Sample sizes and wager sizes

The most difficult task Brady and I face is not figuring out what to bet but figuring out how much to bet.  While the question of “how much to wager” was definitively settled in 1956 by J. L. Kelly Jr, the so-called “Kelly criterion” suffers from drawbacks that make it difficult to implement in practice.  In our case, the most painful of these is that the edge of a wager is very rarely known with any certainty in advance.

Assume for a moment that our spread wagers have 0% edge; our wagers are effectively a coin toss.  Even if we made 100 wagers per year (we don’t) our sample size is so small that we would have an extremely distorted impression of our edge.  Here I show multiple seasons worth of 100 such coin flips.

After season 1, we leave with the impression that we’re pretty unprofitable, losing 55% of our wagers:

We don’t feel any better about ourselves after two seasons:

And despite a nice run in season three, we still think we suck.

To really drive this point home, here’s what we think after a decade:

This simple demo illustrates two points 1) making decisions based on betting records is for fish, and 2) the wager sizing decision is damn near impossible because we can’t compute our edge with any reasonable confidence.

One thing we can do is take Kelly’s concept and apply it to the real world.  Our wager size should scale directly with the strength of the wager, and inversely with the odds.  My original thought process was to scale wagers exponentially with wager strength, but this would result in a handful of wagers totally dominating our performance.  A lucky (or unlucky) run from one team coupled with misappropriation in our power rankings could then result in a very large, very bad wager. 

At the same time, increasing wagers by only 1 unit at key thresholds seems too moderate.  High strength wagers are relatively uncommon, and if we have arrived at them “correctly,” that is, through a well-reasoned process, we should give those wagers more credit than just another unit.

As a reminder, “strength” refers to how many points above or below the consensus Las Vegas spread our model feels the wager should be in order to be “correctly” priced.

I’ve thus settled on the following figures for our wager sizing:

2.0 – 2.9 strength: 1 unit

3.0 – 3.9 strength: 1 unit

4.0 – 4.9 strength: 2 units

5.0 – 5.9 strength: 3 units

6.0 – 6.9 strength: 5 units

Wagers of 7+ points in strength would be exceedingly rare, so I will leave any such instances to our best judgement should the situation arise.  Most likely such a strength would come only from external circumstances/ events, for ex, injuries, sickness, coaching changes, etc. etc.

Starting with wagers at 2pts of strength should have us betting approximately 1/3 of games throughout a season.

The observant among you will notice that this sizing progression is a Fibonacci sequence, and while it holds many uncanny properties, I choose it specifically because the size of each successive wager equals the combined value of the two wagers before it.

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