LA METODOLOGIA DE LA CONTEMPLACION DE LOS MISTERIOS DE LA VIDA DE CRISTO EN LOS EJERCICIOS ESPIRITUALES

Decision analysis – also called decision theory – is one of the main tools used for decision- making in an uncertain environment (Groebner, Shannon et al., Chapters 19, 20). In making a decision, a manager must choose from a number of alternative courses of action that are intended to solve the problem. However, under conditions of uncertainty, the result of choosing any of the alternatives is unknown. Even where the possible outcomes are known, the manager may still be unable to make a prediction because there is insufficient information to allow probabilities to be assigned to the outcomes.

As an example, consider a supplier who has received a telephone order from customer A for 200 widgets. Shortly afterwards, another customer, B, arrives on the premises urgently

requiring 200 widgets. Unfortunately, the supplier has none in stock except for the 200 items reserved for customer A. What should the supplier do? Should he sell the reserved items to customer B and hope that he can get a reorder in time to satisfy customer A? Should he refuse B and lose out on a definite sale? What if customer A decides to cancel the order? What if the reordered widgets are more expensive than current stock? All these questions contain such levels of uncertainty that the decision-maker can only answer them by drawing upon his or her own experience and judgement.

In order to build a model, a decision-maker must distinguish between controllable and uncontrollable variables (see Figure 2.2). In decision-analysis terminology, the uncontrollable variables are called ‘events’ or ‘states of nature’. A future event is a possible outcome over which the decision-maker has no control. In an environment of uncertainty, identifying those future events that can occur is generally easier than identifying events that will occur. Having drawn up a list of possible events, the decision-maker usually constructs a payoff table (or matrix). A payoff table assigns some payoff (i.e., monetary value or weighting factor) to each event in order to quantify them more accurately. An example of a payoff table is given in Example 2.7. The founder of modern decision analysis was an eighteenth-century amateur statistician called the Reverend Thomas Bayes whose best-known contribution is the Bayes rule (or the- orem) for conditional probability. A more recent contribution to decision analysis is Von Neumann and Morgenstern’s theory of games published in 1944. The game theory concept states that the ‘right’ decision in any situation depends upon the objectives of the decision- maker and the likely actions to be taken by the competitor(s), e.g., deciding which piece to move in a chess game. Game theory basically consists of three strategies called the maximax criterion, the maximin criterion, and the minimax regret.

Conditional Probability and the Bayes Rule

Before discussing conditional probability, it is helpful to restate some basic probability defi- nitions. Two events are mutually exclusive if they cannot happen together, i.e., the probability of their occurring together is zero. For example, the probability that a tossed coin will show both a head and a tail simultaneously is zero. Two non-mutually exclusive events can happen together, e.g., the probability of two tossed coins showing a head and a tail is nonzero. Two events are independent if the outcome of one does not affect the outcome of the other, e.g., if a coin is tossed twice and a tail occurs the first time, then the probability of getting a tail on the second toss is still1_/

2. Two events are dependent if the outcome of one affects the outcome of

the second, e.g., the probability of drawing a second consecutive spade from a deck of cards is affected if the first card is not returned to the pack. Two important rules of probability can be applied to independent events, namely the addition rule and the multiplication rule. Addition Rule The probability that one or other of two mutually exclusive events A and B occurring is the sum of their respective probabilities, i.e.,

P (A or B)= P (A) +P (B)

For example, the probability of getting 2 or 5 when a dice is rolled is P(2)+ P(5) =1_/ 6+ 1_/

Multiplication Rule The probability of two independent events A and B occurring together is the product of their respective probabilities, i.e.,

P (A and B)= P (A).P (B)

For example, if two coins are tossed, the probability of a head (H) on the first and a tail (T) on the second coin is P (H and T)= P (H).P (T) =1/

2×1/2=1/4. The probability of two or

more events occurring simultaneously is usually called their joint probability.

Conditional probability refers to two events A and B, where event A has already taken place. It is usually expressed in the form P (B|A). The conditional probability of event B, given that event A has occurred, is equal to the joint probability of A and B, divided by the probability of event A, i.e.,

P(B|A) = P(A and B)

P(A) whereP(A)

≠

When the events are independent, then the conditional probability is the same as the remaining event. This is easily derived by substituting the RHS of the multiplication rule into the above equation, i.e.,

P (B|A) = P(A and B)

P(A) =

P(A).P(B)

P(A) = P(B)

A basic form of Bayes’s rule can be obtained by interchanging the letters in the conditional probability equation to give

P(A|B) = P(B and A)

P(B) whereP(B)

≠

Since P (B and A) is the same as P (A and B), the above equation can then be re-written as P(A|B) = P(A and B)

P(B) =

P(B|A).P(A) P(B)

A more general form of Bayes’s rule for two events, A and B, is given by the equation P(A|B) = P(B|A).P(A)

P(B|A).P(A) + P(B|A).P(A) where A is the converse, i.e., the opposite of A.

Decision-makers are better placed to make a decision if they are able to gather additional information about future events. Potential sources of more information include product sam- pling, materials testing, opinion polls, and updating event probabilities. In other words, as more knowledge about a situation becomes available, the more certain the decision becomes. Statistically, as an event’s certainty increases, its probability moves closer to unity – and vice versa. If an event’s probability can be revised to take account of new information, the revised probability will be greater than the previous value. Conditional probability and Bayes’s rule are important processes in the updating of event probabilities. The following example shows

how Bayes’s rule can be used to derive a revised probability, based on an event that has already taken place.

EXAMPLE 2.5 Using Bayes’s rule

A drilling company reckons that there is a 70% chance of finding a profitable reservoir of oil in a particular area. After drilling a first test bore-hole, the results are found to be favourable. From previous experience, the company estimates the bore-hole results to be 80% accurate. Use Bayes’s rule to find a revised probability of finding oil in economic quantities. Here, the two events are A= oil and B = successful strike. The required conditional probability is given by P(A|B). From the above data, P(A) = P(oil) = 0.7 and P(T) = P(no oil) = 0.3. P(B|A) = P(successful strike| oil) = 0.8 and P(B|T) = P(successful strike | no oil) = 0.2. Applying this data to Bayes’s rule above,

P(A|B) = P(successful strike|oil).P(oil)

P(successful strike|oil).P(oil) + P(successful strike|no oil).P(no oil)

= 0.8 × 0.7

0.8 × 0.7 + 0.2 × 0.3 = 0.56 0.62 = 0.9

(Note that the revised probability of 0.9 is greater than the original probability of 0.7.)

Expected Values and Decision Trees

Under conditions of uncertainty, a decision-maker often has to choose among a number of alternatives. Each alternative usually has some monetary value associated with it. A truer picture of the effects of choosing a particular alternative emerges when these monetary values are taken into account. The expected monetary value (EMV) of an event is simply its weighted average. The weighted average is obtained by multiplying each monetary value by its associated probability and then summing these products. For example, a company requires a new plant facility and must decide whether to choose between type A or type B on the basis of expected annual profits from each plant. The following probabilities and profits (in £000s) have been calculated and are shown in Table 2.4.

On the basis of each plant’s EMV, plant type A would be favourite because of its higher profit returns. The EMV is a special monetary case of the more general expected value concept of a random variable. The expected value is the weighted average of the random variables, where the weights are the probabilities assigned to the values as shown in Table 2.5.

Table 2.4

←Plant type A → ←Plant type B →

Probability Profit × Probability Profit ×

Optimistic 0.2 500 100 0.1 500 50

Most likely 0.5 250 125 0.6 300 180

Pessimistic 0.3 150 45 0.3 50 15

Table 2.5

Random variable, xi 0 1 2 3 4 Totals

Probability, pi 0.1 0.3 0.2 0.3 0.1 1.0

Weighted value, xipi 0.0 0.3 0.4 0.9 0.4 2.0

Expected value= E(x) =xipi= 2.0

Table 2.6

←Revenue (£m.) →

Oil Lease

Drilling options Probability of finding oil

No further testing 50% 140 50

Further tests indicate oil 80% 120 70

Further tests indicate no oil 15% −40 10

A decision tree is a graphical representation of the various alternative courses of action, along with the events and their probabilities which are associated with each alternative. In other words, a decision tree is basically a probability tree applied to the evaluation of expected values. Consider the following situation.

EXAMPLE 2.6 Building a decision tree

An oil company owns exploration rights to a particular field. Several well-bore tests have already been carried out in the area indicating that there is good potential. If further tests are conducted, the company reckons that there is a 70% chance of finding oil in economic quantities. The company has three options available to it: (i) perform further tests before drilling; (ii) start drilling without testing, or (iii) sell the exploration lease. The possible drilling options are listed in Table 2.6, along with their associated revenues (in £m.) from the sale of either oil or the lease.

The option of drilling without further testing will cost the company £32 million if no oil is found. Management has decided to use a decision tree to illustrate the above events and their associated EMVs. It will then evaluate the various options open to it and decide which option represents the best course of action.

The decision tree shown in Figure 2.10 uses standard symbols to represent the two types of nodes. Decision nodes are shown as squares while event nodes are shown as circles. Decision nodes are points where a choice has to be made between several alternatives, e.g., to drill, test, or sell. The event nodes are points from which various branches extend, each branch representing a possible outcome. Each outcome has a probability assigned to it as shown by the figures in parentheses in Figure 2.10. The EMVs associated with each final outcome are written alongside terminal nodes, represented by arrowheads. Instructions for drawing and

120 120 −40 64.6 64.6 ₋₁₆ 140 70 −32 −40 DRILL TEST SELL LEASE Oil (0.5) 50 Oil (0.7) Oil (0.15) SELL LEASE DRILL DRILL No oil (0.3) No oil (0.85) Oil (0.8) No oil (0.2) No oil (0.5) 88 88 54 10 10 SELL LEASE

Figure 2.10 A decision tree for the oil company’s options.

evaluating the decision tree are given below: