Final Exam

Question 1

Cumulative Prospect Theory (CPT)

Answer 1

Issue with original PT is it allowed for the selection of dominated lotteries.

CPT fixes this by imposing ∑𝜋(𝑝_𝑖)=1

Economists usually cite CPT while Psychologists cite PT

Question 2

Nash Equilibrium

Answer 2

Definition: A set of strategies in a game where no player can benefit by changing their strategy while the other players keep theirs unchanged.

Example: In the Prisoner’s Dilemma, both prisoners choosing to squeal is a Nash Equilibrium because neither can unilaterally reduce their prison time by changing their strategy.

Question 3

Allais Paradox

Answer 3

Participants’ choices contradict the independence axiom. They usually prefer a certain gain over a probabilistic gain of higher expected value in gambles 1 and 2, but reverse preference by choosing gamble 4 when comparing 3 and 4.

Gamble 1: (1 million, 1)
Gamble 2: (1 million, .89; 5 million, 0.1; 0, .01)

Gamble 3: (1 million, 0.11; 0, .89)
Gamble 4: (5 million, .1; .9, 0)

Question 4

WARP (Weak Axiom of Revealed Preferences)

Answer 4

Definition: If option A is chosen over B when both are available, then A should always be chosen over B, regardless of other options.

Question 5

Certainty Equivalent

Answer 5

Definition: The guaranteed amount of money that someone would accept instead of taking a gamble with a higher, uncertain payout.

Example: Accepting a guaranteed $40 instead of a 50% chance at $100.

Question 6

DU and Time Consistency

Answer 6

• Intertemporal Choice is Time Consistent if the preference between choices remains consistent through time.
• Time Consistency is a requirement of DU.
• Violating Time Consistency means you can’t be a DU Maximizer!

Examples:
1. For the $100 Sooner or Later scenario, you first state that you prefer foregoing an extra $1 in one week in favor of $100 now, but that 50 weeks from now, you instead prefer waiting one week for the extra $1

2. For the Broken Resolutions scenario, you plan to be healthy, but when the time comes to execute your plan, you succumb to temptation and eat the Cookie

Question 7

Game Theory Assumptions

Answer 7

1. All players are maximizing their payoffs/utilities
2. All players have common knowledge about the rules of the game
3. Each player’s payoff depends on actions taken by all players
4. Each player has Complete Information unless otherwise specified: payoffs are common knowledge among all players
5. Each player has Perfect Information unless otherwise specified: player knows full history of the game (i.e. previous Chess moves)

Question 8

Prisoner’s Dilemma

Answer 8

Concept: A fundamental game in game theory illustrating that two individuals might not cooperate, even if it appears that it is in their best interests to do so.

Example: Two criminals are arrested and must decide independently whether to confess. Confessing (squealing) might benefit them individually but not collectively.

Question 9

Pareto Optimality

Answer 9

Definition: A state of allocation of resources in which it is impossible to make any one individual better off without making at least one individual worse off.

Example: In a different context, allocating hours of operation between two departments in a way that maximizes overall productivity without overburdening any single department.

Question 10

Game Theory

Answer 10

Definition: A branch of mathematics that studies strategic interactions where the outcomes depend on the actions of multiple agents, each trying to maximize their own payoff.

Example: Strategic decisions in business, such as pricing products or choosing production levels, where outcomes depend on competitors’ actions.

Question 11

IA Weaknesses

Answer 11

Inequity Averse Utility does better than Selfish Utility in predicting behavior in social choice games

However, as IA Utility is formulated solely by outcomes, it would not be able to capture certain factors like:

1. Anonymity of participants
2. Source of funds (earned, gift, etc.)
3. Nature of recipient (charity, friend, etc.)

Question 12

Expected Value

Answer 12

Definition: The sum of all possible values each multiplied by the probability of its occurrence.

Example: A lottery ticket offering a 50% chance of $100 and 50% of $0 has an EV of $50.

Question 13

Market Monopsony Game

Answer 13

Each individual in the class chooses how to split $100 with the Professor, but only the best offer accepted.

Selfish NE to offer $100. Best case scenario is you make $1 by offering $99.

Question 14

Evaluating BDU

Answer 14

Pros
1. Future gets discounted to the Present
2. Time further into the future gets discounted more
3. Accommodates time inconsistent behavior

Cons
1. Requires an additional 𝛽 discount term (more complex than DU)

Question 15

Common Ratio Effect

Answer 15

Concept: A violation of the expected utility theory where changing the probabilities of outcomes proportionally across gambles affects preferences.

For higher probabilities of gain, most prefer the larger probability, smaller gain option.

But when the probabilities of gain are low, most prefer the riskier option with the higher gain.

Question 16

Completeness and Transitivity

Answer 16

Completeness: Every pair of alternatives can be compared.

Transitivity: If option A is preferred to B, and B is preferred to C, then A should be preferred to C.

Question 17

Sign Effect

Answer 17

The steeper discounting of gains compared to losses.

Thaler suggests this is related to Loss Aversion: losses are out of pocket, so not willing to pay much more for a delay

Question 18

Inequity Aversion in the Ultimatum Game

Answer 18

Proposer will offer a percentage “s” of their allocation

1. Proposer who very strongly dislikes having more than Receiver (𝛽_1>0.5) will offer exactly 50%
2. Proposer with 𝛽_1<0.5 will offer the minimum acceptable offer

Predictions are consistent with observed behavior of no offers above 50%, many within 40%-50%, and few low offers.

Question 19

Sequences Effect

Answer 19

As it relates to wages, workers prefer an increase year over year vs. a flat structure or decreasing structure even if the wages are exactly the same over a period.

Sequences should not impact preference over time in DU and BDU, just discount each amount and add them up and choose the highest!

Question 20

Splitting $10 Example

Answer 20

Concept: How individuals decide to split a sum of money in social settings, influenced by factors such as fairness, anonymity, reputation, and potential retribution.

Example: Deciding whether to split $10 equally with a professor or take a larger share for oneself based on various personal and social considerations.

Question 21

Dictator Game

Answer 21

Game consists of a Dictator and one or more other Recipients

The Dictator is the only player who can act

The Dictator decides how they want to split an amount of money with the Recipient

Selfish NE is for the Dictator to offer nothing

Question 22

Level-k Reasoning & Keynes Beauty Contest

Answer 22

The idea is people are of different levels of strategic reasoning:

Level-0 player guesses randomly, 50∗0.5^0=50 on average

Level-1 player guesses 50∗0.5^1=25

Level-2 player guesses 50∗0.5^2=12.5

Level-∞ player guesses 50∗〖0.5〗^∞=0, which is NE

Every level player thinks there are no levels above them.

Question 23

Beta-Delta Utility (BDU)

Answer 23

The Beta-Delta Utility (BDU) of an item/money 𝑥 received T periods into the future is

BDU(𝑥, T)=

{ 𝑢(𝑥), T=0
{ 𝛽𝛿^T∗𝑢(𝑥), T>0

where 𝛽<1, 𝛿<1

Decision Rule: Choose the option that maximizes BDU

Question 24

Compound Interest Formula

Answer 24

FV=PV∗(1+𝑟)^T

t in months or years
r = interest rate

Question 25

Normative vs. Positive Theories

Answer 25

Normative Theories: Prescriptive, suggesting how decisions should be made.

Positive Theories: Descriptive, explaining how decisions are actually made in practice.

Question 26

Risky Choice

Answer 26

Definition: Making a decision where outcomes are uncertain but associated probabilities are known.

Example: Tossing a coin where heads might win you $10 and tails wins nothing.

Question 27

NE vs Level-k

Answer 27

NE is viewed as the purely rational solution concept

Level-k may do better in predicting actual behavior when:
1. NE are difficult to compute or reason
2. Players HAVE varied understanding of game or optimal behavior
3. Players ASSUME varied understanding of game or optimal behavior

Question 28

NE and DE

Answer 28

• Every Dominant Equilibrium (DE) is a Nash Equilibrium (NE)
• Not Every NE is a DE
• Every finite game has a NE

Question 29

Risk Preferences

Answer 29

Risk Averse: Preference for certain outcomes over gambles with the same expected outcome.

Risk Neutral: Indifference between certain outcomes and equivalent gambles.

Risk Loving: Preference for gambles over certain outcomes with the same expected outcome.

Question 30

Nash Equilibrium in the Coordination Game

Answer 30

Game: You and professor each pick heads or tails. Values assigned to each of the four possible combos.

Example 1:
(H,H) and (T,T) = 1
(H,T) and (T,H) = 0
Result: NE is either of the options that equal 1, but the outcome could be anything due to the inability to coordinate.

Example 2:
(H,H) = (2,2)
(T,T) = (1,1)
(H,T) and (T,H) = (0,0)
Result: NE is h,h or t,t but outcome will be h,h.

Question 31

Ultimatum vs. Monopsony

Answer 31

The Ultimatum Game results starkly varied from the Selfish NE

The Monopsony Game results were close to the Selfish NE

Question 32

Exponential Discount Factor

Answer 32

PV=FV∗(1/((1+𝑟) ))^T

PV=FV∗𝛿^T

𝛿 (Delta) is the Exponential Discount Factor

Discount Factor = 1/((1+ interest Rate))

Question 33

Delayed Gratification

Answer 33

The most famous and influential time discounting study is likely the Mischel Marshmallow Experiment. Preschoolers could eat one marshmallow or wait 15 minutes for three marshmallows.

Delayed gratification was a strong predictor of:

1. High future self-control and successful pursual of goals
2. High parental ratings of attention and intelligence
3. Higher SAT scores
4. Improved mental health and self-esteem
5. Reduced drug consumption

Question 34

Social Choice

Answer 34

Definition: Social choice theory studies decision-making processes that affect multiple individuals, focusing on how collective decisions are made based on individual preferences.

Example: Choosing how to allocate a budget within a community project, balancing the different needs and preferences of community members.

Question 35

Three Approaches to Choice

Answer 35

Utility-Only Approach: Decisions are made solely based on utility maximization.

Preference Approach: Choices are based on personal preferences, which could be ranked or unranked.

Revealed Preference Approach: Choices are inferred based on observed behaviors.

Question 36

Modifications of Theoretical Models

Answer 36

Concept: Adjusting theoretical models based on empirical data to better reflect observed behaviors.

Example: Modifying preference axioms when real-world data show consistent violations.

Question 37

Risky Lotteries Evaluation

Answer 37

Concept: Analyzing different lotteries based on their expected utility rather than just expected value.

Example: Choosing between a lottery with a high probability of a modest win and a low probability of a large win.

Question 38

Hormones and the Ultimatum Game

Answer 38

Testosterone reduces generosity and oxytocin increases generosity

Question 39

Experimental Tests of Theories

Answer 39

Concept: Validating theoretical models by comparing predictions with actual decision-making behavior in controlled experiments.

Example: Using a lab setting to test if people’s choices align with predicted utility maximization.

Question 40

Certain Choice

Answer 40

Definition: Decision-making process where each option is received with 100% certainty.

Example: Choosing an apple over an orange with no chance of outcome variation.

Question 41

St. Petersburg Paradox

Answer 41

A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The initial stake begins at 2 dollars and is doubled every time tails appears. The first time heads appears, the game ends and the player wins whatever is the current stake. What would be a fair price to pay the casino for entering the game?

Expected value = infinity, highlighting limitations of using expected value to make real-life choices.

Question 42

Independence of Irrelevant Alternatives (IIA) Violation

Answer 42

Concept: Preference between two options should not depend on the presence of a third option.

Example: Choice between gambles is influenced by the introduction or removal of another unrelated gamble.

Question 43

Discounted Utility (DU)

Answer 43

The Discounted Utility (DU) of an item/money 𝑥 received T periods into the future is

DU(𝑥, T)=𝛿^T∗𝑢(𝑥); where delta < 1

Decision Rule: Choose the option that maximizes DU

Question 44

Intertemporal Choice

Answer 44

Definition: Choices involving trade-offs among costs and benefits occurring at different times.

Example: Deciding whether to receive $100 now or $110 one year from now.

Question 45

Preference Reversals

Answer 45

Definition: A situation in which the preference between options changes depending on how they are evaluated (e.g., choice vs. pricing tasks).

Example: Preferring a low-risk option in direct choice but assigning a higher monetary value to a high-risk option when pricing them separately.

Question 46

Ultimatum Game

Answer 46

Concept: A game in which one player, the proposer, offers how to divide a sum of money with another player, the responder, who chooses to accept or reject the proposal. If the responder rejects the offer, both players get nothing.

Example: Offering $5 out of $10 to another player, who must accept for both to receive the money, or reject, resulting in no one receiving anything.

Question 47

Keynes Beauty Contest

Answer 47

Game: Choose a number from 0 – 100. Whoever chooses half the average wins.

NE = everyone choose 0, but no one does this

Game uses level-k reasoning

Question 48

Regret Theory

Answer 48

Definition: Suggests that people anticipate regret from their decisions and choose in a way to minimize this expected regret.

Example: Avoiding a high-risk investment to prevent potential regret of large losses.

Question 49

Examples of Intertemporal Choice

Answer 49

1. Saving for Retirement
2. Education Investment
3. Health Choices with long-term benefits but require immediate effort and sacrifice
4. Saving for larger purchases or investments in the future
5. Environmental Conservation

Question 50

Discounting Anomalies

Answer 50

1. Magnitude Effect
2. Sign Effect
3. Delay-Speedup Asymmetry
4. Hidden Zero
5. Sequences
6. Independence/IIA

Question 51

Discounted Value Limitations

Answer 51

1. What if dollars are not valued equally?
2. What about if the items of choice are NOT money?

Question 52

Loss Aversion

Answer 52

Definition: The tendency to prefer avoiding losses to acquiring equivalent gains.

Example: Being more upset by losing $50 than being pleased by gaining $50.

Question 53

Social Choice

Answer 53

Definition: Decisions that involve multiple individuals’ preferences or social welfare.

Example: Voting on a policy that affects tax distribution.

Question 54

Framing Effect

Answer 54

Concept: Decisions are influenced by the way choices are posed or framed.

Example: Responding differently to a choice depending on whether it is presented as a loss or a gain.

Question 55

Dominant Strategy

Answer 55

Definition: A strategy that results in the highest payoff for a player, no matter what the other players do.

Example: In the Prisoner’s Dilemma, squealing is a dominant strategy because it always leads to a better or equal payoff compared to staying silent, regardless of the other player’s choice.

Question 56

Evaluating DU

Answer 56

Pros
1. Time further into the future gets discounted more
2. Normative: it is an easy, natural extension of Expected Utility
3. Links up nicely with Compound Interest Formula

Cons
1. Cannot accommodate time inconsistent behavior
a. Each time interval is equally
weighted ($100 Sooner or
Later)
b. Doesn’t allow for temptation,
procrastination (Breaking
Resolutions)

Question 57

Ambiguous Choice

Answer 57

Definition: Decisions made under conditions where the probabilities of outcomes are not known.

Example: Investing in a new startup without clear historical performance data.

Question 58

Keys to Willpower

Answer 58

1. Redirect attention
2. Framing: focus on cooler representations of the temptation. Think of a marshmallow as a cloud instead of food.
3. Avoid temptation: remove yourself from situations in which you are exposed to temptation, as willpower consumes costly energy.
4. Give in to minor temptations: if you need to study and not procrastinate, research shows that giving into more harmless temptations (eating chocolate) can reduce procrastination, as maintaining willpower is costly!

Question 59

Marginal Utility

Answer 59

Definition: The additional satisfaction or utility gained from consuming one more unit of a good or service.

Example: The utility gain from consuming a third slice of pizza compared to the second.

Question 60

Expected Utility Theory

Answer 60

Concept: Decision-making model that accounts for risk aversion by using utility values rather than expected payouts.

Example: Preferring a certain payoff of $50 over a 50% chance of $100, despite the same expected value.

Question 61

Now not later: $100 example

Answer 61

Why would you prefer $100 today over $100 a year from now?

1. Death: Chance you won’t live until next year
2. Future: You are not the “one year from now” You
3. Interest: Put the $100 in the bank and get $100 plus Interest
4. Inflation: $100 now can buy more than $100 next year

Question 62

Inequity Aversion

Answer 62

Definition: A preference for fairness and a resistance to inequitable outcomes, even at a cost to oneself.

Example: A person might prefer to receive less money than to see a disproportionate split in an economic game, even if it means receiving less overall.

Question 63

Utility Function

Answer 63

Definition: A mathematical representation that assigns a numerical value to each possible choice to reflect its overall utility to the individual.

Example: U(Apple) = 2; U(Orange) = 1; apples are twice as preferred as oranges.

Question 64

Delay-Speedup Asymmetry

Answer 64

The delay/speed-up asymmetry implies higher discount rates for decisions involving delayed rewards than for decisions involving immediate rewards

Question 65

Coordination Game

Answer 65

Concept: A game where the payoff to each player depends on the actions of both and where multiple Nash Equilibria can exist, often requiring players to coordinate their choices.

Example: Two drivers at an intersection must choose whether to go or stop. Both choosing to go or stop can be equilibrium outcomes, but coordination is required to avoid accidents.

Question 66

Inequity Aversion (IA) in the Monopsony Game

Answer 66

Since IA players prefer to be better off instead of worse off than others (𝛼_𝑖≥𝛽_𝑖), there is an incentive to win

IA players will have the incentive to continually undercut prospective proposals of others to win

This pushes the IA NE proposal equal to the Selfish NE: offer everything!

Question 67

Finding Nash Equilibrium

Answer 67

To identify if a strategy/action set is a Nash Equilibrium, it must be a mutual best response, so no player has an incentive to unilaterally deviate

Unilateral deviation means one player switches action/strategy while holding all other players’ actions/strategies fixed

Question 68

Prospect Theory

Answer 68

Prospect Theory (PT) uses probabilities 𝑝_𝑖 evaluated by a probability weighting function 𝜋(𝑝), and outcomes 𝑥 evaluated by a value function 𝑣(𝑥):

PT(𝐿)= SUM (𝜋(p) * v(x))

Key Concept: Loss aversion - losses are felt more intensely than equivalent gains.

Question 69

Trust Game

Answer 69

• Game consists of a Sender and a Receiver
• Sender can send any of their money to the Receiver
• The Receiver receives x times the money sent (often 3 times)
• The Receiver can then send any amount back to the Sender
• The Sender exhibits Trust by sending money
• The Receiver exhibits Trustworthiness by returning money

Selfish NE is for the Sender to send nothing and Receiver to return nothing

Question 70

Dictator Demographics

Answer 70

1. Sharing increases with age
2. Students most consistent with Selfish NE
3. Elderly never play NE
4. Female Dictators give more than Male Dictators
5. Female Recipients get more than Male Recipients
6. Race has no effects

Question 71

Discounted Value

Answer 71

Similar concept to Expected Value where each dollar is valued equally (Constant Marginal Utility of Wealth)

PV=FV∗𝛿^T

Question 72

Independence/IIA Affecting DU

Answer 72

Each outcome in a sequence should be independently discounted, and the option with the highest discounted sum should be chosen. However, this is not the case in the example of eating at home or at fancy restaurants on different weekends.

Question 73

Hidden Zero Effect

Answer 73

Definition: Representing a single choice as an extended sequence reduces impulsive choice.

Example:
Hidden Zero
$5 today
$7 last day of class
Explicit Zero
$5 today, $0 last day of class
$0 today, $7 last day of class

Question 74

Beta-Delta (Hyperbolic) Discounting

Answer 74

Hyperbolic Discounting means a Varying discount factor with time

Exponential Discounting means a Constant discount factor with time

Question 75

Magnitude Effect

Answer 75

Higher amounts of money are discounted LESS

High Discount Rate 𝑟 ⟺ Low Discount Factor 𝛿 ⟺ High discounting
Low Discount Rate 𝑟 ⟺ High Discount Factor 𝛿 ⟺ Low discounting

Question 76

Present Bias

Answer 76

Present Bias: The Present gets a Premium relative to all Future times

Every time increment gets discounted by the same 𝛿 under DU, but maybe there’s a bigger valuation difference between certain days.

Question 77

Rationality

Answer 77

Definition: If a preference relation satisfies Completeness and Transitivity, then it is Rational

Complete + Transitive = Rational

Question 78

Utility Representation of Preference

Answer 78

Definition: A function 𝑢( .) represents a preference relation ≽ if for all 𝑥,𝑦 ∈ 𝑋,𝑥 ≽𝑦 ⇔ 𝑢(𝑥) ≥ 𝑢(𝑦)

Rationality is a NECESSARY condition for a utility function representation of preferences.

Question 79

Ordinal Utility

Answer 79

Theorem: All positive monotonic transformations of 𝑢 represent the same preference!

Utility functions with this property are called ORDINAL

Question 80

Positive monotonic transformation

Answer 80

A monotonic transformation is a way of transforming one set of numbers into another set of numbers in a way that the order of the numbers is preserved.

The “positive” part means the transformation has a positive slope

Question 81

Criticism of Preference Approach

Answer 81

Preferences are not observable

If we are interested in Choice, make Choice the foundation

Samuelson pioneers the Revealed Preference Approach in 1930s

Are the preferences revealed from observed choice consistent with utility maximization?

Question 82

About Normative Theory

Answer 82

No data, observation or experiments are required for Normative Theory

A useful Normative Theory should be able to be tested and proven false (falsifiable)

Question 83

Econ vs Psych

Answer 83

Historically, Economic Theory of Choice was motivated more by Normative principles (choice should obey WARP, preferences should be transitive, etc.)

Psychology Theory of Choice is arguably more motivated by Positive principles (people’s observed choices obey WARP, people’s preferences are consistent with transitivity, etc.)

Question 84

Normative Features of Choice Theory

Answer 84

1. Completeness (Preference Approach)
   -People SHOULD know how they feel about any two things
   -People SHOULD NOT be able to say they don’t know
2. Transitivity (Preference Approach)
   - People SHOULD exhibit such consistency
   - Otherwise, they could be robbed by a money pump scheme
3. WARP (Revealed Preference Approach)
   - If someone chooses A in a choice scenario in which B was also available, then they should never choose B when A is available in any other scenario

Question 85

Mental Accounting

Answer 85

Mental accounting interprets the tendency of people to mentally segregate their financial resources into different categories. In the event of financial losses or gains in different mental accounts, people will be impacted differently than if the financial loss was integrated across their entire financial portfolio.

Example: losing theater ticket or losing $10 poll

Question 86

Factors Impacting Stability of Preferences

Answer 86

1. Psychological/Mental Accounts
2. Relative Valuations
3. Positive Attribute vs Negative Attribute Framing
4. Separate vs Joint Evaluation
5. Arbitrary Anchoring
6. Endowment Effect/Ownership Rights
7. Defaults
8. Attraction/Decoy/Asymmetric Dominance Effect
9. Compromise Effect

Question 87

Relative Valuation (Relativity Trap)

Answer 87

In a relativity trap, buyers may perceive a relative difference in prices as more important than an absolute difference, even though the absolute difference reflects what they actually save or pay for a good better than the relative difference.

Example: Buying a coat and calculator from the same store or different stores poll

Question 88

Attribute Framing

Answer 88

Subjects told to rate either 75% lean ground beef burger (Positive) or 25% fat ground beef burger (Negative) on a fat-lean scale

These burgers are the same fat content: 75% lean implies the other 25% is fat!

The lean-framed burgers were rated as more than 2 rating levels leaner on a 7-level rating scale

Question 89

Separate vs. Joint Evaluation

Answer 89

Completeness suggests evaluating options separately or jointly should not influence the valuation. However, the resume example (with candidates’ GPA and experience given) proved that the two candidates were judged differently based on separate or joint evaluation.

Question 90

Decoy Effect

Answer 90

Given options by Competitor and Target that have tradeoffs

Add a third Decoy option dominated by Target only

The third Decoy option attracts choices toward the Target option that dominates it and away from the Competitor

Also called Asymmetric Dominance Effect

Question 91

Compromise Effect

Answer 91

• Two options that have tradeoffs, B and C
• Add a third distant option A that is not dominated by B or C
• B now looks like a compromise between the extremes A and C

𝐶(B, C)=C
𝐶(B, C, A)=B
This violates WARP!

Question 92

Condorcet Paradox (Group Intransitivity)

Answer 92

Consider 3 voters with preferences for candidates 𝑥,𝑦,𝑧:
Voter 1: 𝑥≻𝑦≻𝑧
Voter 2: 𝑦≻𝑧≻𝑥
Voter 3: 𝑧≻𝑥≻𝑦

Aggregate:
𝑥≻𝑦 (Voters 1 and 3)
𝑦≻𝑧 (Voters 1 and 2)
𝑧≻𝑥 (Voters 2 and 3)

Transitive individual preferences create intransitive cycles in aggregate!

Question 93

Groups with more intransitivity

Answer 93

1. Elementary school kids
2. Older adults
3. People with mood disorders

Question 94

Cardinal Utility

Answer 94

Under EU, utility functions are no longer Ordinal, they are Cardinal

Cardinal Utility means that utility values have some meaning outside of just Ordinal ranking

Absolute magnitudes are still meaningless:
𝑢^∗ (“Apple” )=3 and 𝑢^(∗∗) (“Apple” )=6 give no information about Apple utility

Differences in utility now have meaning:
(𝑢^(∗∗∗) (“Apple” )− 𝑢^(∗∗∗) (“Orange” )) > (𝑢^(∗∗∗) (“Orange” )−𝑢^(∗∗∗) (“Pear” )) means that an Apple’s preference over Orange exceeds an Orange’s preference over Pear

Question 95

Positive Linear Transformations

Answer 95

Under EU, any Positive Linear Transformation of 𝑢 represents the same preference as 𝑢.

𝑢 is unique up to a positive linear transformation under EU

Question 96

EU Wealth Function

Answer 96

Bernoulli suggests a logarithmic function
It has the diminishing marginal utility property:
log_10 of ⁡10=1, log_10⁡ of 100=2, log_10⁡ of 1,000=3

By cardinality, this means the utility difference between $10 and $100 is the same as the utility difference between $100 and $1,000.

Question 97

Futility of Utility

Answer 97

log base k will not fit all preferences for money – k determined by preferences between (1, 10, 100, 1,000, etc.) – and just one inconsistency falsifies the utility function.

Question 98

Axiomatic Approach to EU

Answer 98

EU Maximizers will follow rules of:

1. Completeness
2. Transitivity
3. Continuity
4. Independence of Irrelevant Alternatives (Independence Axiom)

Question 99

Graphing Riskiness (EU)

Answer 99

x-axis = wealth
y-axis = utility

Risk Neutral = linear (equivalent to maximizing EV)
Risk Aversive = logarithmic (diminishing marginal utility)
Risk Loving = exponential (increasing marginal utility)

Question 100

Rabin’s Critique of Diminishing Marginal Utility

Answer 100

Diminishing Marginal Utility means that under EU, rejecting small bets with slightly positive EV necessitates rejecting large bets with immensely positive EV

The problem is that a Risk Averse EU Maximizer must have Diminishing Marginal Utility of Wealth, and even seemingly moderate Diminishing Marginal Utility means that if small stake, positive EV bets are rejected, large stake, immensely positive or infinite EV bets must also be rejected.

Question 101

Certainty Effect

Answer 101

1% increase of positive gain from 99% to 100% is much more valuable than the 1% increase from 98% to 99%.

EU treats each percent of probability equally (linear in probability)!

Question 102

Prospect Theory Reflection Effect (Fourfold Pattern of RIsk)

Answer 102

1. Risk Averse over high probability gains (Problems 3 and 7)
2. Risk Loving over low probability gains (Problems 4 and 8)
3. Risk Loving over high probability losses (Problems 3’ and 7’)
4. Risk Averse over low probability losses (Problems 4’ and 8’)

Question 103

Isolation Effect

Answer 103

People tend to disregard components/branches that lotteries share and focus on the components/branches that distinguish them.

A single-stage lottery under EU is equivalent to a multi-stage lottery if the final outcomes and probabilities are the same.

Question 104

Problems with EU

Answer 104

1. Rabin’s Critique of Diminishing Marginal Utility of Wealth
2. Allais Paradox/Certainty Effect
3. Common Ratio Effect
4. Reflection Effect
5. Isolation Effect
6. Wealth Changes, not Final States

Question 105

Prospect Theory Value Function Graph (v(x))

Answer 105

1. Should not depend on wealth 𝑤, just outcomes 𝑥
2. Should incorporate Loss Aversion
3. Should display Diminishing Sensitivity: the first dollar gained or lost should impact value MORE than the second dollar, etc.

Result: DMU for positive part, IMU for negative part, and an origin of (0,0) that represents current wealth or expectations.

Question 106

Prospect Theory 𝜋(p) graphed

Answer 106

1. Looks like middle of an x^3 function (rapid increase for first bit, slow upward sloping curve for middle, and rapid increase for last .2)
2. 𝜋(0)=0 and 𝜋(1)=1
3. Overweight small probabilities and underweight large ones
4. Certainty Effect:
   𝜋(1)−𝜋(.95)>𝜋(.55)−𝜋(.5)
5. Common Ratio Effect:
   𝜋(.05)−𝜋(0)>𝜋(.55)−𝜋(.5)

Question 107

First Order Stochastic Dominate (FOSD)

Answer 107

Definition: If for ALL monetary outcomes 𝑥, 𝐿 gives an equal or higher probability than 𝐿′ of receiving something greater than 𝑥.

Normative Rule: You should not choose the dominated lottery 𝐿′.

Question 108

Ellsberg Paradox

Answer 108

Two jars with 100 balls each. Jar A has 50 black and 50 red balls. Jar B has red and black balls in unknown amounts. Pick the right ball, get $100.

Most people prefer jar A.

Demonstrates Ambiguity Aversion.

Question 109

What EU axiom does regret theory violate?

Answer 109

Transitivity