“First let’s change centsports to dollarsports so we can bet pennies instead of fractional pennies (still not perfect but it’s good enough for my purposes here)

“All the rules are the same. You start with a dollar, and if you lose you can reload to a dollar again, no penalty, as many times as you want.

“Now let’s simplify the game, or look at a subgame to prove that kelly is not correct in any of the early bets. You (or Ruspini, whom I also greatly respect) propose that after the first all in you revert to kelly (as I understand) Now the simplified game:

“Let’s make the game to 8 dollars (instead of 100)
and
Let’s look at a series of 50 bets (This is for simplification, neither rule should affect the strategy)

“Strategy a) I go all in every bet. I only need to get 3 consecutive bets right out of 50 to get to 8 dollars. This is rather likely to occur.

“Strategy b) You go all in the first bet (I’ll give you this bet) and you get to 2 dollars and then revert to Kelly. Now Kelly will only let you bet 1% of your bankroll given the edge (51/52% whatever) so after you double, you bet 2 cents 2 cents 2 cents 2.1 cents eventually 3 cents. But you cannot win. Even if you get every single bet right out of 50 (which is astronomically unlikely) you will not ever reach 8 dollars in a series of 50 by following kelly.

“So again, in a simplified game involving one double (i.e. race to 2 dollars) you agree it’s right to go all in, I am proposing that it’s *clearly* right to do it the second time and the third time as well.

“And you can change to a series of 100 bets, or a race to 16 dollars, (4 in a row for me going all in) and you see the all in strategy is still better.

“I won’t go past here but I think you can see how kelly is not appropriate in the beginning stages, and possibly not at all, I admit I don’t know.

“Yes, Kelly optimizes bankroll growth, but this is *largely* because it prevents you from going broke. But there is no penalty for going broke in this game. Furthermore, losses are not cumulative in the same way as they would be in the real world. i.e. if we go broke a million times we are not at negative one million dollars, we are at zero.

“Am I wrong? Did I miss something? Would you care to reconsider this problem?”

Daniel

I like the social features. Good luck.

“take the regular kelly derivation and replace the losing trade term with min(account reset value, losing trade term), then re-maximize the whole thing.”

Basically, you use the regular Kelly strategy with one exception: if Kelly outputs a bet that might take you below the reset value, then go “all in” instead. But if Kelly does not put you in jeopardy of going below the reset value, then just stick with exactly what Kelly says.

Anyone care to prove it formally?

aj’s idea of opening 2^N accounts and Jason’s comments in his email to me that “the account values of the leader board are also important” inspire a further question:

What is the optimal strategy when managing multiple accounts in order to maximize the wealth of the largest account? I’m guessing that any improvement over Kelly will be at the beginning, to give yourself a faster running start, but in the limit, you’ll want to converge back to using Kelly strategy on whatever account is the largest. (Note I’m ignoring the option of “cheating” by transferring money from one account to another.)

The only thing I am sure of, is you *should* go “All-In” on the first bet. If we shrink the time to reload close to zero as well as the time to place a bet close to zero, we are effectively starting with double the bankroll. How is this not preferable? The question for me is when do you *stop* going “all in” and revert back to regular Kelly. In fact, if we shrink all transaction times to zero, then we can perform infinite number of bets, and should move “all in” on every hand.

Regarding Centsports, Since all bets are roughly 50/50 I am pretty sure going all in every bet is correct. I don’t think you will be able to beat this with any kind of modified Kelly.

With the play money prediction exchanges, there are many bets that are significantly better than 50/50. Still, I think the best strategy is to go all in for some set time, and then revert to standard Kelly based on your ultimate goals. Because the edges are bigger with play money exchanges we don’t want to risk our entire bankroll when we expect to double rather fast anyways.

One case when you can obviously improve on the Kelly criterion in the above circumstance: if Kelly says bet f% of your cash, c, then if (100%-f%)*c <= r, betting 100% of your cash is more optimal (you’ll win slightly more if you win, and by restarting with $r, you’ll be no worse off if you lose).

I can’t see any reason why the Kelly criterion would otherwise differ with play money though — it explicitly ignores any fundamental value to how much money you have and just aims to maximise the expected amount you have; and equally, in real life you can declare bankruptcy and sit on the street with a tin can until someone gives you ten cents.

Going all in everytime increases the maximum possible amount you can end up with, but reduces the liklihood of that to nothing. In a play money scenario, you could make use of that by creating 2^N accounts, and having them bet in a combinatorical way, so that you’re guaranteed exactly one of them will have placed the optimal series of bets. If you then pretend all the other accounts never existed, that could work…

I do play a little looser on that site than I do in my investment accounts, but there’s good reason to keep my bets spread around. OTOH, there have been people (outside the top fifty) who did risk everything on one bet, and ask to have their accounts reset to zero. But for the most part, they were better off before they made their big bet than they were a year later, for having started over.

I suspect there are many cases where it pays to play a slow and steady game. If there’s a prize for being in the top 100, and you aren’t close, then maybe you want to play a higher variance game, but otherwise, I’m not sure when you’d want to do that.