Player's Stories In the last article, we addressed some of the issues the small blind must consider in short-handed play. By examining pot odds, it was easy to come to a reasonable conclusion as to the number of hands to play in the small blind. Then, we began examining the reasons to reraise preflop to knock out the big blind. In this article, we examine heads-up versus multiway when the small blind flops a good hand. We will consider both when the small blind is leading after the flop and when the small blind is behind after the flop. Then, we can undertake the arduous yet valuable task of comparing our options and deciding the crucial question: reraise or call. First, let's make some important assumptions, for the purpose of facilitating our discussion. We assume that each opponent will make the decision to play based on their connection with the flop and their relative looseness-so we assign a percentage chance each opponent will call on the flop (e.g. 75%). When we considered pure bluffs, it was enough to check if either opponent would call. If the small blind instead has a good hand, it is necessary to break down what happens when both opponents call versus when one opponent calls. This is a simple question of probabilities. If each opponent calls 75% of the time, then... For the purpose of bluffing, we assumed either the bluff worked and the small blind won the pot, or it didn't work and the small blind lost the pot. For semibluffs (aka drawing hands), we assumed the small blind would earn an average of 2BB each time the suckout was successful, plus the small bet on the flop. Of course, this is a massive oversimplification, but for our purposes, we just need something to help make the math a little closer to reality-not to reflect reality perfectly. Therefore, we can assume after the flop the small blind will win/lose 2.5 BB on average. Example 1. $10/20 game. 3-handed. $60 in pot. Small blind has best hand. Each opponent has an average of 5 outs. We'll presuppose the 5 outs are not the same, so there are essentially 10 bad cards that will cause the small blind to lose the pot and 2.5 big bets. Scenario 1. Both the big blind and button continue past the flop. Note: I calculated when the outs would not hit rather than when the outs would hit. This method is easier because you do not need to account for when both opponents hit their outs or the same opponent hits two outs.} Scenario 2. Only one opponent continues past the flop. Example 2. $10/20 game. 2-handed. $70 in pot. Small blind has best hand. The lone opponent has 5 outs. Scenario 1. Button continues past the flop. We compare how much the small blind wins in each of the scenarios above. Example 1: 3-handed Example 2: Heads-Up Scenario 1. ($120 * 78.8%) + (-$50 * 21.2%) = (94.6 - 10.6) = $84 Examining the numbers so far, it appears that calling is the superior play. Even though the EV on examples above show that the heads-up situations earn more profit, they do not show $10 more profit. This is crucial because we assumed a reraise was necessary from the small blind to make the pot heads-up. But the analysis is incomplete, because there is one final difference between multiway and heads-up play. We must finally analyze the number of times the small blind has an inferior hand that will play to the river. When I say "inferior hand," I mean specifically any hand that the small blind feels is strong enough to play beyond the flop, but is behind another opponent's hand. The small blind has already narrowed its selection of starting hands to some relatively potent holdings. A good selection of hands such as the following: {AA-22, AK-AT, KQ-KT, QJ-QT, JT, A9s-A2s, K9s, Q9s, J9s, T8s, T9s-76s} constitutes only 302 of the 1326 possible starting hand combinations: 22.8% of all starting hands. Article 1 showed us that pot odds are still sufficient for the small blind to call or reraise more than even 25% of the time. But these strong hands are still vulnerable. Hands such as T9s, K9s, or A4s will miss the flop entirely nearly 50% of the time, making any bet a bluff or semibluff with few outs. Even if many of these hands hit the flop, they could still be losing. J9s may flop a Jack or Nine but start behind, maybe even drawing dead to a runner-runner. The example below can help illustrate this concept: Example 3: Heads-Up Small blind has Jh 9h. Notice that the opponent is currently winning with 22, 99-AA, 92, K9, A9, or any hand with a Queen. Even if you assumed the button played 100% of all hands preflop, they are still winning 15.5% of the time {Total Winning Hands (168)/Total Possible Hands (47*46/2)}. This may not seem like a lot, but consider that many of the hands the small blind will play are weaker. Hands such as 22 will always be bottom pair unless a 2 is on the flop. Any pair higher in an opponent's hand will leave the 22 with two outs or less. Let's see how much more likely such a problem would be. Example 4: Heads-Up Small blind has 2c 2h. The button is winning with 44, 66-TT, QQ-AA, and any hand that contains a 3, 5, or J. {Total Winning Hands (450)/Total Possible Hands (47*46/2)} = 41.6%. We're not even considering flopped straights or flushes. A strong preflop hand can become a vulnerable or losing hand on the flop very quickly. But the key is that the small blind will do far less guessing against one opponent than two. Against a single random hand, that J9 is still ahead approximately 85% of the time. We've seen how profitable it can be when the small blind is ahead of one or two opponents on the flop, losing only 41.5% against two drawing opponents. But the EV is much worse when the small blind starts the hand behind. Example 5: Small blind is behind on the flop, but has 5 outs. The small blind loses when it misses its 5 outs on both streets. Scenario 1. Scenario 2. Scenario 3. The button continues past the flop after small blind reraise preflop. ($70 pot) Notice that in all three cases, if the small blind is behind on the flop, there is a negative expectation for the rest of the hand. Of course, the small blind has also lost any money put in preflop in all the scenarios we've examined in the past two articles. That fact leads us to the final analysis we need. If we put it all together and compare the advantages and disadvantages, we can mathematically estimate when a reraise in the small blind is better than a call. Putting it All Together Now, we must compare how often the small blind will make a hand against two active opponents, one active opponent, or when both opponents fold on the flop. What we will find is the key to this whole debate. No matter how loose the opponents, the same number of scenarios exist where the small blind is losing. If we average them all out, we can make some estimates. The small blind will likely have one pair or better or a solid draw 50% of the time on the flop. Against a single opponent with a random starting hand, we can estimate that the small blind will be losing 25% of the time it holds a legitimate hand. Therefore, against a single opponent {who will call with 75% of hands on the flop}, · 1/2 of the time, the small blind is bluffing or (at best) semibluffing. · 3/8 of the time, the small blind is winning. · 1/8 of the time, the small blind is losing. Against two opponents {who call 75% of hands on the flop}, · 1/2 of the time, the small blind is bluffing or (at best) semibluffing. · 1/4 of the time, the small blind is winning. (both opponents fold 1/16 of the time) · 1/16 of the time, the small blind is losing, and one opponent folds. · 3/16 of the time, the small blind is losing, and both opponents continue past the flop. And that is all the information we need to come to a final analysis. With all the assumptions we made, we can compare the final postflop EV in a 3-handed versus heads-up match. We know the reraise costs $10 more preflop, but will it make $10 more in postflop EV by knocking out one opponent? Final Comparison: Multiway: Heads-Up: As you can see, the expectation earned by knocking out the big blind is worth more than $10 postflop. There are settings where this will not be the case. If the opponents' skill is sufficiently higher, the big blind is very loose (and will call a reraise cold), or the big blind is very tight (and will fold to a single bet very often), the value of a reraise is diminished. However, these conditions are rare enough to make a reraise in the small blind profitable in most circumstances in my opinion, and I believe our arithmetic above confirms my advice. Finally, there is one last factor that we did not address. Reraising shows strength. Against most opposition, this display of strength preflop increases the likelihood of the button folding on the flop. Thus, the small blind earns further profit from extra successful bluffs, and the small blind is more likely to know where it stands if the button continues to play, because the small blind has indicated strength early. All in all, the last two articles were rough, complex, and assumptive. But they help illustrate how an understanding of simple probabilities can allow any player with a pad, pen, and a calculator to ascertain the superior of two alternatives. This preparation gives credibility to a strategy-far improved from a writer simply stating "Reraising from the small blind is usually more profitable." And it can hopefully help prepare you to analyze other authors' arguments to come to your own conclusions. Until next month, good luck! If you have any questions or comments, I may still be reached at my email: jason@pokerpages.com.
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Short Handed Poker: The Scary Small Blind
(Part II)
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