CHAPTER 1-4 ALLOWING FOR UNCERTAINTY Uncertainty, in the presence of vivid hopes and fears, is painful, but must be endured if we wish to live without the support of comforting fairy tales. Bertrand Russell, A History of Western Philosophy (New York: Simon and Schuster, 1945, p. xiv Will Chip Lohmiller's kick from the 45 yard line go through the uprights? How much oil can you expect from the next well you drill, and what value should you assign to that prospect? Will you be the first one to discover a workable system for converting speech into computer-typed output? Today's actions often continue to affect events many years later. This perseveration of consequences constitutes a difficulty in decision-making. Chapter 1-2 showed how the mechanism of time-discounting and present-value calculation deals nicely with that difficulty. Inter-relatedness of activities is another difficulty in making decisions. However, the mechanism of tabular analysis and the consideration of each combination of activities handles the difficulty of inter-relatedness with ease, as we saw in Chapter 1-1. Now we come to uncertainty, a third major difficulty in decision-making. When reading the business examples in previous chapters, you certainly realized that you usually cannot know with reasonable certainty just how many sales you will make at each possible price. And often the expenditures you must make at each possible level of sales are quite uncertain, too. This chapter presents to you the intellectual machinery to deal with uncertainty in a systematic fashion when valuing and comparing alternatives. The estimation of probabilities is discussed in Chapter 00, and the combination of probabilities in complex situations is discussed in Chapter 4-2. The central concept for dealing with uncertainty is probability. Philosophers have wrestled long and hard with the nature of probability and its proper interpretation. For decision-making, however, the following uncontroversial interpretation suffices. A probability statement is always about the future. To say that an event has a high or low probability is simply to make a forecast. But one does not know what the likelihoods really are for future events, except in the case of a gambler playing black on an honest roulette wheel, or an insurance company issuing a policy on an event with which it has had a lot of experience, such as a life insurance policy. Therefore, we must make guesses about the likelihoods, using various commonsense gimmicks. All the gimmicks used to estimate probabilities should be thought of as "proxies" for the actual probability. For example, if NASA Mission Control simulates what will probably happen if a valve is turned aboard an Apollo spacecraft, the result on the ground is not the real probability that it will happen in space, but rather a proxy for the real probability. If a manager looks at the last two Decembers' sales of radios, and on that basis guesses the likelihood that he will run out of stock if he orders 200 radios, then the last two years' experience is serving as a proxy for future experience. If a sales manager just "intuits" that the odds are 3 to 1 (a probability of .75) that the main competitor will not meet a price cut, then all his past experience summed into his intuition is a proxy for the probability that it will really happen. Whether any proxy is a good or bad one depends on the wisdom of the person choosing the proxy and making the probability estimates. A probability is stated as an arbitrary weight between 0 and 1. Zero means you estimate that there is no chance of the event happening, and 1 means you are sure it will happen. A probability estimate of .2 means that you think the chances are 1 in 5 (odds of 1 to 4) that the event will happen. A probability estimate of .2 indicates that you think there is twice as great a chance of the events happening as if you had estimated a probability of .1. There is no logical difference between the sort of probability that the life insurance company estimates on the basis of its "frequency series" of past death rates, and the salesman's seat-of-the-pants estimate of what the competitor will do. No frequency series speaks for itself in a perfectly objective manner. Many judgments necessarily enter into compiling every frequency series -- in deciding which frequency series to use for an estimate, and choosing which part of the frequency series to use. For example, should the insurance company use only its records from last year, which will be too few to give as many data as would be liked, or should it also use death records from years further back, when conditions were somewhat different? In view of the inevitably subjective nature of probability estimates, you may prefer to talk about "degrees of belief" instead of probabilities. That's fine, just as long as it is understood that we operate with degrees of belief in exactly the same way as we operate with probabilities. The two terms are working synonyms. A probability estimate for an event that occurs many times - - such as the likelihood of death of a man in the U. S. during his fiftieth year -- is easy to interpret. But the probability of a one-time or first-time event, such as the likelihood of success of the first mission to Mars, is harder to interpret. I view the latter as a representative of that category of events that have some similarity to the event which was to forecast, with the extent of similarity judged on the basis of analogy and theoretical reasonings. ALLOWING FOR UNCERTAINTY WHEN COMPARING ALTERNATIVES The Concept of Expected Value Consider these two alternatives: a) a thousand-dollar bill in hand, or b) a 1/2 chance of two thousand-dollar bills and a 1/2 chance of nothing. It is intuitively clear that if you were to be given a choice between these two alternatives on (say) 5000 occasions, you would be equally well off whichever you consistently choose. The concept of expected value enables us to evaluate and compare the two alternatives formally, leaving aside (for now) any feeling of pleasantness or unpleasantness about the certain and the uncertain choices. The expected value is the combination of the value of each outcome weighted by the probability that the outcome will take place. That is, the expected value is the weighted average obtained by first multiplying the value of each outcome by its conditional probability, and then summing. It is the same as the present value in the single-period context where no discounting need be done. An example: Conditional on the fact that someone offers to gamble double-or-nothing for a dozen apples, using a fair coin, the expected value is: (1) (2) (1)x(2) Outcome Probability Expected Payoff ____________________________________________ No apples .50 0 apples 12 apples .50 6 apples ________ Expected value = 6 apples ____________________________________________ An expected value can be calculated meaningfully for payoffs measured in apples, dollars, happiness points, or whatever. But please notice that expected value is not synonymous with worth. Twice as much money does not necessarily mean twice as much pleasure or utility to you. For example, a 50-50 chance of $l,000 may be worth less to you than a sure $500. We'll deal later with that complication. If you wish to know the present value of an expected value of a set of outcomes at some future time, you may discount the expected values just as if they were sums of money, just like any other present-value calculation discussed earlier. But of course this does not take into account the fact that there is uncertainty and risk in the expected value of a set of possible outcomes, as compared to a sum for sure; this matter will be handled later. The value of a business opportunity is the sum of all the possible outcomes of an alternative choice, each weighted by the probability that it will occur. For example, the expected value of a one-in-ten chance that you'll get $100 if you sing in a contest, plus a nine-in-ten chance of getting nothing, equals (.1 X $100 + .9 X 0 =) $10. This does not mean that the value to you of this contest opportunity is $10; If you need money badly, and this will be your last day on earth, having a one-in-ten chance of $100 may not be worth to you ten times a sure prospect of having $10. Or the chance of $1000 may be worth more, if you desperately need $100 for a ticket out of hell. But over the long run of a good many alternatives in the operation of a business or a live, the expected value is a reasonable way to compute the values of opportunities. Later, we will see how to modify the expected value to take into account the special circumstances of the disadvantages of risk for an individual (and for firms, too). We operate on the basis of expected value literally all the time. When you decide whether to take an umbrella in case of rain, you are implicitly taking into account the probability of rain, together with the costs of carrying the umbrella and of getting wet if it rains and you do not take the umbrella. Without an explicit calculation, your implicit intuitive solution will often be in error. For example, consider that the chance of rain is 1 in 50, your valuation of getting wet is $-10, and your valuation of carrying the umbrella is $-1. The expected value of carrying the umbrella is $-1. and the expected value of not carrying the umbrella is .02 x -$10 + .98 x $0 = -.$20. So the expected value of carrying it is much more negative (less positive) than not carrying it. If you do this calculation for yourself, you may well find yourself not carrying an umbrella in many situations where you would otherwise have carried it for lack of thinking clearly about the matter. (Indeed, an analysis is only useful if it often leads to conclusions other that you would have arrived at in the absence of the analysis.) Another way to do the same calculation: State your values of carrying the umbrella and of getting wet if it rains if you have no umbrella, then figure backwards to the probability of rain that would make it worthwhile to take the umbrella -- that is, the probability that would balance the expected values of the two alternatives. I will leave it to you to check that the probability must be 10 percent (.1) or greater for carrying it to be worthwhile, given the valuations in the paragraph above. Figure of Umbrella Calculation Here Expected value is at the heart of all insurance. The insurer estimates the probability of the insured-against event -- say, the probability of death of a man during the year he is aged 55 in the U. S. at present -- and then multiplies that probability by the value of the insurance to obtain the expected value of the loss. That expected value plus its operating expenses forms the basis for the insurance company calculating the price it will charge for that insurance policy. The expected-value concept is also at the heart of all prices of wagers with bookmakers (in the states and countries where that is legal, of course!). Another example: The concept of expected value underlies the decision to accept an offer of a settlement in a law suit about a patent of yours. Assume that the company you are suing offers you a $200,000 settlement. You figure that you have a .6 chance of winning $1 million in court. (Leave aside for now the complication that you do not know for sure how much you would be awarded if you do win.) The expected value of continuing the suit to a trial is $600,000, and you should therefore turn down the offer unless you are willing to pay a lot to avoid the risk of losing (a matter that will be discussed in the next chapter). In such a case, unfortunately, the calculation of expected value to her of a lawyer working on a contingency basis will differ from the client's expected value, because the lawyer will take into account the costs of her time if the settlement is not accepted; in contrast, the client does not pay those costs, and the appropriate calculation for you therefore does not include them. Hence the lawyer sometimes has a stake in your accepting a settlement even when it is not to the client's interests. (This discrepancy between interests of people on the same side of the table occurs in many circumstances. For example, it is usually to the interest of a publisher to set a higher price for a book than is best for the author.) The choice of a price to bid in a closed auction is another important application of the concept of expected value. The decision hinges on the probabilities of winning the auction at various prices you might bid; the higher your bid, the more you would gain if you win the auction, but the lower the chance of winning because some competitor is more likely to underbid you. You should evaluate alternative bids according to their expected values, which you calculate as the probability of winning multiplied by the gain if the bid is won at that price. Consider for example that you are in the painting business, and your town calls for bids on painting the town hall. You figure that the work will cost you $16,000 if you get the job. The bid prices you are considering, and the probabilities you estimate for winning the auction at each of those prices, are shown in columns 1 and 2 in Table 1-4-1. The expected value for each price is calculated by multiplying the probability of winning by the difference between revenue and expenditures (column 4) if you do get the bid. (For completeness, we also show the probability multiplied by the expected value of $0 if you do not win the auction.) The bid price with the highest expected value in column X is the best alternative. When risk is ignored in a present-value calculation, an expected value in a future period may be treated just like a certain income or outgo. In that fashion the complications of both futurity and uncertainty may be dealt with at once, as long as no decisions need be made in the future. (If they will be, we must resort to the more complex machinery of the decision tree, which we tackle below.) The Decision Tree The situation is more complex when there will be a sequence of choices rather than only one choice. Consider calculating the expected value of this gamble: You flip a nickel. If it falls on its head, I'll give you $240, and you will also get a chance to flip a dime. If the nickel does fall heads and you do get a chance to flip the dime -- a chance you may reject, of course -- I'll give you $250 if the dime falls on its head, but you must give me $300 if it falls on its tail. If the original nickel flip falls on its tail, you get $150 from me plus a chance to flip a quarter. If you get the chance and choose to flip the quarter and it falls on its head, I'll give you $150, but if it falls on its tail, you must give me $100. Would you pay $200 for this gamble which is diagrammed in Figure 1-4-1? It looks easy to evaluate at first, but you soon see how puzzling it is. The heart of the difficulty is that you cannot evaluate the choice of taking the deal now unless you know what you will choose to do after you see whether the coin falls on its head or tail. Figure 1-4-1 Curiously, even though all the necessary elements of knowledge to solve this problem were available 300 years ago, it was only in the past half-century that the solution was discovered, a powerful mathematical technique known as "decision- tree analysis" or "backward induction" -- or more frighteningly, "dynamic programming". The way out of the impasse is to start at the farthest-away point in time and figure the expected values of the farthest-away sets of outcomes. Then you decide which alternatives you would choose in the next-to-last period if you reach those points, and so on, all the way backward to the present. When this process is complete, and only then, you can choose a first-period alternative. The steps in a decision-tree analysis require only simple arithmetic, and can be easily learned when you need to do so. In perhaps 9 of 10 cases, the greatest value of the decision-tree analysis is not the formal calculation, but rather the exercise of forcing yourself to clarify your thinking on paper.1 Consider, for example, the picture of the decision about choosing a college major (Figure 1-4-2). You will find it very difficult to decide on the probabilities, costs, and benefits to put into the picture. You can avoid making these quantities explicit if you avoid putting the analysis on paper. But a sound decision requires that you do make these quantities explicit. And in most cases, the process of making your best guesses about these quantities reveals the best decision without formal analysis.2 Figure 1-4-2 (The value of paper, pencil, and picture-making is brought out by this famous puzzle: A man points to the image of a person and says, "Brothers or sisters have I none. That man's father is my father's son." The puzzle is hard for most of us to solve in our heads. But drawing simple pictures usually reveals the answer immediately. Try it.) Before you can assess its value to you, you must know the chance that an event will occur. But the relevant probability is not obvious in many circumstances. First there is the problem of assessing even the "simplest" likelihood -- for example, the likelihood that it will rain on Sunday. (Estimation of probabilities is discussed in Chapter 00.) Then there is the complication that several probabilities may interact, such as the probability of rain on Sunday and your favorite football team winning the game, a complex probability. Complex probabilities are dealt with in Chapter 4-2. Page # thinking uncrt14% 3-3-4d