Saturday, March 10, 2012

Folding Quad Aces

I started thinking about doing a post on stack-to-pot ratios (SPR), but I quickly realized that before I could do that, I needed to step back and think about the whole notion of commitment in general. Why? Because the main purpose of SPR is to determine (and in fact plan for) commitment in a hand. But this begs the question of what we mean by being committed in the first place. And the place to start this discussion is with a look at "pot commitment."

Pot commitment is a term you often hear bandied about at poker tables. Things like “I was pot committed, so I had to call” or “I made a crying call on the river” or “I knew I was beat, but the pot was so big I had to stay in” are often said by players who don’t fully understand what it means to be committed to the pot. They vaguely understand that as the pot size grows in the middle of the table, they become mathematically more and more compelled to continue in the hand, but they don’t understand when they’ve crossed a threshold and actual commitment takes place. Or why. Too often, players simply use the notion of pot commitment as an excuse to make reckless calls. They throw good money after bad because of a vague idea that the pot is so big that they're not supposed to give up on it. Here’s an extreme example to illustrate the point.

Let’s pretend that you sit down in a $0.50/1.0 cash game. Everyone has $100 stacks at the start of the hand. On the very first hand in the big blind, you are dealt A♣-A. The action folds to the small blind, who raises to $5. You decide to get tricky and just call. The pot is $10, and the flop comes out A-A♠-2♠, giving you quad aces. Your opponent bets $30 into the pot, and again you just call. The pot is now $70. The turn is the 8♠ and your opponent now bets $64 into the pot. Again, you just call. There is now $198 in the pot, and each of you has just $1 left. The river is the 5♠, and your opponent bets his last $1 chip.

As you are preparing to call the $1 bet, your opponent accidently exposes his hand. He has 3♠-4♠, which gives him a straight flush. There is a $199 in the pot and it costs you just $1 to call. Should you? Are you pot committed to call?

Obviously, the answer is no, of course not. The pot could be a million dollars, and you would be wrong to make this call. Hell, it could be ten million dollars and you’d still be wrong to call. Why? Because your expected value of calling would always be a negative number:

EV = (0%)($10,000,000) – (100%)($1) = negative one dollar

You are guaranteed to lose one dollar if you make this call. Said another way, determining whether you are pot committed or not rests entirely on comparing the price that the pot is currently offering you to the probability that your hand can win. Now let’s take this same extreme example above and make it slightly more believable:

Let’s say that you still have quad aces and the board is still A-A♠-2♠-8♠-5♠. This time, however, the pot is $150 and you are deciding whether to call off your last $50. Your opponent exposes his hand again, but this time you only see the 3♠ clearly; the other card is obviously a spade, but you don’t know which spade it is. It’s either the 4♠ or the 6♠. Should you make the call?

To answer this, we need to first think about what the pot is offering us, and compare that to what our likelihood of winning the hand is. Our pot odds are simply $150:$50, which is 3:1, or 25%. We will win half the time (if the unknown card is the 6♠), and lose the other half of the time (if the unknown card is the 4♠). This means we’re even money to win, or 1:1, or 50%. Now, 50% is greater than 25%, so this is a clear call. We can arrive at the same “call” answer by calculating our expected value:

EV = (50%)($150) – (50%)($50) = +$50

Calling seems obvious, doesn’t it? Well, what if the pot had been $100, and we were contemplating calling off our entire $100 stack on the river. The EV would be:

EV = (50%)($100) – (50%)($100) = zero

Said another way, folding quads here is just as correct as calling. You are not pot committed, and folding is mathematically correct.

“But, but, but…”, you’re shouting right now. How can folding quad aces ever be the right play? Quads are so incredibly strong. You’d have to be insane to fold!

Wrong. The math doesn’t lie. The absolute strength of your hand is irrelevant. What matters is its relative strength compared to your opponent’s likely holdings. What matters is whether you’re getting a good enough price to call. And this is the essence of what it means to be pot committed. Are you getting the correct price or not? Do the math, and then decide.

All-in for now...
-Bug

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