Fold equity

When I was studying for my degree, part of my course involved taking classes in business management. Business is well-known for having its buzzwords – unnecessary, complex or roundabout ways of describing simple things. Well, it’s come to my attention that the same thing is happening in the poker world. We have ‘reverse implied odds’, ‘fundamental theorems’ and ‘semi-bluffs’.

One of the most prolific buzz terms of today is ‘fold equity’. But what is it exactly? Is it a useful concept, or is it a waste of time? If it’s so important, how come Chris ‘Jesus’ Ferguson, widely regarded as one of the best game theory experts, doesn’t know what it is? He was asked by viewers of the WSOP live final table in 2006, and replied that he had ‘never heard of it’.

What is fold equity?

Nobody really seems to know who came up with the term ‘fold equity’ (it’s not actually described in any poker book I’ve ever read – and I’ve read a lot), but the intelligent gambler would bet that it was somebody on the TwoPlusTwo internet forums. Those guys love to quantify things; to give trivial issues important-sounding names, and discuss poker minutiae into the ground.

The concept of fold equity is nothing new. Essentially, it’s the money you will win in the long-term when your opponent folds. Let’s start with a simple example: imagine you have 6♠-6♣ and your opponent holds J♠-10♦. It doesn’t get much closer to a coin flip than this; if you were to simply deal out five cards, your pot equity would be 50.2%. In reality, if you are the player making the aggressive move, you’ll usually win more often. If you raise all-in, for example, the player with J♠-10♦ isn’t going to call you all of the time. If he only calls half of the time, your equity leaps up to 75.1%. Here’s how you work it out:

1 – (chance opponent calls x opponent’s equity, if they do call) = our overall equity
1 – (0.5 x 0.498) = 0.751
0.751 x 100 = 75.1%

If your opponent calls less often, your equity increases further still. This extra equity (in this case 24.9%) is your fold equity. You can work out exactly what it is in monetary terms by simply comparing this with the size of the pot. For example, if the pot is £100, we have made a theoretical £24.90. One of the strengths of fold equity as an idea is that you can quantify exactly how much more money we have made by betting, compared to checking. In reality, it’s a little more complicated because we haven’t yet considered that we are risking money to gain the extra equity. The key to successfully using the concept is balancing the amount of money you risk to the amount of equity you gain by forcing a fold.

As a more complicated example, we can find out exactly how often we need our opponent to fold in order to break even. Let’s say it’s the last round of betting, and we are considering raising all-in on a bluff. The pot is £100, and our opponent bets £50. You can raise all-in for £250 more, and if your opponent calls, he will always have you beat. If he folds, you win the £150 pot. If he calls, you lose £250. Essentially, you’re laying £250 to win £150, at odds of 5/3. In order to break even, your opponent must fold five times for every three he calls – or 62.5% of the time. You can confirm this figure is correct by performing a quick calculation. The result should be an expected value (EV) of zero (the same EV as folding):

(chance opponent folds x pot size) – (chance opponent calls x loss when he does) = 0 (0.625 x 150) – (0.375 x 250) = 93.75 – 93.75 = 0

As you might expect, the bigger our raise in relation to the pot, the bigger our fold equity needs to be and the more often our opponent needs to fold. For example, if we raise £1,000 into the £150 pot, we’re laying odds of 1,000 to 150, so our opponent needs to fold a whopping 87% of the time for us to break even.

You can experiment with different pot and bet sizes to find a bet that offers the most fold equity relative to its size (although remember that in general, players will fold to a small bet less often than a big one). Unsurprisingly, you’ll find that bets between half the pot and the full pot are generally the most ‘efficient’ in terms of fold equity, and bets that are extremely large or small are either too risky or too ineffective.

The pitfalls of fold equity

My old karate instructor used to say that the most dangerous rank to be was blue belt. That’s because as a blue belt, you know just enough to get yourself into trouble, but not enough to get yourself out of it.

The same idea applies in poker. If you’re an intermediate player, you can get yourself into trouble by misapplying a concept you don’t fully understand. One of the most common ways I see this arise is when a player uses the concept of fold equity to justify a reckless raise. For example, let’s say that Phil wants to attempt to steal a pot in an online single table tournament. He has a chip stack of 1,500, the blinds are 10/20, the under the gun player raises to 100, and Phil decides to move all-in from the big blind holding J♦-10♦. (If you think this is far fetched just watch a couple of $5 sit&gos.)

Now, there are more problems with this play than I care to mention. But often, you’ll see people try to justify a play like this using fold equity. Phil will do the maths, figure out that his opponent needs to fold x amount of the time, and then decide that his opponent was the type of person who would fold this often – usually ignoring evidence to the contrary – and conclude that he made a good play, only to get unlucky when his opponent flipped over Aces and won.

Mike Caro wrote about this natural optimism years ago, and advised that you should make your estimates before you decide to do the maths. If you do the maths first, there’s a tendency to twist your estimate to suit the figures you calculate.

First things first

If Phil had done things the right way round, he might have decided there was a 50% chance that his opponent would fold. Then, when he did the maths and learned that his opponent needed to fold much more often to make the play profitable, it would have been difficult for him to conclude that he made the right play.

Another problem with fold equity as an idea is that it forces you to make a lot of assumptions; consequently it isn’t of much use in the heat of the moment. When you’re sat at the table, for example, how can you reasonably figure out the exact chance that your opponent will fold, or do the calculations I have just shown you in 20 seconds or less?

Nonetheless, fold equity is a handy way of putting an otherwise difficult-to-explain concept into words, and it can be a useful tool in the postmortem process of analysing hands and improving your game. But before you apply fold equity (or any poker concept), make sure you’re not twisting the figures to suit your own ends. If you do, you’ll simply reinforce the errors you’re trying to eliminate.