Gamble with time
- Journal name:
- Nature Physics
- Volume:
- 9,
- Page:
- 3
- Year published:
- (2013)
- DOI:
- doi:10.1038/nphys2520
- Published online
Date: circa 1950. Topic: information theory. Location: AT&T Bell Laboratories. Ask any physicist, computer scientist or electrical engineer to add another item to this list and they will probably say “Person: Claude Shannon”, naming the engineer who famously laid the foundations of information theory in his landmark paper, 'A Mathematical Theory of Communication', published in 1948 in The Bell System Technical Journal.
But if Shannon is the most obvious response, it isn't the only one that makes sense. In 1956, a physicist colleague of Shannon's from Texas, John Larry Kelly, published a lesser-known but equally profound paper looking at how a gambler — facing a series of risky bets — could optimize his winnings in the long run and avoid ruin along the way. Kelly offered a concise answer: the gambler should at each stage wager a specific fraction of his or her current wealth, the fraction determined by the odds and potential winnings.
Today, more than half a century later, Kelly's solution — now known as the Kelly Criterion — finds wide use in finance as a tool for guiding investments over time. But the deeper meaning of Kelly's perspective, and its relation to other ideas about optimal behaviour in the presence of risk, remain controversial. This is clear from the polarized responses to the recent work of physicist Ole Peters, who has re-visited Kelly's thinking — and applied it to resolve a centuries' old paradox (Philos. Trans. R. Soc. A 369, 4913–4931; 2011).
Consider how much you might be willing to pay to play a lottery based on a coin flip. If the first flip is heads, you win $1. If tails, you flip again. Heads on the second toss and you win $2, otherwise you flip again, with heads on the third toss giving $4 and so on. The lottery pays out 2n dollars if the first head comes up on the nth roll. An easy calculation shows that the expected payout of the lottery is actually infinite, as the size of the payout grows just as fast as its likelihood decreases.
There's nothing paradoxical in this, of course, but what has seemed contradictory — since the eighteenth century, when Nicolas Bernoulli proposed the puzzle — is that no sensible person would pay much to play this game, despite the infinite expected pay-off. Real people do not find this lottery appealing and generally offer less than $10 or so to play. Maximizing expected return is, in this case, just not what people do.
Of course, this is still only strange if you believe for some reason that people should act to maximize expected return — a notion first proposed by Pierre de Fermat and often taken for granted in economics. However, this is where Peters, inspired by Kelly's perspective, suggests that probabilistic thinking has gone awry and stands in need of correction. The way to make sense of the 'paradox', he argues, is to think in terms of time.
After all, the familiar calculation based on expected value actually entails supposing that the gamble plays out simultaneously in several parallel worlds, one for each possible outcome. The result of the calculation is influenced by every one of these no matter how unlikely. This is the essence of probability, of course, yet it clearly introduces an artificial element into the situation. An alternative way to treat the problem — Kelly's way — is instead to calculate the expected pay-off from a string of such wagers actually playing out in real time, as a real person would experience if trying to learn how to play the game by trial and error.
Mathematically, this way of thinking leads Peters to consider the time average of the growth rate (log return) of the wealth of a player who begins with wealth W and plays the gamble over N periods, in the limit as N goes to infinity. A simple calculation leads to a formula for this growth rate that gives more sensible guidance. The rate is positive when the cost C of playing is sufficiently low, relative to a player's wealth, and negative when C becomes too high. Hence, how much you ought to be willing to pay depends on your initial wealth, as this determines how much you can afford to lose before going broke. Plug in real numbers, and the results predict fairly well what real people feel about the bet.
This aspect — the dependence of the calculation on the gambler's initial wealth — doesn't figure in the usual ensemble average in any way. Coincidentally, this result is identical to a solution to the paradox, originally proposed by Daniel Bernoulli — that people don't care about the absolute pay-off, but about the logarithm of the pay-off. But Bernoulli offered that solution without any fundamental justification; it emerges more naturally if one simply imagines playing in time, not in parallel worlds.
Some people argue that this in-time perspective still doesn't really solve Bernoulli's paradox (known more usually as the St. Petersburg paradox). After all, the original question asks what to do in one play, not in repeated play, which is required to calculate the time average. To my mind, this objection doesn't really hold. After all, any gamble has to be situated in time. You only care about winning or losing a gamble because you intend to go on living afterwards, and can profit from the extra wealth, facing future challenges in a more secure position. Psychologically, it's more or less impossible to consider any gamble as happening outside of time, because we live in time (and living in time is part of what makes us averse to risk, because we actually have to live with the consequences).
There is one other aspect to Peters's resurrection of Kelly's way of thinking that holds special interest for physicists. We're familiar with the notion of ergodicity — the property of the dynamics of a system that makes a time average equal to an ensemble average. This is a neat trick, hugely useful, as ensemble averages are typically much easier to calculate. The assumption of ergodicity lies at the basis of statistical mechanics.
And it is the failure of this assumption that distinguishes time and ensemble averages in gambling problems also. Economists have long relied on the equality of ensemble and time averages, assuming that the probabilities they deal with often have this feature. But the multiplicative growth process involved in any situation of repeated gambles is necessarily not ergodic. Go broke at one time step, and you are permanently out of the game, stuck at wealth = 0, a situation never captured by the ensemble average, which assumes continued exploration of the space of outcomes.
It's curious that a Bell Labs physicist hit on this idea so long ago, and we still can't quite fathom its full implications.
Buchanan科学时评之带着时间赌博(下) [ witten1 ]
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But if Shannon is the most obvious response, it isn't the only one that makes sense. In 1956, a physicist colleague of Shannon's from Texas, John Larry Kelly, published a lesser-known but equally profound paper looking at how a gambler — facing a series of risky bets — could optimize his winnings in the long run and avoid ruin along the way. Kelly offered a concise answer: the gambler should at each stage wager a specific fraction of his or her current wealth, the fraction determined by the odds and potential winnings.
Today, more than half a century later, Kelly's solution — now known as the Kelly Criterion — finds wide use in finance as a tool for guiding investments over time. But the deeper meaning of Kelly's perspective, and its relation to other ideas about optimal behaviour in the presence of risk, remain controversial. This is clear from the polarized responses to the recent work of physicist Ole Peters, who has re-visited Kelly's thinking — and applied it to resolve a centuries' old paradox (Philos. Trans. R. Soc. A 369, 4913–4931; 2011).
Consider how much you might be willing to pay to play a lottery based on a coin flip. If the first flip is heads, you win $1. If tails, you flip again. Heads on the second toss and you win $2, otherwise you flip again, with heads on the third toss giving $4 and so on. The lottery pays out 2n dollars if the first head comes up on the nth roll. An easy calculation shows that the expected payout of the lottery is actually infinite, as the size of the payout grows just as fast as its likelihood decreases.
“Probabilistic thinking has gone awry and stands in need of correction.”
Of course, this is still only strange if you believe for some reason that people should act to maximize expected return — a notion first proposed by Pierre de Fermat and often taken for granted in economics. However, this is where Peters, inspired by Kelly's perspective, suggests that probabilistic thinking has gone awry and stands in need of correction. The way to make sense of the 'paradox', he argues, is to think in terms of time.
After all, the familiar calculation based on expected value actually entails supposing that the gamble plays out simultaneously in several parallel worlds, one for each possible outcome. The result of the calculation is influenced by every one of these no matter how unlikely. This is the essence of probability, of course, yet it clearly introduces an artificial element into the situation. An alternative way to treat the problem — Kelly's way — is instead to calculate the expected pay-off from a string of such wagers actually playing out in real time, as a real person would experience if trying to learn how to play the game by trial and error.
Mathematically, this way of thinking leads Peters to consider the time average of the growth rate (log return) of the wealth of a player who begins with wealth W and plays the gamble over N periods, in the limit as N goes to infinity. A simple calculation leads to a formula for this growth rate that gives more sensible guidance. The rate is positive when the cost C of playing is sufficiently low, relative to a player's wealth, and negative when C becomes too high. Hence, how much you ought to be willing to pay depends on your initial wealth, as this determines how much you can afford to lose before going broke. Plug in real numbers, and the results predict fairly well what real people feel about the bet.
This aspect — the dependence of the calculation on the gambler's initial wealth — doesn't figure in the usual ensemble average in any way. Coincidentally, this result is identical to a solution to the paradox, originally proposed by Daniel Bernoulli — that people don't care about the absolute pay-off, but about the logarithm of the pay-off. But Bernoulli offered that solution without any fundamental justification; it emerges more naturally if one simply imagines playing in time, not in parallel worlds.
Some people argue that this in-time perspective still doesn't really solve Bernoulli's paradox (known more usually as the St. Petersburg paradox). After all, the original question asks what to do in one play, not in repeated play, which is required to calculate the time average. To my mind, this objection doesn't really hold. After all, any gamble has to be situated in time. You only care about winning or losing a gamble because you intend to go on living afterwards, and can profit from the extra wealth, facing future challenges in a more secure position. Psychologically, it's more or less impossible to consider any gamble as happening outside of time, because we live in time (and living in time is part of what makes us averse to risk, because we actually have to live with the consequences).
There is one other aspect to Peters's resurrection of Kelly's way of thinking that holds special interest for physicists. We're familiar with the notion of ergodicity — the property of the dynamics of a system that makes a time average equal to an ensemble average. This is a neat trick, hugely useful, as ensemble averages are typically much easier to calculate. The assumption of ergodicity lies at the basis of statistical mechanics.
And it is the failure of this assumption that distinguishes time and ensemble averages in gambling problems also. Economists have long relied on the equality of ensemble and time averages, assuming that the probabilities they deal with often have this feature. But the multiplicative growth process involved in any situation of repeated gambles is necessarily not ergodic. Go broke at one time step, and you are permanently out of the game, stuck at wealth = 0, a situation never captured by the ensemble average, which assumes continued exploration of the space of outcomes.
It's curious that a Bell Labs physicist hit on this idea so long ago, and we still can't quite fathom its full implications.
Buchanan科学时评之带着时间赌博(下) [ witten1 ]
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带着时间赌博（下）
Gamble with time
在数学上，Ole Peters的想法就是假定一个参与这赌博的人初始财富为W，玩了N把，在N趋于极限的情况一下，把这个参赌的人的财富的增长率对时间做平均。一个简单的计算最终给出了一个关于财富增长率的公式（对数回报），这个公式对于实际情况有着更大指导价值。公式表明，如果玩的成本相对于玩家的财富的比例C足够低，那么总财富的平均增长率是正的，而当C太高的时候，这个增长率变为负。所以你想花多少钱来玩这个游戏取决于你的初始财富，因为这决定了在输得倾家荡产之前能承受的损失。所以当加入了实际情况，最终结果非常好的反应了人们关于这样子抛硬币赌博的真实感受。
依赖于玩家的初始财富的这一样的一种方式在任何情形下都不会出现在通常的系综平均的方法里（就是前面形象的平行世界的方法）。巧合的是，这个结果和三四百年前Daniel Bernoulli(D.伯努力)用直觉得到答案是一样的。当年Bernoulli认为人们不会关心绝对得收益，而是会关心收益的对数值！但是bernoulli的回答并没有任何fundamental的正当的原因。现在当我们放入时间这个因素后，我们自然的得到了我们需要的答案而无需要求助于平行世界！
一些人会争辩这个实时的看法并没有真的解决了Bernoulli佯谬（更多的时候被称作St.Petesburg佯谬）。毕竟，原始的问题是在单把之中怎么玩，而不是重复多把怎么玩，而重复多把的玩是时间平均的先决条件。对我来说，这样的反对是站不住脚的。毕竟，任何赌博都是在一定的时间范围内玩的。你仅仅关心赢或输一把，因为你过后是想要继续生活的，并且能从额外的财富中获得利润，以及为了让自己在社会竞争中能处于一个更安全的位置所要面对的未来的挑战。从心理意义上来说，考虑上在时间之外进行赌博应当是不可能的，因为我们生活在时间之中（生活在时间之中正是时间可以让我们规避风险，因为我们事实上不得不生活在一系列的“结果”（因果）之中）。
对于物理学家来说，Peter对Kelly的思考方式的再现是有着特别意义的。我们熟悉遍历性----一个（假定的）动力学体系对时间的平均可以等价于系综平均（这里作者的说法有欠缺，应当是平衡态体系，对非平衡态体系，不能这样做）。这近乎就是一个小把戏(其实就是一个很难推翻的假定),非常有用,因为系统平均会容易处理得多(相对于实时的处理).遍历性的假设是基于统计力学基础的。
然而也正是这个假定的失效区分了在博弈问题中的“时间”平均和“系综平均＂！经济学家们长久以来一直依赖于系综等价和时间平均，假定他们所处理的概率自然会有这样的特点．然而包含在多次博弈中的倍增过程必然不是遍历的－－一旦在其中一步破产了，你就永远出局了，停在财富为零的位置， 一个从未被系综平均表现出来的点。（后面一句评论关于状态空间的在这时就是“结果”组成的空间，但是过于专业，我就不写出来到）
我觉得奇怪的是，一个Bell实验室的物理学家在几十年就有了这样的想法，为何直到现在仍然不能很好的去领会其全部含义？（我来试着回答这个问题，其实就是人们一旦抛弃了系综平均，问题往前一步将变得极为困难，这里的赌博的例子毕竟只是一个相对简单的例子，别一个原因我猜出是人的惰性吧，哈哈，翻译完这起头，俺要休息了）
Gamble with time
在数学上，Ole Peters的想法就是假定一个参与这赌博的人初始财富为W，玩了N把，在N趋于极限的情况一下，把这个参赌的人的财富的增长率对时间做平均。一个简单的计算最终给出了一个关于财富增长率的公式（对数回报），这个公式对于实际情况有着更大指导价值。公式表明，如果玩的成本相对于玩家的财富的比例C足够低，那么总财富的平均增长率是正的，而当C太高的时候，这个增长率变为负。所以你想花多少钱来玩这个游戏取决于你的初始财富，因为这决定了在输得倾家荡产之前能承受的损失。所以当加入了实际情况，最终结果非常好的反应了人们关于这样子抛硬币赌博的真实感受。
依赖于玩家的初始财富的这一样的一种方式在任何情形下都不会出现在通常的系综平均的方法里（就是前面形象的平行世界的方法）。巧合的是，这个结果和三四百年前Daniel Bernoulli(D.伯努力)用直觉得到答案是一样的。当年Bernoulli认为人们不会关心绝对得收益，而是会关心收益的对数值！但是bernoulli的回答并没有任何fundamental的正当的原因。现在当我们放入时间这个因素后，我们自然的得到了我们需要的答案而无需要求助于平行世界！
一些人会争辩这个实时的看法并没有真的解决了Bernoulli佯谬（更多的时候被称作St.Petesburg佯谬）。毕竟，原始的问题是在单把之中怎么玩，而不是重复多把怎么玩，而重复多把的玩是时间平均的先决条件。对我来说，这样的反对是站不住脚的。毕竟，任何赌博都是在一定的时间范围内玩的。你仅仅关心赢或输一把，因为你过后是想要继续生活的，并且能从额外的财富中获得利润，以及为了让自己在社会竞争中能处于一个更安全的位置所要面对的未来的挑战。从心理意义上来说，考虑上在时间之外进行赌博应当是不可能的，因为我们生活在时间之中（生活在时间之中正是时间可以让我们规避风险，因为我们事实上不得不生活在一系列的“结果”（因果）之中）。
对于物理学家来说，Peter对Kelly的思考方式的再现是有着特别意义的。我们熟悉遍历性----一个（假定的）动力学体系对时间的平均可以等价于系综平均（这里作者的说法有欠缺，应当是平衡态体系，对非平衡态体系，不能这样做）。这近乎就是一个小把戏(其实就是一个很难推翻的假定),非常有用,因为系统平均会容易处理得多(相对于实时的处理).遍历性的假设是基于统计力学基础的。
然而也正是这个假定的失效区分了在博弈问题中的“时间”平均和“系综平均＂！经济学家们长久以来一直依赖于系综等价和时间平均，假定他们所处理的概率自然会有这样的特点．然而包含在多次博弈中的倍增过程必然不是遍历的－－一旦在其中一步破产了，你就永远出局了，停在财富为零的位置， 一个从未被系综平均表现出来的点。（后面一句评论关于状态空间的在这时就是“结果”组成的空间，但是过于专业，我就不写出来到）
我觉得奇怪的是，一个Bell实验室的物理学家在几十年就有了这样的想法，为何直到现在仍然不能很好的去领会其全部含义？（我来试着回答这个问题，其实就是人们一旦抛弃了系综平均，问题往前一步将变得极为困难，这里的赌博的例子毕竟只是一个相对简单的例子，别一个原因我猜出是人的惰性吧，哈哈，翻译完这起头，俺要休息了）
文章出处：Gamble with time
时间：约在1950年。主题：信息理论。位置：AT&T Bell实验室。如果你问任何一个物理学家，计算机科学家或者电气工程师在实验室的单子上再添一样东西的话，最应当加怎么的时候，他们很有可能都会回答“人：Claude shannon”--信息理论奠基者--其标志性的著名文章发表在1948年的"The Bell System Technical Journal"，标题为“A Mathematical Theory of Communication”(通信的数学理论)。
如果说Shannon是最明显的回答的话，那么他并不是唯一的的回答。在1956年，来自美国德州的Shannon的物理同事，John Larry Kelly，发表了名声要小一些的但是同样是杰出的工作。其工作试图回答，一个赌徒，在面对一系列的有风险的压注的时候，如何最优化他的长期胜率同时避免在中途破产。Kelly给出了简洁的回答：这个赌徒在每一步都应当拿出他当前财富的一个比例做为保证金，这个比例取决于胜算和可能的赢面。
今天，在逾半世纪之后，Kelly解----现在被称作Kelly判据----广泛的应用于引导金融中的投资。而Kelly的见解是有着更深层的内涵，它与其他的在风险存在时的最优行为的观点依然存在着争论。这些争论被一个叫Ole Peters的物理学家----在重新思考Kelly的想法之时----在最近的工作予以澄清，并且解决了一个几个世纪的佯谬。（Pilos.Trans.R.Soc.A 369, 4913-4931;2011）.
设想你愿意来玩基于抛硬币的赌博，你愿意投多少钱来玩这样的一玩法？如果第一抛之后是头朝上，你赢一块钱。如果是尾巴，你就再抛。在第二次抛硬币中头才朝上，那么你赢两块钱，否则你就再抛一次，如果在第三次头才朝上那么你赢四块钱，并以此类推。坐庄的将付你2^n块钱如果在第n次抛硬币之中才出现头朝上。一个简单的计算表明坐庄的付出期望值将达到无穷大，这个趋于无穷大的速度就像第一次头朝上出现在第n次抛投中的下降的几率一样快！
当然，这里面没有任何佯谬。然而与该理想情况相矛盾的是，自从Nicolas Bernoulli在十八世纪提出来了这一玩法以来，没有任何一个人愿意付太多的钱这样玩。人们并不觉得这种赌法有吸引力并且一般只愿意玩不超过10块钱来玩这个游戏。在这个赌法里，尽管有最大的期望回报（直至无穷），然而人们却不以为然。
当然，这仍然仅仅只是奇怪如果你相信由于特定原因人们应当总是去追求回报最大化----一个由Pierre de Fermat最早提出的概念，经常在经济学里不加思索的就直接拿来用了。然而，这正是Ole Peters被Kelly的见解所启发的地方，Ole Peters指出概率性的思考扭曲了实际情况应当予以修正，并进一步指出澄清这里的佯谬的方式是“时间”。
毕竟，那些基于期望值的大家熟悉的计算已经预先假定了所有的赌博（所有次的抛硬币）是在多个平行的世界中同时进行的，每一个对应于一个可能的结果！计算的结果被每一个平等世界中的结果所影响，无论这发生在平等世界中的结果是多么不可能。当然，我们知道这是概率论的精髓，但是很显然在这样的概率论之中已经有了这个人为的不自然的设定！一个可选择的方式来解决这个问题----Kelly的方式----所有的这些赌博都是实时进行的，所以替代一下子计算全部赌博的期望回报的更好的方法是按照时间演化来进行，因为做为一个真实的人所经历的也是在一次一次的试错中来学习如何玩这个游戏
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