Discuss.

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!

Quote:MichaelBluejayBloomberg 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!'

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.

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.

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.Quote:ssho88ev= +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

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.

Quote:MentalTechnically, I thought the ROR is zero.

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

Quote:ssho88ev= +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 BR

^{2}.

N | Ave BR | VAR |
---|---|---|

1 | $104.99 | 0.2025 |

3 | $115.76 | 2941.066 |

10 | $162.87 | 11672.05 |

30 | $432.50 | 992336 |

100 | $11613.87 | 258954827488 |

Anything seem out of place here?

EDIT: I used the wrong normalization for the variance. Ave BR is not changed.

N | Ave BR | VAR |
---|---|---|

1 | $105.00 | 0.2025 |

3 | $115.76 | 0.8823 |

10 | $162.89 | 11.66 |

20 | $265.33 | 199.25 |

30 | $431.50 | 2,841 |

40 | $704.43 | 33,101 |

50 | $1143.30 | 271,264 |

75 | $3849.67 | 29,796,153 |

100 | $13737.59 | 5,186,109,723 |

I have no problem with what ssho88 is saying about ROR.Quote:WizardI was tempted to post that, but wasn't really sure what the post in question was saying.

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.