## Small Edges and Win Rate

### Economics, Gambling (general)

January 19, 2009

I would venture a guess that even most winning poker players don’t put enough thought into the ultimate effect of their win rate on their gambling career.  Some work I did on sports betting recently went a long ways towards clarifying this for me personally.  I realize this isn’t a sports betting blog, but I want you to follow along anyways because I promise this will get back to poker eventually.

The issue at hand was a comparison of two methods of handicapping professional football games against the spread.  For those not familiar with sport betting, the spread adds points to one side’s results and thereby “evens out” the contest so that in theory either side is equally likely to win.  You can then wager 11 betting units to win 10 (i.e. you give the book 11 when you place the bet and get 21 back if you win) on either side.  Of course, the spread isn’t always exactly right. so it may not really be 50:50 as to which side wins, and that’s where there’s money to be made betting sports.

In any case, one method of selecting picks selected a subset of the games where it felt the line was wrong and accurately predicted the outcome against the spread 55% of the time on those games.  The other method worked similarly, but was more like 57% accurate.  My immediate thought was that there wasn’t much practical difference between the two – 55% looks a lot like 57%.  But one of my gambling friends who I was discussing this with said, “no, do the math – that 2% makes all the difference in the world”.  The process of doing the math turned out to be highly enlightening.

The first step is to figure your raw edge in each case:

Edge = money won divided by money wagered

55% case:

Edge = ((.55 * 10) – (.45 * 11)) / 11

Edge = .05 or 5%

57% case:

Edge = ((.57 * 10) – (.43 * 11)) / 11

Edge = .088 or 8.8%

Now already, you should see that something is afoot.  A 2% difference in the raw win rate on our wagers results in a near-doubling of our edge.  I certainly wasn’t expecting that.  But it turns out it gets far more interesting.  If you’ll recall the article on the Kelly Criterion, the amount of money you should be willing to wager on a given gamble is a function of the win rate and odds.  So how much should of our roll should we wager on  each method?

Fraction of roll to wager = F = (RP-Q)/R

where R is the amount of our wager we get back if we win (the odds), P is the probability we win, and Q = 1 – P

55% case:

F = ((0.909 * .55) – .45) / 0.909

F = 0.055 = 5.5%

57% case:

F = ((0.909 * .57) – .43) / 0.909

F = 0.097 = 9.7%

Now, you’ll notice that having 57% accurate picking method, we should wager almost twice as much of our bankroll vs. the 55% method.  Also, as an aside, notice that the Kelly amount to wager is very similar to our edge.  They would be exactly the same if the wager were even money instead of 11 to win 10.

Now the next question is how much should we expect to win per bet.  For simplicity’s sake I’ll put it in units of a percentage of our bankroll.

Expected win = money wagered * edge

55% case:

Expected win = .055 * .05 = .00275 of our bankroll per bet

Expected win = 0.275% of our bankroll per bet

57% case:

Expected win = .097 * .088 = .00854 of our bankroll per bet

Expected win = 0.854% of our bankroll per bet

Now things have gotten even more interesting.  Our 2% increase in win rate has more than tripled the size of our expected win on each bet.  That small difference at the start is looking like a pretty big difference now.  However, it turns out to be even more dramatic than these numbers indicate.  The whole purpose of Kelly bet sizing is that you intend to make proportional wagers in the future.  So when you make more money, on average, on the first wager, you have more money to wager on the second wager and so on.  It works in a way very similar to compounding interest.

Each method selects about 4 games per week to bet on, and there are 16 weeks of the pro football season of which the first 3 were used to gather data without placing bets.  That means that there were roughly 52 betable games per season with each method.  That’s a lot of opportunities to compound your interest.  So the obvious next question is what a “simulated” season of 52 bets with compounding bet size would look like.  The math for this is a bit of a pain in the rear, involving lots of logarithms, so I’m not going to bother writing it up.  But at the end of the simulated season the 55% method will have increased our bankroll by about 15%.  That’s only a tiny bit more than we would have gotten if we hadn’t compounded our bets (it would have been about 14% without compounding).  The 57% picks, however, increase our bankroll by about 55% over a season.  That’s quite a big difference – more than the 3:1 difference in expected wins.  This is the compounding effect at work, and the more time (games) it’s given to work, the bigger the effect is.  If you were to simulate 5 seasons (260 games) the better 57% picks ramp the bankroll up to over 9x it’s starting value in 5 years.  The 55% picks just manage to double it.  Now the ratio of money made between the two methods over that 5 year period is more like 8:1.

What’s the point of all this?  What does it have to do with poker?  What I’m trying to illustrate here is that the small stuff matters.  It matters a lot.  In this case a 2% difference in the quality of some football picks results in a 8x increase in the money won betting sports over 5 years.  That could easily be the difference between a wildly successful gambler and one who can barely make his rent.

The same sort of situations arise in poker, but they’re slightly more disguised.  There are plenty of situations where betting your hand right, or applying some extra little piece of poker knowledge, or reading your opponent right, can increase your edge by a few percent.  It might not seem like these things matter that much, but over time they’re the only thing that matters.  The reason is that poker compounds just like the sports betting example above.  If you do a little better, you have a little more money.  Which means that you move up to bigger stakes sooner, and if you keep doing this well you get further and further ahead of the curve.  Meanwhile other players who are playing just a few % worse can’t get any traction and end up stuck struggling to meet expenses.  One guy ends up being a superstar with a million dollar bankroll and the other guy’s still playing the same 20/40 game he played in 1982.  And the difference between the two was just a couple of percent.

Food for though.