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).