## A Mathematical Example of Why “Small Ball” Tournament Play Works

### Mathematics, Tournaments

May 5, 2008

For those not familiar with the terms, “small ball” tournament play is a style of play in NL tournaments (usually holdem) where you avoid large confrontations unless you believe you have a huge edge – a small positive expectation in tournament chips isn’t enough to justify going all in against someone.

For some reason, small ball is a concept that a lot of people have a hard time grasping, or believing is correct. They believe any edge, however small, should be taken. In cash games, assuming you have plenty of reloads in your pocket, this is true. But in certain circumstances in tournaments, it’s provable that you should pass on a small edge. The example below will illustrate this:

Hero is a very skilled tournament player that has a 4x overlay in big 10k live MTTs ie. he expects to claim on average 50k of prizes. Let’s imagine our tournament, for convenience, has 1024 entrants and is winner take all for just over 10 million. Now, suppose our hero has the option of playing the tournament in the normal way, or instead taking a series of coin flips for his entire stack until he has all the chips or busts out. Fortunately for Hero, he has a weighted coin, so he wins 60% of his flips. Is he better off flipping, or playing? What if the coin is less weighted, and he only has a 55% chance of winning his flips?

The answer is surprising. Either way he’ll have to make 10 flips to get all the money if he goes the flip route. With the better coin, his probability of winning is 0.6^10=.0060 which gives him an equity in the tournament of 60k – that’s 10k better than he would have gotten by playing. But with the weaker coin, his chance of surviving the flips is .55^10=.00253 for an equity of just over half what he would expect if he just played the tournament.

Moral of the story: If you’re a skilled player with a big overlay, and are offered a flip for your entire stack, and that flip is weighted only slightly in your favor, it may well be correct to pass. The idea that a winning player should never pass up an opportunity to move in with a small edge is simply wrong. You want either bigger edges, or bets that are a smaller portion of your stack. Furthermore, there’s a key point as the odds of winning your all-in confrontation pass about 60% or so. Much past that, and no one has a big enough overlay to justify not taking the flips.