To Flip or Not to Flip

It's the first day of a five-figure buy-in no-limit hold'em tournament. You've gotten a good night's sleep. You feel alert.



You wade through all the railbirds and all the media and finally locate your seat. Just as you do, the tournament director announces, "Shuffle up and deal!" It's your big blind, and you toss two of your green chips onto the felt. You've now got $9,950 in chips in your stack. Everyone folds around to the small blind, who shoves all in for $10,000. You haven't even taken your chip protector out of your pocket yet, but you figure you probably won't need it on this hand. You're going to fold, unless you look down at aces. But there's a problem. The small blind doesn't have a protector on his cards, either, and when he looks down at them one more time, he accidentally exposes his hand. He has the A K. You look down at your hand, and find two black queens. You've done your research. You know you have a 53.8 percent chance of winning if you call. But should you?



This is a classic hypothetical question, and it creates raging arguments almost every time it's discussed. I believe there is a right answer to this question, one that doesn't depend on how skillful the player is, or what the player is hoping to get from the tournament. I hope by the end of this column that I will have convinced you.



First, let's look at the common arguments I hear for folding:



(1) If you're a good player, you want to use your skill to find a better spot to get your chips in.



(2) You don't want to risk your entire tournament on one hand, especially in a coin-flip situation.



(3) I don't play these big buy-in tournaments very often, so I want to get some experience playing them.



In case you haven't guessed, I strongly believe all of these arguments are specious. Here's a quick thought experiment: Let's say you're playing in a tournament with 1,024 people. How do you win this tournament? By getting all the chips, of course! This means that if you calculate your chances of doubling up 10 times, you will calculate your chances of winning the event. Now let's say that you have a 53.8 percent chance of doubling up whenever you get all in for your stack. This means that your chance of winning the tournament is .538 to the 10th power, or about 0.203 percent. The average player's chance of winning the tournament is 1÷1,024, or about 0.098 percent. So, if you consistently get your chips in with a 53.8 percent chance of winning, you will be more than twice as likely as an average player to win the event. It gets better.



Let's say you choose to fold the queens, thinking you have a better than 53.8 percent chance to double up in this event. If you decline the "coin flip," you're stuck with your initial starting stack, as you're expecting to have a better than 53.8 percent chance of doubling up at some point later in the tournament. If you accept, and win, the coin toss, you double up immediately.



You need to estimate, then, the expected value (EV) of your brand-new $20,000 stack size at a later point in the tournament – the hypothetical point at which you'd eventually double up after declining the "coin flip." Let's reasonably (conservatively, actually) say that when you double up right away, your stack will be worth $22,000 at that hypothetical future point at which you would've found your better spot.



It's time to do the math. If taking the "coin flip" gives you a 53.8 percent chance to have a stack of $22,000 later in the tournament, how likely do you have to be to double up later in order to fold your pocket queens? Well, you can answer that by solving this equation: x(20,000) = (.538)(22,000).



Do the algebra and you get x = .5918, or 59.18 percent. So, do you think you're good enough to have a 59.18 percent chance of doubling up later on? If you said yes, you're wrong. Go back to our thought experiment. If you could consistently have a 59.18 percent chance of doubling up, you'd win a 1,024-player tournament more than five times as often as an average player. Trust me, you're not that good. I don't think it's possible to be that good. I'm certainly not that good.



Here's one more way to look at it: Let's say you're a very good player. You win a no-limit hold'em tournament twice as often as an average player – which is a spectacular rate. You win the 1,024-player tournament one time in 512. Now we can work backward and figure out our chance of doubling up. We do this by solving the equation 1÷512 = (chance of doubling up) to the 10th power.



And we get the chance of doubling up, .536, or 53.6 percent (note that this is smaller than the chance of your two queens beating the A-K suited).



Using the same equation as above, it turns out that we would take any edge greater than 48.63 percent. Yes, that's right. I just made the argument that very good players should actually take slightly negative EV situations early in a tournament, because if they win the hand, they get to use their skill with their new stack. And that's more important than waiting around for a slightly better situation – much more important. Have you seen a lot of successful players using the "get chips or go broke" strategy early? This is part of the reason why.



Some say calling with the queens would amount to a good player letting his skill go to waste. Here's the thing about poker – the skill is about finding edges. And edges are precious. Think about it; on most hands, we fold before the flop. It's very hard to find a way to get our chips in profitably. And here, we have a known edge. We know that in the long run, we'll earn $810 by calling with the Q-Q. That's not a small edge. Folding here would be akin to flushing an hour's work down the toilet. Calling here doesn't negate our skill over the field. Calling here is our skill over the field.



You don't want to risk your whole tournament on one hand? Then you shouldn't be in the tournament. The only question you should be asking yourself is, "Will I make more money in the long run by calling here?" And even if it's "the experience" you're after, wouldn't the experience of a final-table run be much more valuable than the experience of playing for a day or so and then busting out near the bubble?



If you don't believe all of this math mumbo jumbo, I suggest a little record-keeping experiment. For every tournament you play, write down whether you double your stack or bust out before doing so. I did this for a little while, and I doubled up 67 times in 127 tournaments. That's about 52.8 percent of the time. I think that's pretty good! If, after a thousand tournaments, you find that you're doubling up more than 59 percent of the time or so, congratulations – you might be good enough to fold queens in the above situation. In the meantime, stick to getting your chips in with an edge. That's how poker tournaments are won in real life.

Matt Matros is the author of The Making of a Poker Player, which is available at www.CardPlayer.com. He is grateful to Dr. Bill Chen for first presenting the above argument as The Theory of Doubling Up.