Kelly for the cowardly

Kelly staking is - as we've seen before - the mathematically optimal way to grow your bankroll. It has one glaring problem, though: it's horrifically volatile. Let's imagine we make 100 bets which we know are 50-50 shots but the bookies insist on pricing at 2.10. Our Kelly stake is (po-1)/(o-1) = 4.55%.

Now, when we win, we tend to win big - about a quarter of the time, we'd get 53 or more correct. That would net us at least a 49% profit. The flip-side of that is, a quarter of the time we get 46 or fewer and lose a quarter of our bankroll. There's a one in twenty chance that we'll lose half of our bankroll (although one in ten that we'll double it). With bigger edges or shorter odds, the fluctuations can be terrifying.

Is there a way to reduce them? Well, obviously, if you don't bet so much, your bankroll is steadier. But let's say you're still pretty greedy, and want to maximise your worst plausible outcome.

How do you even define that? Well, given that we're looking at a binomial distribution, we can use stats to help us. If we look at N identical bets with probability p, we know that 97.7% of the time* we'll win at least W_min = Np - 2 sqrt(Np(1-p)) of them. Bumping up the 2 to 3 gives us 99.87% confidence.

Whatever value we choose - I'm happy enough with two - outlines our worst plausible set of results over N trials**. We can then calculate our worst plausible outcome, which is B₀ (1 + k(o-1))^Wmin(1 - k)^(N-Wmin).

The trick now is to maximise this with respect to each k. It turns out, if we define p* as Wmin/N, that our optimal Kelly stake in this sense is (p*o-1)/(o-1). And if it's less than zero, we don't bet.

This is quite restrictive - in the case above, with N = 100 we simply wouldn't bet - p* is 40%, far too low to allow us to meet our minimum. N = 1000 isn't that much better - p* = 46.8%, where we need 47.6%. N = 2500 is just about enough.

Here are the results of running 2500 bets 1000 times over (using the two staking patterns on the same events):

           Pure Kelly     Modified
Stake        4.55%           0.73%
AROI***     26.07%           3.81%
SD          64.52%          23.58%
Worst      -79.85%         -11.44%

So, on average, Kelly outperforms the modified version by some way - but at the cost of much higher risk. The modified stakes 'guarantee' that the lowest plausible value is as large as possible.

It is possible to make up the discrepancy to a fair degree by increasing N, because the larger N is, the closer p* is to p (the square root term ends up getting very small).

Modified Kelly staking is worthwhile for bets with sufficiently large edges, or over sufficiently long runs. If you plan to make only 100 bets, you would need odds of at least 2.5 on a 50-50 shot before the modified stakes allowed you to bet.

I just typed bed, which is probably a Freudian slip. It's getting late.

* Look it up in a normal distribution table.

** We needn't assume the bets are identical. In general, we can replace Np with sum(p) and the bit inside the square root would be sum( p(1-p) ). But that complicates things a bit more than we need for the proof of concept.

*** Average Return on Investment

The Gambler's Fallacy and the Law of Large Numbers

In its simplest form, the gambler's fallacy is to believe that, just because something hasn't happened for a while, it's due to happen soon. The usual example is a coin that comes up heads ten times in a row. The fallacious gambler thinks it has to come up tails soon - "the law of averages says it must!" But of course it needn't. The chances of the next toss being a tail remain 50-50, as they ever were.

There are occasions in which the gambler's reasoning might not be fallacious - for instance, if you were drawing cards from a pack without shuffling, each non-ace that was drawn would increase the chances of the next card being an ace - but in almost all gambling scenarios (at least in the casino and possibly excluding blackjack), there is no memory involved.

It's possible that the fallacious gambler is misremembering the Law of Large Numbers. This is a much misunderstood mathematical theorem - I once spent an entertaining breakfast arguing with a friend about this, much to the bemusement of everyone else in the backwater Wyoming diner. The Law says that, if you look at a random event for enough trials, the number of successes divided by the number of trials will get as close as you like to the probability. If you toss a million coins, you should end up with an observed probability of almost exactly 0.50. Unfortunately, that doesn't quite mean that you can put the bank on there being exactly 500,000 heads, though. In fact, the Law of Large Numbers even says how far away you can expect to be - in this case, on average you'll be within 500* tosses of that. I just ran a quick test, emptying my piggybank and tossing 1,000,000 coins a hundred times over**. I found:

• the average set of tosses contained 499,978 heads
  
• the standard deviation was about 460 - pretty close to what the Law predicts.
  
• 97% of the observations were within 1000 of the real mean
  
• 71% were within 500.
  
• None were exactly 500,000 (or even within 10).
  

So the experiment conforms pretty much exactly with what the Law predicts.

* in general, the expected number of successes, each with probability p in N trials, is pN +/- sqrt(Np(p-1)). The square root is a standard deviation; 68% of the time, the answer will lie within one SD of the mean (pN); about 95% of the time, it'll be within two.

** My piggybank is entirely virtual. Code and results are available on request; if there's enough demand I might make a little javascript to demonstrate.