Prediction Markets and the Kelly Criterion, Part 3

Let me continue with my example from Part 2. Yes, this example is a toy. But I believe that studying simple cases can help to understand complex ones.

To recap, we have a game where you place a bet that you will win with probability \(p = \frac{2}{3}\) and that pays off 1:1. You have a $1000 bankroll to play this game once per day for two days. You may compound (roll) any win/loss from the first day into the second.

We compared three betting strategies:

• Strategy A (“Rock”): Go all in, always
• Strategy Z (“Paper”): Bet nothing, always
• Strategy ZA (“Scissors”): Bet nothing on the first day and go all in on the second

Changing terminology slightly, let’s say that one strategy “beats” another if it is more likely to leave you with more money in a head-to-head comparison.

We saw last time that — for this two-day game — Paper beats Rock, and Scissors beats Paper, and Rock beats Scissors.

Consider one more strategy:

• Strategy K: Bet \(\frac{1}{3}\) of your current bankroll, always

This is the Kelly bet for this game. The math is simple. When the payoff is 1:1, the Kelly formula reduces to \(p-q\). For this game, \(p = \frac{2}{3}\) and thus \(q = 1-p = \frac{1}{3}\), so Kelly says to bet \(\frac{2}{3}-\frac{1}{3} = \frac{1}{3}\) of your bankroll.

(Note: This hypothetical game has positive expectation; that is, the payoff is more than sufficient to compensate for your chance of losing. If you study any actual casino game and plug its numbers into the Kelly formula, you will get a negative answer, which is Kelly’s way of telling you to take the other side of the bet.)

You can check for yourself that strategy K beats A, and it beats Z, but it loses to ZA. The latter is easy to see since ZA leaves you with $2000 six times out of nine, while the best Kelly can do is win twice leaving you with \($1000 * \frac{16}{9} = $1777\). I suppose this makes it “Dynamite”, blowing up Rock and Paper while having Scissors cut its fuse. And we will pretend I designed the example this way on purpose.

Now wait a minute… Did we just beat the Kelly Criterion?

Yes. Yes, we did. For the two-day version of this game.

But look at Strategy ZA and tell me how to extend it to three days. Or 10 days, 1000 days, 1 million days… You will find it becomes harder and harder to develop any strategy to beat Kelly’s simple “always bet \(\frac{1}{3}\)”. This includes adaptive approaches that change strategy based on your win/loss record.

I want to mention again that, in all cases, Strategy A (good old Rock) still has the highest expectation value. For example, if you come back every year for 100 years and play the 10-day game with Strategy A, you will probably win the $1 million once or twice, which is enough to outrun Kelly’s expected ~$2900 per year. You will still go bust the other 98 years, of course.

And if we extend the game to 100 days, and you stick with Strategy A, you have to come back for something like \(10^{17}\) years for a decent shot at seeing your astronomical payoff and pulling ahead.

I believe I have now beaten this example into the ground, and I am debating what direction to head. Tune in next time to find out.