Some ten years ago I ran Gambol, a short-lived gambling fund that invested money in statistical sports betting. I even managed to convince gullible friends to invest. Eventually, the fund lost all its money to the online bookmakers, and I tried to figure out what went wrong. I ran lots of simulations to better understand what had happened, and in one of them I encountered something remarkable. Recently, I was indirectly reminded about the paradox I had discovered, which I still have a hard time to understand intuitively. Here's the story of our gambling hero Andrew, who despite being an intelligent gambler, here takes a tumble and loses his entire bankroll.

When Bob offers Andrew to flip coins for even money, Andrew wrongly assumes that this couldn't possibly be a losing proposition. The catch is that Andrew needs to bet x% of his bankroll on each coin flip. He is free to choose the value of x himself, provided that x stays the same throughout the game.

Theoretically, the expected value of each betting round is +-0 for Andrew. His chances of winning each coin flip is 50%. When he loses a flip, his bankroll decreases by x% and if he wins he gains the same amount.

Andrew decides to bet 10% of his bankroll, which is $100. Let's take a look at possible scenarios for the outcome of the first rounds. After the first betting round Andrew's new bankroll will be either $90 or $110. If he ends up losing the first bet, his bet for the second round will be $9 (10% of his new bankroll of $90), and if he wins the first bet, his second bet will be $11. After round two he will end up in one out of the following four scenarios:

- 1st round lost + 2nd round lost: $81
- 1st round lost + 2nd round won: $99
- 1st round won + 2nd round lost: $99
- 1st round won + 2nd round won: $121

It is important to note that in 3 out of 4 cases Andrew is a loser after round #2. Theoretically, each bet is even money, and the average bankroll in the four scenarios remains $100 - but still Andrew is likely to be a loser in the long run. The more betting rounds there are, the more likely Andrew is to eventually lose his entire bankroll. There is also a chance that he will win a lot of money, but that chance is getting slimmer and slimmer for each betting rounds he participates in. After the third round Andrew happens to be back at a 50% chance of being a winner, but this is just a temporary fluctuation as he is again a likely loser after the fourth round.

Below is a graph showing how Andrew's bankroll develops in 50 simulations of 2,500 betting rounds, betting 10% of an initial $100 bankroll. After 2,500 bets the best case out of the 50 simulations has Andrew's bankroll at less than $60.

The blue line in graph below shows how many of the 50 initial scenarios are in the black (with a bankroll not smaller than the initial $100). The red lines displays the average bankroll over the 50 scenarios. The average bankroll should have stayed around the initial bankroll, since the expected value of the bet is +-0, but due to the limited number of simulations (50 in this case), we eventually run out of winning scenarios. No matter how many scenarios I choose to run, I can always make the average bankroll go down towards 0 by running enough betting rounds.

This bet, which theoretically is even money, makes you lose your money in real life - a fascinating paradox! Note that the final outcome where the bankroll dwindles towards 0 does not change, no matter which value is chosen for x (the ratio of your bankroll wagered on each round).

A useful lesson. Of course, one could argue that Andrew has no business betting anything on a proposition with no edge. I notice that you tagged the post with "kelly", which leaped into my mind while reading this. Applying the Criterion to the example provided, the optimal bet size would, of course, be zero.

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