You start with with 10 bucks, I start with 10 bucks. You wager any amount up to a hundred times, each time doubling it 60% of the time and losing it 40% of the time, until one of us is bankrupt or you stop. If you wager it all, I have a 40% chance to win. If you wager one buck at a time, you win almost certainly.
If you wager one buck at a time, you win almost certainly.
But that isn’t the Kelly criterion! Kelly would say I should open by betting two bucks.
In games of that form, it seems like you should be more-and-more careful as the amount of bets gets larger. The optimal strategy doesn’t tend to Kelly in the limit.
EDIT: In fact my best opening bet is $0.64, leading to expected winnings of $19.561.
EDIT2: I reran my program with higher precision, and got the answer $0.58 instead. This concerned me so I reran again with infinite precision (rational numbers) and got that the best bet is $0.21. The expected utilities were very similar in each case, which explains the precision problems.
EDIT3: If you always use Kelly, the expected utility is only $18.866.
I believe you missed one of the rules of Gurkenglas’ game, which was that there are at most 100 rounds. (Although it’s possible I misunderstood what they were trying to say.)
If you assume that play continues until one of the players is bankrupt then in fact there are lots of winning strategies. In particular betting any constant proportion less than 38.9%. The Kelly criterion isn’t unique among them.
My program doesn’t assume anything about the strategy. It just works backwards from the last round and calculates the optimal bet and expected value for each possible amount of money you could have, on the basis of the expected values in the next round which it has already calculated. (Assuming each bet is a whole number of cents.)
You start with with 10 bucks, I start with 10 bucks. You wager any amount up to a hundred times, each time doubling it 60% of the time and losing it 40% of the time, until one of us is bankrupt or you stop. If you wager it all, I have a 40% chance to win. If you wager one buck at a time, you win almost certainly.
But that isn’t the Kelly criterion! Kelly would say I should open by betting two bucks.
In games of that form, it seems like you should be more-and-more careful as the amount of bets gets larger. The optimal strategy doesn’t tend to Kelly in the limit.
EDIT: In fact my best opening bet is $0.64, leading to expected winnings of $19.561.
EDIT2: I reran my program with higher precision, and got the answer $0.58 instead. This concerned me so I reran again with infinite precision (rational numbers) and got that the best bet is $0.21. The expected utilities were very similar in each case, which explains the precision problems.
EDIT3: If you always use Kelly, the expected utility is only $18.866.
Does your program assume that the Kelly bet stays a fixed size, rather than changing?
Here’s a program you can paste in your browser that finds the expected value from following Kelly in Gurkenglas’ game (it finds EV to be 20)
https://pastebin.com/iTDK7jX6
(You can also fiddle with the first argument to
experimentto see some of the effects when 4 doesn’t hold)I believe you missed one of the rules of Gurkenglas’ game, which was that there are at most 100 rounds. (Although it’s possible I misunderstood what they were trying to say.)
If you assume that play continues until one of the players is bankrupt then in fact there are lots of winning strategies. In particular betting any constant proportion less than 38.9%. The Kelly criterion isn’t unique among them.
My program doesn’t assume anything about the strategy. It just works backwards from the last round and calculates the optimal bet and expected value for each possible amount of money you could have, on the basis of the expected values in the next round which it has already calculated. (Assuming each bet is a whole number of cents.)
I did indeed! So I guess this game fails (5) out of Zvi’s criteria.