Expected Utility Theory

Question 1

Expected Value EV

Answer 1

If we play a game over and over again, this value would be the average

Question 2

CALCULATION: Expected Value

Answer 2

EXAMPLE: 80% of winning 25€, else 0
- EV = 0.825 + 0.20 = 20

Question 3

Expected Utility Theory

Answer 3

• Future hold different possibilities but only one outcome will become a reality
• My decision may affect the probabilities for the different outcomes
• In a risky world I know how likely each outcome is and what exactly each outcome implies and how actions affect the probabilities of outcomes
• In an uncertain world I do not have knowledge about the probabilities and/or exact implications of the outcomes

Question 4

CALCULATION: Expected Utility

Answer 4

Insert value into utility function and weighted with probabilities

–> EU = p1 * u(x1) + (p1-1) * u(x2)

Question 5

DEFINTION: Utility Function

Answer 5

• Mathematical tool to represent preferences
• Assigns a number value to each possible outcome
• Different values imply a strict preference relationship
• Unit of Measure: Utils

Question 6

Certainty equivalent C

Answer 6

Single amount we would accept, in order to not have any risk –> we would trade certainty equivalent against the gambling

Question 7

CALCULATION: Certainty equivalent

Answer 7

utility function of c must equal utility function of gamble

EXAMPLE: 80% of winning 25€, else 0
- u(x) = x^0.5
- EU(x) = 0.8u(25) + 0.2u(0) = 0.8u(25^0.5) = 0.8 * 5 = 4
- c: 4 = x^0.5 –> c=16

Question 8

Risk premium

Answer 8

Maximum amount, the decision maker is willing to give up in order to avoid the risk that comes with gamble –> describes how risk averse somebody is

Question 9

CALCULATION: Risk premium

Answer 9

Expected value - Certainty Equivalent

EXAMPLE: 20 (EV) - 16 (c) = 4

Question 10

Risk aversion

Answer 10

• EU(g) < u(EV(g))
• Certainty equivalent of gamble is less than EV(g)
• Risk premium is a positive number (c < w)
• Concave utility function

Question 11

Risk neutrality

Answer 11

• EU(g) = u(EV(g))
• Certainty equivalent of gamble is equal to EV(g)
• Risk premium is zero (c = w)
• Linear utility function

Question 12

Risk proneness

Answer 12

• EU(g) > u(EV(g))
• Certainty equivalent of gamble is greater than EV(g)
• Risk premium is a negative number (c > w)
• Convex utility function

Question 13

Independence Axiom

Answer 13

If you prefer g1 over g2, you must also prefer the compound of g1 & g3 over the compound of g2 & g3

Question 14

CALCULATION: Independence Axiom

Question 15

Common Consequence Effect

Answer 15

Safe option (1.000.000€ with 100%) is preferred over gambling (1.000.000€ with 85%, 0€ with 5%, and 5.000.000€ with 10%) when one is safe and one gambling, but if both are gambling, the higher possible profit is preferred (5.000.000€ with 10% over 1.000.000€ with 15%).

–> Violates EUT (equations don’t match)

Question 16

DEFINTION: Prospect Theory

Answer 16

Denison makers evaluate outcomes against a reference point –> outcomes are perceived as gains or as losses relative to that

• Decision makers are assumed to be loss averse (individuals suffer more from loss than enjoy equally sized gain)

Value Function: in the loss region steeper than in the gains region –> Overweighting of small probabilities and underweighting of moderate/high probabilities

Question 17

CALCULATION: Prospect Theory

Answer 17

Values are difference between reference point and actual value –> than normal utility calculation

Question 18

DEFINTION: St. Petersburg Paradox

Answer 18

• Utility increase from one extra monetary unit is the smaller the more money a person has (Jeff Bezos Example)

Coin Game: EU = E((1/2)^i * u(2^i))

Question 19

QUESTION: St. Peterburg Paradox - How much is somebody willing to pay? –> Expected Utility at Coin Toss game, 1st time 1€ 2nd time 2€ 3rd time 4€ 4th time 8€…, utility function ln(x)

Answer 19

EU = 1/2 * u(1) + 1/4 * u(2) + 1/8 * u(4) + 1/16 * u(8)

–> replace denominator and utility function with multitude from 2

–> Sum function with i: E(1/2^i * ln(2^i-1))

–> E((i-1)/2^i * ln(2))

–> ln (2) * E((i-1)/2^i)

–> ln(2)

Question 20

QUESTION: What can we conclude concerning risk attitude?

Answer 20

If Expected Value of chosen gamble is higher than the alternative–> risk loving

If Expected Value of chosen gamble is lower than the alternative –> risk averse