[G10] Formal decision theory

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.