Deck 9: Decision Under Risk and Uncertainty

Question 1

Decisions under risk have two components: preferences over outcomes and perceptions of probabilities.

Answer 1

True

Question 2

A decision-maker has more information under Knightian risk than Knightian uncertainty.

Answer 2

True

Question 3

When trying to forecast the weather, a forecaster's situation is best described as Knightian risk.

Answer 3

False

Question 4

The order axiom is to choices under uncertainty as rationality is to choices under certainty.

Question 5

Regret aversion always allows for violations of transitivity.

Question 6

Figure 9.1 reflects the following idea: as $x$ increases above $y$ then the amount of utility I get from choosing $x$ over $y$ increases at an increasing rate; as $x$ decreases below $y$ then amount of utility I lose from choosing $x$ over $y$ decreases at a decreasing rate.

Question 7

The Independence Axiom describes how people should value a single gamble with complex probabilities.

Question 8

The common outcome effect is an example of the violation of the independence Axiom.

Question 9

The certainty effect is a rational preference for gambles with a certain outcome.

Question 10

Both the outcome effect and the certainty effect violate the independence axiom.

Question 11

Under $\alpha-\operatorname{maxmin}$ expected utility theory, the decision-maker is more ambiguous averse the more weight he puts on the highest probability of the preferred outcome occuring.

Question 12

With probability weighting, the weighting function $\pi(p)$ is convex below some threshold $\bar{p}$ and concave above some threshold $\bar{p}$ .

Question 13

Ambiguity aversion occurs when individuals show a preference for gambles in which the outcome and the probabilities are known over those gambles for which they are unknown.

Question 14

Knightian risk refers to situations in which

A) The probabilities are not known with certainty, but the set of possible outcomes are known.
B) The outcome is not known with certainty, but the probabilities of each outcome and the set of outcomes are known.
C) The probability and the outcome are known with certainty.
D) Neither the probabilities nor the set of outcomes are known with certainty.

Question 15

The heart of Bernoulli's solution to the Saint Petersburg paradox is

A) Convexity of the utility function.
B) Violation of transitivity of preferences.
C) Violation of the independence axiom.
D) Risk aversion.

Question 16

Consider three gambles: $\mathrm{A}, \mathrm{B}$ , and $\mathrm{C}$ . Write out the Order Axiom using the " $>$ " notation.

Question 17

Consider gamble A: $\$ 80$ with probability .4 and $\$ 100$ with probability .6. Which of the gambles stochastically dominate gamble $\mathrm{A}$ .

A) $\$ 50$ with probability .4 and $\$ 90$ with probability .6.
B) $\$ 80$ with probability .3 and $\$ 100$ with probability .7.
C) $\$ 70$ with probability .6 and $\$ 110$ with probability .4 .
D) $\$ 0$ with probability .5 and $\$ 180$ with probability .5.

Question 18

Regret aversion is offered as an alternative to Expected Utility Theory in order to understand which of the following phenomena:

A) The Allais Paradox.
B) The Saint Petersburg Paradox.
C) The Ellsburg Paradox.
D) The Reversal Paradox.

Question 19

Regret theory differs from Expected Utility theory in which way?

A) Expected Utility theory assumes the decision-maker is a rational utility maximizer, whereas regret theory does not.
B) Expected Utility Theory assumes risk aversion, not regret aversion.
C) Expected Utility Theory relies on the independence axiom. Regret Theory relaxes the independence axiom.
D) Regret theory is a form of Expected Utility Theory.

Question 20

Consider three outcomes: $\mathrm{a}, \mathrm{b}, \mathrm{c}$ such that $a>b>c$ . Regret aversion implies which of the following?

A) $U(a, b)<U(b, c)+U(a, b)$
B) $U(c, a)<U(b, a)+U(c, b)$
C) $U(b, a)<U(b, b)+U(c, c)$
D) $U(c, a)<U(a, b)+U(c, b)$

Question 21

Consider the three gambles presented in Table 9.1. What are the three conditions implied by regret aversion?

Question 22

Similarity is

A) A heuristic used to simplify complex gambles in order to help the individual make a choice.
B) A heuristic used to judge when an individual is indifferent between two choices.
C) A heuristic used to assess the expected utility of a single gamble.
D) A heuristic used to assess the probability of a gamble.

Question 23

Consider four gambles: A,B, C, D. If $A>B$ and $C>D$ and my preferences satisfy the independence axiom, then what must be true for any $p \in[0,1]$ ?

A) $p A+(1-p) B>p A+(1-p) C$
B) $A>C$
C) $p C+(1-p) B>p D+(1-p) B$
D) $p C+(1-p) B>(1-p) D+p B$

Question 24

Maurice has a utility function such that $u(250)-u(125)<u(125)-u(0)$ and Lucy has a utility function such that $u(250)-u(125)=u(125)-u(0)$ . Who is more risk averse?

Question 25

Which of the following allows for the slope of indifference curve to depend on the level of utility?

A) Expected Utility Theory.
B) Regret Theory.
C) Loss Aversion.
D) Weighted Expected Utility Theory.

Question 26

All of the following theories seek to explain deviations from the rational model of behavior at extreme probabilities, except

A) Weighted Expected Utility Theory
B) Probability Weighting
C) Expected Utility Theory.
D) Rank Dependent Expected Utility.

Question 27

Which of the following scenarios describe a situation in which ambiguity is present?

A) The probabilities are known and the set of outcomes are known.
B) The probabilities are known and the outcome is known.
C) The probabilities are unknown and the outcome is known.
D) The probabilities are unknown and the set of outcomes are known.

Question 28

Suppose there are two jars. In Jar $1 \mathrm{I}$ know that there are 10 white balls and either 5 or 20 black balls. In Jar 2 I know that there are 10 red balls and either 10 or 30 black balls. I receive $\$ 5$ if a white ball is drawn from jar 1 and $\$ 10$ if a red ball is drawn from jar 2 and 0 otherwise.
a. Suppose that I am risk-neutral. Using maxmin expected utility theory, do I draw from jar 1 or jar 2 and what is the value of my maxmin expected utility?
b. Now suppose I am risk-averse, with a utility function given by $U(x)=\ln (x)$ . Do I choose Jar 1 or Jar 2 ?

Question 29

Consider two gambles: Gamble A: $a=\$ 100$ with probability $p_{A}=.4$ and $\$ 0$ with probability .6. Gamble B: $b=\$ 75$ with probability $p_{B}=.45$ and $\$ 0$ with probability .55 . Use the similarity algorithm to indicate whether you would choose Gamble A or Gamble B.

Question 30

Consider two possible outcomes, $\mathrm{a}$ and $\mathrm{b}$ and a utility function over these two outcomes $U(a, b)$ . What must be true about $U(a, b)$ if the utility function is skew-symmetric?

Question 31

Using the gambles in Example 3, use the common outcome effect to show how the following preferences: $A>B, D>C$ violates the independence axiom.