Fallacy of Large Numbers

无聊又再次想到这个问题~

Samuelson (1963) told a story inwhich 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.”

虽然samuelson名声在外,但第一次看到这个问题的时候还是觉得他的colleague很rational啊,而且还想股票市场其实也可以这样减少风险的 (i call it "diversification across time", 呃,有点像长期持有/定投)

quote一个叫Erol A. Peköz的人的一篇论文:

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 suf ciently
long nite 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 + );
for a suf ciently small > 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...
 
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 suf ciently long nite sequence of good
bets. Interestingly, the strategy of quitting when you get ahead does not perform
well, but quitting when you get behind does.
 
原因如下:
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 gameun acceptable.Oneuses the strategy of quitting whenever the
gambler's wealth goes below the starting wealth, and for sufficiently long sequences
of good bets, the bene t of large gains always eventually overwhelms the risk of
losses.

需要注意的是,股票投资每个时段不是independent的,所以这个fallacy不大适用.
假设适用的话,如果不是封闭基金的话,是有option随时quit的,因此输的人会继续
投资,赢的人会减少, 即止赚不止蚀, 这倒是很符合散户行为.