data/en/crtw_121.txt

Formal decision theory might be considerd to be a branch of mathematics. It
provides a more precise and systematic study of the formal or abstract
properties of decision-making scenarios. Game theory concerns situations where
the decisions of more than two parties are involved. Decision theory considers
only the decisions of a single individual. Here we discuss only some very
basic aspects of decision theory.

The decision situations we consider are cases where a decision maker has to
choose between a list of mutually exclusive decisions. In other words, from
among the alternatives, one and only one choice can be made. Each of these
choices might have one or more possible consequences that are beyond the
control of the decision maker, which again are mutually exclusive.

Consider an artificial example where someone, say Linda, is thinking of
investing in the stock market. Suppose she is considering four alternatives :
investing $8000, investing $4000, investing $2000, or not investing at all.
These are the four choices that are within her control. The consequences of
her investment, in terms of her profit or loses, are dependent on the market
and beyond her control. We might draw up a _payoff table_ as follows :

Choices | Profit  
---|---  
| Strong market | Fair market | Poor market  
invest $8000 | $800 | $200 | -$400  
invest $4000 | $400 | $100 | -$200  
invest $2000 | $200 | $50 | -$100  
invest $1000 | $100 | $25 | -$50  
  
Although the possible returns of the investment are beyond the control of the
decision maker, the decision maker might or might not be able or willing to
assign probabilities to them. If no probabilities are assigned to the possible
consequences, then the decision situation is called " _decision under
uncertainty_ ". If probabilities are assigned then the situation is called "
_decision under risk_ ". This is a basic distinction in decision theory, and
different analyses are in order.

## §1. Decision under uncertainty

### Maximin

The Maximin decision rule is used by a pessimistic decision maker who wants to
make a conservative decision. Basically, the decision rule is to consider the
worst consequence of each possible course of action and chooses the one thast
has the least worst consequence.

Applying this rule to the payoff table above, the maximin rule implies that
Linda should choose the last course of action, namely not to invest anything.

Choices | Profit  
---|---  
| Strong market | Fair market | Poor market  
invest $8000 | $800 | $200 | -$400  
invest $4000 | $400 | $100 | -$200  
invest $2000 | $200 | $50 | -$100  
invest $1000 | $100 | $25 | -$50  
  
Maximin tells Linda to consider the worst possible consequence of her possible
choices. These are indicated by the orange boxes here. Among the worst
consequences of the four choices, the last one is the best of the worst. So
that would be choice to make.

### Maximax

Choices | Profit  
---|---  
| Strong market | Fair market | Poor market  
invest $8000 | $800 | $200 | -$400  
invest $4000 | $400 | $100 | -$200  
invest $2000 | $200 | $50 | -$100  
invest $1000 | $100 | $25 | -$50  
  
Whereas minimax is the rule for the pessimist, maximax is the rule for the
optimist. A slogan for maximax might be "best of the best" - a decision maker
considers the best possible outcome for each course of action, and chooses the
course of action that corresponds to the best of the best possible outcomes.
So in Linda's case if she employs this rule she would look at the first column
and picks the fist course of action and invest $8000 since it gives her the
largest possible return.

### Minimax regret

This rule is for minimizing regrets. Regret here is understood as proportional
to the difference between what we actually get, and the better position that
we could have got if a different course of action had been chosen. Regret is
sometimes also called "opportunity loss".

Choices | Regret  
---|---  
| Strong market | Fair market | Poor market  
invest $8000 | 0 | 0 | 350  
invest $4000 | 400 | 100 | 150  
invest $2000 | 600 | 150 | 50  
invest $1000 | 700 | 175 | 0  
  
In applying this decision rule, we list the maximum amount of regret for each
possible course of action, and select the course of action that corresponds to
the minimum of the list. In the example we have been considering, the maximum
regret for each course of action is coloured orange, and the minimum of all
the selected values is 350. So applying the minimax regret rule Linda should
invest $8000.

## §2. Decision Making Under Risk

When we are dealing with a decision where the possible outcomes are given
specific probabilities, we say that this a case of decision making under risk.
In such situations the _principle of expected value_ is used. We calculate the
expected value associated with each possible course of action, and select the
course of action that has the higest expected value. To calculate the expected
value for a course of action, we multiple each possible payoff associated with
that course of action with its probability, and sum up all the products for
that course of action.

Choices | Profit | expected value  
---|---|---  
| Strong market  
(probability = 0.1) | Fair market  
(probability = 0.5) | Poor market  
(probability = 0.4) |  
invest $8000 | $800 | $200 | -$400 | $800x0.1+$200x0.5+(-$400)x0.4  
= **$20**  
invest $4000 | $400 | $100 | -$200 | $400x0.1+$100x0.5+(-$200)x0.4  
= **$10**  
invest $2000 | $200 | $50 | -$100 | $200x0.1+$50x0.5+(-$100)x0.4  
= **$5**  
invest $1000 | $100 | $25 | -$50 | $100x0.1+$25x0.5+(-$50)x0.4  
= **$2.5**  
  
Since the first course of action has the highest expected value, the principle
of utility implies that Linda should invest $8000. For further discussion
about expected value, see the corresponding section in statistical reasoning.

In the example here, it is assumed that the probabilities assigned to
different market conditions are independent of Linda's decisions. Is this a
reasonable assumption to make? answer

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