## Friday, March 01, 2013

### Equity Squared

Until recently, there have been two flavors of hand rankings provided by ProPokerTools. The first kind is created using a  program (described in previous posts) to 'evolve' an ordering over time. The second kind is simply all-in equity versus a random hand. Equity-squared is a third way to order hands, one that has been used extensively by the poker artificial intelligence community.

The easiest way to see how equity-squared works is to compare it to all-in equity.  One way to compute all-in equity vs. a random hand is to generate all possible boards, count our wins and ties vs. every possible opposing hand, and divide by the total number of matchups. This gives us a number between 0 and 1, where 0 always loses and 1 always wins. For example, in the case of AsTd, the answer turns out to be 0.62722, which means it has 62.722% all-in equity.

When we compute equity-squared, we again generate all possible boards. For each board, we compute our equity vs. every possible opposing hand. Then, we square the result (hence the name 'equity squared'). Our total equity squared is the average of all the equity-squared values for every board.

Why is this interesting, you may ask? The answer has to do with the power of strong draws. Imagine a theoretical hand that will make the nuts 50% of the time but make essentially nothing the other 50% of the time. Now imagine a second hand that always ends with exactly average  equity. It should be clear that while both hands have the same average all-in equity, they have a very different quality; the first hand is a very strong draw, while the second is an average value hand.

The effect of using equity-squared is to increase the value of hands that can make strong draws. For a hand like JsTs, it has a good all-in equity rank, placing 24th out of 100. But using equity squared, its place moves up to 15. Conversely, a hand like As8h has a significantly better all-in equity rank, scoring a 16. But it doesn't make a very strong hand nearly as often as the JsTs, so its equity-squared rank is only 19.

You can view the complete equity-squared ranking for hold'em here: http://propokertools.com/orderings/heequitysquaredordering.txt