Core course Session 1

1
Core course Session 1

• Decision Theory

KW 23
2
A simple brewery case A globally operating beer
brewery is reconsidering its position in the
Nigerian market. Currently, the company exports
to a moderately developed market while
extensively cooperating with local distributors.
A serious alternative, however, is to invest in a
local brewery installation and to meet the
Nigerian demand from this plant. This has the
advantage of relatively low cost of labor and
other resources, and of a reduction of the
transport costs associated with exporting.
However, local conditions are not particularly
optimal as a result of the not too stable
political situation and the fact that the
production site is far removed from the consumer
areas. Also, the development of future demand is
quite uncertain. Economic analysts of the firm
estimate that the foreign investment option would
induce a loss of m 10 when demand is low, and a
m 2 and m 20 profit when demand will be
moderate or high. Continuing the current export
strategy would lead to a zero profit situation at
low demand, while the profit levels are estimated
at about m4 and m8 in the case of moderate or
high demand, respectively. Withdrawing from the
market is not an option, since the firms
ambition is to establish a truly global brand.
But also it does not wish to give in on competing
breweries that are also actively present in this
market. Prior discussions with local agents in
Nigeria suggest that the probability of a low
demand is about 15, that of a moderate demand
65 and a high demand 20. The request is to
provide the brewery with a solid advise about the
most desirable course of action.
3
Brief intro

• Situations like these are characterized by
  uncertainty throughout
• uncertainty about the outcomes
• uncertainty about the possible alternatives from
  which to choose
• uncertainty about the circumstances that
  influence the outcomes
• uncertainty about the structures that govern
  particular outcomes or events
• One way to handle this uncertainty is to ignore
  it. Another way is to develop a theoretical
  framework to analyse the relations between the
  different uncertain elements of the decision
  problem and to further support the decision
  making
• Note beforehand that all theorizing occurs at the
  expense of simplification. Management should be
  particularly aware of the relevance and impact of
  these simplifications

4
Agenda Session 1

• Introduction and case
• Anatomy of decisions
• actions, events, payoffs and opportunity losses
• instruments payoff tables and decision trees
• Non-probabilistic decision strategies
• maximin, maximax, minimax regret
• Probabilistic decision strategies
• maximum likelihood, expected monetary value,
  backward induction
• Using additional information
• Bayesian analysis, a posteriori probabilities,
  preposterior analysis
• The cost of information
• EPPI, EVPI, EVSI, and ESI
• Utility of money
• St Petersburg paradox
• the utility of money
• attitudes towards risk

5
Anatomy of decisions
KW 23.1, 23.2
6
Anatomy of decisions
Decision problems like these contain several
common elements, notably actions, events (states
of nature) and some payoff (or loss)
export
Actions, are the (managerial) alternatives from
which the decision maker is to choose, A1, ...,
An. The brewery has but two alternatives
continue exporting (A1) or local investment (A2)
invest
Events or states of nature, are the conditions
that affect the outcomes of actions, but which
are beyond the immediate (complete) control of
the decision maker. Usually these events are of a
stochastic nature, S1, ..., Sm. For the brewery,
the firms demand is the conditioning variable
with 3 possible outcomes low, moderate or high.
low
mod
high
Payoffs are the positive or negative benefits
associated with each combination of action and
state of nature. These payoffs maybe monetary or
different. In the brewery example, the payoffs
are in net cashflow (m) .
7
Anatomy of decisions
Using the notational conventions, the decision
problem may be summarized by means of a decision
tree and/or a payoff table
Actions
Events
Payoffs
A decision tree is a tree-like illustration of
the (possibly alternating) sequence of actions
and events that are characteristic of the
decision problem with the payoffs at the end
points of the branches. In the brewery example,
this would look like the following...
m 0
export
m 4
m 8
m -10
invest
m 2
Implicit in this formalization is the assumption
that the decision maker wishes to maximize
payoffs (or minimize losses). This will be made
more specific later on.
Payoff table
m 20
8
Anatomy of decisions
The information from the decision problem may
also be summarized by means of a payoff table or
by an opportunity loss table
Payoff table (in m )
Opportunity loss table (in m )
Actions
Actions
A1 export
A2 invest
A1 export
A2 invest
Events
Events
Maximum or Opportunity
S1 low
S1 low
m 0
m -10
m 0
m 10
m 0
S2 mod
S2 mod
m 4
m 2
m 0
m 2
m 4
S3 high
S3 high
m 8
m 20
m 12
m 0
m 20
... in terms of losses ...
... in terms of benefits ...
the highest payoff for each state of nature
9
Anatomy of decisions
By now, we have structured the decision problem.
This is important because it provides insight
into the main elements of the problem. But even
more importantly, it helps to organize ones mind
about the decision problem (and prevents against
mixing up details, forgetting relevancies, or
drawing incorrect conclusions about the logic)
The next step is to use these conventions to
support the decision making. The problem is that
different decision strategies may be employed
possibly suggesting different optimal actions.
Two broad classes of strategies are the
non-probabilistic and the probabilistic
strategies. Both will be addressed here, starting
with the former.
10
Non-probabilistic decision strategies
KW --
11
Non-probabilistic strategies

• Non-probabilistic decision strategies are
  decision strategies that ignore the probabilistic
  nature of the events that follow the decisions
  taken. Three examples are discussed here (we skip
  throwing darts)
• maximin (or minimax)
• maximax (or minimin)
• and minimax regret

Maxi refers to maximizing some sort of benefit,
while Mini refers to minimizing a loss. The
latter may be interpreted as a Maxi problem by
considering losses as negative benefit.
Hereafter, we only mention the Maxi problems.

• These methods are best explained with the payoff
  table (maximin and maximax) and the opportunity
  loss table (minimax regret)

Maximin and maximax
12
Non-probabilistic strategies
MAXIMIN
MAXIMAX
Maximin choose the action with the highest
minimum benefit
Maximax choose the action with the highest
maximum benefit
Conservative view of nature
Optimistic view of nature
Minimum
m 0
m -10
Maximum
m 8
m 20
13
Non-probabilistic strategies
MINIMAX REGRET
If one had perfect foresight regarding the event
occurring, then one would choose the action with
the highest payoff. In the absence of such
perfect foresight, one would regret the
opportunity loss, when the action chosen does not
yield the highest payoff.
Minimax regret choose the action with the lowest
maximum regret
Particularly for those who are easily moved by
opportunity losses
Maximum regret
m 12
m 10
14
Non-probabilistic strategies
Pros and cons of non-probabilistic decision rules

• The advantage of non-probabilistic strategies is
  that they are simple to understand and to apply
• The disadvantages of these methods are that ...
• the decisions taken vary with the mood and nature
  of decision makers
• prior information about the likeliness of the
  events is not taken into account
• adding irrelevant alternatives may alter the
  decision (see next slide)

Illustration
15
Non-probabilistic strategies
Example. In a subsequent meeting about the issue,
it is suggested to license a Nigerian partner to
produce the beer. This alternative has payoffs m
3, 5, and 6. Analyze the impact of this action
(A3)
Impact of irrelevant alternatives
m 0
m 3
m 13
m 0
m 1
m 3
m 0
m 12
m 14
Maximum regret (old)
m 12
m 10
--
Minimum
m 0
m -10
m 3
Maximum regret (new)
m 12
m 13
m 14
Maximum
m 8
m 20
m 6
The optimal choice changes from A2 to A1 due to
A3 (which is called irrelevant because it is
itself not chosen)
So, the maximin strategy changes from A1 to A3
(in which case A3 is not irrelevant!) the
maximax strategy is unchanged
16
Probabilistic decision strategies
KW 23.2
17
Probabilistic strategies

• Usually some (prior) idea about the likeliness of
  the events (states of nature) is available. Using
  this information leads to the so-called
  probabilistic decision strategies
• maximum likelihood decision rule
• expected monetary value (EMV) rule
• This prior idea about the probability of events
  may either have a subjective origin (thumb,
  feeling, etc) or an objective origin (based on
  current facts, last years results, etcetera)
• The prior probabilities may either be
  uninformative (we simply dont know, all events
  are considered equally likely) or informative
  (some events are considered more likely than
  others)
• Both strategies (maximum likelihood and EMV) are
  illustrated...

18
Probabilistic strategies
Maximum likelihood method
The maximum likelihood method simply attaches all
the weight to the most likely event after which
the action with the highest payoff (or lowest
opportunity loss) is taken
Prior probability distribution
Disadvantage a large amount of information is
ignored and the associated probability may itself
be very small (in case of many events) or very
close to probabilities of other events
... since exporting (A1) has the highest payoff
for the event with the largest probability, this
is the optimal decision in a maximum likelihood
approach
m 4
EMV
19
Probabilistic strategies
Maximize expected monetary value (EMV)
The EMV-method determines the optimal action as
the alternative with the highest expected payoff
(or lowest expected opportunity loss)
Let X denote the payoff, then for each separate
action we find
A1 export
EMV1
E(XA1)

0
?
0.15

4
?
0.65

8
?
0.20
m 4.2

A2 invest
EMV2
-10
?
0.15

2
?
0.65

20
?
0.20
m 3.8

EMV
m 4.2
m 3.8
... since exporting (A1) has the highest expected
monetary value, this is the optimal decision in
an EMV-approach
Tree
20
Probabilistic strategies
Maximize expected monetary value (EMV)
The decision tree can, of course, also be used to
perform the same EMV-calculations. The
calculations involved are denoted as backward
induction, rollback technique, folding back and
likewise expressions...
Backward induction, averaging out-and-rollback
At every chance node , we average the payoff
and at every decision node , we choose the
action with the highest (expected) payoff
m 0
m 4.2
m 4
m 8
m 4.2
The decision tree easily handles more complex
decision problems (for which the simple payoff
table is no longer suited)
m -10
m 3.8
m 2
Pruning, the branches (actions) that are not
optimal are cut away from the tree
Illustration
m 20
21
Probabilistic strategies
This sounds a little bit speculative. Is there a
less ad hoc way to incorporate additional
information in the decision problem?
Maximize expected monetary value (EMV)
The decision tree easily handles more complex
situations...
Imagine that one of the brewery directors claims
that foreign investment would stimulate demand
making a poor performance extremely unlikely.
Instead, demand is moderate or high with
probabilities 0.7 and 0.3, respectively. Payoffs
are assumed to remain the same. Redo the
analysis...
m 4.2
m 7.4
m 7.4
Note that the plot can easily be extended to
include subsequent decisions and associated
events. For instance, the low demand event may
invite the brewery to set up a promotional
campaign which has more or less favorable (though
uncertain) outcomes.
22
Using additional information
KW 23.3
23
Using additional information

• The decision problem discussed sofar proceeds as
  if the only knowledge available about the
  uncertain events is the prior probability
  function p(S1) 0.15, p(S2) 0.65, and p(S3)
  0.20
• In practice, decision makers often launch testing
  stages, expert meetings or market research to
  increase their insights into the likeliness of
  the uncertain states of nature. The question is
  how this information R ( results) can be used to
  enhance the decision.

... the answer is by using the rule of Bayes
(Refresher Session 1)

• The decision problem is then solved based on the
  posterior probability function for the events p(S
  R)

24
Using additional information

• Next, we show
• how expert information may affect the prior
  probabilities leading to the posterior
  probability function
• how information about the quality of market
  research reports may enter the problem
• how the decision tree should be adjusted to take
  the additional information into account
• how the outcome of the decision problem may
  change (or not) due to the available information

The impact of expert info on prior
probabilities
25
Using additional information
We start this part of decision theory by
illustrating the impact of additional information
on the prior probabilities. The aim of this
exercise is to provide an extra illustration of
Bayes rule.
Following the original decision problem,
management decides to search for some expert
information. It calls the Nigerian Embassy in The
Hague to invite 20 Nigerian expats to a (free and
perfectly decent) beer tasting party. During the
party these guests were asked whether they were
likely to buy this brand of beer on a regular
base, of which 5 responded positively (and the
rest not). How likely is this outcome in light
of the prior probabilities (P(low) 0.15, P(mod)
0.65, P(high) 0.20 and considering that the
state of demand is associated with the proportion
of Nigerian households regularly buying beer
0.05 (low), 0.10 (mod), 0.20 (high)? How does
this information affect the (prior) probability
distribution ?
Calculations
26
Using additional information
We start with the prior
Likelihood of the results
Demand
Prior
Joint results
Posterior
Low
P(S0.05) 0.15
P(R5 S0.05)
0.002
0.002?0.15 0.0003
0.0003/0.05600.0060
P(R5 S0.10)
0.032?0.65 0.0207
0.0207/0.05600.3705
Mod
P(S0.10) 0.65
0.032
P(R5 S0.20)
0.175
0.175?0.20 0.0349
0.0349/0.05600.6235
P(S0.20) 0.20
High
0.0560
1
1
Probability of 5 positive answers (out of 20), R
Bin(n20, p si)
27
Using additional information
Impact of the incorporation of the costless
beer tasting event
Sometimes information is not without cost also,
the decision to collect the information is itself
part of the decision problem. An example is given
in the next slide.
m 6.5
m 13.2
m 13.2
Costless is put in between quotes. Although the
tasting event involves no (or negligible)
immediate costs, the impact of making an
investment decision on this sort of information
may be huge
28
Using additional information
Example. The probability that the research
institute will predict a low demand when in fact
demand is low, is 70. The research institute
seems to have a problem predicting low demand ...
Management, after thorough discussion, begins to
understand that the beer tasting experiment might
not just be the most valid approach towards
understanding the market. It therefore decides to
hire a famous but expensive research institute to
explore the Nigerian market at the cost of m
0.5. The beer brewery no longer has any control
over the experiment (so, no direct observations).
However, it does have information about the track
record of the company (that is, its ability to
predict market demand correctly). This
information is summarized in the table stating
the conditional probabilities of a particular
research outcome (Rj) given the true state of
nature (Si). Note that this information is
available before the desired market analysis is
performed, and that a decision about whether to
hire the research company still has to be taken.
How does hiring the institute affect the decision?
29
Using additional information
We have (a) the prior probabilities, p(S) and
(b) the past record of the institute p(R S)
We want (c) posterior probabilities p(S R)
Institute predict low demand, R low
P(R l S l)
0.70
0.70?0.15 0.105
0.105/0.144 0.732
P(R l S m)
0.05?0.65 0.033
0.05
0.033/0.144 0.227
P(R l S h)
0.03
0.03?0.20 0.006
0.006/0.144 0.042
0.144
1
P(R low)
Institute predict moderate demand, R mod
P(R m S l)
0.20
0.20?0.15 0.030
0.030/0.564 0.053
P(R m S m)
0.80?0.65 0.520
0.80
0.520/0.564 0.922
P(R m S h)
0.07
0.07?0.20 0.014
0.014/0.564 0.025
0.564
P(R mod)
1
Institute predict high demand, R high
P(R h S l)
0.10
0.10?0.15 0.015
0.015/0.293 0.051
P(R h S m)
0.15?0.65 0.098
0.15
0.098/0.293 0.333
P(R h S h)
0.90
0.90?0.20 0.180
0.180/0.293 0.615
0.293
P(R high)
1
30
All payoffs are minus the cost of research, m
0.5
m 0.7
Consequences for the decision (tree)
m 0.7
Caution. Expected monetary value of the
research-option (EMVS) after paying for the
research to enable direct comparison with the EMV
of the original no research option ( m 4.2)
m -6.5
m 3.4
m 3.4
m 5.5
research
m 1.3
In this particular case, the investment in
research seems to be worthwhile, but how much
would one be willing to pay in general, though?
m 5.8
m 5.5
m 12.0
m 12.0
cf. slide 20
no research
Hence, the brewery should hire the research
institute. When the results predict high demand,
then the brewery should invest, otherwise he
should continue exporting
31
The value of information
KW 23.3
32
The value of information

• Information (test results, market research, etc.)
  is usually not for free the value of the
  information therefore becomes an important
  consideration in the decision process

• Two situations are discerned
• the presence of perfect foresight (or perfect and
  free information) this is not entirely
  realistic, but it sets a benchmark for the
  maximum that a decision maker should be willing
  to pay to reduce uncertainty
• the availability of imperfect information from
  samples and tests, which is usually available at
  a cost
• The question in both instances is what the
  expected value of the information is

33
The value of information
The (expected) value of perfect information (EVPI)
If the brewery has perfect foresight, it chooses
exports in case of low or moderate demand, and
foreign investment in case of high demand
If no other information is available than the
prior probabilities, the best we can do is to
maximize the expected monetary value
The (expected) payoff associated with this
strategy is the so-called expected payoff of
perfect information (EPPI)
EPPI
0
?
0.15

4
?
0.65

20
?
0.20

m 6.6
The expected value of this perfect information
(EVPI) is the difference between EPPI and the
expected monetary we already had without any
foresight (EMV)
EMV
m 4.2
m 3.8
Exporting (A1) is preferred when having no
insight into the state of nature
EVPI
EPPI - EMV
m 6.6 - m 4.2
m 2.4
34
The value of information
The (expected) value of (imperfect) sample
information (EVSI)
The expected payoff of sample information (EPSI)
is the payoff that can be expected from the use
of imperfect (sample) information before
considering the cost of research. By definition,
EPSI EMVS (cf slide 30) research costs.
EPSI
m 6.0
m 5.5 m 0.5
The expected value of sample information (EVSI)
is the amount of money a decision maker can be
expected to pay for the imperfect (sample)
information to reduce (not remove) uncertainty
before making the decision to research.
EVSI
EPSI - EMV
m 6.0 - m 4.2
m 1.8
The expected gain from the sample information
(EGSI) is the expected benefit of the research
decision (EMVS) vis-a-vis the expected monetary
value we have without foresight (EMV). By
definition, EGSI EVSI research cost
EGSI
EVSI - cost
m 1.8 - m 0.5 m 1.3
EMVS - EMV
... if the institute had known this beforehand,
they might have charged just a little bit more
m 5.5 - m 4.2 m 1.3
35
The value of information
The efficiency of sample information (ESI)

• The use or efficiency of the results from market
  research or product tests is sometimes expressed
  as a percentage of the expected value of perfect
  information. The result is denoted as the
  efficiency of sample information (ESI)

The market analysis by the research institute
provided 75 of what perfect information would
have been worth. If this ratio is considered
low, then the beer brewery should look for
another research company. If it is considered
high enough, the brewery could farm out the order
to the institute and await the results.
EVSI
ESI
? 100

EVPI
1.8

? 100
2.4

75.0
36
The utility of money
KW --
37
The utility of money

• The framework presented so far heavily emphasizes
  the (linear) importance of money. Actions with
  the highest expected monetary value are always
  considered preferable.
• In a way, the framework can be easily amended by
  reverting to other units of measurement, for
  instance employees, soldiers, m2 areas of nature
  reserve, etcetera.
• In another way, the framework should somehow
  account for the fact that an additional monetary
  unit payoff does not have the same meaning
  (utility) for each and every decision maker.
• We shall subsequently discuss the St. Petersburg
  paradox and attitudes towards risk to make the
  point.

38
The utility of money
St. Petersburg paradox

• A valid coin is tossed in game between two people
  A and B.
• If head appears in the first toss, A pays B 1.
  
• If the first head appears in the second toss, A
  pays B 2.
• If the first head appears in the third toss, A
  pays B 4.
• If the first head appears in the nth toss, A
  pays B 2n-1.
• The question now is, how much money is person B
  willing to pay to play this game?
• Daniel Bernouilli (1700-1782)

Payoff table for the gamble
P(first head)
Toss
Payoff
EMV
1
0.5
0.5
1
2
(0.5)2
2
0.5
3
(0.5)3
4
0.5
4
(0.5)4
8
0.5
EMV ?
39
The utility of money
?800 28.28
Bernouilli explained the unwillingness to
participate in this game with such a large stake
by the fact that people do not maximize expected
monetary values but rather the expected utility
(overall welfare) of money. Also, this utility
of money was assumed to increase as a function of
money, but at a decreasing rate. For example, U
?
?500 22.36
increase of 5.92
decrease of 8.22
?200 14.14
500
200
800
If we take this utility function as a starting
point for the payoffs of the St. Petersburg game,
then it can be shown that the expected utility of
the game is 1.707 ( 1 ½?2) utility units, which
is equivalent to 2.91 (lt?!)
40
The utility of money
Different types of decision makers may have
different attitudes towards risk. They may be
risk averse, risk neutral, or risk seeking.
What to do? Imagine that you have 10.000 to
spend on either of the following two options.
First, you may invest it in a brand new fund
which is predicted to generate another 6000 when
the market rises, but leads to a loss of 4000
when the market goes down. Second, you may invest
it in a venerable type of government bond that
generates a 1000 regardless the development of
the market. The institute that offers the brand
new fund is a little bit secretive about their
expectations regarding the market. But some
investigation on your own account leads to expect
a 50-50 chance of a high or a poor market.
41
The utility of money
The decision problem may be illustrated by the
following payoff table, which lets you easily
calculate the expected monetary values and to
make the decision
Which option would you choose?
1000
1000
EMV
42
The utility of money
Different types of decision makers may have
different attitudes towards risk. They may be
risk averse, risk neutral, or risk seeking.
Risk aversity
Risk neutrality
Risk seeking
U(6)
U(1)
U(-4)
1
-4
6
1
-4
6
1
-4
6
E(U(-4)U(6))
The utility of a certain 1000, U(1), is valued
over the expected value of the utilities
associated with an equally likely but uncertain
4000 (loss) and 6000 (gain), E(U(-4)U(6)).
Risk aversity
The utility of the certain amount U(1) is equally
valued as the uncertain alternative
E(U(-4)U(6)). Risk neutrality
The utility of the certain amount U(1) is valued
less than the expected utility of the
alternative, E(U(-4)U(6)). Risk seeking
43
to conclude
Course Organization Next Week
44
End of Session 1

• A word on organization
• course manual
• weekly self-study exercises (own responsibility)
• weekly assignment (obligatory, group work)
• one larger assignment (obligatory, group work)
• mid-term exam and final exam (obligatory,
  individual efforts)
• various support (see manual)
• Next time
• sampling, surveying, measurement scales
• Ch.5 KW, Powell(1995)