8. Decision Making Under Uncertainty - Games Against
Nature
Up to now randomness (uncertainty) has not been directly taken
into account (except in life insurance).
For example, the risk of a loan not being repaid was measured in
terms of an appropriate increase in the interest.
The resulting calculations were carried out as if everything were
deterministic, but by applying a different interest rate.
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Decision Making Under Uncertainty - Games Against
Nature
When a problem is assumed to be deterministic, it is clear that we
should maximise profits or minimise costs.
However, when profits (or losses) are random, then various criteria
are used for decision making, which reflect individuals’ attitude
towards risk.
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Criteria for Decision Making Under Uncertainty
1. The minimax criterion.
2. The Savage (regret) criterion.
3. The Hurwicz criterion.
4. The Laplace criterion.
5. Maximisation of expected payoff.
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Criteria for Decision Making Under Uncertainty
Suppose one should choose between
A. $40 for sure.
B. $100 if a coin toss results in heads, otherwise $0
Hence, the payoff of the individual my depend on the outcome of a
random experiment, here the result of a coin toss.
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The Minimax Criterion
Using the minimax criterion, we make the decision which gives us
the optimal ”worst case scenario”.
Thus if xmin,D is the minimum payoff that is possible under action
D, then an individual makes the decision that maximises this
payoff.
In this case, by taking decision A, the decision maker’s minimum
(guaranteed) payoff is $40.
By taking decision B, the worst possible outcome is that the
decision maker gets nothing.
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The Minimax Criterion
Hence, using the minimax criterion, the decision maker should take
decision A.
Note: When costs are considered, let cmax,D be the maximum
possible cost when decision D is taken (i.e. the worst possible
scenario).
Using the minimax criterion, the decision maker takes the decision
which minimises this ”worst case scenario” cost.
The minimax criterion is generally very conservative (risk averse).
For example, if the payoff under decision A were changed to $0.01,
the minimax criterion still recommends decision A.
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The Savage (Regret) Criterion
This is a less conservative approach to decision making.
When payoffs are considered, the regret from making a decision
given the outcome of an experiment is the different between the
maximum payoff obtained given the outcome of the experiment
and the payoff actually obtained.
For example, when the coin toss results in tails, if the decision
maker takes the decision A (and thus gets $40), this is the
maximum possible payoff (i.e. the regret in this case is 0).
In this case, when the decision maker takes the decision B (and
thus gets $0), his regret is 40 (the difference between the payoff he
gets and the maximum possible given the result of the experiment).
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The Savage (Regret) Criterion
On the other hand, when the result of the coin toss is heads, B is
the best possible decision, obtaining a payoff of $100.
Hence, in this case the regret from making decision B is 0.
In this case, by taking the decision A, the decision maker gets $40
rather than the maximum possible $100.
Hence, the regret from taking decision A in this case is 60.
The following two slides illustrate the minimax and Savage criteria.
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Minimax Criterion
A
B
Tails
40
0
Heads
40
100
Min.
40
0
Note that since we are considering payoffs, the worst situation for
a given decision corresponds to the lowest payoff.
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Savage Criterion
A
B
Tails
40
0
Heads
40
100
Regret (T)
0
40
Regret (H)
60
0
Max. Regret
60
40
Note that regret is a cost, the worst situation for a given decision
corresponds to the highest regret.
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The Savage (Regret) Criterion
Under the Savage (Regret) Criterion, a decision maker should
make the decision which minimizes the maximum possible regret.
The maximum possible regret given decision A is 60.
The maximum possible regret given decision B is 40.
Hence, under the Savage (Regret) criterion, the decision maker
should take action B.
Note: It should be noted that when costs are considered, the
regret felt when an individual takes an action is equal to the cost
actually incurred minus the minimum costs possible given the
outcome of the experiment.
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Relation Between the Savage (Regret) Criterion and the
Minimax Criterion
It should be noted that whatever way the original problem is
formulated in (using payoffs or costs), regret is a ”cost”.
The Savage criterion suggests minimising the maximum possible
regret.
Hence, the Savage criterion applies the minimax criterion to the
regret scores.
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The Hurwicz Criterion
The Hurwicz criterion takes into account both the best and worst
possible outcomes.
The best outcome is weighed by a factor of α, 0 < α < 1, where α
can be interpreted as the index of optimism.
The worst outcome is weighed by a factor of 1 − α.
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The Hurwicz Criterion
When payoffs are considered, a decision maker should maximise
the Hurwicz score of the decision taken, where the score for
decision D is given by
HD = αxmax,D + (1 − α)xmin,D ,
where xmax,D and xmin,D are the maximum and minimum,
respectively, possible payoffs under the action D.
Under action A, the decision maker always gets $40, thus
xmax,D = xmin,D = 40.
Under action B, xmax,D = 100, xmin,D = 0.
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The Hurwicz Criterion
Suppose the index of optimism is α = 0.3. It follows that
HA =0.3 × 40 + (1 − 0.3) × 40 = 40
HB =0.3 × 100 + (1 − 0.3) × 0 = 30.
Hence, the decision maker should take action A.
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The Hurwicz Criterion
When costs are considered, the Hurwitz score of decision D is
given by
HD = αcmin,D + (1 − α)cmax,D ,
where cmax,D and cmin,D are the maximum and minimum,
respectively, possible costs under the action D.
Note that the coefficient α is always associated with the best
situation (maximum payoff or, as in this case, least costs).
In the case of costs, the decision maker should minimise the
Hurwicz score of the decision taken.
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The Hurwicz Criterion
When α = 0 (i.e. the decision maker is very pessimistic), the
Hurwicz criterion corresponds to the minimax criterion.
When α = 1 (i.e. the decision maker is very optimistic), the
Hurwicz criterion corresponds to choosing the action which gives
the maximum possible payoff of all those possible (minimum
possible cost).
In this way, we can model a range of decision makers, from the
most pessimistic to the most optimistic.
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The Laplace Criterion
The Laplace criterion states that in the absence of any other
information, we should assume that the possible outcomes of the
experiment are equally likely.
This is the principle of insufficient information
In this case, we maximise the average of the possible payoffs for a
given decision (this is the expected payoff under the above
assumption).
In this case the expected payoff from decision B is
0+100
2
= 50.
The guaranteed payoff from taking decision A is 40.
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The Laplace Criterion
Hence, the decision maker should take decision B.
When costs are considered, under the Laplace criterion, we
minimise the average of the possible costs.
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Maximisation of Expected Payoff
However, in many cases we may have sufficient information to
estimate the probabilities of various outcomes from previous
experience
e.g. the results of coin tosses, die rolls, the probability of rain on a
day in June.
In this case, it makes sense to use this information, in order to
make a more informed decision.
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Maximisation of Expected Payoff
Given that the coin is fair, then the Laplace criterion is equivalent
to the maximisation of the expected payoff (or minimisation of
expected costs, when costs are considered).
More generally, suppose that the probability of heads is p.
The expected reward from decision B is thus
100p + 0(1 − p) = 100p.
Thus the decision maker should take decision B when 100p > 40,
i.e. p > 0.4.
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Decision Making Under Uncertainty
When the possible results of an experiment form a discrete set
(e.g. the result of a die roll, coin toss, winner of an election), then
the possible payoffs can be presented in a matrix.
Each row corresponds to a decision.
Each column corresponds to an outcome of the experiment.
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Example 8.1
Suppose the time taken to travel to work using one of three modes
of transport: A, B, C , in three different traffic conditions: Good,
Average, Poor, are given below
A
B
C
Good
10
30
40
Average
30
40
40
Poor
90
60
40
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Example 8.1
Derive the decision that should be taken under
1. The minimax criterion.
2. The Savage (regret) criterion.
3. The Hurwicz criterion with optimism index α = 0.4.
4. The Laplace criterion.
5. Minimisation of expected cost, given that the
probability of good and poor conditions are both
equal to 0.2.
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Example 8.1
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Example 8.1
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Example 8.1
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Example 8.1
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Example 8.1
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Example 8.1
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Example 8.1
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Example 8.1
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