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ABSTRACT
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 AND 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^sup -x^ and bets giving a 50 percent chance...