This column includes excerpts from the book The Math of Hold’em, by Collin Moshman and Douglas Zare (available on Amazon).
In my last column, we looked at set-mining and what could go wrong. You may call, only to see someone three-bet behind you, forcing you to fold. When you are up against an overpair, you could flop a set and see your opponent fl op a better set about 12 percent of the time.
In many cases, you won’t get paid off. In most poker situations, you should prefer to be up against a weaker range, but this is often reversed when you rely on implied odds. If you are playing mainly for the chance of flopping a set, you want your opponent to be on a tighter range, so that he is more likely to pay you off in a big pot when you do hit your set.
Finally, you could be outdrawn. An overpair has at least 9 percent equity to make a set on the turn or river. However, overpairs often have more ways to win from flushes or straights, or from counterfeiting the set. On average, you get outdrawn one way or another about 12 percent of the time that you fl op a set against an overpair.
All of these are negative outcomes of set-mining, but set-mining can be highly profitable under the right conditions.
Let’s suppose that you have a low pocket pair and call a raise preflop against one opponent with an overpair. Assume that if you miss the flop, you will have to fold. Assume further that if you flop a set, you can get all in against your opponent. How large can the initial raise be so that you still have a profitable call?
If you are sure that your opponent has A-A, your implied odds are ideal, since when you hit, your opponent will often have a strong second-best hand.
A card of your rank comes on the flop 12.2 percent of the time when you are up against an overpair. That means that you miss 7.2 times for every time that you hit. To make the preflop call profitable, you need to win 7.2 times as much when you hit as you lose when you miss.
When you hit, your exact equity depends on how high your pair is and which overpair you face. You can get a rough approximation with the equity calculator PokerStove by choosing only one community card to match the rank of your pair. The average equity when you flop at least one card of your rank is about 81 percent.
Since we assume that all of the money goes in when you have about 81 percent equity, that means you gain about one stack 81 percent of the time and lose one stack 19 percent of the time, for a net gain of about 0.62 stacks. This means that you can afford to lose only 0.62 ÷ 7.2 = 8.6 percent of a stack when you miss, even when assuming that every set will get paid off.
If you call more than 8.6 percent of the effective stack depth with a low pocket pair, you need to count on something better than what we describe as ideal conditions. For example, some opponents will let you see a free turn card, and will still stack off if you spike a set on the turn. You might find ways to win pots unimproved. If your opponent prides himself on folding A-A, you might be able to find some profitable bluffs. However, many situations are less favorable for set-mining than these ideal conditions.
Most opponents will not pay off all sets. Let’s give your opponent a more realistic range that includes non-pair hands that usually will not stack off unimproved. Let’s still give your opponent a tight range of 10-10+, A-K, and A-Q, which consists of 30 combinations of high pairs and 32 combinations of overcards. Your opponent will make many continuation bets, including with A-K unimproved, but will not stack off with A-K unimproved.
Let’s suppose that the effective stack depth is 100 big blinds. You raise preflop to three big blinds, and an opponent three-bets to 10 big blinds from the small blind, seven big blinds more. If you call, the pot will have 21 big blinds, and your opponent will make a two-thirds pot continuation-bet of 14 big blinds.
Suppose that you push with your sets, and your opponent calls with any overpair, top pair, any nut-flush draw, or a rare open-end straight draw (impossible when you have a low pair, but relevant if you have 9-9; with A-Q, the only open-ender is J-10-9). Is it profitable to call the reraise for set value?
You can choose to push with sets alone, but you also have the option of semibluffing with no set, such as betting an open-end straight draw like the 6 6 on a flop of 8 7 5. This will not actually get any better hand to fold, but it avoids giving up the pot of 35 big blinds against overcards. When you have 10 outs to a straight or set against an overpair, you have around 39 percent equity, and it isn’t that expensive to get all in as a slight underdog: You will pay 90 big blinds to get back about 78 big blinds, for a net loss of 12 big blinds. This loss is acceptable if it gains 35 big blinds against A-K or A-Q slightly more frequently. Similarly, it is profitable to push with a combination of a flush draw and a gutshot.
Pure bluffs with no draw on low flops are not a good gamble. You risk paying 90 big blinds to get back about 20 big blinds against an overpair, a cost of 70 big blinds. Losing 70 big blinds against the 30 combinations of pocket pairs tens and higher outweighs gaining 35 big blinds against the 32 combinations of A-Q and higher.
Semibluffs with either a gutshot straight draw or a flush draw (which might not be good) are slightly unprofitable on most flops, even with backdoor draws. So, we’ll assume that you semibluff only with open-end straight draws and combination draws.
This pair of strategies can be set up in the program Cardrunners EV. The low pair raises on 13.5 percent of flops, including 12.2 percent sets or better, and 1.27 percent draws. The reraiser calls the push 49 percent of the time, including 47.1 percent made hands of top pair or better, with only 22.3 percent equity. The result is that all pairs 2-2 to 8-8 lose about 0.4 big blinds by calling the reraise, with the lower pocket pairs performing slightly worse.
Even though the reraise was only pot-sized, and was only 7 percent of a stack, it was still slightly unprofitable to call the reraise. Only about 6.5 percent can be called profitably under these conditions, although this also depends on the size of the continuation bet. Meanwhile, when your opponent is on a looser range, set-mining becomes much less profitable.
In my next column, I’ll provide some set-mining tips and look at strategies for combating set-miners. Spade Suit
This article first appeared in Card Player magazine.
Matthew is the owner of Dimat Enterprises, “Publishing Today’s Best Poker Books”. The Math of Hold’em by Collin Moshman and Douglas Zare is available now at pokerbooks.InternetTexasHoldem.com.