01 Expected Utility Theory- Business Economics 1 PDF

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Business Economics erste Vorlesung zu decision making under risk, diminishing marginal utility, expected utility theory und prospect theory...

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BE 510 Business Economics 1

1. Expected utility theory Prof. Dr. Henrik Orzen

Office hour: Tuesdays, 16:00-17:00 (by appointment) Room 4.01 (Department of Economics, L7, 3-5) E-Mail: [email protected]

BE 510 Business Economics 1 • Bayes Nash Equilibrium

Decision-making under risk What does ‘risk’ mean? 

Risky world: Future holds different possibilities but only one outcome will become a reality.



My decisions 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 (full) knowledge about the probabilities and/or exact implications of the outcomes.

BE 510 Business Economics 1 • Bayes Nash Equilibrium

Decision-making under risk Simplest application: Monetary gambles 

Good outcomes yield higher monetary payoffs, bad outcomes yield lower monetary payoffs.



Once taken for granted that rationality requires expected payoff maximization.



Example: 

Gamble A: €25 with probability 0.4; €0 otherwise.



Gamble B: €9 with certainty.



𝐸𝑉(𝐴)

𝐸𝑉(𝐵)

Gamble C: €100 with probability 0.05; €0 otherwise. 𝐸𝑉(𝐶)

=

=

€10

€9

= €5

BE 510 Business Economics 1 • Bayes Nash Equilibrium

Decision-making under risk A coin game 

A fair coin is tossed repeatedly until “Heads” comes up.



If “Heads” appears… …the first time, you receive: …the second time (after 1 “Tails”), you receive: …the third time (after 2 “Tails”), you receive: …the fourth time (after 3 “Tails”), you receive: …the fifth time (after 4 “Tails”), you receive: …and so on.

€2 €4 €8 €16 €32

H 1/2 TH 1/4 TTH 1/8 TTTH 1/16 TTTTH 1/32



If you play this game many times, how much money can you expect to earn on average?



How much would you be willing to pay (as an entry fee) to be allowed to play this game?

BE 510 Business Economics 1 • Bayes Nash Equilibrium

Decision-making under risk A coin game 

How much would you be willing to pay (as an entry fee) to be allowed to play this game? 



Probability of winning up to €16:

1 1 1 15 = 0.9375 = = + + + 2 4 8 16 16 Probability of winning more than €64: Pr

𝑋 ≤ 16

Pr

𝑋 > 64

1 1 11 1 1 = 1 −+ + + + + 2 4 8 16 32 64

1 = = 0.0156 64

BE 510 Business Economics 1 • Bayes Nash Equilibrium

Resolving the St. Petersburg Paradox Diminishing marginal utility 



Utility function in general: 

Mathematical tool to represent preferences.



Assigns a number value to each possible outcome.



Different values imply a strict preference relationship.

𝑢(𝑥)

Utility increase from one extra monetary unit is the smaller the more money a person has: 𝑢′ 𝑥



Utility (Utils)

> 0 and 𝑢′′

𝑥

< 0.

Consequence for the coin game: 𝐸𝑉 = 𝐸𝑈 =

∞ 𝑖=1 ∞

𝑖=1

1 2

1 2

𝑖

𝑖

2𝑖 = ∞ 𝑢 2𝑖 = Some finite value

0

1

2

3

4

5

6

7

8

9

10

Monetary units (𝑥 )

BE 510 Business Economics 1 • Preferences, utility and optimization • 22

The DMU principle implies risk aversion A choice problem 

Option 1: Get €100 with probability 0.5 (else get nothing).



Option 2: Get €50.



An expected utility maximizer characterized by DMU must pick option 2.

Utility (Utils)

𝑢(100)

𝐸𝑈

𝑢(0)

= 0.5𝑢

100 + 0.5𝑢(0)

0 10 20 30 40 50 60 70 80 90 100 𝑥 (€)

BE 510 Business Economics 1 • Preferences, utility and optimization • 22

The DMU principle implies risk aversion A choice problem 

Option 1: Get €100 with probability 0.5 (else get nothing).



Option 2: Get €50.



An expected utility maximizer characterized by DMU must pick option 2.

Utility (Utils)

𝑢(100)

𝑢(𝑥)

𝐸𝑈

𝑢(0)

0 10 20 30 40 50 60 70 80 90 100 𝑥 (€)

BE 510 Business Economics 1 • Preferences, utility and optimization • 22

The DMU principle implies risk aversion A choice problem 

Option 1: Get €100 with probability 0.5 (else get nothing).



Option 2: Get €50.



An expected utility maximizer characterized by DMU must pick option 2.

Utility (Utils)

𝑢(100)

𝑢(𝑥)

𝑢(50)

𝐸𝑈

𝑢(0)

0 10 20 30 40 50 60 70 80 90 100 𝑥 (€)

BE 510 Business Economics 1 • Preferences, utility and optimization • 22

The DMU principle implies risk aversion A choice problem 

Option 1: Get €100 with probability 0.75 (else get nothing).



Option 2: Get €75.



An expected utility maximizer characterized by DMU must pick option 2.

Utility (Utils)

𝑢(100)

𝑢(75) 𝐸𝑈

𝑢(0)

𝑢(𝑥) = 0.75𝑢

100 + 0.25𝑢 (0)

0 10 20 30 40 50 60 70 80 90 100 𝑥 (€)

BE 510 Business Economics 1 • Preferences, utility and optimization • 22

The DMU principle implies risk aversion A choice problem 

Option 1: Get €100 with probability 0.25 (else get nothing).



Option 2: Get €25.



An expected utility maximizer characterized by DMU must pick option 2.

Utility (Utils)

𝑢(𝑥)

𝑢(100)

𝑢(25) 𝐸𝑈 𝑢(0)

= 0.25𝑢

100 + 0.75𝑢 (0)

0 10 20 30 40 50 60 70 80 90 100 𝑥 (€)

BE 510 Business Economics 1 • Preferences, utility and optimization • 22

The DMU principle implies risk aversion A choice problem 

Option 1: Get €100 with probability 0.25 (else get nothing).



Option 2: Get €25.



An expected utility maximizer characterized by DMU must pick option 2.

Utility (Utils)

𝑢(100)

𝑢(0)

𝑢(𝑥)

0 10 20 30 40 50 60 70 80 90 100 𝑥 (€)

BE 510 Business Economics 1 • Preferences, utility and optimization • 22

The DMU principle implies risk aversion A choice problem

Utility (Utils)



Option 1: Get gamble 𝑔 = 𝑝 𝑥2 ⊕



Option 2: Get monetary amount 𝑤 = 𝑝𝑥2 + 1 − 𝑝 𝑥1 .



1 − 𝑝 𝑥1 .

𝑢(𝑥2 )

𝑢(𝑥)

An expected utility maximizer characterized by DMU must pick option 2.

𝑢(𝑥1 )

𝑥1

𝑥2 𝑥 (€)

BE 510 Business Economics 1 • Preferences, utility and optimization • 22

Certainty equivalent and risk premium A choice problem

Utility (Utils)



Option 1: Get gamble 𝑔 = 𝑝 𝑥2 ⊕



Option 2: Get monetary amount 𝑤 = 𝑝𝑥2 + 1 − 𝑝 𝑥1 .



Certainty equivalent of 𝑔: Get amount 𝑐 such that 𝑢 𝑐 = 𝑝𝑢 𝑥2 + 1 − 𝑝

1 − 𝑝 𝑥1 .

𝑢 𝑥1 = 𝑢(𝑔).

𝑢(𝑥)

𝑢(𝑥2 ) 𝑢(𝑤)

𝑢(𝑔)

Risk premium

𝑢(𝑥1 )

=𝑤−𝑐

𝑥1

𝑐

𝑤

𝑥2 𝑥 (€)

BE 510 Business Economics 1 • Bayes Nash Equilibrium

EUT can accommodate different risk attitudes Three basic categories 

Risk aversion 



.



Certainty equivalent of a gamble is less than 𝐸𝑉(𝑔). Risk premium is a positive number.



Concave utility function.

Risk neutrality 



A person is risk averse in the context of a particular gamble if 𝐸𝑈(𝑔) < 𝑢 𝐸𝑉(𝑔)

A person is risk neutral in the context of a particular gamble if 𝐸𝑈

𝑔 = 𝑢 𝐸𝑉(𝑔)



Certainty equivalent of a gamble is equal to 𝐸𝑉(𝑔). Risk premium is zero.



Linear utility function.

.

Risk proneness 

A person is risk prone in the context of a particular gamble if 𝐸𝑈

𝑔 > 𝑢 𝐸𝑉(𝑔)

.



Certainty equivalent of a gamble is greater than 𝐸𝑉(𝑔). Risk premium is a negative number.



Convex utility function.

BE 510 Business Economics 1 • Bayes Nash Equilibrium

EUT can accommodate different risk attitudes Being an expected value maximizer 

Example from before: 

Gamble A: €25 with probability 0.4; €0 otherwise.



Gamble B: €9 with certainty.



𝐸𝑉(𝐴)

𝐸𝑉(𝐵)

Gamble C: €100 with probability 0.05; €0 otherwise. 𝐸𝑉(𝐶)

=

€10

= €9

= €5

EUT can accommodate different risk attitudes Allowing heterogeneity in risk attitudes 

Person 1: 𝑢

€9 = 13 utils; 𝑢

𝑢

A = 0.4×15 utils + 0.6×10 utils



𝑢

C = 0.05×20 utils + 0.95×10 utils



𝑢

A = 0.4×625 utils + 0.6×0 utils

𝑢

C = 0.05×10,000 utils + 0.95×0 utils

 



€0 = 10 utils; 𝑢

𝑢

B =𝑢

Person 2: 𝑢  

𝑢

€9

€0 = 0 utils; 𝑢

B = 81 utils

€25

=

𝟏𝟐 uti utils ls

=

𝟏0.5 0.5 ut utils ils

=

=

€100 = 20 utils

𝟏𝟑 uti utils ls

€9 = 81 utils; 𝑢 =

= 15 utils; 𝑢

€25

𝟐𝟓𝟎 utils

𝟖𝟏 utils

= 𝟓𝟎0 0 utils

= 625 utils; 𝑢

€100 = 10,000 utils

Can actual choice behavior be inconsistent with EUT? How EU maximizers evaluate gambles: Von Neumann/Morgenstern utility  

Consider a choice between a safe outcome (𝑥2 ) and a gamble (50-50 chance of either 𝑥1 or 𝑥3 ). An expected utility maximizer evaluates these options as follows:  



𝑢(Safe) = 𝑢(𝑥2 ).

𝑢 Gamble = 𝑢 0.5

𝑥1 ⊕ 0.5

𝑥3

=

0.5𝑢(𝑥1) + 0.5𝑢(𝑥3 ).

Von Neumann/Morgenstern expected utility function

In EUT a number of assumptions are made implicitly. 

For example: Rationality.



Or: More of something is preferred to less of the same thing (ceteris paribus).



Most of these assumptions are very conventional in economics.



However, it does not follow from the conventional assumptions that people must evaluate gambles in the von Neumann/Morgenstern way.



Thus, the von Neumann/Morgenstern expected utility function requires an additional assumption about human choice behavior.



This extra assumption is called the independence axiom.

Oskar Morgenstern 1902 – 1977

John von Neumann 1903 – 1957

The independence axiom Definition If 𝑔1 and 𝑔2 are two gambles such that 𝑔1 ≿ 𝑔2 , then for any 𝛼 ∈ [0,1] and any gamble 𝑔3 : 𝛼 𝑔1 ⊕

1 − 𝛼 𝑔3 ≿ 𝛼

𝑔2 ⊕

1 − 𝛼 𝑔3

In words, if you construct two compound gambles by (1) combining in some way gamble 𝑔1 with gamble 𝑔3 and by (2) combining 𝑔2 with 𝑔3 in the same way, then the preference ordering of the two compound gambles is identical to the preference ordering of 𝑔1 and 𝑔2 . Thus, the preference ordering is independent of the characteristics of the third gamble.

The independence axiom An example Situation 1

Situation 2

Option A: A packet of nuts.

Option A: Throw a coin. Heads: You get the nuts. Tails: You get a glass of beer.

Option B: A pot of strawberry yoghurt.

Option B: Throw a coin. Heads: You get the yoghurt. Tails: You get a glass of beer.



Suppose you prefer A to B in Situation 1.



Then, says the independence axiom, you should also prefer A to B in Situation 2.



Key argument: 

The beer on the one hand and the nuts/yoghurt on the other hand are mutually exclusive events.



Therefore, your preference ranking between nuts and yoghurt should be independent of the presence or absence of the beer.



Likewise, the probability (here: 50-50) and the nature of the common outcome (beer) should not matter.

The common consequence effect The idea 

Originally described by Maurice Allais in 1953, and also known as the “Allais Paradox”.



Choice problem 1:  



Alternative A: 1.00[€1,000,000] Alternative B: 0.85 €1,000,000

Choice problem 2:  

Alternative C: 0.85 Alternative D: 0.90

⊕ 0.05

€0 ⊕ 0.10[€5,000,000]

€0 ⊕ 0.15[€1,000,000] €0 ⊕ 0.10[€5,000,000]



A reasonable response: Choose A in choice problem 1, D in problem 2.



But this violates EUT.  

Problem 1: 𝑢(1m) > 0.85𝑢 (1m) + 0.05𝑢(0) + 0.10𝑢(5m)

⇔

Problem 2: 0.85𝑢(0) + 0.15𝑢(1m) < 0.90𝑢(0) + 0.10𝑢(5m) ⇔

0.15𝑢(1m) >

0.15𝑢(1m) <

Maurice Allais 1911 – 2010

0.05𝑢(0) + 0.10𝑢(5m)

0.05𝑢(0) + 0.10𝑢 (5m)

The common consequence effect Experimental evidence 

For example Starmer and Sugden (1991), AER.



Choice problem 1:  



Alternative A: 1.00[£7] Alternative B: 0.75 £7

Choice problem 2:  

⊕ 0.05

Alternative C: 0.75[£0] ⊕ 0.25[£7] Alternative D: 0.80[£0] ⊕ 0.20[£10]

£0

⊕ 0.20

£10

D

D

C C

A Choices

B Choices

Consequences of EUT violations in the lab Normative interpretation of EUT as a way out? 

EUT provides sensible guidance to decision-making under risk.



Violations are essentially mistakes.



If this was properly explained to subjects they would correct their behavior.

Accepting the findings and moving forward 

Used to be a minority view. Now more or less mainstream.



Development of alternative theories of decision-making under risk.



Dialog between theoretical development and empirical/experimental tests.

Prospect theory Modifying EUT in several ways 



Two foundation papers: 

D. Kahneman and A. Tversky (1979). Prospect Theory: An Analysis of Decision under Risk. Econometrica, 47(2), 263-292.



A. Tversky and D. Kahneman (1992). Advances in Prospect Theory: Cumulative Representation of Uncertainty. Journal of Risk and Uncertainty, 5, 297-323.



Plus: Further work on this theory by other authors.

In PT decision makers evaluate outcomes against a reference point. Outcomes are perceived as gains or as losses relative to that.



Daniel Kahneman 1934 –

Amos Tversky 1937 – 1996

Decision makers are assumed to be loss averse. Losses loom larger than gains, i.e. incurring a loss reduces the DM’s utility more than an equally sized gain lifts it.



Also incorporates a switchover in risk attitudes at the reference point: Risk aversion in the domain of gains and risk proneness in the domain of losses (reflection effect).

Prospect theory Value function 

Based on these ideas the utility function (referred to as value function in PT) looks like this: Psychological value 𝜋

Monetary value minus reference value



Finally, PT also assumes a transformation of probabilities: Decision makers have a distorted perception of probabilities: 

There is overweighting of small probabilities



There is underweighting of moderate/high probabilities.

𝑝

1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 .0

.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 𝑝

Prospect theory Back to the common consequence effect 

Recall the two choice problems:  



Choice problem 1: A = 1.00[€1m]

Choice problem 2: C = 0.85





⊕ 0.15[€1m]

Utility function (“value function”): 𝑢 𝑥 =



€0

versus B = 0.85

𝑥 0.4 −2.5 −𝑥

Reference point: Receive €1m. Problem 1: 𝜋 1

Problem 2: 𝜋

𝑢

0.85

±0

𝑢

> 𝜋

⊕ 0.05

versus D = 0.90

€0

€0

⊕ 0.10 [€5m]

⊕ 0.10[€5m]

Probability weighting function:

if 𝑥 ≥ 0 if 𝑥 < 0

0.4

€1m

0.85

−1m + 𝜋

𝜋

𝑝0.6 𝑝 = 𝑝0.6 + 1 − 𝑝

𝑢(±0) + 𝜋

0.15

𝑢

±0

0.05

< 𝜋

0.6 0.6

𝑢(−1m) + 𝜋

0.90

𝑢

1

0.10

𝑢(+4m)

−1m + 𝜋

0.10

𝑢(+4m)

Summary EUT: Decision making under risk

Violations of EUT



What is “rational” depends on a person’s risk attitude.



E.g. the common consequence effect.



Expected value maximization is a special case.



Not random but systematic.

Risk aversion/neutrality/proneness

Prospect theory



Concave/linear/convex utility function 𝑢(𝐸𝑉(Gamble)) ⋛ 𝑢(Gamble)



Changes EUT in fundamental ways.

 

Certainty equivalent ⋛ 𝐸𝑉(Gamble)



Decision makers evaluate outcomes as gains or losses relative to a reference point.



Risk premium is pos./zero/neg.



Assumes loss aversion and a change of risk attitudes at the reference point.



Decision makers are modelled as perceiving probabilities in a distorted way.



Is compatible with some behavior that is not compatible with EUT.

Independence axiom 

The assumption that characterizes EUT specifically.