Week 2
Decision Making Under Risk and Uncertainty
Introduction
Risk- we can accurately calculate the probability that something will happen, the probability
distribution of outcomes is known objectively (eg. Lottery)
Uncertainty- the probability distribution of outcomes is unknown (eg. Horse racing)
Lotteries
1) Simple lottery
N possible outcomes
X G xn
with a probability distribution over outcomes
can be represented by a lottery Pi Pn whereEntpn l
L ien
eg. X 100,107,4 025,075 X 100,101,4p105.0.5
2) Compound lottery
a list Li Leia ae
where is the probability of lottery k occurring andEYand
ar
eg. L L20.2.0.8
This can be transformed into a reduced lottery a tank
Eg. X 100,107,410.250.751 x 100,101,2210.50si
Lit
4,2202,08
o.zxo.zsto.sxo.so.es l ottery
Reduced
p o.zxo.zsto.sxo.s o.se x100,101,210.450.54
B
Expected value of a lottery:111
11 II pix exittenXn
St. Petersburg Paradox
• You have an initial endowment of £1
• You ip a coin until it is tails, if it is heads your endowment doubles
• You win the endowment when it comes up tails for the rst time
How much would you pay to play:
EV 42 2 444thnk H n n
I 42
Royd
2 Y Pyg's
114
I
k i Yar
As a result of this, Bernoulli proposed that people should maximise expected utility not expected
value
Preferences over lotteries
Let
E denote the decision maker’s preference relation on L (The set of all simple lotteries)
The decision maker has preferences over lotteries on some nite price space X
Only the reduced form lottery over nal outcomes is if relevance to the decision maker
Decision Making Under Risk and Uncertainty
Introduction
Risk- we can accurately calculate the probability that something will happen, the probability
distribution of outcomes is known objectively (eg. Lottery)
Uncertainty- the probability distribution of outcomes is unknown (eg. Horse racing)
Lotteries
1) Simple lottery
N possible outcomes
X G xn
with a probability distribution over outcomes
can be represented by a lottery Pi Pn whereEntpn l
L ien
eg. X 100,107,4 025,075 X 100,101,4p105.0.5
2) Compound lottery
a list Li Leia ae
where is the probability of lottery k occurring andEYand
ar
eg. L L20.2.0.8
This can be transformed into a reduced lottery a tank
Eg. X 100,107,410.250.751 x 100,101,2210.50si
Lit
4,2202,08
o.zxo.zsto.sxo.so.es l ottery
Reduced
p o.zxo.zsto.sxo.s o.se x100,101,210.450.54
B
Expected value of a lottery:111
11 II pix exittenXn
St. Petersburg Paradox
• You have an initial endowment of £1
• You ip a coin until it is tails, if it is heads your endowment doubles
• You win the endowment when it comes up tails for the rst time
How much would you pay to play:
EV 42 2 444thnk H n n
I 42
Royd
2 Y Pyg's
114
I
k i Yar
As a result of this, Bernoulli proposed that people should maximise expected utility not expected
value
Preferences over lotteries
Let
E denote the decision maker’s preference relation on L (The set of all simple lotteries)
The decision maker has preferences over lotteries on some nite price space X
Only the reduced form lottery over nal outcomes is if relevance to the decision maker
