Weirich on Conditional and Expected Utility Wayne A. Davis The Journal of Philosophy, Vol. 79, No. 6. (Jun., 1982), pp. 342-350. Stable URL: http://links.jstor.org/sici?sici=0022-362X%28198206%2979%3A6%3C342%3AWOCAEU%3E2.0.CO%3B2-B The Journal of Philosophy is currently published by Journal of Philosophy, Inc..
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One of the advantages of V-maximization is that its prescriptions are robust under transformations of the way possible outcomes are represented. We do not need Horgan's arguments to show that the theory of U-maximization is in hot water. Indeed, we do not need to discuss the probability of subjunctive conditionals at all in this connzction. That is a comfort to those of us who are convinced that conditionals lack truth values and, hence, are not fit objects for probabilistic appraisale6 ISAAC LEV1
Columbia University
WEIRICH O N CONDITIONAL AND EXPECTED UTILITY
I
N "Conditional Utility and Its Place in Decision Theory" Paul Weirich* tries to show that conditional utilities are not the utilities of conditionals, that common formulations of the expected utility law are inadequate, and that better formulations of the law use conditional utilities in uncommon ways. I argue that all three attempts fail. I
Weirich first argues that conditional utilities are not just the utilities of the corresponding conditionals, on the grounds that "the utility of an action given a condition usually has the same value whether or not one thinks one will perform the action, whereas the utility of the corresponding conditional often does not" (704). It seems to me, on the contrary, that the utility of the corresponding conditional is similarly independent of whether one expects to perform the action. Weirich supports his premise as follows: Suppose that John has asked a n out-of-state friend Mary to go to a neighboring town to help the citizenry there with a problem o n which she is a n expert. If she will go, he wants to g o there to see her. Unfortunately, John cannot reach her to find out whether she is going; and since he thinks that she will not be able to go, he has decided not to g o himself. Now take the conditional that if Mary goes, By 'conditionals' I mean both subjunctive conditionals and indicative conditionals insofar as the latter are not understandable as material conditionals. See my "Subjunctives, Dispositions and Chances," Synthese, xxx~v,4 (April 1977): pp. 423-455, and T h e Enterprise of K n o w l e d g e (Cambridge, Mass.: MIT Press, 1980), 11.9 and 12.12. LXXVII, *This JOURNAL, 11 (November 1980): 702-715. 0022-362X/82/7906/0342$00.90
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1982 The Journal of Philosophy, Inc.
M'EIRICH O N UTILITY
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John will go. Since John is not going, this conditional is true only if Mary does not go. Since J o h n wants her to go, the . . . utility for h i m of the conditional is negative. On the other hand, John still wants to go if Mary is going. So given that Mary will go, the . . . utility for John of going is positive. Hence the conditional utility is positive, although the utility of the corresponding conditional is negative (704; my emphasis).
I have emphasized what I take to be the crucial inference. If it is valid, so is the following: the conditional "If Mary goes, John will go" is true only if John is alive; since John wants to be alive, the utility for John of the conditional is positive. Clearly, something is amiss. I believe the principle behind Weirich's inference is his definition of utility: "the utility of a proposition is roughly the rational degree of desire for the situation or outcome that would obtain if the proposition were to obtain" (702/3). Since many things would be true if any given proposition were true, the uniqueness of "the situation" is problematic. One solution is to interpret the principle as referring to the total situation that would obtain. Then Weirich's mistake was to equate part of the total situation that would obtain if "John will go if Mary goes" were true, namely Mary's not going, with the total situation that would obtain, which also includes John's being alive and countless other things. I1
Weirich's main concern is to show that conditional utilities can be used to solve various problem's facing standard formulations of the expected-utility law. One standard version of the law goes as follows. Take a set of logically exclusive and exhaustive states sl, s2, . . . and obtain the probability of each state given the action, and obtain the absolute utility of the action conjoined with each state. The absolute utility of the action is the probability-weighted average of the absolute utilities of the conjunctions. Symbolically, A U ( a )= C P(si/a)A U ( a iL sf) (707).
Weirich argues against this "conjunction" version of the expected utility law as follows:
. . . consider the following case. John goes to a garage sale where some jewelry is being sold. One ring in particular catches his eye, for he thinks that it might be gold, and its price is only $10. It is rational for him to buy the ring, since even if the ring is not gold, it is worth $10 to him. The "conjunction" version of the expected-utility law leads to the opposite result, however. Let us suppose that the following partial decision tree represents John's situation.
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1 not gold
.99
\
not gold
To complete the tree we need the absolute utility of John's buying the ring and its being gold. Since in decision contexts utilities are o b tained by subjunctive supposition, this absolute utility depends on what John thinks of his situation if the ring were gold and he were to buy it. And since, so we suppose, he thinks that if the ring were gold, it would be for sale not for $10 but for some unaffordable price, he thinks that he would be in financial trouble if the ring were gold and he were to buy it. Consequently the missing absolute utility is lower then the absolute utility of not buying, i.e., lower than 1. And so the conjunction version leads to a recommendation not to buy, which is a mistake. The mistake arises because one ought to consider the outcome of buying the ring if the ring is gold rather than if the ring were gold. That is, one ought to suppose the condition indicatively. In general, the problem is that in decision making, as we have seen, actions and conditions are supposed in different ways. Since the conjunction of an action and a condition can be supposed only in one way, the utility of the conjunction will involve supposing either the action or the condition in an inappropriate manner (707/8).
Weirich's rejection of the "conjunction" version of the expectedutility law depends on his principle that i n decision contexts utilities are obtained by subjunctive supposition. Weirich had earlier distinguished between indicative and subjunctive supposition, on the basis of the difference pointed out by Ernest Adams between "If Oswald did not shoot Kennedy, someone else did" and "If Oswald had not shot Kennedy, someone else would have." The principle in question is embodied in Weirich's definition of the utility of an action as "the rational degree of desire for the situation that would obtain if the action were performed." But Weirich gave no reason at all for using a subjunctive rather than an indicative conditional to define utility. Why not define utility as the desirability of the situation that will obtain if the action is performed? With this def-
W E I R I C H ON U T I L I T Y
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inition, the problem of "mixed" conjunctions disappears: both conjuncts will be supposed indicatively. Insistence on the subjunctive definition would seem especially difficult to defend, given the fact that in decision making we always consider future actions and given David Lewis's observation that indicative and subjunctive conditionals pertaining (exclusively) to the future seem equivalent.' Another way to avoid the above problem is to replace ' a cL s' in the expected utility law with 'a/s': Thus one might propose, as in other common formulations of the expected-utility law, that for an action-state pair we use the utility of the outcome of the action given the state, i.e., the absolute utility of the action given the state. This replacement yields the law A U ( a )= C P ( s i / a ) A U ( a / s i ) . One might also propose using the utility of the consequences of the action given the state, i.e., the relative utility of the action given the state. This replacement yields the analogous law R U ( a ) = C P ( s i / a ) R U (a/si) (708/9).
We may refer to these formulations, following Weirich, as the '(straightforward conditional versions of the expected utility law." Before going into Weirich's reasons for rejecting these versions, the distinction between absolute and relative utility needs to be explained. There are two closely related but distinct kinds of nonconditional utility found in the literature on decision theory. They stand to one another as wealth to profit. According to the first, the utility of a proposition is roughly the rational degree of desire for the situation or outcome that would obtain if the proposition were to obtain; it gives utility levels. According to the second, the utility of a proposition is roughly the rational degree of desire for the change in one's situation that would occur, or the consequences that would ensue, if the proposition were to obtain; it gives utility gains. . . . Let us call the first "absolute utility," or " A U," since it measures utility without regard to present circumstances. And let us call the second "relative utility," or " R U," since it uses the status quo, or one's position if one "does nothing," as a baseline. . . . To illustrate the distinction between A U ( a / c ) and R U ( a / c ) , suppose a corporation has twenty-five million dollars and that utility for the corporation is equal to money in millions. Then for the corporation, the absolute utility of gaining one million, given that it has gained two, is 28, whereas the relative utility is 1. The relative utility of an action equals the absolute utility of the action minus the absolute utility of the status quo (702/3).
' Counterfactuals (Cambridge, Mass.: Harvard, 1973), p. 4. I have tried to explain this apparent equivalence in "Indicative and Subjunctive Conditionals," Philo4 (October 1979): 544-564,p. 555ff. sophical Review, LXXXVIII,
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Weirich's first problem concerns the law for relative utility. Suppose that a n extortion ring prevails in a certain city. A house there has a good chance of burning down, a 99% chance, unless its owner buys "insurance" from the ring. If the owner buys insurance, the chance that his house will burn down is only 1%. T h e insurance costs 1/3 the value of the house, and it pays 2/3 the value of the house if the house burns down. Although the insurance is expensive, buying insurance is clearly rational under the circumstances. T h e straightforward use of relative utilities, however, goes awry in this case. Not buying insurance is "doing nothing." So given that there will be a fire, the relative utility of not buying insurance is 0; and given that there will be n o fire, the relative utility of not buying insurance is also 0. T h e relative utility of insuring given that there will be a fire is positive, since i n that case insuring means a gain (with respect to not insuring) of 2/3 the value of the house. T h e relative utility of insuring given that there will be n o fire is negative, since i n that case insuring means a loss (with respect to not insuring) of 1/3 the value of the house. Suppose, then, the following decision tree represents the situation.
fire
[-A1
insure no fire
.Ol
\
no fire
1
-5
[-31
0
[OI
As is apparent from the decision tree, according to the straightforward approach, the relative utility of insuring is approximately -1/3, and the relative utility of not insuring is 0. T h u s the straightforward a p proach leads to the recommendation not to insure, which is a mistake (709/10).
There is a minor mistake here that is probably typographical. In order to determine the gain with respect to not insuring in case of fire, the cost of the insurance must be subtracted from the payout. If the gain is to be 2/3, and the cost is 1/3, the payout must be 1 times the value of the house. According to Weirich, given that there will be a fire, the relative utility of not insuring is 0. How can this be when under these con-
WEIRICH ON UTILITY
347
ditions the individual loses his house? Currently the individual has a house. This, surely, is the status quo. If he does not insure and there is a fire, he will have no house. His wealth will decrease by an amount equal to the value of his house. So the relative utility of not insuring given fire should be - 1. If there is no fire and he does not insure, the individual retains his house and pays no insurance premiums, so his net gain is 0. Now suppose the individual does insure. If there is a fire, he loses his house (-I), loses the cost of the insurance (-1/3), and gains a sum equal to the value of the house ( + I ) . So the relative utility of insuring given fire is -1/3. If there is no fire, he loses only the cost of the insurance, so the relative utility is again -1/3. Therefore, the expected utility of insuring is -.33 while the expected utility of not insuring is -.99, which leads to the conclusion that insuring is preferable. Once the relative utilities are properly evaluated (see numbers in brackets), the straightforward conditional version of the expected utility law leads to the right results. Weirich's reason for assigning 0 utility to not insuring is that "Not buying insurance is 'doing nothing'." Earlier, Weirich had identified the status quo with "one's position if one 'does nothing'," which leads to the conclusion that "if a is doing nothing, then R U(a/c) is zero whatever c is." This identification is erroneous. If relative utility is to absolute utility as profit is to wealth, then the status quo should represent one's current utility level. Since doing nothing may increase or decrease one's utility level, the relative utility of doing nothing is not always 0. Just imagine stockholder reaction if the chairman reported that, even though the company's net worth declined a billion dollars last year, the company did not lose any money because the decline was a result of doing nothing! Weirich's identification also faces uniqueness problems. As his fire example shows, doing nothing ("not insuring") can have more than one possible outcome. In one outcome ("no fire") the individual has a house, in another ("fire") he does not. How can Weirich identify both these outcomes with the status quo? How can two outcomes in which the individual's utility levels differ by a house have the same relative utility? The status quo could be better defined as one's position before doing anything, that is, one's position before performing any of the actions (including perhaps doing nothing) that define the decision problem. Weirich might defend his choice of the status quo and his assignment of utilities by pleading vagueness: In using relative utilities one must, of course, keep in mind that the status q u o and doing nothing are vague notions. If in some applica-
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tion the vagueness causes trouble, it will be best to specify the action
taken as doing nothing. The status quo will then be one's position if
one performs that action (703).
I believe the terms 'status quo' and 'doing nothing' are sufficiently clear to enable us to see that the status quo cannot be equated with "one's position if one does nothing." But suppose I am wrong. Let us assume that 'the status quo' and therefore 'relative utility' are so vague that Weirich's choices are no less reasonable than mine. Weirich still has no argument against the straightforward conditional version of the expected-utility law. For a defender of that version could simply insist on using my resolution of the vagueness rather than Weirich's. Weirich next argues that the straightforward conditional version of the expected utility law also leads to trouble when absolute utilities are used. There is a reward for bringing to justice a criminal whom John has
just seen on the streets. Getting the reward is all that John cares about,
and he is entitled to it only if he is responsible for the criminal's being
apprehended. Reporting the criminal's whereabouts is a small nui-
sance, but chances are that he will get the reward if he reports, and so
the absolute utility of his reporting is higher than the absolute utility
of his not reporting. However, the probability-weighted averages of
the straightforward absolute conditional utilities are oppositely re-
lated. Let us suppose the following partial decision tree. (The
numbers are somewhat arbitrary, but agree with the story sketched
above.)
apprehended .9 7
"+<-
apprehended
not report
not apprehended
lo
The missing utility is the absolute utility of reporting given that the
criminal will be apprehended. It turns out to be 9. For if the criminal
will be apprehended, then John's reporting would be superfluous. So
if he were to report, he would not get the reward. And hence the abso-
lute utility of reporting given that the criminal will be apprehended is
WEIRICH ON UTILITY
349
the same as the absolute utility of reporting given that he will not be apprehended, viz., 9. Given this value for the missing utility, the straightforward use of absolute utilities recommends not reporting, which is a mistake (710/1).
The mistake here is the fatalistic claim that if the criminal will be apprehended then John's reporting would be superfluous, which is like saying that it would be superfluous for a woman to get pregnant, given that she will have children. "[Iln the context imagining that the criminal will be apprehended amounts to imagining that he is apprehended regardless of the action performed" (711). This fatalistic principle is not something the traditional decision theorist has to accept. It therefore cannot be used to discredit the straightforward version of the expected utility law. John's utility level if the criminal is apprehended depends on whether or not John is responsible for the apprehension. Hence the state of apprehension must be partitioned. Two more branches should be added to each "apprehended" branch of Weirich's tree, . " ~ need one labeled "Responsible" the other "Not r e ~ ~ o n s i b l e We then to evaluate the conditional probability that John is responsible for the criminal's being apprehended given that the criminal was apprehended and that John reported the criminal's whereabouts. The utility of reporting turns out to be 9 only if this probability is zero, as it would be if fatalism were true. Weirich stipulated, however, that "chances are that he will get the reward if he reports." A moment's thought and calculation will reveal countless assignments of numbers to the probability that John will be responsible and to the utility of the reward (such as .7 and 12) that both "agree with the story sketched" and make the expected utility of reporting greater than the expected utility of not reporting (10.89 versus 10). Again, once the situation is properly analyzed, the straightforward conditional version of the expected utility law prescribes the correct decisions. Weirich might argue that the analysis I have given
. . . would be incompatible with the view that the actions under consideration are future contingencies-i.e., that for each action it is neither true nor false before the decision that the action will be performed. For supposing a condition requires supposing that the condition is true. Then given that it is not true before the decision that a certain action will be performed, one cannot suppose that the condition is true because the action will be performed (71 1/2). 'Alternatively, each "apprehended" branch could be replaced by two branches labeled "apprehended and responsible" and "apprehended but not responsible," with Weirich's probabilities .9 and .8 suitably partitioned.
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The last sentence here is incorrect. The fact that something is not true does not prevent us from supposing that it is true. Weirich proposes modifying the straightforward conditional version of the expected utility law by replacing the utility of a given si with the utility of a given that si would obtain if a were performed, which yields U ( a )= C P ( s i / a ) U ( a / s iif a ) . This is no help. "If A occurred, B would occur" may be true even though "If A occurred, it would cause B to occur" is false. It is true, for example, that if I parted my hair on the right, I would play tennis just as well as I do now; but parting my hair on the right would not cause me to play tennis as well as I do now. Similarly, it may be true that if I saw lightning I would soon hear thunder; but it would not be my seeing the lightning that would cause my hearing the thunder. It could perfectly well be true, therefore, that the criminal would be apprehended if John reported the criminal's whereabouts, even though John's report would not be responsible for the apprehension, in which case John would still not get the reward. Weirich concedes that the conditional utilities his version of the expected utility law calls for are "complex and not easy to evaluate directly" (713). He therefore devotes a section to showing how they can be derived from simpler utilities. If my arguments have been sound, these are complexities we can safely avoid. WAYNE A. DAVIS
Georgetown University
NOTES AND NEWS The Philosophy Institute of the University of Chieti announces an international colloquium on phenomenology to take place on October 5-7, 1982. The colloquium is entitled "La fenomenologica nella filosofia del Novecento." Participants will include F. Barone, A. Caracciolo, I. Mancini, A. Rigobello, E. Fink, H.-G. Gadamer, L. Landgrebe, E. Levinas, X. Tilliette, and A. Tymieniecka. Further information can be obtained from Giuseppe Beschin, Direttore, Istituto di Filosofia, FacoltB di Lettere e Filosofia, via Discesa della Carceri 1, 66100 Chieti, Italy. The editors welcome the appearance of the Popper Newsletter, which has been launched at the University of Guelph. Its purpose is to improve international and interdisciplinary communication among those interested in the philosophy of Karl Popper. Sample issues are available upon request. Further information can be obtained from the editor, Fred Eidlin, Department of Political Studies, University of Guelph, Ontario N1G 2W1.