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This example assumes you only have $1 to bet, and if you lose it, you’re out of the game. I wonder what happens to the simulated outcome if you can keep betting

This example assumes you only have $1 to bet, and if you lose it, you’re out of the game. I wonder what happens to the simulated outcome if you can keep betting, even if you lose.

vitus 6y ago
Not quite. > Everyone starts with $1, gets 50% profit if they win, and pays 40% of their bet if they lose. From the wealth-over-time graph, it looks like the bet is sized such that it's always 100% of what you have (per some trajectories going as small as 10^-7). My read is that while each individual bet looks good in isolation, the fact is that one win and one loss puts you in the red overall -- when dealing with this iterated experiment you want to look not at E[X] = 0.05, but at E[log(X)] = -0.05 to get a sense for how your assets evolve each round. For some intuition: if you win once and lose once, your net result is 1.5 * 0.6 = 0.9, so you've lost 10% of your starting money.
- vitus 6y ago
  To expand on this further, and to plug the Kelly Criterion which the author mentions having written a former blog post on: Suppose instead of betting all your money every round, you instead decide to bet 10 cents each time. Now, instead of being essentially guaranteed long-term ruin, you can and most likely will be able to continue making money indefinitely. (In fact, your chance of ever dipping below, say, $0.50 is finite even when extending your rounds played arbitrarily.) The Kelly Criterion for this scenario actually dictates that you should bet 25% of your money each round. Using this betting strategy, somewhere around 70% of people end up making money off this game when run for 100 rounds (1% end up ending up with a respectable $25 or more, while about as many end up with <$0.15). You even have an opportunity for redemption -- when we drag out the horizon to 5000 rounds played, somewhere around 90% of individuals become billionaires, even as 30% of people were behind after 100 rounds (so 2/3 of those redeemed themselves). On the other hand, with the all-or-nothing solution outlined in the article, about 13% of the population coming out ahead (around 1% of the population gets really rich, ending up with >$200, while more than half end up with less than a penny). Meanwhile, the odds get worse as the game goes on, as at just 500 rounds, >80% of players have been reduced to less than a penny. That's a long-winded way of saying that the amount you bet is really important.

This example assumes you only have $1 to bet, and if you lose it, you’re out of the game. I wonder what happens to the simulated outcome if you can keep betting, even if you lose.

vitus 6y ago

Not quite. > Everyone starts with $1, gets 50% profit if they win, and pays 40% of their bet if they lose. From the wealth-over-time graph, it looks like the bet is sized such that it's always 100% of what you have (per some trajectories going as small as 10^-7). My read is that while each individual bet looks good in isolation, the fact is that one win and one loss puts you in the red overall -- when dealing with this iterated experiment you want to look not at E[X] = 0.05, but at E[log(X)] = -0.05 to get a sense for how your assets evolve each round. For some intuition: if you win once and lose once, your net result is 1.5 * 0.6 = 0.9, so you've lost 10% of your starting money.

vitus 6y ago
To expand on this further, and to plug the Kelly Criterion which the author mentions having written a former blog post on: Suppose instead of betting all your money every round, you instead decide to bet 10 cents each time. Now, instead of being essentially guaranteed long-term ruin, you can and most likely will be able to continue making money indefinitely. (In fact, your chance of ever dipping below, say, $0.50 is finite even when extending your rounds played arbitrarily.) The Kelly Criterion for this scenario actually dictates that you should bet 25% of your money each round. Using this betting strategy, somewhere around 70% of people end up making money off this game when run for 100 rounds (1% end up ending up with a respectable $25 or more, while about as many end up with <$0.15). You even have an opportunity for redemption -- when we drag out the horizon to 5000 rounds played, somewhere around 90% of individuals become billionaires, even as 30% of people were behind after 100 rounds (so 2/3 of those redeemed themselves). On the other hand, with the all-or-nothing solution outlined in the article, about 13% of the population coming out ahead (around 1% of the population gets really rich, ending up with >$200, while more than half end up with less than a penny). Meanwhile, the odds get worse as the game goes on, as at just 500 rounds, >80% of players have been reduced to less than a penny. That's a long-winded way of saying that the amount you bet is really important.