Course support for the DIA module at the ESC Pau

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Risk and
Uncertainty
January 25 2017
Prof. M. MINHAJ
Introduction
Managerial Decision Making
http://Dsign4.biz

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What is a risk ?
(In French : Risqué)

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Are you a risk taker or
averse ?

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Imagine that you are offered two
options. Option 1 involves a guaranteed
payout of a negotiable amount. Option 2
is a payout based on a coin toss. You
have a 50/50 chance of winning. If you
do, you receive €100. If you lose, you
receive nothing.

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Decisions under risk
vs.
decisions under uncertainty

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Choose between two lotteries
(Problem L 1)
• A. Lottery that gives
you €0 with
probability 0.5 and
€1,000 otherwise.
• B. Lottery that gives
you €500 with
probability 1.
A < B or A > B or A ~ B (you are indifferent between
the two)

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Choose between two lotteries
(Problem L 2)
• A. Lottery that gives
you €0 with
probability 0.2, €400
with probability 0.6
and €1,000 otherwise.
• B. Lottery that gives
you €400 with
probability 0.6 and €
500 otherwise.
A < B or A > B or A ~ B (you are indifferent between
the two)

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Draw Decision Tree for the following
Assume that nature first decides whether,
with probability 60%, you get €400 for sure,
or with probability 40 %, you get to choose
between the alternatives in problem L1

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Problem L 1:
Problem L 2 :
400 P 40
0
Q
0.60 0.600.40 0.40
P Q
P = (1,500)
Q = (0.5, 0 ; 0.5, 1,000)

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The independence axiom
The choice
Should be the same as
P Q
R P R Q
1 – α 1 –
α
α α
The preference between two
lotteries P and Q is the same as
between
(α, P ; (1 – α), R)
and
(α, Q ; (1 – α), R)

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Von Neumann–Morgenstern’s
theorem
• The decision maker can be viewed as if they
were maximizing the expectation of a utility
function.
• There exists an assignment of utility numbers
to outcomes such that for any pair of lotteries
P and Q, the decision maker will always prefer
the one that has a higher expected utility.

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Expected utility
• Suggested by Daniel Bernoulli in the
eighteenth century
• A lottery (p1, x1; … ; pn, xn)
is evaluated by the expectation of the utility:
p1*u(x1) + … + pn *u(xn)

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For example, assume that utility function is :
U (€ 0) = 0
U (€ 400) = 0.5
U (€ 500) = 0.6
U (€ 1000) = 1
For the problem given below :
A : €0 0.5 B : € 500 1
€ 1,000 0.5
The expected utility of option A is
0.5 * U (€0) + 0.5 * U(€1000) = 0.5 * 0 + 0.5 * 1 = 0.5
The expected utility of option B is
1 * U (€500) = 1 * 0.6 = 0.6
According to Von Neumann–Morgenstern’s theory, we would prefer B to A.

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Prospect Theory
Problem 1 :
A : € 0 0.2 B: € 3,000 1
€ 4,000 0.8
Problem 2 :
A : € 0 0.8 B: € 0 0.75
€ 4,000 0.2 € 3,000 0.25
Many people choose B in Problem 1 and A in Problem 2

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Prospect Theory
Prospect theory is a behavioural economic theory that describes the way people
choose between probabilistic alternatives that involve risk, where the probabilities of
outcomes are known.
The theory states that people make decisions based on the potential value of losses
and gains rather than the final outcome, and that people evaluate these losses and
gains using certain heuristics. The model is descriptive: it tries to model real-life
choices, rather than optimal decisions, as normative models do