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赌博是最古老的人类活动,几千年来的发展主要由那些想发财的科学家特别是数学家推动。Fortune's Formula《财富公式》,讲几名当代科学家的努力:发明信息论的香农(Shannon)、香农在贝尔实验室的同事凯利(John Kelly)、在MIT的同事Ed Thorp,成果是凯利判据。

    Thorp在三人中最热中发财。他带人用凯利判据扫荡了21点赌场,著书Beat the Dealer《战胜庄家》。21点不许赌客算牌,应该跟他有关。他创建了最早的量化对冲基金,追逐涡轮、可转债的错价,十余年间稳定地获取绝对收益,又著书Beat the Market《战胜市场》。Black-Scholes期权定价公式在学术杂志上发表之前,Thorp就在实战中自己发明并运用了同样的定价方法。他不在乎credit,在乎赢钱。萨缪尔逊及其得意弟子罗伯特·默顿与凯利判据派打了十几年的笔仗,互斥对方为愚不可及。在学界,Thorp等人无法与萨缪尔逊的威望对抗,但赌博这种事最终由实绩说话。默顿与Scholes加入长期资本管理公司,两人因期权定价公式上的贡献获得1997年诺贝尔奖,但长期资本次年爆仓。Thorp说:“默顿这帮人没按凯利判据下注,赌得太大。”
    下面是用iPad+ibooks+evernote做的零星摘录。有兴趣者应读全书,William Poundstone是个很好的作者。


William Poundstone
Bernoulli’s thesis was that risky ventures should be evaluated by the geometric mean of outcomes.
There is a deep connection between Bernoulli’s dictum and John Kelly’s 1956 publication. It turns out that Kelly’s prescription can be restated as this simple rule: When faced with a choice of wagers or investments, choose the one with the highest geometric mean of outcomes. This rule, of broader application than the edge/odds Kelly formula for bet size, is the Kelly criterion.
When the possible outcomes are not all equally likely, you need to weight them according to their probability. One way to do that is to maximize the expected logarithm of wealth. Anyone who follows this rule is acting as if he had logarithmic utility.
In short, the Kelly criterion may risk money you need for gains you may find superfluous; it may sacrifice welcome gains for a degree of security you find unnecessary. It is not a good fit with people’s feelings about the extremes of gain and loss.
The promises of the Kelly criterion recall those tales of mischievous genies granting wishes that never turn out as planned. Before you wish for maximum long-term return and zero risk of ruin, Samuelson is saying, you had better make sure that is exactly what you want—because you may get it.
There is a catch. Life is short, and the stock market is a slow game. In blackjack, it’s double or nothing every forty seconds. In the stock market, it generally takes years to double your money—or to lose practically everything. No buy-and-hold stock investor lives long enough to have a high degree of confidence that the Kelly system will pull ahead of all others. That is why the Kelly system has more relevance to an in-and-out trader than a typical small investor.
The bankroll fluctuations in Kelly betting obey a simple rule. In an infinite series of serial Kelly bets, the chance of your bankroll ever dipping down to half its original size is…½.
This is exactly correct for an idealized game in which the betting is continuous. It is close to correct for the more usual case of discrete bets (blackjack, horse racing, etc.). A similar rule holds for any fraction 1/n. The chance of ever dipping to 1/3 your original bankroll is 1/3. The chance of being reduced to 1 percent of your bankroll is 1 percent.
The good news is that the chance of ever being reduced to zero is zero. Because you never go broke, you can always recover from losses.
The bad news is that no matter how rich you get, you run the risk of serious dips. The 1/n rule applies at any stage in the betting.
One is to stake a fixed fraction of the Kelly bet or position size. As before, you determine which opportunity or portfolio of opportunities maximizes the geometric mean. You then stake less than the full Kelly bet(s). A popular approach with gamblers is “half Kelly.” You consistently wager half of the Kelly bet.
This is an appealing trade-off because it cuts volatility drastically while decreasing the return by only a quarter. In a gamble or investment where wealth compounds 10 percent per time unit with full-Kelly betting, it compounds 7.5 percent with half-Kelly.
The core of John Kelly’s philosophy of risk can be stated without math. It is that even unlikely events must come to pass eventually. Therefore, anyone who accepts small risks of losing everything will lose everything, sooner or later. The ultimate compound return rate is acutely sensitive to fat tails.
Shannon was a buy-and-hold fundamental investor. From the late 1950s through 1986, Shannon’s return on his stock portfolio was about 28 percent.
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