Challenge: Derive the Kelly criteria for play money

The Kelly criteria is a money management strategy for gamblers and investors. The strategy says that, when faced with a positive-expectation bet, you should invest a fraction of your budget that is proportional to your expected profit. The more your expect to gain, the more you should risk, but you never risk your entire budget.

The Kelly strategy is optimal in several senses: (1) it minimizes your “doubling time”, or the time it takes to go from having X dollars to having 2X dollars; (2) it minimizes the time it takes to achieve any given level of wealth; (3) it maximizes your long-run wealth.

(It turns out that the Kelly strategy is equivalent to maximizing a logarithmic utility function.)

A key reason the Kelly strategy is optimal is that it is very careful to never take you completely bankrupt: you spend only a fraction of your money, always reserving a bit for tomorrow, however small. This is sound advice when dealing with real money. (Aside: this all assumes you have a strict budget cap, which is not entirely realistic: you can almost always borrow at least some amount, even in today’s economy.)

But what about maximizing your virtual “wealth” inside a play-money game like NewsFutures, InklingMarkets, HubDub, or MediaPredict? The problem is not quite the same, precisely because you cannot really go bankrupt. Almost every game offers an option to “recharge” your account if you go bust. Even if the option is not explicit, you can always just abandon your account and start a new one with a fresh initial bankroll they typically give to new players.

So what is the Kelly criteria for play money? What is the optimal strategy that minimizes your doubling time when you’re always allowed to recharge back to a fixed starting value any time you go bankrupt? The answer is not obvious to me, so I’m crowdsourcing the problem: can readers derive the right rule?

My only conjecture is that it might become optimal to go “all in” on every single bet. But I’m not sure. [Update: I’ve convinced myself this is not optimal. Imagine two sequential bets, the first with minuscule expected profit and the second with huge expected profit: surely you should not go “all in” on the first.]

Note that finding the optimal solution may not just help you win more bragging rights in online games. There is a fascinating sports betting site called CentSports that gives everyone ten real cents to start with. If you can turn that ten cents into twenty dollars, they’ll cut you a check. Moreover, if you ever go to zero, they’ll restore you right back to ten cents. In other words, the system works just like play-money games except the potential for profit is real. So another way to phrase the challenge question is: what strategy in CentSports minimizes the time it takes you to go from ten cents to twenty dollars?

12 thoughts on “Challenge: Derive the Kelly criteria for play money”

  1. Interesting question. I think it would be difficult to answer for a pure play money site (such as the Inkling public markets). What is an ‘inkling’ worth? What does losing an ‘inkling’ really cost? Hard to say, yet a lot of people try to gain inklings and try to avoid losing them, so at least the sign on the value is the same as with real money.

    As you suggest, CentSports may be an interesting place to start looking for an answer, with real money at stake but in small enough amounts that you are much closer to play money values, whatever they are.

  2. You really have to take the time invested in the account so far before you can push in the direction of saying “you could just start over”. I’m in the

    top 50 players on the foresight exchange, and I’ve been playing on that site for more than 10 years.

    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.

  3. Thanks Michael. Maybe a clarification is in order. I didn’t mean to ask about how to maximize real value. I just meant, within the game itself, what is the optimal strategy to maximize your balance in play-money currency. In other words, maximize your Inklings, irrespective of what they are actually worth in the “real world”. I tweaked the post a little bit to try to clarify.

  4. Ah,,, thanks for the clarification.
    Even “play money” has some “real” value if not exactly monetary value. That “one thin dime” for free is worth far more than ten cents! Its similar to that Hollywood bar wherein any actor with a royalty check for 99 cents or less will get it tacked up on the wall and ten bucks will be taken off his bar tab.
    I would think that Play Money can make one engage in riskier behavior but will be watching this thread to see just how one should calculate conserving capital that is not real.

  5. Hrm, interesting… One thing: the Kelly criterion does actually risk your entire budget if your bet has 100% offs of succeeding (and non-zero payoff).

    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…

  6. Hi Dave –

    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.

  7. Thanks everyone. I believe a conjecture is emerging. aj does a good job of explaining it and Jason Ruspini sums it up nicely in an email to me:

    “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.)

  8. I don’t want to mess with all the theory, but flashing back to reality, centsports doesn’t allow you to bet fractional pennies, so Kelly can’t really be applied for some time, unless you happen to have a series of abnormally high edge bets. So, moving all in on our first few bets is surely the right play.

  9. Hey I just found your blog through Google and am interested in your discussion here. I am the founder of BettorFan.com, which is another free sports betting website, with more social networking features. Since we start you with $.25 and if you lose, we restart you with another $.25, the optimal bet would to go all in every bet.

  10. Daniel has convinced me that the conjecture to “go all in at first, then revert to Kelly” is incorrect. So the challenge is still open. Here is Daniel’s email:

    “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

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