MichaelBluejay
MichaelBluejay
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unJon
December 16th, 2020 at 9:46:32 PM permalink
Bloomberg has a recent article about Ole Peters, who challenges traditional economic theory. One of his ideas is a coin flip game where you win +50% of your bankroll for heads, and -40% of your bankroll for tails. Peters posits that this is a bad game, even though it has a positive expected value, because (perhaps surprisingly), the most common result is losing over 99% of the starting bankroll.

Discuss.
odiousgambit
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December 17th, 2020 at 3:03:00 AM permalink
Perhaps intuition tells people the problem is the violation of the Kelly Criterion, the article notes most people decline the bet. Putting up your entire bankroll to start betting is asking for it, and each bet afterwards has to be the entire bankroll [according to the graph that illustrates the below]

Quote:

Suppose in the same game, heads came up half the time. Instead of getting fatter, your $100 bankroll would actually be down to $59 after 10 coin flips. It doesn’t matter whether you land on heads the first five times, the last five times or any other combination in between



This is the most interesting part to me. Winning exactly half the time doesn't cut it, evidently. Not if you keep putting it all on the line. The article does go on to say some people would win big if allowed to keep winning, which is another deviation from reality that every gambler should know: no one is allowed to just keep winning!
the next time Dame Fortune toys with your heart, your soul and your wallet, raise your glass and praise her thus: “Thanks for nothing, you cold-hearted, evil, damnable, nefarious, low-life, malicious monster from Hell!” She is, after all, stone deaf. ... Arnold Snyder
Mental
Mental
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December 17th, 2020 at 4:13:22 AM permalink
Quote: MichaelBluejay

Bloomberg has a recent article about Ole Peters, who challenges traditional economic theory. One of his ideas is a coin flip game where you win +50% of your bankroll for heads, and -40% of your bankroll for tails. Peters posits that this is a bad game, even though it has a positive expected value, because (perhaps surprisingly), the most common result is losing over 99% of the starting bankroll.



In a Kelly world, the proportion of bankroll risked would be much smaller and the most likely case closer to breakeven. People should intuitively understand this.

What happens in the real world is that wealthy people invest in diversified government and corporate bonds with low risk and +EV. Low wealth people invest in bankrupt companies on Robinhood and lottery tickets with negative EV and a chance for a big score. This is an anti-Kelly world. 'People Are Idiots and I Can Prove It!'
Wizard
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Wizard
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MichaelBluejay
December 17th, 2020 at 5:28:48 AM permalink
The author suggests a 50/50 bet that pays 6 to 5.

If find the optimal Kelly bet is 1/12 of bankroll.

Should you take the bet if forced to bet half your bankroll, thus increasing wealth by 60% or losing wealth by 50%? The Kelly Criterion says definitely not as there is a drop in expected utility. I simulated doing this 1,000 times many times and every time a starting bankroll of $1,000,000 was less than a penny by the end, which illustrates why.

I majored in math and economics. The economics half of that never did me much good. However, every teacher I had taught that intelligent people act in a way to maximize expected utility, not expected value. This explains why people buy insurance, despite being a bad bet.

The topic goes to show the danger of over-betting your bankroll.
It's not whether you win or lose; it's whether or not you had a good bet.
ThatDonGuy
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MichaelBluejay
December 17th, 2020 at 8:13:00 AM permalink
Since you are increasing your bets with each win and decreasing them with each loss, this is a Reverse D'Alembert, which normally results in losing money whenever the wins and losses balance, but that assumes the bets are even money.

In this case, after N wins and N losses, in any order, a bankroll of 1 becomes 1.5^N x 0.6^N = 0.9^N.
ssho88
ssho88
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December 17th, 2020 at 9:42:42 AM permalink
ev= +5%, Var =0.2025. If BR = bet size, Kelly ratio = 0.2025/0.05 = 4.05, ROR = e^(-2/4.05) = 0.61

So your risk of ruin = 61%.

I am not sure, please correct me if you don't agree
Mental
Mental
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December 17th, 2020 at 9:54:34 AM permalink
Quote: ssho88

ev= +5%, Var =0.2025. If BR = bet size, Kelly ratio = 0.2025/0.05 = 4.05, ROR = e^(-2/4.05) = 0.61

So your risk of ruin = 61%.

I am not sure, please correct me if you don't agree

Technically, I thought the ROR is zero. Perhaps your ROR is a reasonable estimate or is truncated at some minimum bankroll. For example, if you can't bet less than a penny, you are are ruined.

Is the variance of the outcome of the infinite bet stream infinite when expressed as starting bankroll? I would think you would need to do MC sims to get the variance of a finite number of flips based on min and max bankroll stopping conditions. I am not a math guy, so I don't know if you can do it analytically for N flips.

Your variance agrees with mine, assuming you mean single flip variance stated in units of BR^2 . The problem does not state how much each wager is. One could use different units.
Last edited by: Mental on Dec 17, 2020
Wizard
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December 17th, 2020 at 11:25:21 AM permalink
Quote: Mental

Technically, I thought the ROR is zero.



I was tempted to post that, but wasn't really sure what the post in question was saying.
It's not whether you win or lose; it's whether or not you had a good bet.
Mental
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December 17th, 2020 at 11:35:19 AM permalink
Quote: ssho88

ev= +5%, Var =0.2025. If BR = bet size, Kelly ratio = 0.2025/0.05 = 4.05, ROR = e^(-2/4.05) = 0.61

So your risk of ruin = 61%.

I am not sure, please correct me if you don't agree



Monte Carlo sims don't converge very well because the distribution of results for N flips is horribly skewed toward zero bankroll with a long tail to very high bankrolls. After 100M trials, the results for N=100 flips is nowhere near converging. Treat these as estimates. This is the average BR after N flips based on a $100 initial BR. VAR is the variance of the final BR after N flips in units of initial BR2.
NAve BRVAR
1$104.990.2025
3$115.762941.066
10$162.8711672.05
30$432.50992336
100$11613.87258954827488


Anything seem out of place here?

EDIT: I used the wrong normalization for the variance. Ave BR is not changed.
NAve BRVAR
1$105.000.2025
3$115.760.8823
10$162.8911.66
20$265.33199.25
30$431.502,841
40$704.4333,101
50$1143.30271,264
75$3849.6729,796,153
100$13737.595,186,109,723
Last edited by: Mental on Dec 17, 2020
Mental
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December 17th, 2020 at 11:48:41 AM permalink
Quote: Wizard

I was tempted to post that, but wasn't really sure what the post in question was saying.

I have no problem with what ssho88 is saying about ROR.

I would consider that turning $100 into 0.1 pennies in a few minutes to be be ruin, especially if that was my entire BR. At that point, I will just steal the coin that Ole is flipping and walk out without being technically ruined.

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