Can you make money betting on random coin flips?

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Beliavsky
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Re: Can you make money betting on random coin flips?

Post by Beliavsky » Thu Jan 08, 2015 12:54 pm

nisiprius wrote:P.P.S. Maybe maximizing expected(log of wealth) is a sensible thing to do--maybe a logarithmic utility function for wealth is reasonable--but I'd like to see some actual evidence from perceptual psychology or sociology or behavioral economics that it really is a decent quantitative approximation to how human brains work. And even if it is a "not crazy" assumption, it is one heck of an assumption to smuggle in without explicitly stating in in boldface, since the Kelly criterion is often presented as if it were objective and psychology-free.
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Re: Can you make money betting on random coin flips?

Post by tadamsmar » Thu Jan 08, 2015 1:06 pm

Consider the Kelly Black Box.

You put a dollar in the black box and let N serial Kelly bets operate on the dollar. The you open the box.

If N is small, then you might find less than a dollar.

But, if N is large, then you will open the box and you will find a fortune. The likelihood of not finding a fortune is very small, smaller than lots of risks that you never worry about.

Anyway, you can bury the risk with a large finite N. So making a rule against infinity does not seem to skewer use of the Kelly Criteria.

The problem is that there are few practical applications where you can make N large in the real world. It's hard to even approximately equate Boglehead investing to a series of bets, and it's harder to still to get a large N. For instance, a value bet on PE10 looks like a favorable bet, but it pays off in a decade or two, if ever. You will die before N>10, so there is really no capital management strategy that allows you to exploit this seemingly favorable bet to get a reliable return.

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Re: Can you make money betting on random coin flips?

Post by Browser » Thu Jan 08, 2015 1:28 pm

In investing, you not only have the problem of infinity, you have the problem that it is highly unlikely that the outcome distributions are stable over time. At least with the coin flips, you don't have that problem. Maybe betting on coin flips is more reasonable. :)
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Re: Can you make money betting on random coin flips?

Post by 555 » Thu Jan 08, 2015 1:33 pm

tadamsmar wrote:
555 wrote:In other words, always bet half your bankroll in this specific scenario, which is also what the Kelly criterion tells you. In other words your optimal "asset allocation" is 50% "cash" and 50% "bet" and you "rebalance" at each step.
This Kelly Calculator:

http://www.albionresearch.com/kelly/default.php

shows 25% of your bankroll to be the optimal.

I used 2:1 odds offered and a 50% win probability.
You bet 50% for the "double-or-halve" bet.
You bet 25% for the "triple-or-nothing" bet.
It's basically the same thing.

This observation has already been made many times in this thread, and has been fully understood for decades at least.

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Re: Can you make money betting on random coin flips?

Post by 555 » Thu Jan 08, 2015 1:56 pm

nisiprius wrote:The thing is, I have a TERRIBLE problem with saying it maximizes something that is "really very likely and gets more and more likely the longer the sequence is," because you have to have some sensible agreed-on way to account for the unlikely occurrences and you can't just throw them away--not even if their combined probability becomes arbitrarily small.

The example of this is the gambler's Martingale. This is the evil twin of the presumably virtuous Kelly bet. In one form, it proposes a series of fair even-money bets in which the bettor follows the sequence of betting $1, $3, $7, $15, $31, ... bet(n) = 2 x bet(n-1) + 1... until he wins. It seems perfectly clear that he must win eventually, and when he does, his final wealth is his original wealth plus $1 per bet. Now, the flaw in the reasoning is that for any finite sequence, the expectation continues to be zero, because there is a low-probability high-consequences case--no wins--in which the loss precisely balances all the cases with wins. NEVERTHELESS, the most probable scenario is a win of $1 per bet, and the probability that this happens can be made arbitrarily large by arbitrarily increasing the number of bets.

We all know that this is fallacious, so why aren't we worried by any formulation in terms of "it optimizes the most likely outcomes, and the probability of the unlikely outcomes becomes arbitrarily small if you extend the sequence indefinitely?"
The Kelly situation, and the standard constraint on the Martingale situation is that you have a finite bankroll, and you can't ever bet more than you have, so you can't go negative. So the situation you describe simply doesn't happen.

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Re: Can you make money betting on random coin flips?

Post by tadamsmar » Thu Jan 08, 2015 2:40 pm

Browser wrote:In investing, you not only have the problem of infinity, you have the problem that it is highly unlikely that the outcome distributions are stable over time. At least with the coin flips, you don't have that problem. Maybe betting on coin flips is more reasonable. :)
I don't think the distributions are stable in either case.

The issue is the probability that the outcome falls below a threshold by a deadline. You need to fund retirement.

Even with coin flips, you may have to be concerned about a bad run, depending on the threshold and the deadline. In this case the deadline is a finite number of coin flips.

In both cases, you don't have any real worries if the deadline is far enough out and/or the threshold is low enough.

Also, the coin flip scenario is for a perfectly stable system, stable over time, it's just a thought experiment after all. The stock market is apparently not as stable with respect to time.

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Re: Can you make money betting on random coin flips?

Post by tadamsmar » Thu Jan 08, 2015 3:05 pm

nisiprius wrote: 555 says it does not and many or most presentations suggest that it is pure objective math and that it is best by almost any reasonable criterion. I continue to think this just can't be true, there must be some assumption about value of risk, or utility of distribution of outcomes (wider is worse)--
There is at least one good man who claims that we should not make mean log of wealth big though years to act are long. He said why he thinks that here:

http://www-stat.wharton.upenn.edu/~stee ... on1979.pdf

So, can one claim one should make mean log of wealth big when years to act are long in words so short?

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Re: Can you make money betting on random coin flips?

Post by chaz » Thu Jan 08, 2015 3:34 pm

For some, gambling is fun, but not for me. Too much risk.
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Re: Can you make money betting on random coin flips?

Post by Browser » Thu Jan 08, 2015 3:48 pm

tadamsmar wrote:
nisiprius wrote: 555 says it does not and many or most presentations suggest that it is pure objective math and that it is best by almost any reasonable criterion. I continue to think this just can't be true, there must be some assumption about value of risk, or utility of distribution of outcomes (wider is worse)--
There is at least one good man who claims that we should not make mean log of wealth big though years to act are long. He said why he thinks that here:

http://www-stat.wharton.upenn.edu/~stee ... on1979.pdf

So, can one claim one should make mean log of wealth big when years to act are long in words so short?
This, of course, relates to Samuelson's paper: Risk and Uncertainty: A Fallacy of Large Numbers. In that paper he shows that if you would not accept a single bet based on the odds of winning, then you should not accept a series of such bets no matter how long. This applies when you are betting your entire stake with each bet - the "all in" strategy. In this instance, the size of N makes no difference in expected utility.
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Re: Can you make money betting on random coin flips?

Post by nisiprius » Thu Jan 08, 2015 9:07 pm

Beliavsky wrote:The triple-or-nothing bet is a just a leveraged version of double-or-halve, since investing 50% of your wealth in the former bet gives the same results as investing 100% in the second bet. If one starts with $2, 1 + 3*1 = 4 = 2*2, and 1 + 0*1 = 2*0.5 = 1.
So, you can use this principle to derive the second formula in Wikipedia,
1. The probability of success is p.
2. If you succeed, the value of your investment increases from 1 to 1+b.
3. If you fail (for which the probability is q=1-p) the value of your investment decreases from 1 to 1-a. (Note that the previous description above assumes that a is 1).
In this case, the Kelly criterion turns out to be the relatively simple expression
f* = p/a - q/b
In order to convert the failure to total loss, we need to apply leverage of 1/a. So, we take the formula (p(b + 1) - 1) / b, we plug in (1/a) * b instead of b to get the Kelly bet for the leveraged situation, and then we multiply that by 1/a, and simplify.

For the case where p = 0.5, b = 27% and a = 13%, which has approximately the same mean and standard deviation as the REAL (inflation adjusted) annual return of the U.S. stock market, we get:

f* = .5/.13 - .5/.27 = 3.846 - 1.851 = 1.994.

This seems to suggest that according to the Kelly criterion you should use 2X leverage when investing in the stock market, or am I missing something? This isn't in very good accord with tadasmar's citing of a paper by Rotando and Thorpe, The Kelly Criterion and the Stock Market, that says 17% leverage is optimum...
Last edited by nisiprius on Fri Jan 09, 2015 10:12 am, edited 1 time in total.
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Re: Can you make money betting on random coin flips?

Post by nisiprius » Thu Jan 08, 2015 9:09 pm

555 wrote:
nisiprius wrote:The thing is, I have a TERRIBLE problem with saying it maximizes something that is "really very likely and gets more and more likely the longer the sequence is," because you have to have some sensible agreed-on way to account for the unlikely occurrences and you can't just throw them away--not even if their combined probability becomes arbitrarily small.

The example of this is the gambler's Martingale. This is the evil twin of the presumably virtuous Kelly bet. In one form, it proposes a series of fair even-money bets in which the bettor follows the sequence of betting $1, $3, $7, $15, $31, ... bet(n) = 2 x bet(n-1) + 1... until he wins. It seems perfectly clear that he must win eventually, and when he does, his final wealth is his original wealth plus $1 per bet. Now, the flaw in the reasoning is that for any finite sequence, the expectation continues to be zero, because there is a low-probability high-consequences case--no wins--in which the loss precisely balances all the cases with wins. NEVERTHELESS, the most probable scenario is a win of $1 per bet, and the probability that this happens can be made arbitrarily large by arbitrarily increasing the number of bets.

We all know that this is fallacious, so why aren't we worried by any formulation in terms of "it optimizes the most likely outcomes, and the probability of the unlikely outcomes becomes arbitrarily small if you extend the sequence indefinitely?"
The Kelly situation, and the standard constraint on the Martingale situation is that you have a finite bankroll, and you can't ever bet more than you have, so you can't go negative. So the situation you describe simply doesn't happen.
The Kelly situation puts the problem on the other side of the bet. As I understand it, the Kelly situation assumes that the bettor has found

a) a bet that is his favor, that

b) can be repeated indefinitely, at the same payoff odds and same win probabilities, for as long as the bettor wishes... for long enough for the "law of large numbers" to iron out any wrinkles.

If the bettor is winning indefinitely large amounts of money, then someone else is losing indefinitely large amounts of money, and if we assume that the other side of the bet also has "a finite bankroll" then the assumption (b) does not apply.
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Re: Can you make money betting on random coin flips?

Post by Epsilon Delta » Thu Jan 08, 2015 10:15 pm

Browser wrote: This, of course, relates to Samuelson's paper: Risk and Uncertainty: A Fallacy of Large Numbers. In that paper he shows that if you would not accept a single bet based on the odds of winning, then you should not accept a series of such bets no matter how long. This applies when you are betting your entire stake with
each bet - the "all in" strategy. In this instance, the size of N makes no difference in expected utility.
I don't see where you get the "all in" strategy from these. I'll also let Samuelson take issue with himself:
Risk and Uncertainty wrote:I should warn against undue extrapolation of my theorem. It does not say one must always refuse a sequence if one refuses a singe venture.
Last edited by Epsilon Delta on Fri Jan 09, 2015 12:13 am, edited 1 time in total.

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Re: Can you make money betting on random coin flips?

Post by 555 » Thu Jan 08, 2015 10:26 pm

Here's a semi-realistic example of a triple-or-nothing bet (with 50:50 win:lose) chance.

A risky company could offer a junk bond, where you invest $10,000 now, and in return they give you a single lump sum of $30,000 two years in the future, if they are still in business, and let's suppose there is 50% chance of that, otherwise you get nothing.

The crucial point is that there is a time delay between the bet/investment and the potential payoff.

When you make an investment with positive expected reward, someone might say who's the fool on the other side of the bet making the losing bet, but the (potential) payoff is in the future, not now.

ETA: I'll just add that this comment is, amongst other things, a rebuttal of nisiprius's previous comment, and also to various other comments that dismiss the connection between gambling theory and investing theory.

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Re: Can you make money betting on random coin flips?

Post by tadamsmar » Fri Jan 09, 2015 8:27 am

I think 555 is correct. There is no distinction between gambling and investing that is based on the issues being discussed here. The government makes a distinction between gambling and investing: Investing provides an economic benefit by providing liquidity for capital, mere gambling does not. The distinction has nothing to do with any of the matters we are discussing.

However, there is a distinction between a thought experiment about gambling and real-world gambling or investing. For instance, Thorpe developed the theory of card counting in Black Jack, but when he used it for real in Los Vegas the casinos deployed all sorts of counter-measures. Card counting was discovered before Thorpe and the casinos had been dealing with it for years. My favorite counter-measure was the "mechanic". If someone seemed to be successfully card-counting , the casino switched out the current dealer and replaced him with a dealer that was an accomplished magician who cheated, called a "mechanic". Thorpe said that these guys were so good that you could never see them cheat even if you had your own watcher with you, you had to infer that you were being cheated by statistically estimating that your edge had evaporated. Thorpe had to be cautious after every dealer change, particularly if it was an unscheduled change.

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Re: Can you make money betting on random coin flips?

Post by Browser » Fri Jan 09, 2015 3:13 pm

Epsilon Delta wrote:
Browser wrote: This, of course, relates to Samuelson's paper: Risk and Uncertainty: A Fallacy of Large Numbers. In that paper he shows that if you would not accept a single bet based on the odds of winning, then you should not accept a series of such bets no matter how long. This applies when you are betting your entire stake with
each bet - the "all in" strategy. In this instance, the size of N makes no difference in expected utility.
I don't see where you get the "all in" strategy from these. I'll also let Samuelson take issue with himself:
Risk and Uncertainty wrote:I should warn against undue extrapolation of my theorem. It does not say one must always refuse a sequence if one refuses a singe venture.
Samuelson's argument applies only to the situation where the bettor allows his winnings to roll over -- which is the same as the "all in" strategy of betting 100% of your stake with each bet.
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Re: Can you make money betting on random coin flips?

Post by tadamsmar » Fri Jan 09, 2015 4:14 pm

Browser wrote:
Epsilon Delta wrote:
Browser wrote: This, of course, relates to Samuelson's paper: Risk and Uncertainty: A Fallacy of Large Numbers. In that paper he shows that if you would not accept a single bet based on the odds of winning, then you should not accept a series of such bets no matter how long. This applies when you are betting your entire stake with
each bet - the "all in" strategy. In this instance, the size of N makes no difference in expected utility.
I don't see where you get the "all in" strategy from these. I'll also let Samuelson take issue with himself:
Risk and Uncertainty wrote:I should warn against undue extrapolation of my theorem. It does not say one must always refuse a sequence if one refuses a singe venture.
Samuelson's argument applies only to the situation where the bettor allows his winnings to roll over -- which is the same as the "all in" strategy of betting 100% of your stake with each bet.
Where did you get that idea? It applies to serial bets, but there is no "all in" requirement, you resize your bet based on your total capital for the next bet, you always hold back some capital if you are not 100% certain of winning the bet. No "all in" betting strategy would maximized log growth, so how could he even be debating the merits of log growth?

Perhaps you mean something different by "all in". In the OP, none of the double or half bet was really not "all in", since half was not a risk, half was not really bet. If that your definition, then you can structure any bet to be not "all in" by just making an "all in" bet with a bookie and then handing him a penny that he is suppose to return regardless of whether you win or lose. That definition of "all in" is meaningless.

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Re: Can you make money betting on random coin flips?

Post by nisiprius » Fri Jan 09, 2015 10:06 pm

Thorp's article, Understanding the Kelly Criterion, says:
So, given any admissible strategy, there is a fractional Kelly strategy with which has a growth rate that is no lower and a variance of the growth rate that is no higher.
1) Is this the definitive statement of exactly what it is that the Kelly strategy optimizes?

2) Does he mean "variance of the growth rate" or "variance of log(1 + growth rate)?

3) Earlier, tadasmar cited a paper by Rotando and Thorpe, The Kelly Criterion and the Stock Market, that says 117% stocks, i.e. 17% leverage is the Kelly strategy for the stock market.

Let us suppose that I am a conservative investor who would like a 30/70 allocation; but, in order to apply the Kelly criterion directly, I am willing to make that 30% stocks, 70% cash.

Does the Kelly strategy force me to invest in 117% stocks?

Or is 30% stocks, 70% cash, rebalanced annually, already the Kelly strategy?

If not, what is the Kelly strategy that has "a growth rate that is no lower and a variance of the growth rate that is no higher" than 30% stocks, 70% cash, rebalanced annually?
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Re: Can you make money betting on random coin flips?

Post by 555 » Sat Jan 10, 2015 12:01 am

I'm guessing "fractional Kelly strategy" means Kelly strategy diluted with cash to lower risk and reward. Your link has mathematical symbols missing.

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Re: Can you make money betting on random coin flips?

Post by Epsilon Delta » Sat Jan 10, 2015 12:13 am

nisiprius wrote:
Let us suppose that I am a conservative investor who would like a 30/70 allocation; but, in order to apply the Kelly criterion directly, I am willing to make that 30% stocks, 70% cash.

Does the Kelly strategy force me to invest in 117% stocks?

Or is 30% stocks, 70% cash, rebalanced annually, already the Kelly strategy?

If not, what is the Kelly strategy that has "a growth rate that is no lower and a variance of the growth rate that is no higher" than 30% stocks, 70% cash, rebalanced annually?
Readers may prefer to go to the .doc instead of the Google cache. Googles text recognition leaves strange lacunas in place of the formulas.
http://edwardothorp.com/sitebuildercont ... erion2.doc

My read of that paper:

If you choose a model for stock market returns that meets the conditions for the paper, the rebalanced portfolio with 117% stocks is the full Kelly Portfolio and has the maximum mean growth. Any rebalanced portfolio with 0 to 117% stock is a fractional Kelly portfolio and, by the paper, on the efficient frontier. This includes your 30/70 preference so it is efficient and there is no other portfolio with both a higher growth rate and a lower variance.

You can quibble about the continuous v. annual rebalancing, but that really depends on the model of the market and nobody really knows what that should be.


This paper is addressing a slightly different problem than an infinite series of bets. Instead of saying the Kelly criterion results in a portfolio that dominates with probability 1 as N goes to infinity, it's looking at the distribution at the end of a finite period. The Kelly criterion seems to be the solution to several different but related problems. I don't think you can say one is definitive, but you can have a strong preference.

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Re: Can you make money betting on random coin flips?

Post by itstoomuch » Sat Jan 10, 2015 12:24 am

So, Browser, Are you making Money?
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Re: Can you make money betting on random coin flips?

Post by guymo » Sat Jan 10, 2015 3:01 am

nisiprius wrote:Thorp's article says:
So, given any admissible strategy, there is a fractional Kelly strategy with which has a growth rate that is no lower and a variance of the growth rate that is no higher.
1) Is this the definitive statement of exactly what it is that the Kelly strategy optimises?
This is really the statement that the efficient frontier is made up of the fractional Kelly strategies. It is definitive for the restricted assumptions of the paper. As I understand it, the fact that the full Kelly strategy maximises growth rate holds for more general cases than considered in this paper.
2) Does he mean "variance of the growth rate" or "variance of log(1 + growth rate)?
All values in the paper are in the form log(1+return) so it is the latter. For small values of x, log (1+x) is very close to x, so for the case of investments, where you don't actually know the probabilities in advance, these are not meaningfully distinguishable.
3) Earlier, tadasmar cited a paper by Rotando and Thorpe, The Kelly Criterion and the Stock Market, that says 117% stocks, i.e. 17% leverage is the Kelly strategy for the stock market.

Let us suppose that I am a conservative investor who would like a 30/70 allocation; but, in order to apply the Kelly criterion directly, I am willing to make that 30% stocks, 70% cash.

Does the Kelly strategy force me to invest in 117% stocks?

Or is 30% stocks, 70% cash, rebalanced annually, already the Kelly strategy?
Yes, that is the fractional Kelly strategy and sits on the efficient frontier, as Epsilon Delta said.

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Re: Can you make money betting on random coin flips?

Post by selftalk » Sat Jan 10, 2015 6:12 am

Of course but the secret is to know when to quit which is largely impossible.

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Re: Can you make money betting on random coin flips?

Post by tadamsmar » Sat Jan 10, 2015 8:56 am

555 wrote:I'm guessing "fractional Kelly strategy" means Kelly strategy diluted with cash to lower risk and reward. Your link has mathematical symbols missing.
For example, in the bet offered in the OP the optimal strategy was to bet 0.25 of your current bankroll. Full Kelly is 0.25. But if you cannot tolerate that much risk or you are not sure of the size of your edge, then you can bet half Kelly which is 0.125 of your current bankroll. Or you can bet any fraction of full Kelly, that's the general notion of fractional Kelly.

PS: In the OP, it appears that you bet 0.5 of your current bankroll, but you always get half of it back, so really, you are only betting (ie. putting as risk) 0.25 of your current bankroll.

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Re: Can you make money betting on random coin flips?

Post by tadamsmar » Sat Jan 10, 2015 9:14 am

One interesting thing about the efficient frontier is that if you go too far to the right you can end up having a sub-optimal AA.

You can create an efficient frontier for the bet in the OP. But AAs riskier than Full Kelly are sub-optimal in the sense that you are doing something worse than taking on uncompensated risk, you are losing long-term expected returns while taking on more risk. In the plot of the efficient frontier, the return keeps going up but the long-term return is going down. The Y axis is expected bet-wise or yearly arithmetic return, not expected CAGR.

The efficient frontier plots you typically see in investing usually don't extend that far to the right, they truncate the right end of the frontier. Sometimes 100% of the riskiest asset will still not be too far to the right, you'd have to leverage to get too far to the right.

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Re: Can you make money betting on random coin flips?

Post by tadamsmar » Sat Jan 10, 2015 9:41 am

This means that the efficient frontier plots can be considered visually misleading. If they plotted CAGR on the Y axis then the frontier would be flatter representing the fact that you are not getting as big a risk premium.

This was discussed here:

http://www.bogleheads.org/forum/viewtop ... 0&t=120591

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Re: Can you make money betting on random coin flips?

Post by nisiprius » Sat Jan 10, 2015 1:59 pm

So, then... is all this saying (merely?) that the Kelly criterion puts an upper limit on how much you should bet... that below that limit, it is a matter of personal choice, you are using a "fractional Kelly strategy" and are on the efficient frontier, but not above it? (Perhaps there is also the implication that you should always bet a constant percentage of your current holding).
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Re: Can you make money betting on random coin flips?

Post by tadamsmar » Sat Jan 10, 2015 2:54 pm

nisiprius wrote:So, then... is all this saying (merely?) that the Kelly criterion puts an upper limit on how much you should bet... that below that limit, it is a matter of personal choice, you are using a "fractional Kelly strategy" and are on the efficient frontier, but not above it? (Perhaps there is also the implication that you should always bet a constant percentage of your current holding).
Bettors use something like half Kelly when the win probability is not certain. In betting you typically know the odds offered but you only have an estimate of the win probability.

Investor's may use fractional Kelly because they are trying to reach some goal other that maximum log wealth at infinity, for instance the goals of the sort used in the Trinity study. They cannot tolerate a run of bad luck.

Retirement investors tend to use a declining percentage because the deadline for consumption is approaching, but I don't know the math that backs that up.

Betting the same fraction of full Kelly does not imply betting the same fraction of your current holdings if all the bets are not the same. In the OP thought experiment, each bet had the same odds offered and win percentage so the full Kelly was constant over time. Should one assume that full Kelly is a constant for the stock market? (I think the Boglehead philosophy tends toward assuming it's constant.)

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Re: Can you make money betting on random coin flips?

Post by 555 » Sat Jan 10, 2015 3:00 pm

nisiprius wrote:So, then... is all this saying (merely?) that the Kelly criterion puts an upper limit on how much you should bet... that below that limit, it is a matter of personal choice, you are using a "fractional Kelly strategy" and are on the efficient frontier, but not above it? (Perhaps there is also the implication that you should always bet a constant percentage of your current holding).
Basically if you have a strategy with a risk/reward tradeoff that might make sense, then for some people it might make sense to have another strategy with more risk and more reward, and for some people it might make sense to have another strategy with less risk and less reward, but it doesn't make sense to accept more risk and less reward.

The Kelly criterion maximizes reward. But it's perfectly sensible for some people to accept less risk and less reward, e.g. if you have enough saved up.

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Re: Can you make money betting on random coin flips?

Post by nisiprius » Sat Jan 10, 2015 5:24 pm

555 wrote:
nisiprius wrote:So, then... is all this saying (merely?) that the Kelly criterion puts an upper limit on how much you should bet... that below that limit, it is a matter of personal choice, you are using a "fractional Kelly strategy" and are on the efficient frontier, but not above it? (Perhaps there is also the implication that you should always bet a constant percentage of your current holding).
Basically if you have a strategy with a risk/reward tradeoff that might make sense, then for some people it might make sense to have another strategy with more risk and more reward, and for some people it might make sense to have another strategy with less risk and less reward, but it doesn't make sense to accept more risk and less reward.

The Kelly criterion maximizes reward. But it's perfectly sensible for some people to accept less risk and less reward, e.g. if you have enough saved up.
A big problem with this kind of discussion is that the precise statements are awkwardly long and everyone starts to shorten them... and changing their meaning. As I presently understand it, the Kelly criterion does not "maximize reward." It maximizes the expectation of logarithm(terminal wealth). I haven't quite made the jump yet to how you make a precisely equivalent statement that includes the phrase "growth rate," but I think it maximizes the expectation of the logarithm of the CAGR. And, indeed, it does do that, see chart below.

However, the choice of fraction to bet also affects the dispersion of results. So Thorp is trying to get at some analog of the efficient frontier for compounded situations, and he says about "So, given any admissible strategy, there is a fractional Kelly strategy with 0 ≦ c ≦ 1 which has a growth rate that is no lower and a variance of the growth rate that is no higher." And I think but I am not sure that he is misspeaking slightly and means "variance of the LOGARITHM of the growth rate."

So, there is some kind of risk-return tradeoff, and I am trying to understand what it is. We are already mixing a riskless asset--literal cash, 0% return, 0% standard deviation--with a risky asset. I'm not sure how we draw the efficient frontier, what it looks like, where one would put the riskless asset... In the MPT world, when you mix something with the riskless asset you get a straight line, and all points on that line are equally "good," all have the same Sharpe ratio. Obviously this is not true in the Kelly world, but I'm not succeeding in getting the mental map.

This is for the coin-flip 50%-win bet that is triple or nothing. It's a Pascal's triangle direct calculation using binomial percentages of each number of wins or losses. The red curve is the logarithm of the mean final wealth (i.e. the expectation). The data points are shown at X positions that are denser in "interesting" regions. To avoid negative logarithms we don't go all the way to 100%; the highest value shown is for the case of betting 99.92%, and the mean payoff, i.e. the expectation after 100 flips, is something like $10^45. The mean includes those superskew extreme blazing-white-swan lottery wins. Obviously the expected simply increases with increasing bet size, and is far greater at the high values than at the 25% Kelly value.

Here comes the chart, and all I have is return. I need to do something to get dispersion of return (variance of log of compound growth rate IF that is in fact that is what we are walking about).

The green curve is the mean of the logs, that is to say the expectation of the log of final wealth, and it indeed has its maximum (a very very shallow, broad maximum) at 25%.

These results aren't going to surprise anyone but seeing it on a chart makes me more comfortable, and underlines the enormous difference between log of mean wealth and mean of log wealth... and I am still struggling with why we want the log. (Yes, I get it that it's more germane to compounding, but if we are not doing "utility functions" we want to get back to actual wealth someday, and I am uncomfortable with the idea that we really can just do it all in logs and exponentiate at the end).

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Re: Can you make money betting on random coin flips?

Post by 555 » Sat Jan 10, 2015 6:08 pm

The point is that "log of wealth" is the thing that is binomially distributed, and hence approximately normally distributed (after enough flips, due to central limit theorem). The (approximate) normal distribution is fully determined by its mean ("return") and standard deviation ("risk").

So "wealth" is approximately lognormally distributed after enough flips.

Those are just the brute facts, and they don't involve any utility considerations at all.

Now, different strategies will give different approximate normal distributions of "log of wealth", so you have some choice over "return" and "risk", and that's where ones personal utility would enter the discussion.

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Re: Can you make money betting on random coin flips?

Post by Bob.Beeman » Sat Jan 10, 2015 8:15 pm

Sorry, but if the bettor persists he will ALWAYS lose in the end.

This is due to the fact that, ultimately on some level, all bet amounts are integers. You can't bet 1/2 cent, or 1/10 of a Share of Apple. Thus it is inevitable that with random coin flips the bettor's balance will hit 1 unit of whatever the minimum possible bet is. Then he must either fold, and end the game with the minimum possible balance, or make a "double or nothing" bet.

Further, when the balance is an odd number of units the bet must be rounded up to the next full unit. If you have a balance of 7 your bet must be 4, not 3.5.

I have written a simulator for this using the JavaScript Math.random() function. This is not a function that should be used for cryptography, but it is probably more random than the best coin.

Try betting a small amount, say 100 units and use the "Bet" button to follow the action. When you want to run some big scenarios press the "Auto Bet" button to see how long it takes to bust. You will always bust in the end.

I might add the opinion that this game has a failure mechanism similar to "doubling down". Eventually the betting will reach the value of a gold sphere the size of the moon, and where do you go from there?

Coin Toss Calculator

Let us know your results.

- Bob Beeman.

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Re: Can you make money betting on random coin flips?

Post by 555 » Sat Jan 10, 2015 8:23 pm

Bob.Beeman wrote:Sorry, but if the bettor persists he will ALWAYS lose in the end.
Nonsense.
Bob.Beeman wrote:This is due to the fact that, ultimately on some level, all bet amounts are integers.
Irrelevant.

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Re: Can you make money betting on random coin flips?

Post by robert88 » Sat Jan 10, 2015 9:11 pm

nisiprius wrote: 3) Earlier, tadasmar cited a paper by Rotando and Thorpe, The Kelly Criterion and the Stock Market, that says 117% stocks, i.e. 17% leverage is the Kelly strategy for the stock market.

Let us suppose that I am a conservative investor who would like a 30/70 allocation; but, in order to apply the Kelly criterion directly, I am willing to make that 30% stocks, 70% cash.

Does the Kelly strategy force me to invest in 117% stocks?

Or is 30% stocks, 70% cash, rebalanced annually, already the Kelly strategy?

If not, what is the Kelly strategy that has "a growth rate that is no lower and a variance of the growth rate that is no higher" than 30% stocks, 70% cash, rebalanced annually?
No discussion of Thorpe is complete without what he actually did and Princeton/Newport Partners never had a year in which it lost money.

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Re: Can you make money betting on random coin flips?

Post by Bob.Beeman » Sat Jan 10, 2015 9:23 pm

555 wrote:
Bob.Beeman wrote:Sorry, but if the bettor persists he will ALWAYS lose in the end.
Nonsense.
Bob.Beeman wrote:This is due to the fact that, ultimately on some level, all bet amounts are integers.
Irrelevant.
Please respond with meaningful comments. "Nonsense" and "Irrelevant" are not meaningful comments by themselves.

Did you actually follow through a simulation and find the errors, if any?

Thanks:

- Bob Beeman.

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Re: Can you make money betting on random coin flips?

Post by 555 » Sat Jan 10, 2015 9:32 pm

@ Bob.Beeman
I think you must have missed the point that the OP's payout is "double-or-halve", and also "triple-or-nothing" is being discussed.

Perhaps you are thinking of "double-or-nothing", but that has non-positive expectation, so is not germaine to this thread.

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Re: Can you make money betting on random coin flips?

Post by Bob.Beeman » Sat Jan 10, 2015 10:04 pm

555 wrote:@ Bob.Beeman
I think you must have missed the point that the OP's payout is "double-or-halve", and also "triple-or-nothing" is being discussed.

Perhaps you are thinking of "double-or-nothing", but that has non-positive expectation, so is not germaine to this thread.
I think you are correct. I misunderstood the OP.

-Bob Beeman.

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Re: Can you make money betting on random coin flips?

Post by guymo » Sun Jan 11, 2015 5:35 am

555 wrote: The reason to optimize geometric mean, is that (1) it is easy to calculate and more importantly (2) it simultaneously optimizes a bunch of other important things including things that can be loosely described as "maximizing the probability of success" or "minimizing the probability of failure".
For me, the above statement (2) is the thing that I would like to understand more explicitly.

555, or anyone, can you explain, hint or provide references as to why optimising the geometric mean maximises probability of success, please? I guess the first thing to ask is what precisely the statement means.

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Re: Can you make money betting on random coin flips?

Post by guymo » Sun Jan 11, 2015 8:56 am

Aha, I have found something, from Thorp's paper "The Kelly Criterion in Blackjack, Sports Betting and the Stock Market".
http://www.eecs.harvard.edu/cs286r/cour ... on2007.pdf

Theorem 1 in that paper explains what happens when you adopt the Kelly strategy vs any other strategy, and in particular parts (i) and (iii) cover the difference between the Kelly strategy for the double-or-half coin flip game, and the all-in strategy. Wealth tends to grow with the Kelly strategy but to oscillate between arbitrarily high and low values with the all-in strategy. The explanation in the article is pretty clear I think.

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Re: Can you make money betting on random coin flips?

Post by tadamsmar » Sun Jan 11, 2015 10:55 am

guymo wrote:
555 wrote: The reason to optimize geometric mean, is that (1) it is easy to calculate and more importantly (2) it simultaneously optimizes a bunch of other important things including things that can be loosely described as "maximizing the probability of success" or "minimizing the probability of failure".
For me, the above statement (2) is the thing that I would like to understand more explicitly.

555, or anyone, can you explain, hint or provide references as to why optimising the geometric mean maximises probability of success, please? I guess the first thing to ask is what precisely the statement means.
It starts with the idea that losses are relative to your wealth. A $1000 loss is a bigger deal for a poor man than a rich man, a given sum has more utility the poorer you are. A simple function that captures this general notion is utility(w) = log(w) where w is wealth, but it's not the only function that captures the idea.

Log utility is discussed here:

http://en.wikipedia.org/wiki/Risk_avers ... k_aversion

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Re: Can you make money betting on random coin flips?

Post by guymo » Sun Jan 11, 2015 12:16 pm

tadamsmar,

Thanks for your reply. However, I think there is more to it than log utility. In particular 555's statement says that maximising log of wealth, i.e. adopting a Kelly strategy, also happens to optimise other things; specifically, 555 stated "probability of success" is maximised.

As I mentioned, Theorem 1 of Thorp's paper linked above has some statements that help in that direction, especially for the asymptotic case. It seems pretty clear that a Kelly strategy is preferable to an expectation maximising strategy for these coin-flip games, even for someone who values every dollar equally: going "all in" results in wealth that bounces around indefinitely, while a Kelly strategy grows wealth over time. You don't have to have a logarithmic utility for your wealth to prefer that.

It is starting to look like a precise understanding of 555's statement is quite tricky. For instance the Breiman article cited in Thorp's paper does a bit of analysis of what "optimal strategy" really means, and the answers seem to depend heavily on the detail of the situation under consideration.

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Re: Can you make money betting on random coin flips?

Post by Jacotus » Sun Jan 11, 2015 1:24 pm

nisiprius wrote: ... [lots of good stuff] ...
These results aren't going to surprise anyone but seeing it on a chart makes me more comfortable, and underlines the enormous difference between log of mean wealth and mean of log wealth... and I am still struggling with why we want the log. (Yes, I get it that it's more germane to compounding, but if we are not doing "utility functions" we want to get back to actual wealth someday, and I am uncomfortable with the idea that we really can just do it all in logs and exponentiate at the end).
You are right in that the mathematically precise statements are clear and it's the mystifying cloud of words when trying to translate to English that muddy the issue. Here is my take.

While expected value is something we often want to maximize through our behavior, other times it is not. We might not want to maximize expected value because of outlier events (See the St. Petersburg paradox or lottery) or because of the way people often do not appreciate what truly low probability actually means (see the lottery). In a lottery-like game where you can pay $1 for a one-in-one-billion chance to win $2 billion, you really shouldn't play (unless you have many many billions to throw at it). Your expected value is positive, so if you had enough money you would come out ahead, but even if you put $10 million into this game, you have roughly a 99% chance of losing all your money. So I would not behave to maximize expected value of wealth in this instance. If, however, I change the rules of the game, and instead of having to bet $1, I were allowed to bet billionths of $1 with the same odds, then suddenly the game becomes desirable, intuitively. And that is exactly what the Kelly criterion tells me to do.

Expected value is one particular linear function on your probability distribution. Why maximize it? Why not some other value, such as growth rate, that might involve a nonlinear transform of your probability distribution? There is no mathematical proof that behaving to maximize expected value is optimal in every respect. I am not appealing to a log-utility type of argument, but rather simply saying that plain expected value is not always the best intuitive measure of results.

Approximated to a continuous function with constant growth rate, with wealth W(t) and growth rate r, your wealth is:
W(t) = W(0) e^rt
log W(t) = log W(0) + rt
Assuming a fixed amount of time t, should we behave so as to maximize the expected value of W, or maximize the expected value of log W (equivalent to maximizing the expected value of r)? Or should we do something else?

One takeaway from the above is this: when serious outliers can occur, such that expected value of W is a poor measure of results, one might prefer behavior that maximizes the expected value of log W. Why the logarithm? Its mathematical behavior is to "squish" or "compress" large ranges of values into a shorter range, so outliers play less of a role in distorting expected value.

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Re: Can you make money betting on random coin flips?

Post by tadamsmar » Sun Jan 11, 2015 6:22 pm

nisiprius wrote:To avoid negative logarithms we don't go all the way to 100%; the highest value shown is for the case of betting 99.92%, and the mean payoff, i.e. the expectation after 100 flips, is something like $10^45.
I get e^45 instead of 10^45 or thereabouts.
The mean includes those superskew extreme blazing-white-swan lottery wins.
Yes, the chance a losing it all is something like .999999999999999999999999999999

Using Full Kelly the chance of losing it all is zero, but it's still a bit of a roller coaster ride.

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Re: Can you make money betting on random coin flips?

Post by tbradnc » Sun Jan 11, 2015 6:31 pm

Here in Appalachia where I live scratch off lottery tickets are very popular with the poor.

I was talking to a guy playing them the other day and he thought that 1:4 odds meant that if bought 4 tickets one of them should be a winner.

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Re: Can you make money betting on random coin flips?

Post by Awsi Dooger » Mon Jan 12, 2015 12:47 am

tbradnc wrote:Here in Appalachia where I live scratch off lottery tickets are very popular with the poor.

I was talking to a guy playing them the other day and he thought that 1:4 odds meant that if bought 4 tickets one of them should be a winner.
I see that throughout my travels. In Miami I can't believe the lottery lines at some spots, like Publix. You can tell based on the clothes they are wearing and cars they are driving that the majority have no business playing into those pools, but they feel they have no other choice to escape their financial tier.

Anyway, Kelly Criterion is very popular among the sharpest guys in Las Vegas. I mentioned that several months ago to another gambler here. He described the amounts he was playing, and his strategy. They made absolutely no sense based on Kelly. He was being far too conservative, if his quoted bankroll and percentages were accurate.

Still, the most successful guys I know take advantage of the middling opportunities, not grind aspects like Kelly. There was a massive example on Sunday. One guy I know made the biggest score of his life. He took Dallas +6.5 early in the week and then played back Packers -4.5 when that line dropped in the late going a couple of hours prior to gametime. I knew he had that +6.5 and thought it was risky to expect it to drop. Besides, 5 is normally a dead number in football. Unbelievable that such a big payoff was accomplished on a game falling 5. My friend interrupted his celebrating tonight to describe how he sweated out the failed 2-point conversion, the play that enabled the middle to hit. Aaron Rodgers hit every pass late in the game. Except that one.

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Re: Can you make money betting on random coin flips?

Post by Rob Bertram » Mon Jan 12, 2015 1:12 am

nisiprius wrote:
555 wrote:
nisiprius wrote:So, then... is all this saying (merely?) that the Kelly criterion puts an upper limit on how much you should bet... that below that limit, it is a matter of personal choice, you are using a "fractional Kelly strategy" and are on the efficient frontier, but not above it? (Perhaps there is also the implication that you should always bet a constant percentage of your current holding).
Basically if you have a strategy with a risk/reward tradeoff that might make sense, then for some people it might make sense to have another strategy with more risk and more reward, and for some people it might make sense to have another strategy with less risk and less reward, but it doesn't make sense to accept more risk and less reward.

The Kelly criterion maximizes reward. But it's perfectly sensible for some people to accept less risk and less reward, e.g. if you have enough saved up.
A big problem with this kind of discussion is that the precise statements are awkwardly long and everyone starts to shorten them... and changing their meaning. As I presently understand it, the Kelly criterion does not "maximize reward." It maximizes the expectation of logarithm(terminal wealth). I haven't quite made the jump yet to how you make a precisely equivalent statement that includes the phrase "growth rate," but I think it maximizes the expectation of the logarithm of the CAGR. And, indeed, it does do that, see chart below.

However, the choice of fraction to bet also affects the dispersion of results. So Thorp is trying to get at some analog of the efficient frontier for compounded situations, and he says about "So, given any admissible strategy, there is a fractional Kelly strategy with 0 ≦ c ≦ 1 which has a growth rate that is no lower and a variance of the growth rate that is no higher." And I think but I am not sure that he is misspeaking slightly and means "variance of the LOGARITHM of the growth rate." This is my understanding as well. Thorp says variance but means variance of the log.

So, there is some kind of risk-return tradeoff, and I am trying to understand what it is. We are already mixing a riskless asset--literal cash, 0% return, 0% standard deviation--with a risky asset. I'm not sure how we draw the efficient frontier, what it looks like, where one would put the riskless asset... In the MPT world, when you mix something with the riskless asset you get a straight line, and all points on that line are equally "good," all have the same Sharpe ratio. Obviously this is not true in the Kelly world, but I'm not succeeding in getting the mental map. I think he gets tricky with the efficient frontier. The traditional MPT efficient frontier uses arithmetic mean. Thorp is using geometric mean (compound growth), so higher variance degrades returns at about -0.5 * variance^2.
My comments are in blue.

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Re: Can you make money betting on random coin flips?

Post by nisiprius » Mon Jan 12, 2015 7:21 am

Brainlock again. I am getting ready to calculate something-or-other about the compound growth RATE per flip, in place of final wealth, because I get it that for some purposes it makes more sense that way. And I'm thinking about Thorp actually talking about the mean of the log of that rate. And I'm saying, OK, but I still have a little problem relating to the log of a growth rate, even if I call it "decikels" in my mind, so after I've gotten that number, why not exponentiate it and subtract one to get it back to a familiar growth percentage?

And I say "Whoa... that's a geometric mean taken across parallel scenarios over the same period of time, not consecutively."

Now, I'm all for using the geometric mean when one is concatenating consecutive financial events in time, but this is using it for parallel hypothetical scenarios all occurring at the same time.

Suppose that over a 15-year period three investors, A, B, and C, have their investments remain constant, double, and quadruple, respectively. 1X, 2X, and 4X. Those are CAGR's of 0%, 4.73%, and 9.68% respectively. Add one and take logs, 0, 0.0201, and 0.0401 respectively. Mean of logs, 0.0201. 10^(mean of logs), 4.73% = 2X over 15 years.

(Notice that "rebalancing" here is a red herring. We are talking about three separate investors' experiences. Each of them rebalances or does whatever they like in their own accounts, we are looking at a figure of merit for their collection of independent investing experiences).

I can't wrap my head around the idea that, for the three investors taken together, their group experience amounted to "doubling." Looking at the group, nothing meaningful "doubled" here... did it?

Even less so if you consider A, B, and C to be three equally probable hypothetical scenarios that could be experienced by an individual investor.

(Once again, I insist that "rebalancing" is a red herring. The hypothetical strategy may well involve rebalancing. That strategy, whatever it may be, plays out in three different ways. There is no way to "rebalance" across the strategies; the "me" living in parallel universe C can't stop five years into the period and exchange some probability across the universe with the "me" living in universe B).

The best I can come up with is that if you imagine A, B, and C racing along an uphill racetrack whose height follows a compound interest law, then the relative speeds of the three investors are 0, 1, and 2 somethings per year, but I can't see how that is relevant other than for bragging rights.

I really do NOT see how geometric means can be relevant for parallel histories occurring simultaneously over the same time period.

(And yes, I know the BLS calculates CPI by taking the geometric mean of price growth within each specific narrow category... and that seems bogus to me, too!)
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Re: Can you make money betting on random coin flips?

Post by tadamsmar » Mon Jan 12, 2015 9:41 am

This explains the ABC thing:

https://en.wikipedia.org/wiki/Expected_ ... hypothesis

In particular, the Axiom of Continuity.

Log utility is the simplest utility function that follows from the assumption of relative risk aversion, so it tends to pervade economics as the plain vanilla option. We directly or indirectly use things like the Sharpe Ratio without even thinking about the logic behind it.

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Re: Can you make money betting on random coin flips?

Post by guymo » Mon Jan 12, 2015 10:44 am

nisiprius wrote:Brainlock again. I am getting ready to calculate something-or-other about the compound growth RATE per flip, in place of final wealth, because I get it that for some purposes it makes more sense that way. And I'm thinking about Thorp actually talking about the mean of the log of that rate. And I'm saying, OK, but I still have a little problem relating to the log of a growth rate, even if I call it "decikels" in my mind, so after I've gotten that number, why not exponentiate it and subtract one to get it back to a familiar growth percentage?

And I say "Whoa... that's a geometric mean taken across parallel scenarios over the same period of time, not consecutively."
I see where you are coming from here. A friend just showed me the following argument which, although it does not address this point directly, may convince you that the log of the growth rate per flip is important anyway. It's a mathematical argument rather than an intuitive one and I do not think it maps onto the intuition you have in mind. There's also a step I don't quite trust. Anyway, here goes.

Consider the compound growth rate after n periods. For the coin-flip game, you called this Compound Flipual Growth Rate so I will write it as CFGR(n). Obviously,

CFGR(n) = ((1 + r_1) x (1 + r_2) x … x (1 + r_n) ) ^(1/n) - 1

where r_i is the growth rate on the i-th flip.

CGFR(n) = e^((1/n) x ( log (1+r_1) + log (1 + r_2) + … + log (1 + r_n))) - 1

Letting n tend to infinity:

CFGR(infty) = e^(E(log ( 1 + r) ) - 1.

On the left we have long run CFGR. On the right, the limit of (1/n) x ( log (1+r_1) + log (1 + r_2) + … + log (1 + r_n)) is the expectation of log (1 + r) [think of the formula we're taking the limit of as an estimate of this expectation based on a single n-period trial, so as the number of periods grows to infinity, the figure should converge to the expectation.] Therefore e^E(log (1 + r)) - 1 is the long run growth rate.

I think there is something fishy about the part in [brackets] above, but nevertheless it just about convinces me that maximising log of growth per flip is the way to maximise long run growth, and it doesn't require you to think about geometric mean of parallel trials.

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Re: Can you make money betting on random coin flips?

Post by guymo » Mon Jan 12, 2015 10:50 am

tadamsmar wrote:This explains the ABC thing:

https://en.wikipedia.org/wiki/Expected_ ... hypothesis

In particular, the Axiom of Continuity.

Log utility is the simplest utility function that follows from the assumption of relative risk aversion, so it tends to pervade economics as the plain vanilla option. We directly or indirectly use things like the Sharpe Ratio without even thinking about the logic behind it.
Can you square this view --- which seems to place log utility at the centre of the discussion --- with what Kelly wrote in his original paper, quoted by nisiprius above?
At every bet he maximizes the expected value of the logarithm of his capital. The reason has nothing to do with the value function which he attached to his money, but merely with the fact that it is the logarithm which is additive in repeated bets and to which the law of large numbers applies.

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Re: Can you make money betting on random coin flips?

Post by Epsilon Delta » Mon Jan 12, 2015 11:39 am

guymo wrote: Can you square this view --- which seems to place log utility at the centre of the discussion --- with what Kelly wrote in his original paper, quoted by nisiprius above?
At every bet he maximizes the expected value of the logarithm of his capital. The reason has nothing to do with the value function which he attached to his money, but merely with the fact that it is the logarithm which is additive in repeated bets and to which the law of large numbers applies.
Suppose we have the triple or nothing scenerio, but instead of tossing coins after the bet I pick the sequence ahead of time. I tell you there are H heads and T tails. You have to pick the fraction of your wealth to bet at each step ahead of time, after which I will open the envelope and settle up. What do you pick?* There is no probability involved in this one, but the result looks a lot like the Kelly criterion, and depends only on the ratio of heads and tails (i.e. it's the same for 1 head 1 tail as for 100 heads and 100 tails)

Now for a long enough random sequence we know the ratio of heads and tails with probability one. (More precisely we know it within a smidgen with probability arbitrarily close to one for a long enough sequence.) Hence the Kelly criterion is optimal with probability one. No utility function needed.

* Solution
Let f be fraction bet at each turn
1 be initial wealth
b be odds paid on success (2 in the case of triple or nothing)
W be the value after the game,
W = (1+bf) ^ H * (1- f) ^T

Find the maximum by differentiating by parts, setting to zero and solve for f

dW/df = H(1+bf)^(H-1) * b * (1-f)^T + (1+bf)^H * T(1-f)^(T-1) * (-1)

0 = (1+bf)^(H-1) * (1-f)^(T-1) *{ H*b*(1-f) - (1+ bf) * T }

0 = { H*b*(1-f) - (1+ bf) * T }

f = {(H/(H+T)) * (b + 1) -1 }/b

Now verify that this is a maximum and we haven't divided by zero etc. i.e. continuity at f = 1.

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