The St. Petersburg gamble and risk

PAUL W E I R I C H T H E ST. P E T E R S B U R G GAMBLE AND RISK ABSTRACT. Pursuing a line of thought initiated by Maurice AUais (1979), I consider whether the mean-risk method of decision making introduced by Harry Markowitz (1959) and other resolves Karl Menger's (1934) version of the St. Petersburg paradox. I provide a conditional answer to this question. I demonstrate that given certain plausible assumption about attitudes toward risk, a certain plausible development of the mean-risk method does resolve the paradox. My chief premiss is roughly that in the St. Petersburg gamble the small chances for large prizes create big risks. The St. Petersburg gamble poses a long standing problem for decision rules that utilize expected valuesJ Here I will investigate the possibility of resolving the St. Petersburg paradox by adopting the so-called mean-risk method of evaluating options, espoused, for example, by Harry Markowitz (1959, Chapter XIII). First I will briefly review the St. Petersburg gamble. Then I will advance a version of the mean-risk method that evaluates an option using both the expected or mean utility of the causal consequences of the option and the utility o f the risk involved in the option. Finally 1 will show that given certain assumptions about attitudes toward risk, our mean-risk method of evaluation does indeed resolve the St. Petersburg paradox. I do not claim to provide a complete resolution of the St. Petersburg paradox. For my version o f the mean-risk method and my assumptions about attitudes toward risk are somewhat controversial. My purpose is mainly to show that the mean-risk method has good prospects for resolving the paradox. 1 think that my approach to the paradox is sufficiently plausible and suggestive to do this. I. THE ST. P E T E R S B U R G GAMBLE The St. Petersburg gamble awards cash prizes of various amounts depending on the outcome of a series of tosses of a fair coin. Letting the prizes be amounts of dollars, the gamble pays 2*n dollars if the first heads of the series Theory and Decision 17 (1984) 193-202. 9 1984 D. Reidel Publishing Company. 0040-5833/84/0172-0193~01.00. 194 PAUL WEIRICH occurs on the nth toss. (2*n is 2 raised to the nth power.) The expected payoff of the gamble is therefore 2;n--~n=l(1/2*n x 2*n), or 2;n=1 1, which is infinite. However it would be irrational to pay more than a modest sum of money for the gamble. Evidently, the expected payoff of the gamble is not a good measure of the value of the gamble. In order to bring out what I take to be the main problem raised by the gamble, I assume some additional background conditions. First, I suppose that there are no obstacles to offering the gamble. In particular, there is a limitless supply of money for payoffs and a limitless amount of time for coin tosses. Second, I suppose that there is no barrier to enjoying the large prizes that are possible. More specifically, the utility of money is unbounded. 2 Third, I suppose that conditions are ideal for evaluating the gamble. For example, there are no cognitive limitations that justify neglecting small probabilities. And fourth, I suppose that the gamble is offered to an individual, not a group, and that it is offered just one time, not repeatedly. Given these background conditions, the St. Petersburg gamble challenges the method of using expected payoffs to evaluate options, and not just the application of this method to real life cases. Thus attempts to resolve the paradox by appealing to realism - such as the attempts of Leonard Savage (1954), Richard Jeffrey (1965), Maurice Allais (1979), and Samuel Gorovitz (1979) - are not effective here. The best known attempts to resolve the St. Petersburg paradox, taken as a methodological problem, is due to Daniel Bernoulli (1738). He claims that the expected payoff of the gamble is the wrong quantity to use to evaluate the gamble. He recommends using the expected utility of the payoff, and observes that given his law of the diminishing marginal utility of money, the expected utility of the payoff is finite. However, BernouUi's attempt to resolve the paradox is ultimately unsatisfactory. As Karl Menger (1934) points out, if the cash prizes are adjusted for the diminishing marginal utility of money so that the utility of the prize for no heads until the nth toss is 2*n, the expected utility of the payoff is infinite. But the value of the gamble is still not very large. Apparently, the failure to use expected utilities is not the main source of trouble in evaluating the St. Petersburg gamble. The main problem seems to be that the probabilities of the prizes make the value of the St. Petersburg gamble finite no matter what (finite) values THE ST. P E T E R S B U R G GAMBLE AND RISK 195 the prizes have. To resolve the problem, one has to find a method of evaluating options that shows why the probabilities have this effect. II. MEAN-RISK EVALUATION Maurice AUais has long argued that the expected value of (causal) consequences misevaluates an option because it neglects attitudes toward the risk involved in the option. Recently he has argued, in particular, that the expected value of consequences misevaluates the St. Petersburg gamble because it neglects attitudes toward the risk involved in the gamble (1979, p. 502). His arguments suggest that we might resolve the St. Petersburg paradox by adopting a method of evaluating options that is more sensitive to attitudes toward risk than using expected values of consequences of options. In this section I will advance such a method, and in the next section I will show that it does indeed correctly evaluate the St. Petersburg gamble given some plausible assumptions about attitudes toward risk. The best known risk sensitive method of evaluating options is the meanrisk method of Markowitz (1959) and others. This method takes the value of an option to be a function of the expected or mean value of its (causal) consequences and one's attitude toward the risk it involves. I will advance a version of this mean-risk method of evaluation. Let us begin by saying a bit about risk. In the mean-risk school risk is taken broadly so that a risky option is any option that involves an element of chance, even an option that involves no chance of a loss. Given this broad sense of risk, which we adopt, aversion to risk is equivalent to aversion to chance. Three factors are generally acknowledged to influence the size of the risk involved in an option. First, there is the dispersion of the values of the possible consequences of the option. The greater the dispersion, the greater the risk. (See Allais (1953).) 3 Second, there is the weight of the evidence for one's assignment of probabilities to the possible states of the world that determine the consequences of the option. The greater the weight of the evidence, the smaller the risk. (See Daniel Eilsberg (1961).)And third, there is one's wealth, or more accurately, one's utility level. The greater one's utility level, the smaller the risk. (See Ralph Keeney and Howard Raiffa (1976, Chapter 4).) 196 PAUL WEIRICH The main task in developing our mean-risk method of evaluating an option is to find a way of calculating the value of an option from the expected value of its consequences and an assessment of the risk it involves. I will not argue in detail for the method that I advance. I hope that it has enough native plausibility to arouse interest in the kind of resolution of the St. Petersburg paradox to which it leads. Our first step is to adopt a uniform criterion of value for assessing possible consequences and risks. It is convenient to choose utility since one can assign utilities to both possible consequences and risks. Second, for simplicity, we restrict our method of evaluation to options where the only factors that matter are the expected utility of consequences and the utility of the risk involved. Since satisfaction of the restriction is standardly presumed in discussions of gambles such as the St. Petersburg gamble, the restriction is not an impediment from our point of view. Third, we assume that the expected utility of the consequences of an option does not reflect the utility of the risk involved in the option. This is plausible since the risk involved in the option is part of the option itself and not a (causal) consequence of the option. Given the assumption, we do not have to worry about double counting risk when we combine the expected utility of consequences with the utility of the risk involved. And fourth, we assume the independence of the utility of the risk involved in an option and the utilities of the possible consequences of the option. This assumption is plausible since once the consequences of an option are obtained, the risk taken in realizing the option seems to be so much water under the bridge. Given the assumption, we can obtain the utility of an option by simply adding the expected utility of its consequences and the utility of the risk it involves. In virtue of the foregoing, we formulate our mean-risk rule as follows. Let P stand for probability, U stand for utility, C stand for consequences, and R stand for risk. Then if sl, s2, 9 9 are mutually exclusive and exhaustive states of the world that are independent of an option o, U(o) = ~., P(sn)U(C[o, snl) + U(R[ol). 4 In the St. Petersburg gamble (spg) the dispersion of the utilities of the possible consequences is large. Given some standard measures of the dispersion, such as the standard deviation, the dispersion is infinite. This large THE ST. P E T E R S B U R G GAMBLE AND RISK 197 dispersion makes the risk involved in the gamble large. It seems possible that aversion to the large risk is the cause of the gamble's finite utility. In particular, although ~ P(Sn)U(C[spg, Sn]) is infinite in Menger's version of the St. Petersburg gamble, if U(R [spg]) is negatively infinite, then U(spg) may be finite. To investigate this possibility using our mean-risk method, we will have to rearrange the mean-risk equation for the St. Petersburg gamble to avoid addition of infinite factors. We can do this in the following way. Consider a series of gambles involving a limited number of coin tosses. For the ruth gamble of the series a fair coin is tossed exactly rn times. If the first heads comes up on the nth toss, the gamble pays 2*n dollars. If heads never comes up, the gamble pays nothing. The St. Petersburg gamble is the limit of this series of gambles. The increase in risk as one moves from gamble m -- 1 to gamble m can be thought of as part of the risk involved in the St. Petersburg gamble. One can think of it as the risk added by the possibility that the first heads will be on the ruth toss. Let us call it Rm(spg). Analogously, the difference in the utility of the risk involved in gamble m and gamble m -- 1 can be though of as part of the utility of the risk involved in the St. Petersburg gamble. One can think of it as the utility of Rm(spg), i.e., U(Rm[spg]). According to these conventions, U(R [spg]) = 2; U(Rn [spg]). Hence, associating U(Rn [spg]) with U(C[spg, sn ]), the mean-risk equation for the St. Petersburg gamble becomes U(spg) = ~ [1/2*n x U(C[spg, Sn]) + U(Rn [spg])]. Given aversion to risk, U(Rn [spg]) is negative for eachn. And asn increases, aversion to risk makes U(Rn[spg]) decrease. If the decrease is very rapid, 1/2*n x U(C[spg, sn])+ U(Rn[spg]) may approach 0 as n increases. And if it approaches 0 very fast, 2; [1/2*n x U(C[spg, Sn]) + U(Rn[spg])] may be finite. In this way it is possible for U(spg) to be finite. IIl. ASSUMPTIONS ABOUT RISK As we saw in Section I, the problem raised by the St. Petersburg gamble is to show that the gamble has finite value regardless of the values of possible consequences. Given the mean-risk method of evaluation of Section II, the problem is, more specifically, to show that 2; [1/2*n x U(C[spg, s~ ]) + U(Rn [spg])l 198 PAUL WEIRICH is finite regardless of the values of U(C[sgp, sad for n = 1, 2 . . . . . I will now introduce some plausible assumptions about attitudes to risk that lead to the desired result. Since my goal is just to present a program for resolving the St. Petersburg paradox, I will not argue for the assumptions in detail. Let us begin by extending our terminology a little. Consider again the series of gambles introduced above. The difference between the utility of gamble m and the utility of gamble rn -- 1 can be though of as part of the utility of the St. Petersburg gamble. One can think of it as the utility added by the possibility that the first heads will be on the ruth toss. Let us call it Um(spg). According to this definition, U(spg) = ~ Un(spg ). Now let us focus on the relationship for each n between increases in U(C[spg, Sn]) and increases in Un(spg). As U(C[spg, sn ]) increases, the rate of increase in Un(spg) for further increases in U(C[spg, Sn]) changes. The instantaneous rate of increase in Un(spg) for a given value of U(C[spg, sn]) is the sum of the instantaneous rate of increase in 1/2*n x U(C[spg, sn]), or 1/2*n, and the instantaneous rate of increase in U(Rn[spg]). Because greater stakes create greater risk, the instantaneous rate of increase in U(Rn [spg]) is negative given aversion to risk. s So the instantaneous rate of increase in Un(spg) is less than 1/2*n. Our special assumptions about attitudes to risk take the form of assumptions about other features of the instantaneous rate of increase in U.(spg). The firstassumption is motivated by three considerations. First, it seems that the greater the stakes, the greater the rate of increase in risk for subsequent increases in the stakes. Hence as U(C[spg, Sn]) increases, the rate of increase in U(Rn[spg]) drops, and, as a result, the rate of increase in Un(spg) drops as well. s Second, the inverse of the rate of increase for 1/2*n x U(C[spg, Sn]), viz., -- 1/2*n, is a lower bound for the rate of increase for U(Rn[spg]). And it seems that it is the least lower bound. That is, it seems that the rate of increase in U(Rn[spg]) approaches - 1 / 2 * n as U(C[spg, sn ]) approaches infinity so that decreases in U(Rn [spg]) eventually come close to cancelling increases in U(C[spg, Sn]). Hence it seems that the rate of increase in Un(spg) approaches 0 as U(C[spg, sn ]) approaches infinity. Third, it seems that aversion to risk puts a limit on the attractiveness of gambles. That is, more picturesquely, there is some number of birds in hand worth more than any number of birds in the bush. Hence it seems that after a point, not matter how much U(C[spg, sn]) increases, there is no THE ST. PETERSBURG GAMBLE AND RISK 199 appreciable increase in Un(spg). In other words, it seems that after a point, the area under the rate curve from 0 to U(C[spg, Sn]) is not appreciably greater as U(C[spg, sn ]) increases. The rate of increase in Un (spg) 1/2"n 0 ~-Z'- U(C [spg, Sn ]) Put more precisely, it seems that the integral of the rate function is finite. Now functions of U(C[spg, Sn]) of the form 1/2*n x k* -- U(C[spg, sn]), where k > 1, decrease from 1/2*n to 0 and have finite integrals. So bearing the foregoing in mind, some such function will provide an upper bound for the rate function. This leads us to our first assumption. (1) For each n there is a k > 1 such that the rate of increase in Un(spg) with respect to U(C[spg, Sn]) is less than 1/2*n x k*-U(C[spg, s, ]). Our second assumption is motivated by two considerations. First, it seems that as n increases and the probability that the first heads will be on the nth toss decreases, the drop in the rate of increase in U(Rn [spg]) from 0 to -- I/2*n will be more rapid. Next, since smaller probabilities of success make ventures riskier, it seems that as n gets very large, and the probability that the first heads will be on the nth toss gets very small, the rate of increase in U(R, [spg]) will approach --1/2*n almost immediately. This leads to the following assumption. (2) As n increases, the smallest k satisfying the condition in assumption (1) increases without limit. From our two assumptions we obtain the desired result about the St. Petersburg gamble straightforwardly. By (1), Un(spg ) is less than the integral of 1/2*n x k*-- U(C[spg, Sn]) with respect to U(C[spg, Sn]). Hence by 200 PAUL W E I R I C H calculus, Un(spg)< 1/2*n x 1~Ink. By (2), there is some N such that for n > N , k > e , where e is the base of natural logarithms. Hence, for n > N , Un(spg)< 1/2*n. It follows by numerical analysis that Z~=~ Un(spg)< zn=~n=N1/2*n < 1. This entails that Z Un(spg) is finite. Therefore, U(spg) is finite regardless of the values for all n of U(C[spg, sn] ). In this way our assumptions provide a resolution of the St. Petersburg paradox. IV. C O N C L U D I N G R E M A R K S The assumptions of the preceding section are similar to ones that other authors have made in attempting to resolve the St. Petersburg paradox. Assumption (1), that the rate of increase in Un(spg) is bounded by an exponential function, is similar to Savage's (1954) assumption that the utility of prizes is bounded. However assumption (1) is more plausible since aversion to risk provides a reasonable explanation of assumption (1), whereas the assumption that the utility of prizes is bounded seems ad hoc. Assumption (2), that the rate of increase in Un(spg) becomes negligible almost immediately for large values of n, is similar to Gorovitz's (1979) assumption that small probabilities do not count. However assumption (2) is more plausible since it seems that small probabilities cannot be discounted entirely. In the light of such comparisons, I conclude that we have made progress with the St. Petersburg paradox, even if our assumptions have not been sufficiently supported for us to claim to have completely resolved the paradox. NOTES i Recently, the problem has been one of the foci of discussions about rationality triggered by Daniel Kahneman and Amos Tversky (1974 and 1979). Contra Kahneman and Tversky, Lola Lopes (1982) and Henry Kyburg (1983), for example, defend the rationality of people who offer versions of the St Petersburg gamble in violation of the canon to decide according to the expected values of (causal) consequences. 2 Given standard utility theory in the tradition of John Von Neumann and Oscar Morgenstern (1944), typical preferences concerning the St. Petersburg gamble entail that the utility of money is bounded. Hence this supposition requires some revision of standard utility theory. There are many revisions that would suffice. For the sake of definiteness, I will assume the revision proposed by Maurice Allais (1953). T H E ST. P E T E R S B U R G GAMBLE AND RISK 201 3 This assumes that there is a natural way of partitioning the possible consequences, and that the dispersion is measured with respect to the natural partition of consequences. Characterizing the natural partition of consequences is, of course, a difficult task. But we need not undertake it here since it is clear what the natural partition is for the St. Petersburg gamble. 4 One might conjecture that the sum on the right hand side of the equation is equal to the expected value of some utility encompassing both the utility of the consequences of o and the utility of the risk involved in o, perhaps the expected value of the utility o f o itself, i.e., E P(sn)U(o,sn). If this conjecture were correct, our mean-risk method of evaluation would be compatible with an expected utility method of evaluation. I will not explore this issue here, however. s I am ignoring some local aberrations that occur when U(C[spg, sn]) is close to U(C[spg, Sm]) for some m not equal to n. These local abberrations do not affect the plausibility of the assumptions introduced in this section. REFERENCES Allais, Maurice: 1953, 'Fondements d'une th~orie positive des choix comportant un risque et critique des postulats et axioms de l'~cole Am~ricalne', t~conom~trie, CNRS, Paris, vol. 40, translated in Maurlce AUais and Ole Hagen (eds.), Expected Utility Hypothesis and the Allais Paradox, Reidel, D ordrecht, 1979. Allals, Maurice: 1979. 'The So-Called Allals Paradox and Rational Decisions Under Uncertainty', Expected Utility Hypothesis and the Allais Paradox. Bernoulli, Daniel: 1738, 'Exposition of a New Theory on the Measurement of Risk', translated by Louise Sommer in Econometrica 22 (1954), 22-36. Ellsberg, Daniel: 1961, 'Risk, Ambiguity, and the Savage Axioms', Quarterly Journal of Economics 7 5 , 6 4 3 - 6 6 9 . Gorovitz, Samuel: 1979, 'The St. Petersburg Puzzle', Expected Utility Hypothesis and the Allais Paradox. Jeffrey, Richard: 1965, The Logic of Decision, McGraw-Hill, New York. Kahneman, Daniel and Tversky, Amos: 1979, 'Prospect Theory', Econometrica 47~ 263-291. Keeney, Ralph and Ralffa, Howard: 1976, Decisions with Multiple Ob/ectives, Wiley, New York. Kyberg, Henry: 1983, 'Rational Belief', The Behavorial and Brain Sciences 6~ 2 3 1 245. Lopes, Lola: 1982, 'Decision Making in the Short Run', Journal of Experimental Psychology 7 , 3 7 7 - 3 8 5 . Markowitz, Harry: 1959, Portfolio Theory, Yale UP, New Haven. Menger, Karl: 1934, 'Das Unsicherheitsmoment in der Wertlehre', Zeitschrift far National6konomie. Savage, Leonard: 1954, The Foundations of Statistics, Wiley, New York; 2rid edn, Dover, New York, 1972. Tversky, Amos and Kahneman, Daniel: 1974, 'Judgment Under Uncertainty', Science 185, 1124-1131. 202 PAUL WEIRICH Von Neumann, John and Morgenstern, Oscar: 1944, Theory of Games and Economic Behavior, Princeton UP, Princeton; 2rid ed., 1947. Department of Philosophy, The University of Rochester, River Campus Station, Rochester, N Y 1462 7, U.S.A.