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Fold Equity - Buzzword or Critical Concept?
When I was studying for my degree (before I became a degenerate gambler and spent all my waking hours playing or working in poker), part of my course involved taking classes in Management and Business. Business is well known for having its buzzwords – unnecessary complex or roundabout ways of describing simple things. One of the most familiar to us all is ‘Downsizing’ – a nice, polite way of saying ‘firing people’!
Well, it’s come to my attention that the same thing is happening to poker. We have ‘reverse implied odds’, ‘fundamental theorems’ and ‘semi-bluffs’. One of the most prolific buzz terms of today is ‘Fold Equity’. So what exactly is Fold Equity? 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’ in the business, 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 Two Plus Two internet forums. Those guys really love to quantify things, to give trivial issues important-sounding names, and discuss poker minutiae into the ground.
The concept of fold equity, or FE for short, 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 we have two black sixes, and our opponent holds J♠ 10♦. It doesn’t get much closer to a coin flip than this – our pot equity if we were to simply deal out five cards is 50.2%. In reality, if we are the player making the aggressive move, we’ll usually win more often. If we raise all-in for example, the player with J♠ 10♦ isn’t going to call us all the time. If he only calls half of the time, our equity leaps up to 75.1%:
1 - (Chance opponent calls * Opponent’s equity if they do call) = Our overall equity
1 – (0.5 * 0.498) = 0.751 = 75.1%
If he calls less often, our equity increases further still. This extra equity, in this case 24.9%, is our fold equity. We 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 we 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 on a play. As an example, 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. We can raise all-in for £250 more, and if our opponent calls he will always have us beat.
If he folds, we win the £150 pot. If he calls however, we lose £250. Essentially we’re laying 250 to win 150, or odds of 5 to 3. In order to break even, our opponent must fold five times for every three he calls – or 62.5% of the time.
We can confirm that this figure is correct by performing a quick calculation. The result should be an EV of zero (the same EV as folding):
(Chance opponent folds * Pot size) - (Chance opponent calls * Loss when he does) = 0
(0.625 * 150) - (0.375 * 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 £1000 into the £150 pot, we’re laying odds of 1000 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 it’s 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 that 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, and not enough to get yourself out of it.
The same idea applies to poker. If you’re an intermediate player, you can get yourself into trouble by misapplying a concept that 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 1500 stack, the blinds are 10 and 20, the under the gun player raises to 100, and he moves all-in from the big blind holding J♦ 10♦ (think this is far fetched? 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 and got 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 do the maths. If you do the maths first, there’s a tendency to twist your estimate to suit the figures you calculate. If he’d done things the right way round, Phil 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 and guesses, and 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?
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 post-mortem 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.
A Real Life Bluff in FE Terms
This hand is taken from the 2006 World Series of Poker Tournament of Champions final table.
It’s three handed with blinds of 5,000 / 10,000, and everybody has plenty of chips. Mike Matusow limps in first position with K♥ 2♥. Daniel Negreanu completes from the small blind with 8♦ 7♥. Mike Sexton checks the big blind with 7♦ 3♠.
The pot is 30,000 and the flop comes Q♠ 9♥ 3♥. Negreanu checks, Sexton checks, and Matusow checks.
The turn is the A♦. Negreanu checks, Sexton checks, and Matusow bets 20,000. Negreanu folds and Sexton calls.
The pot is 70,000 and the river brings the J♦. Sexton checks and Matusow bets 60,000.
This bet is a pure bluff, and an excellent one at that. Matusow bets 60,000 to win 70,000, meaning that Sexton only needs to fold about 46% of the time for him to make a profit on the play (we can safely assume that no hand worse than Matusow’s will call).
Given Sexton’s actual holding, it’s very likely that he’ll fold often enough to make this a profitable bet for Matusow.
This is an excellent bet by Matusow which is almost certainly profitable in terms of fold equity, but unfortunately it goes awry. Sexton actually calls and wins – one of the best calls of the entire 2006 World Series. Sometimes you can make the right move and still lose out in the short term!
