The Tipping Point

One of the first pieces of poker advice I received was this. “Mix up your play. Never do the same thing every time.”

It’s a pretty decent piece of advice to get for your first one. It conveys a key pillar of poker strategy—you have to keep your opponents guessing.

But today, I find that advice a little ironic. When you analyze the math of the game, you could easily draw precisely the opposite conclusion—the best way to play often is to do the exact same thing every time. Here’s what I mean.

The math of poker is centered on finding so-called equilibrium strategies. An equilibrium strategy is actually a pair of strategies, one for each player, that balance one another. Neither player can alter their own strategy and improve their performance. Any change, necessarily, makes things worse. Here is a simple example:

It’s the river. The first player checks with a bluff-catching hand. The second player, based on the play of the hand, has either a very strong hand (that beats the bluff-catcher) or air. Furthermore, there is $100 in the pot and $100 remaining in the stacks. The second player, for simplicity, decides that if he’s going to bet, he’ll go all-in for the full $100. What is the equilibrium strategy?

The second player is betting $100 to win $100. When he’s bluffing, the math is simple. If it works, he wins $100. If it doesn’t, he loses $100. This bluff bet breaks-even when he gets called 50 percent of the time. Therefore, the first player should call with half of his hands. This makes the second player indifferent to bluffing or not bluffing.

Now, turn it around. The first player gets $200-to-$100 pot odds on his call. This breaks-even when he wins one-third of the time after calling. This means that the second player should bluff with half as many hands as he value bets.

This is the equilibrium strategy. Say the second player has 20 possible value hands. He should bet those, and an additional 10 bluffing hands. This 20-to-10 ratio makes the first player indifferent to calling.

Against a bet, the first player should call half the time. This makes the second player indifferent to bluffing.

Simple enough. But what happens when one player decides not to play an equilibrium strategy? Let’s say that the second player doesn’t have his ratios quite right. He knows he’s supposed to bet the 20 value hands, but instead of 10 bluffing hands, he chooses only nine. It’s a small change, so presumably it shouldn’t change the first player’s strategy very much either.

Since the second player is now bluffing slightly less often, the first player should call less than the 50 percent of the time before. But how much less? Is it 45 percent now? Or 40 percent?

The answer might surprise you. It’s zero percent. When the second player removes just a single bluffing hand, the first player goes from calling 50 percent to never calling. Why?

To understand, we have to do a few expected value (EV) calculations. First, here are the EVs of calling versus folding at equilibrium (10 bluffing hands).

EV_call = (20/30) (-$100) + (10/30) (+$200) = $0
EV_fold = $0

Since they are the same, the player can choose either a call or fold without giving up anything. Now, here are the EVs of calling versus folding against the off-equilibrium strategy.

EV_call = (20/29) (-$100) + (9/29) (+$200) = -$6.90
EV_fold = $0

The EVs are different now, with the EV of a call being negative, and therefore, worse than a fold. This means you should always fold.

It’s a tipping point. As long as both players play exactly at equilibrium, they mix their strategies. If one player strays even a tiny bit off equilibrium, however, the other player’s best strategy is to collapse his strategy one way or the other, making the same play 100 percent of the time.

You’ll find the same sort of result no matter which way you perturb the equilibrium strategy. When player two bluffs with 11 hands, now player one should call 100 percent of the time. When player one calls only 48 percent of the time instead of 50 percent, player two should now bluff with all bluffing hands, and so forth.

Moreover, this is a fundamental property of poker strategies. It tends to work like this. If your opponent is off even a tiny bit from an equilibrium strategy, you should abandon a mixed strategy and go completely one way or the other.

How many people play the perfect equilibrium strategy at all points in a hand? (Answer: Zero)

“Ok,” you say, “Fine, that’s the math. But that’s not how it works in reality. If you do the same thing every time, they’ll adjust, and you’ll get killed.”

Sure, maybe. Some people adjust, sometimes. But I’ve been in plenty of games where one guy was just bluffing hand after hand after hand, and he got away with it for hours. Sure, he got caught sometimes, but people were still folding too much (compared to equilibrium frequencies), so he was right to keep on bluffing.

And here’s the thing. If you aren’t completely obvious about it—if you disguise it just a little bit—often you can get away with an exploitative counter-strategy like this for months and years.

The first step to getting away with it is to play tight preflop. If you aren’t in every other hand, it’s much harder for your opponents to figure out what you’re doing. Beyond that, sure, you can give them one every once in a while.

But the most important point, to me at least, is that “mixing it up” is not a virtue. It’s not the goal. It’s not what the good players do. The good players systematically identify where their opponents are playing off-equilibrium strategies, and they play the counter-strategy as often as possible.

Here’s an example. Say I believe that you don’t bluff frequently enough when you bet $200 on the river. I will never call you with a bluff-catcher. Ever. I’m not going to call you 40 percent of the time or 25 percent of the time. I will never call until I believe that your strategy has changed.

Say I think you play too many hands to the turn, and that you fold too much to turn bluffs. I will bluff every turn against you. Every single one. Sometimes you will see me check a turn, but it will be because I have showdown value. I will have a hand I don’t consider a bluff. You won’t ever see me check down a bluff against you, because to do so would be to throw money away.

At equilibrium, poker is a game where “mixing it up” is at the very heart of good play. But you have to mix it just right. If you get it even slightly wrong, your opponent should not mix at all against you. I hope this idea gives you some food for thought. ♠

Ed’s newest book, Poker’s 1% The One Big Secret That Keeps Elite Players On Top is available now at his website edmillerpoker.com. You can also find original articles and instructional videos by Ed at the brand new site redchippoker.com.