Samuelson ' S Fallacy of Large Numbers and Optional Stopping

Accepting a sequence of independent positive mean bets that are individually unacceptable is what Samuelson called a fallacy of large numbers. Recently, utility functions were characterized where this occurs rationally, and examples were given of utility functions where any finite number of good bets should never be accepted.1 Here the author shows how things change if you are allowed the option to quit early: Subject to some mild conditions, you should essentially always accept a sufficiently long finite sequence of good bets. Interestingly, the strategy of quitting when you get ahead does not perform well, but quitting when you get behind does. This sheds some light on more possible behavioral reasons for Samuelson's fallacy, as well as strategies for handling a series of sequentially observed good investments. INTRODUCTION A D OVERVIEW Samuelson (1963) told a story in which he offered a colleague a better than 50-50 chance of winning $200 or losing $100. The colleague rejected the bet, but said he would be willing to accept a string of 100 such bets. Samuelson argued that the colleague was irrationally applying the law of averages to a sum, and this perhaps has led to a more widely held perception that accepting a sequence of good bets when a single one would be rejected is a "fallacy of large numbers." Since then a number of authors have studied this phenomenon. Samuelson (1989) gave examples of utility functions where a single bet is unacceptable but a sufficiently long finite sequence of good bets will be accepted. Also given were utility functions where a long sequence of good bets is never acceptable: Consider the utility function U(x) = -2-x and bets giving a 50 percent chance of losing $1 or winning $(1 + E), for a sufficiently small e > 0. It can be shown that expected utility decreases with each additional bet made, even though the bets are favorable and the utility function is increasing. Pratt and Zeckhauser (1987) studied the related property they labeled "proper risk aversion," where investors unwilling to make a single bet will also be unwilling to make more than one independent bet of the same type. Erol Pekoz is associate professor in the School of Management at Boston University. Thanks are due for helpful comments from Zvi Bodie, Stephen Ross, Paul Samuelson, and an anonymous referee. i Ross, 1999. 2 THE JOURNAL OF RISK AND INSURANCE Nielsen (1985) found necessary and sufficient conditions for a gambler with a concave utility function to eventually accept a sequence of bounded good bets, and Lippman and Mamer (1988) extended this to unbounded, identically distributed bets. Recently Ross (1999) extended this to independent but nonidentically distributed bets. The essential idea given in Lippman and Mamer (1988) and Ross (1999) is that if the utility function decreases faster than exponentially in the negative direction, the small risk of a loss can be magnified enough to overwhelm the benefits of a gain even for arbitrarily long sequences of good bets. Gollier (1996) gave some related results on how the availability of future optional bets can increase the attractiveness of a current bet, but the eventual attractiveness of a sufficiently large number of optional good bets has not been directly studied. See Bodie (1995) for a discussion of related phenomena surrounding long-term stock market investing. Here the author shows what happens when the gambler has the option to quit early: A sufficiently long sequence of good bets should always be accepted, meaning that given a sequence of positive mean bets, for sufficiently large n you should always agree to sequentially make the first n of them with the option to quit early. This holds provided the gambler's utility function is not bounded from above, the expected utility of a single bet is finite, and a condition on the bet means and variances holds. This result does not hold in the setting of Ross (1999) without a stopping option. There the total number of bets to be made is viewed as fixed in advance, while in the setting herein it is viewed as variable up to some maximum number that is fixed in advance. Note that the gambler is not allowed to play as long as it takes to get ahead, but is only allowed a maximum of n bets, which must be fixed in advance. It is interesting to note that the strategy of quitting when you reach some large wealth level does not perform well. This is because even with arbitrarily long sequences of good bets, there can always be some small chance that the game ends with a very large loss, and a utility function can always be found that magnifies this loss more than enough to make the game unacceptable. One uses the strategy of quitting whenever the gambler's wealth goes below the starting wealth, and for sufficiently long sequences of good bets, the benefit of large gains always eventually overwhelms the risk of losses. As a final note, much controversy exists over Samuelson's fallacy and the behavioral issues surrounding it. Benartzi and Thaler (1999), for example, used it as an example illustrating the limitations of expected utility theory in explaining behavior. This article shows that if faced with the opportunity to play a long sequence of favorable bets with the option to quit early, accepting the game is rational from an expected utility point of view. The organization of the article is as follows. The next section contains the main result. The "Multiplicative Gambles" section has an analogous result for multiplicative payoffs. The "Summary" section summarizes the conclusions, and the "Appendix" section provides the proofs. MAIN RESULT Consider an infinite sequence of available bets, and let Sn be the wealth of a gambler who makes the first n bets. Ross (1999) defines a utility function U to have the Eventual SAMUELSON'S FALLACY OF LARGE NUMBERS AND OPTIONAL STOPPING 3 Acceptance Property (EAP) if E[U(S,)] > U(0) for some n > 0, meaning for some n the gambler will be willing to make the first n bets. Expanding on this notion, say that the utility function has the Eventual Acceptance with Stopping Option Property (EASOP) if E[U(STAn)] > U(0) for some n > 0 and stopping time T, where x A y denotes min(x,y), meaning that for some fixed n the gambler will be willing to make the first n bets with the option to stop early at any time, knowing the outcomes of all previous bets. Note that in the former case the gambler is required to agree to make all n bets in advance, whereas in the latter case the gambler agrees to make up to n sequential bets with the option to stop at any time along the way. Here the concern is only with sequences of positive mean bets, which are represented as a sequence of independent random variables X1,X2,... with means g/i satisfying /i = infi [/i > 0, and first let Sn = 1-inl Xi denote the wealth of a gambler after the nth bet is made. Ross (1999) studied sequences of such positive mean bets and showed, subject to some additional side conditions, that the EAP holds if and only if a concave utility function, U, satisfies lim U(x)eYx 0 X-00+ -O for all y > 0. This type of condition was also previously given in Lippman and Mamer (1988, Theorem 4). The related side condition requiring finite bet variance is discussed in Lippman and Mamer (1988, Example 2). More examples where the EAP does not hold can be easily created from this, such as when U(x) = -e-x on (-oo,0) and is arbitrary above 0, preserving concavity. Letting, for example, Xi N(, 1), a straightforward computation (see Ross, 1999), shows that the expected utility of the sequence Sn approaches -oc exponentially in n. A consequence of the main result below is that the story is different for EASOP: With a sequence of good bets (and some side conditions), essentially all unbounded utility functions satisfy the EASOP. For sufficiently large n a gambler should always be willing to sequentially make the first n bets with the option to stop early. The author now formally states the main result. Theorem 1. Let X1,X2, ... be a sequence of independent random variables with means Ai and variances ai2, and let the wealth after the nth bet be Sn = 1i=_ Xi. Suppose