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1、Chapter 4 Decision Analysis(2010),Decisions often must be made in environments that are much more fraught with uncertainty.Example:The GOFERBROKE COMPANY owns a tract of land that may contain oil.A consulting geologist has reported to management that she believes there is 1 chance in 4 of oil.Becaus
2、e of this prospect,another oil company has offered to purchase the land for$90,000.However,Goferbroke is considering holding the land in order to drill for oil itself.The cost of drilling is$100,000.If oil is found,the resulting expected revenue will be$800,000,so the companys expected profit(after
3、deducting the cost of drilling)will be$700,000.A loss of$100,000(the drilling cost)will be incurred if the land is dry(no oil).,How to approach the decision of whether to drill or sell based just on these data.(We will refer to this as the first Goferbroke Co.problem.)However,before deciding whether
4、 to drill or sell,another option is to conduct a detailed seismic survey of the land to obtain a better estimate of the probability of finding oil.,1.A manufacturer introducing a new product into the marketplace.What will be the reaction of potential customers?How much should be produced?Should the
5、product be test marketed in a small region before deciding upon full distribution?How much advertising is needed to launch the product successfully?,2.A financial firm investing in securities.Which are the market sectors and individual securities with the best prospects?Where is the economy headed?H
6、ow about interest rates?How should these factors affect the investment decisions?,3.A government contractor bidding on a new contract.What will be the actual costs of the project?Which other companies might be bidding?What are their likely bids?,4.An agricultural firm selecting the mix of crops and
7、livestock for the upcoming season.What will be the weather conditions?Where are prices headed?What will costs be?,5.An oil company deciding whether to drill for oil in a particular location.How likely is oil there?How much?How deep will they need to drill?Should geologists investigate the site furth
8、er before drilling?,These are the kinds of decision making in the face of great uncertainty that decision analysis is designed to address.Decision analysis provides a framework and methodology for rational decision making when the outcomes are uncertain.,Frequently,one question to be addressed with
9、decision analysis is whether to make the needed decision immediately or to first do some testing(at some expense)to reduce the level of uncertainty about the outcome of the decision.,For example,the testing might be field testing of a proposed new product to test consumer reaction before making a de
10、cision on whether to proceed with full-scale production and marketing of the product.Therefore,decision analysis divides decision making between the cases of without experimentation and with experimentation.,The first section introduces a prototype example that will be carried throughout the chapter
11、 for illustrative purposes.Sections 4.2 and 4.3 then present the basic principles of decision making without experimentation and decision making with experimentation.We next describe decision trees,a useful tool for depicting and analyzing the decision process when a series of decisions needs to be
12、made.,4.1 A PROTOTYPE EXAMPLE,The GOFERBROKE COMPANY owns a tract of land that may contain oil.A consulting geologist has reported to management that she believes there is 1 chance in 4 of oil.Because of this prospect,another oil company has offered to purchase the land for$90,000.However,Goferbroke
13、 is considering holding the land in order to drill for oil itself.The cost of drilling is$100,000.If oil is found,the resulting expected revenue will be$800,000,so the companys expected profit(after deducting the cost of drilling)will be$700,000.A loss of$100,000(the drilling cost)will be incurred i
14、f the land is dry(no oil).,Table 4.1 summarizes these data.Section 4.2 discusses how to approach the decision of whether to drill or sell based just on these data.(We will refer to this as the first Goferbroke Co.problem.)However,before deciding whether to drill or sell,another option is to conduct
15、a detailed seismic survey of the land to obtain a better estimate of the probability of finding oil.Section 4.3 discusses this case of decision making with experimentation,at which point the necessary additional data will be provided.,4.2 DECISION MAKING WITHOUT EXPERIMENTATION,In general terms,the
16、decision maker must choose an action from a set of possible actions.The set contains all the feasible alternatives under consideration for how to proceed with the problem of concern.This choice of an action must be made in the face of uncertainty,because the outcome will be affected by random factor
17、s that are outside the control of the decision maker.These random factors determine what situation will be found at the time that the action is executed.Each of these possible situations is referred to as a possible state of nature.,For each combination of an action and a state of nature,the decisio
18、n maker knows what the resulting payoff would be.The payoff is a quantitative measure of the value to the decision maker of the consequences of the outcome.If the consequences of the outcome do not become completely certain even when the state of nature is given,then the payoff becomes an expected v
19、alue(in the statistical sense)of the measure of the consequences.A payoff table commonly is used to provide the payoff for each combination of an action and a state of nature.,The decision analysis framework can be summarized as follows:,1.The decision maker needs to choose one of the alternative ac
20、tions.2.Nature then would choose one of the possible states of nature.3.Each combination of an action and state of nature would result in a payoff;which is given as one of the entries in a payoff table.4.This payoff table should be used to find an optimal action for the decision maker according to a
21、n appropriate criterion.,One additional element needs to be added in the decision analysis framework.The decision maker generally will have some information that should be taken into account about the relative likelihood of the possible states of nature.Such information can usually be translated to
22、a probability distribution,acting as though the state of nature is a random variable,in which case this distribution is referred to as a prior distribution.Prior distributions are often subjective in that they may depend upon the experience or intuition of an individual.The probabilities for the res
23、pective states of nature provided by the prior distribution are called prior probabilities.,4.2.1 kinds of Decision Analysis,1.Deterministic Decision Analysis2.Indeterminable Decision Analysis3.Probability Decision Analysis,4.2.2 Characteristics of Decision Analysis,1.Deterministic Decision Analysis
24、2.Indeterminable Decision Analysis3.Probability Decision Analysis,(1)Goal(2)Actions(3)Certain Nature(4)Payoff,Goal(2)Actions(3)Uncertain Nature(4)No knowledge about probability of nature(5)Payoff,(1)Goal(2)Actions(3)Nature(4)Payoff(5)Probability,4.2.3 Formulation of the Prototype Example in This Fra
25、mework,As indicated in Table 4.1,the Goferbroke Co.has two possible actions under consideration:drill for oil or sell the land.The possible states of nature are that the land contains oil and that it does not,as designated in the column headings of Table 4.1 by oil and dry.Since the consulting geolo
26、gist has estimated that there is 1 chance in 4 of oil(and so 3 chances in 4 of no oil),the prior probabilities of the two states of nature are 0.25 and 0.75,respectively.Therefore,with the payoff in units of thousands of dollars of profit,the payoff table can be obtained directly from Table 4.1,as s
27、hown in Table 4.2.,We will use this next to find the optimal action according to each of the four criteria described below.(Indeterminable Decision Analisys),(1)The MaxiMax Payoff Criterion,Maximax payoff criterion:For each possible action,find the maximum payoff over all possible states of nature.N
28、ext,find the maximum of these maximum payoffs.Choose the action whose maximum payoff gives this maximum.,Action:Drill for Oil,(2)The Maximin Payoff Criterion,Maximin payoff criterion:For each possible action,find the minimum payoff over all possible states of nature.Next,find the maximum of these mi
29、nimum payoffs.Choose the action whose minimum payoff gives this maximum.Thus,since the minimum payoff for selling(90)is larger than that for drilling(-100),the former alternative(sell the land)will be chosen as the action to take.,Action:Sell the Land,The rationale for this criterion is that it prov
30、ides the best guarantee of the payoff that will be obtained.Regardless of what the true state of nature turns out to be for the example,the payoff from selling the land cannot be less than 90,which provides the best available guarantee.Thus,this criterion takes the pessimistic viewpoint that,regardl
31、ess of which action is selected,the worst state of nature for that action is likely to occur,so we should choose the action which provides the best payoff with its worst state of nature.,This rationale is quite valid when one is competing against a rational and malevolent opponent.However,this crite
32、rion is not often used in games against nature because it is an extremely conservative（保守的）criterion in this context.In effect,it assumes that nature is a conscious opponent that wants to inflict as much damage as possible on the decision maker.,Nature is not a malevolent opponent,and the decision m
33、aker does not need to focus solely on the worst possible payoff from each action.This is especially true when the worst possible payoff from an action comes from a relatively unlikely state of nature.Thus,this criterion normally is of interest only to a very cautious decision maker.,(3)The Savage ru
34、le Criterion（后悔值法）,Criterion:For each possible nature state,find the maximum payoff over all possible alternative actives.Next,find the difference of each active payoff comparing with the maximum payoff.Choose the maximum difference of every active,after that select the action whose difference gives
35、 the minimum.,Action:Drill for Oil,(4)The Equivalent Probability Criterion,Solution of Probability analysis:The Maximum Likelihood Criterion,Maximum likelihood criterion:Identify the most likely state of nature(the one with the largest prior probability).For this state of nature,find the action with
36、 the maximum payoff.Choose this action.,The appeal of this criterion is that the most important state of nature is the most likely one,so the action chosen is the best one for this particularly important state of nature.Basing the decision on the assumption that this state of nature will occur tends
37、 to give a better chance of a favorable outcome than assuming any other state of nature.Furthermore,the criterion does not rely on questionable subjective estimates of the probabilities of the respective states of nature other than identifying the most likely state.,The major drawback of the criteri
38、on is that it completely ignores much relevant information.No state of nature is considered other than the most likely one.In a problem with many possible states of nature,the probability of the most likely one may be quite small,so focusing on just this one state of nature is quite unwarranted.In e
39、ffect,the criterion does not permit gambling on a low-probability big payoff,no matter how attractive the gamble may be.,Action:Sell the Land,Solution of Probability analysis:Bayes Decision Ruler,Bayes decision rule:Using the best available estimates of the probabilities of the respective states of
40、nature(currently the prior probabilities),calculate the expected value of the payoff for each of the possible actions.Choose the action with the maximum expected payoff.,For the prototype example,these expected payoffs are calculated as follows:E Payoff(drill)=0.25(700)+0.75(-100)=100.E Payoff(sell)
41、=0.25(90)+0.75(90)=90.Since 100 is larger than 90,the alterative action selected is to drill for oil.,The big advantage of Bayes decision rule is that it incorporates all the available information,including all the payoffs and the best available estimates of the probabilities of the respective state
42、s of nature.It is sometimes argued that these estimates of the probabilities necessarily are largely subjective and so are too shaky to be trusted.There is no accurate way of predicting the future,including a future state of nature,even in probability terms.This argument has some validity.The reason
43、ableness of the estimates of the probabilities should be assessed in each individual situation.,Nevertheless,under many circumstances,past experience and current evidence enable one to develop reasonable estimates of the probabilities.Using this information should provide better grounds for a sound
44、decision than ignoring it.Furthermore,experimentation frequently can be conducted to improve these estimates,as described in the next section.Therefore,we will be using only Bayes decision rule throughout the remainder of the chapter.,4.3 DECISION MAKING WITH EXPERIMENTATION,Frequently,additional te
45、sting(experimentation)can be done to improve the preliminary estimates of the probabilities of the respective states of nature provided by the prior probabilities.These improved estimates are called posterior probabilities.,We first update the Goferbroke Co.example to incorporate experimentation,the
46、n describe how to derive the posterior probabilities,and finally discuss how to decide whether if is worthwhile to conduct experimentation.,4.3.1 Continuing the Prototype Example,As mentioned at the end of Sec.4.1,an available option before making a decision is to conduct a detailed seismic survey o
47、f the land to obtain a better estimate of the probability of oil.The cost is$30,000.,A seismic survey obtains seismic soundings that indicate whether the geological structure is favorable to the presence of oil.We will divide the possible findings of the survey into the following two categories:USS:
48、Unfavorable seismic soundings;oil is fairly unlikely.FSS:Favorable seismic soundings;oil is fairly likely.,Based on past experience,if there is oil,then the probability of unfavorable seismic soundings is P(USS|State=Oil)=0.4,so P(FSS|State=Oil)=1-0.4=0.6.Similarly,if there is no oil,then the probab
49、ility of unfavorable seismic soundings is estimated to be P(USS|State=Dry)=0.8,so P(FSS|State=Dry)=1-0.8=0.2.,We soon will use these data to find the posterior probabilities of the respective states of nature given the seismic soundings.,Prior Probabilities,Conditional Probabilities,Joint Probabilit
50、ies,Posterior Probabilities,Fig 4.6 probability tree diagram showing all the probabilities leading to the calculation of each posterior probability of the state of nature given the finding of the seismic survey,4.4 DECISION TREES,Decision trees provide a useful way of visually displaying the problem
