even if he dislikes risk, he might prefer a riskier option if it has a higher expected value.

  19. Expected Utility • John Von Neumann and Oskar Morgenstern (1944) propose a standard utility model to incorporate risk. • Treat utility as a cardinal measure (not an ordinal measure like we did before). • A rational person maximizes expected utility

  20. Expected Utility

  21. Expected Utility • For example, EU of an outdoor concert is • EU=Pr(no rain)U(Vno rain)+Pr(rain)U(Vrain)= .5U(15)+.5U(-5) Expected value is a probability weighted average of a monetary value, whereas expected utility is a probability weighted average of the utility from that monetary value.

  22. Risk Attitudes • What is a fair bet? • A fair bet is a lottery with expected value of zero. • For example, pay $1 if a coin comes up H and win $1 if a coin comes up T (expected value zero) • In contrast a bet in which you pay $1 if you lose and receive $2 if you win is an unfair bet that favors you.

  23. Risk attitudes • Someone is unwilling to make a fair bet is risk averse. • A person who is indifferent about making a fair bet is risk-neutral • A person who makes a fair bet is risk preferring.

  24. Utility Function for Wealth • U(W) is concave • U’(W)>0 • U’’(W)<0 • Diminishing marginal utility of wealth. • A person whose utility function is concave is risk averse (picks a less risky choice if both choices have the same expected value).

  25. Example • Status Quo • W=$40 • U(40)=120 • Or buy a vase which has V=10 with probability .5 and V=$70 with probability .5 • EV=.5(10)+.5(70)=40 • Buying a vase is a fair bet because has the same EV as status quo

  26. Example • U(10)=70, U(70)=140 • EU=.5(70)+.5(140)=105 • U(40)=120>105 (prefers status quo) • Risk premium is an amount a risk averse person would be willing to pay to avoid risk. • U(26)=105. therefore indifferent between buying a vase and having $26 for sure. Risk premium is $40-$26=$14 to avoid bearing a risk of buying a vase.

  27. Risk Aversion

  28. A risk averse person chooses a riskier option only if it has a sufficiently higher expected value. • Risk neutral person has a constant marginal utility of wealth (linear utility) • A risk neutral person chooses an option with the highest expected value

  29. A person with an increasing marginal utility of wealth is risk-loving.

  30. Arrow Pratt Measure of Risk Aversion

  31. Experiments to Measure Risk AttitudesHolt and Laury (AER, 2002)

  32. Results with Incentive Effects

  33. Hypothetical Payoffs

  34. Expected Utility

  35. If choose A1 • U(3000)>.8U(4000)+.2U(0) • If choose C2, • .25U(3000)+.75U(0)<.2U(4000)+.8U(0) • But first choice of A1 implies that: • .25U(3000)>.8*.25U(4000)+.2*.25U(0) • A contradiction…

  36. Allais Paradox • Allais paradox is by far the most severe evidence against Expected Utility Theory.

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