Expected Utility Theory Flashcards
Expected Value EV
If we play a game over and over again, this value would be the average
CALCULATION: Expected Value
EXAMPLE: 80% of winning 25€, else 0
- EV = 0.825 + 0.20 = 20
Expected Utility Theory
- 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
CALCULATION: Expected Utility
Insert value into utility function and weighted with probabilities
–> EU = p1 * u(x1) + (p1-1) * u(x2)
DEFINTION: Utility Function
- Mathematical tool to represent preferences
- Assigns a number value to each possible outcome
- Different values imply a strict preference relationship
- Unit of Measure: Utils
Certainty equivalent C
Single amount we would accept, in order to not have any risk –> we would trade certainty equivalent against the gambling
CALCULATION: Certainty equivalent
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
Risk premium
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
CALCULATION: Risk premium
Expected value - Certainty Equivalent
EXAMPLE: 20 (EV) - 16 (c) = 4
Risk aversion
- 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
Risk neutrality
- EU(g) = u(EV(g))
- Certainty equivalent of gamble is equal to EV(g)
- Risk premium is zero (c = w)
- Linear utility function
Risk proneness
- 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
Independence Axiom
If you prefer g1 over g2, you must also prefer the compound of g1 & g3 over the compound of g2 & g3
CALCULATION: Independence Axiom
Common Consequence Effect
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)
DEFINTION: Prospect Theory
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
CALCULATION: Prospect Theory
Values are difference between reference point and actual value –> than normal utility calculation
DEFINTION: St. Petersburg Paradox
- 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: 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)
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: What can we conclude concerning risk attitude?
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
