Bubble Math

This article includes excerpts from the latest Dimat release, The Math of Hold’em (available on Amazon), by Collin Moshman and Douglas Zare.

The bubble of a tournament occurs when the number of players left is one more than the number who gets paid. Many players play tightly in hopes of making the bubble, while other players will try and take advantage of this and start abusing the players who are folding frequently.
Some of these adjustments are correct, while others are misguided.

Example No. 1. It is the bubble of a 9-man SNG.

Blinds: 200-400
Cutoff: 1,400
Button: 2,600
Small blind: 5,100
Big blind (Hero): 4,400

Action: Everyone folds to the small blind, who shoves. Hero has a stack of 4,400 and holds A-8. He knows the small blind is shoving a very wide range of hands, but reasons that it’s not worth putting his stack at risk on the bubble.

Question: Is Hero using correct reasoning?

Answer: Yes, the general idea is correct. While A-8 beats the range of a wide-shoving player, and the pot is laying odds, the big blind must be tight. This reasoning is incomplete however: the same argument would tell you to fold AA in that spot, since you’d similarly be putting your stack at risk when you could fold into the money. However, you should call with aces because you would have a sufficient edge to justify the risk of busting out of the money.

You can use an Independent Chip Model (ICM) calculator like ICM Explorer to calculate the equity you need against the small blind’s range in order to call. While you would gain chips if you called with 45.5% equity, you need 69.72% equity against the small blind’s range for the call to gain equity according to the ICM. Folding leaves you in good shape, with 29.31% of the prize pool, while calling and winning is only worth 42.04%. To have 70% equity against a random hand, you have to fold even A-Ks, and you need pocket nines or better to call, even though you know the small blind is pushing a lot of trash.

You should combine your understanding of the mathematics with reads on your opponents. If the small blind is a casual player who would not know that it is correct to push hands like 7-5 from the small blind, you may have to fold TT. Similarly if you’re the small blind, you can often shove a wide range into a risk-averse big blind that has you covered, despite strongly wanting not to get called.

Example No. 2. Later in the same SNG, the cutoff has been eliminated.

Blinds: 200-400, 3 players
Button: 2,700
Small blind: 7,200
Big blind (Hero): 3,600 As 8d

Action: The button folds, and the small blind shoves. Hero reasons, “Now that the bubble has burst, I’m in the money and can call this guy since I’m ahead of his range.”

Question: Is hero’s reasoning correct?

Answer: This reasoning is wrong, and raises an important point about the bubble. Mathematically, there is no difference between the bubble of a tournament which pays two places 75-25, and after the bubble bursts in a 50-30-20 structure. Once it’s three-handed, each player will win a minimum of $20, so it’s as if that $20 has already been put in everyone’s account, and it’s the bubble of a tournament with three players left, top two paid $30 and $10. Every time you are one elimination away from a pay increase, there is effectively a new bubble.

How this affects your strategy is a function of the other players at the table. If they were all playing rationally, you would want to follow normal bubble guidelines: generally play tighter as a mid stack since you are risking so much equity if you play for your stack, and looser as a big stack since you can threaten the mid-stacks with elimination.

For many players however, the psychology causes them to behave differently. Rather than look at how their stack compares to the remaining active stacks as the primary guideline for deciding whether to loosen up, they simply play tighter before the actual money bubble, and looser once they’re in the money. The flawed reasoning is that once you’re “in the money,” you’re free-rolling in a sense.

You must force yourself not to reason like this. Once you’re in the money, then you already have the minimum prize in your pocket. You are always playing to maximize equity, whether or not you already have a win locked-up. If you play for the win too much, ignoring the prize for second, you will take first place more often. However, you will win less money overall, since you will give too much of your share of second place to the player who isn’t involved in the hand.

Example No. 3. It is the final table of the 2010 WSOP main event.

Blinds: 500,000-1,000,000-150,000

Mizrachi: 62,475,000
Cheong: 58,950,000
Duhamel: 41,425,000
Candio: 27,175,000
Racener: 16,275,000
Dolan: 13,300,000


First place:      $8,944,138
Second place: $5,545,855
Third place:    $4,129,979
Fourth place:   $3,092,497
Fifth place:     $2,442,960
Sixth place:     $1,772,939

Action: Everyone folds to John Dolan in the small blind, who shoves. You are playing as Duhamel in the big blind with 4♦ 4♠. Call or fold?

Answer: This is a question for ICM. According to an ICM calculator, Duhamel’s current equity in the tournament if he folds and forfeits his 100k blind (and his ante) is approximately $4,665,000.

If he calls, his expected equity is:

EV = P(win) x Equity(Win) + P(Lose) x Equity(Lose)

To determine P(Win), we must decide on a shoving range for Dolan, and then can use a poker equity calculator to determine the equity that pocket fours has against that range. Since Dolan is an experienced tournament player, he will presumably know to shove wide as a short stack in the small blind. (Sure enough, in the telecast we see that he only looks at the 5♦ before shoving.) So Duhamel might reasonably assume that Dolan is shoving the top two-thirds or so of starting hands; in this case, Pokerstove says that the 4♦ 4♠ has 52.4% equity.

When Duhamel calls and loses, his stack decreases by 12,150,000 relative to folding, since he has already posted his blind and ante. His equity therefore falls to around $3,990,000. If he calls and wins, his stack increases by 14,300,000 plus 900,000 in antes relative to folding, and his equity increases to $5,365,000. Therefore the expected value of his equity when he calls is:

EV = 52.4% x $5,365,000 + 47.6% x $3,990,000 ~ $4,710,000.

Compared to the $4,665,000 equity he retains if he folds, his expected equity increases slightly by calling. While this result would quickly change if we narrowed Dolan’s shoving range, ICM shows us that it is quite reasonable for Duhamel to call off a significant portion of his stack with a low pocket pair in this situation.

In the actual hand, Duhamel did make this difficult call, and his low pair held up.