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8. Expected Utility representation of

preferences over lotteries; Allais and
related paradoxes
Bibliography: Just, ch. 9

Decision Making 2024/25

Underneath the (expected) utility
function
 Now let us look at the utility function
 The basic idea that individuals maximize their expected
utility when making choices between lotteries is grounded
on the notion that an extra €1 may not have the same
value to everyone
 It should typically depend on the wealth level
 Why has it become used as the prototype of ‘rational
decision makers’?
 Because it is supported by ‘rational’ axioms
 Let A, B and C be lotteries

Decision Making: 8. Expected Utility Theory 2

Underneath the (expected) utility
function
 Order axiom
 Preferences must be complete
 Consider A and B: either 𝐴 ≻ 𝐵, 𝐴 ≺ 𝐵 or 𝐴 ∼ 𝐵
 Preferences must be transitive
 Consider A, B and C: if 𝐴 ≻ 𝐵 and 𝐵 ≻ 𝐶, then 𝐴 ≻ 𝐶
 Similar requirements to choice under certainty

Decision Making: 8. Expected Utility Theory 3

Underneath the (expected) utility
function
 Continuity axiom
 If 𝐴 ≻ 𝐵 ≻ 𝐶, there exists a single value of 𝑟 such that r𝐴 +
1−𝑟 𝐶 ∼𝐵
 If 𝑝 > 𝑟, then 𝐴 ≻ 𝐵; if 𝑝 < 𝑟, then 𝐴 ≺ 𝐵
 Underlying logic:
 A small increase in the probability of the preferred gamble
must increase the utility of that gamble!

Decision Making: 8. Expected Utility Theory 4

Underneath the (expected) utility
function
 Independence axiom
 If 𝐴 ≻ 𝐵, then 𝑝𝐴 + 1 − 𝑝 𝐶 ≻ 𝑝𝐵 + 1 − 𝑝 𝐶
 Underlying logic:
 My preferences over A and B should not be affected by a
possible third alternative C!
 Think about this example
 1. Flip a coin; if it lands tails, you get C
 2. If it lands heads, you choose between A or B
 Independence implies that the answer to:
 “Before flipping the coin, do you prefer A or B?”
 Must be equivalent to the answer you give in 2. [where C is, in fact, a
counterfactual world that did not materialize]
 Choices must be dynamically consistent

Decision Making: 8. Expected Utility Theory 5

Implications of Expected Utility Theory

 Stochastically dominant lotteries should always be

chosen!
 Consider the choice between:
 Lottery X: 40% probability of winning €10 and 60% probability
of winning €3
 Lottery Y: 50% probability of winning €11 and 50% probability
of winning €3
 Convince yourself that Y is always better!
 Higher probability of winning a higher best prize
 Lower probability of winning a similar worst prize
 That is, lottery Y stochastically dominates lottery X

Decision Making: 8. Expected Utility Theory 6

The Marschak-Machina triangle

 The main issues with Expected Utility Theory

 Transitivity
 Independence
 Many (but not all) alternatives theories to Expected
Utility Theory ‘relax’ these axioms

 Let us carefully understand how

Decision Making: 8. Expected Utility Theory 7

The Marschak-Machina triangle

 Let us introduce the Marschak-Machina triangle

 Consider lotteries over three outcomes:
 €100, €50 and €0
 Draw a triangle where the x-axis is the probability of the worst
outcome (in this case, €0) – call it 𝑥
 The y-axis is the probability of the best outcome (in this case,
€100) – call it 𝑦
 The probability of the intermediate outcome is ‘hidden’ from
sight, but can be inferred: 1 − 𝑥 − 𝑦
 Note that there are 3 ‘certainty’ points in the triangle
 Carefully interpret the x-axis, the y-axis and the hypotenuse

Decision Making: 8. Expected Utility Theory 8

 P(100)=1
Best outcome
(€100) received
with certainty

Probability of €100
Best outcome

Worst outcome
(€0) received with
Intermediate certainty
outcome (€50)
received with
certainty
P(100)=0
p(0)=0 p(0)=1
Probability of €0

Worst outcome

Decision Making: 8. Expected Utility Theory 9

The Marschak-Machina triangle

 What would indifference curves look like here?

 A ‘general’ lottery would yield the following expected utility
 𝑦. 𝑢 100 + 𝑥. 𝑢 0 + 1 − 𝑥 − 𝑦 . 𝑢 50 = 𝑘
 Rearrange this and obtain:
𝑘−𝑢 50 𝑢 50 −𝑢 0
 𝑦 = 𝑢 100 −𝑢 50 + 𝑥. 𝑢 100 −𝑢 50
 K is a constant, which implies that this is a straight line with
slope equal to:
𝑑𝑦 𝑢 50 −𝑢 0
 𝑑𝑥
=
𝑢 100 −𝑢 50

 Main implication: under expected utility, all indifference curves

for a given individual are straight lines with the above slope

Decision Making: 8. Expected Utility Theory 10

The Marschak-Machina triangle

 Let us carefully interpret this slope:

 It is the ratio between:
 the difference in utility level of receiving 50 and receiving zero
and…
 the difference in utility level of receiving 100 and receiving 50
 So the slope is really related to how an individual
perceives or values a marginal increase in wealth
 In other words, it is definitely related to the concavity of its
utility function

Decision Making: 8. Expected Utility Theory 11

The Marschak-Machina triangle

 Consider a risk neutral individual

 In that case, the utility function is a straight line
 For simplicity, just assume it equal to 𝑢 𝑥 = 𝑥
 Then we would get:
𝑘−50 50−0 𝑘−50
 𝑦 = 100−50 + 𝑥. 100−50 = 50
+ 𝑥
 The indifference curve’s slope, in this case, is equal to :
𝑑𝑦
 𝑑𝑥
=1

Decision Making: 8. Expected Utility Theory 12

The Marschak-Machina triangle

 Let us plot these indifference curves

 If indifference curves are straight lines, all we need are two
points
 Consider the lottery ‘receiving 50 for sure’ (𝑥 = 0; 𝑦 = 0)
 Recall 𝑦. 𝑢 100 + 𝑥. 𝑢 0 + 1 − 𝑥 − 𝑦 . 𝑢 50 = 𝑘
 And thus obtain 𝑘 = 50
1
 Now consider a 50-50 lottery over 100 and 0 only (𝑥 = 2 ; 𝑦 =
1
; 𝑥 + 𝑦 = 1)
2
 Recall 𝑦. 𝑢 100 + 𝑥. 𝑢 0 + 1 − 𝑥 − 𝑦 . 𝑢 50 = 𝑘
1
 And thus obtain 𝑘 = 2 . 100 = 50
 We now have two points of an indifference curve and we can
plot them in the Marschak-Machina triangle!

Decision Making: 8. Expected Utility Theory 13

 P(100)=1

Higher utility

Probability of €100
E[u(x)]=50=1/2.100+1/2.0
Best outcome
[risk neutral]
1/2

P(100)=0
p(0)=0 1/2 p(0)=1
Probability of €0

Worst outcome

Decision Making: 8. Expected Utility Theory 14

The Marschak-Machina triangle

 We know that indifference curves are straight lines with

the same slope (parallels to one another)
 But which direction yields higher utility?
 Think about it
 Take any point in the interior of the triangle
 Going ‘up’ means higher probability of 100, same probability
of 0 and, necessarily, lower probability of 50
 Better or worse?
 Going ‘left’ means lower probability of 0, same probability of
100 and higher probability of 50
 Better or worse?

Decision Making: 8. Expected Utility Theory 15

The Marschak-Machina triangle

 So going ‘up and left’ is the direction of higher utility!

 Indifference curves to the left of 𝑘 = 50 yields higher utility
 Indifference curves to the right of 𝑘 = 50 yield lower utility
 You could actually get this in two other ways:
 Just look at the certainty points and ‘combine’ them with
the knowledge that indifference curves are straight lines
(in the case of a risk neutral individual, with a slope of 1)
 ‘Visually’ look at your most preferred point or your most
preferred lotteries
 Certainty point 100 or lotteries where y is as high as possible
and x as low as possible

Decision Making: 8. Expected Utility Theory 16

The Marschak-Machina triangle

 Now consider a risk averse individual

 Recall the slope of the indifference curves
𝑑𝑦 𝑢 50 −𝑢 0
 𝑑𝑥
=
𝑢 100 −𝑢 50

 For risk averse individuals, the utility function is concave

 This implies that the numerator is higher than the denominator
 Or, in other words, it conveys the notion of diminishing
marginal utility of wealth
 The ‘jump’ from 0 to 50 is worth more than the ‘jump’ from 50 to
100

Decision Making: 8. Expected Utility Theory 17

 P(100)=1

Risk averse indifference curves

Probability of €100
Risk neutral indifference curve
Best outcome

1/2

P(100)=0
p(0)=0 1/2 p(0)=1
Probability of €0

Worst outcome

Decision Making: 8. Expected Utility Theory 18

The Marschak-Machina triangle

 The actual slope of the indifference curves will vary

across individuals (each will have their own utility
function), but it will be larger than 1!
 And the more risk averse an individual is, the more
concave the utility function is and the larger the slope is

 Intuitively, you can ‘test’ this by drawing a close to

vertical straight line that crosses the certainty point 50
 Receiving 50 for sure yields the same utility as a lottery
with a very high probability of receiving 100, and a very
low probability of receiving 50 or 0!
 This individual must really be very, very risk averse!

Decision Making: 8. Expected Utility Theory 19

The Marschak-Machina triangle

 The same reasoning (in reverse) applies to risk lovers

 Recall the slope of the indifference curves
𝑑𝑦 𝑢 50 −𝑢 0
 𝑑𝑥
=
𝑢 100 −𝑢 50

 For risk lovers, the utility function is convex

 This implies that the numerator is lower than the denominator
 Or, in other words, it conveys the notion of increasing
marginal utility of wealth
 The ‘jump’ from 50 to 100 is worth more than the ‘jump’ from 0 to
50

Decision Making: 8. Expected Utility Theory 20

 P(100)=1

Risk lover indifference curves

Probability of €100
Risk neutral indifference curve
Best outcome

1/2

P(100)=0
p(0)=0 1/2 p(0)=1
Probability of €0

Worst outcome

Decision Making: 8. Expected Utility Theory 21

The Marschak-Machina triangle

 The actual slope of the indifference curves will vary across

individuals (each will have their own utility function), but it will
be lower than 1!
 And the more risk loving an individual is, the more convex the
utility function is and the lower the slope is

 Intuitively, you can ‘test’ this by drawing a close to horizontal

straight line that crosses the certainty point 50
 Receiving 50 for sure yields the same utility as a lottery with a
very low probability of receiving 100, and a very high probability
of receiving 0 or 50!
 This individual must really be very, very risk loving!

Decision Making: 8. Expected Utility Theory 22

Allais’ paradox

 Allais’ paradox (1953)

 Choose between:

 Lottery A: 100% probability of winning €100

 Lottery B: 10% probability of winning €500, 89% probability of

winning €100 and 1% probability of winning €0

Decision Making: 8. Expected Utility Theory 23

Allais’ paradox

 Allais’ paradox (1953)

 Now choose between:

 Lottery C: 11% probability of winning €100 and 89%

probability of winning €0

 Lottery D: 10% probability of winning €500 and 90%

probability of winning €0

Decision Making: 8. Expected Utility Theory 24

Allais’ paradox

 Allais’ paradox (1953)

 Most people prefer:
 A over B
 D over C
 But this violates expected utility!
 Notice that if 𝐴 ≻ 𝐵 then:
 𝑢 100 > 0.1𝑢 500 + 0.89𝑢 100 + 0.01𝑢 0
 This is equivalent to
 0.11𝑢 100 > 0.1𝑢 500 + 0.01𝑢 0

Decision Making: 8. Expected Utility Theory 25

Allais’ paradox

 Allais’ paradox (1953)

 Now just add 0.89𝑢 0 to both sides and obtain
 0.11𝑢 100 + 0.89𝑢(0) > 0.1𝑢 500 + 0.9𝑢 0
𝐶 𝐷
 Why?
 Maybe A is chosen to avoid regret if choose B and get €0
 This regret does not occur between C and D because most likely
outcome is €0 anyway
 ‘extra’ 1% probability of winning something in C does not ‘compensate’
for payoff reduction from €500 to €100

Decision Making: 8. Expected Utility Theory 26

Allais’ paradox: common outcome
effect
 Allais’ paradox (1953)
 Maurice Allais was one of the earliest critics of expected
utility theory
 The paradox illustred above is also known as the ‘common
outcome effect’
 Quite clearly, as we will shortly see, this is related to the
independence axiom (which is violated when people have
these preferences)

Decision Making: 8. Expected Utility Theory 27

Allais’ paradox: common outcome
effect
 Allais’ paradox (1953)
 The lotteries can be seen in the following way
 Lotteries A and C are equivalent to 11% probability of receiving
€100 and 89% probability of receiving X
 In lottery A, X=€100
 In lottery C, X=€0
 Lotteries B and D are equivalent to 10% probability of receiving
€500, 89% probability of receiving X and 1% probability of
receiving €0
 In lottery B, X=€100
 In lottery D, X=€0

Decision Making: 8. Expected Utility Theory 28

Allais’ paradox: common outcome
effect
 Allais’ paradox (1953)
 X is a common outcome in both choices
 It has the same value and same probability
 Expected utility theory, namely its independence axiom, says
that this common outcome should not affect choices
 But it does affect choices!
 Violation of the independence axiom
 Carefully understand it

Decision Making: 8. Expected Utility Theory 29

Allais’ paradox: common outcome
effect
 Allais’ paradox (1953)
 Choice between A and B is the choice between
 A: 11% probability of receiving €100 and 89% probability of
receiving X
 B: 10% probability of receiving €500, 89% probability of receiving
X and 1% probability of receiving €0
 That is, in both cases, you receive X with 89% probability
 So your choice is whether you prefer:
 A: 11% probability of receiving €100
 B: 10% probability of receiving €500 and 1% probability of receiving €0

Decision Making: 8. Expected Utility Theory 30

Allais’ paradox: common outcome
effect
 Allais’ paradox (1953)
 Now choice between C and D is the choice between
 C: 11% probability of receiving €100 and 89% probability of
receiving X
 D: 10% probability of receiving €500, 89% probability of receiving
X and 1% probability of receiving €0
 That is, in both cases, you receive X with 89% probability
 So your choice is whether you prefer:
 C: 11% probability of receiving €100
 D: 10% probability of receiving €500 and 1% probability of receiving €0

Decision Making: 8. Expected Utility Theory 31

Allais’ paradox: common outcome
effect
 Allais’ paradox (1953)
 Framed in this way, the choices between A and B, on the one
hand, and C and D, on the other, are indistinguishable!
 So why do people prefer A over B and D over C?
 Clearly the X matters!
 When choosing A or B, X=100
 When choosing C or D, X=0
 And this is an obvious violation of the independence axiom!
 Let us frame the problem in the Marschak-Machina triangle

Decision Making: 8. Expected Utility Theory 32

 p(500)=1

Probability of €500
Best outcome

B D

p(500)=0 A C
p(0)=0 p(0)=1
Probability of €0
Worst outcome

Decision Making: 8. Expected Utility Theory 33

Allais’ paradox: common outcome
effect
 Allais’ paradox (1953)
 The lotteries C and D are obtained by ‘sliding’ the lotteries A
and B rightward
 Increasing the probability of €0 by 0.89
 The line CD has the same slope and length as the line AB
 Regardless of the individual’s risk attitude, under expected
utility theory:
 When A is preferred to B, C should be preferred to D
 When B is preferred to A, D should be preferred to C
 In this case, A (and C) should be the preferred choices only
when the individual is rather risk averse!

Decision Making: 8. Expected Utility Theory 34

 p(500)=1

Probability of €500
Best outcome

B D

p(500)=0 A C
p(0)=0 p(0)=1
Probability of €0
Worst outcome

Decision Making: 8. Expected Utility Theory 35

Allais’ paradox: common outcome
effect
 Allais’ paradox (1953)
 And yet A is commonly chosen
 Suggesting that an individual is very risk averse
 And D is commonly chosen
 Suggesting that an individual is not too risk averse
 Clear contradiction!
 What may be happening is that the slope of (whatever)
indifference curves the individual has, they have different
slopes in different parts of the triangle!!

Decision Making: 8. Expected Utility Theory 36

Common ratio effect

 Common ratio effect

 This was also found by Maurice Allais
 Consider the following two lotteries

 Lottery A: receive €3,000 with certainty

 Lottery B: probability 80% of receiving €4,000 and 20%

probability of receiving €0

Decision Making: 8. Expected Utility Theory 37

Common ratio effect

 Common ratio effect

 Now consider the following two lotteries

 Lottery C: probability 25% of receiving €3,000 and 75%

probability of receiving €0

 Lottery D: probability 20% of receiving €4,000 and 80%

probability of receiving €0

Decision Making: 8. Expected Utility Theory 38

Common ratio effect

 Common ratio effect

 Kahneman and Tversky found that 80% of people choose
A and 65% choose D
 And again this violates expected utility!
 If A is preferred then we must have:
 𝑢 3000 > 0.8. 𝑢 4000 + 0.2. 𝑢 0
 And if D is preferred then we must have
 0.25. 𝑢 3000 + 0.75. 𝑢 0 < 0.2. 𝑢 4000 + 0.8. 𝑢 0
 Which is equivalent to
 𝑢 3000 < 0.8. 𝑢 4000 + 0.2. 𝑢 0 !!!

Decision Making: 8. Expected Utility Theory 39

Common ratio effect

 Common ratio effect

 How does this differ from the common outcome effect?
 Lottery C is essentially 25% probability of playing lottery A
and 75% probability of receiving €0
 Lottery D is 25% probability of playing lottery B (€4,000 is
received with 0.8x0.25=0.2, as in lottery D) and 75%
probability of receiving €0
 In other words, the ratios of probabilities in C and D are
equal to A and B
 Only difference is the common outcome added
 Independence axiom is violated

Decision Making: 8. Expected Utility Theory 40

Common ratio effect

 Common ratio effect

 As practice, plot the Marschak-Machina triangle and
confirm that A and D cannot be simultaneously chosen
under expected utility theory
 Again, this may be because of the certainty effect associated
with A
 The possible explanations for the common outcome effect
may apply here too
 Probability weighting may explain what is going on…
 … or it may be regret aversion
 Two alternative and competing theories

Decision Making: 8. Expected Utility Theory 41

Indifference curves in the triangle

 A lot of research has gone into trying to depict individuals’

indifference curves inside the Marschak-Machina triangle

 The general shape found is:

 Steeper than 45 degree line in the top-left
 Flatter than 45 degree line in bottom-right
 They appear to fan out near the axis
 But are close to parallel in the centre
 Also some evidence of fan in around the hypotenuse
 That is, expected utility violations appear more likely near the
edges, where either very high or very low probabilities are
associated with lotteries

Decision Making: 8. Expected Utility Theory 42

 P(100)=1

Probability of €100
Best outcome

1/2

P(100)=0
p(0)=0 1/2 p(0)=1
Probability of €0

Worst outcome

Decision Making: 8. Expected Utility Theory 43

Indifference curves in the triangle

 In lotteries involving losses, the indifference curves

become ‘mirror images’ of the above!
 One of the implications is that people appear to be more
risk loving (flatter curves) in the centre of the triangle…
 … whereas when gains are involved they are more risk
averse

Decision Making: 8. Expected Utility Theory 44

 P(100)=1

Probability of €0
Best outcome

1/2

P(100)=0
p(0)=0 1/2 p(0)=1
Probability of -€100

Worst outcome

Decision Making: 8. Expected Utility Theory 45