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Math
Corner
Inducing the Bluff, Redux
Monday, January 24, 2005 We PokerStars players who were lucky enough to spend a week on Paradise Island in the Bahamas witnessed a striking phenomenon there--a loungeful of people playing online poker over the Atlantis's WiFi connection. "Have you ever got the message, 'You are not allowed to play on this table'?" one guy asked. No, buddy, but I know what it means. It means some other clown in this room is already playing the game you want to play. I wasn't in the lounge for the big Sunday tournaments, but I can only imagine the temptations of collusion that abounded. My favorite moment came when my buddy Chris decided to play a late-night Sit and Go on PartyPoker, with a whole bunch of veteran players standing around watching him and giving commentary. Of course Chris won it, but I bring it up because there was one hand I found particularly interesting that I wanted to look at a little more closely. I don't remember the exact numbers so I'll fudge them a little. Three players left. Chris has about T2800 on the button, and the small blind has about T2400. Blinds are T100-T200. Chris opens for T600 with Ac8c. The small blind (a loose, bad player) calls. The big blind mucks. The flop comes AdKhQd. The small blind, who has shown a propensity for making small bets, leads out for T300. What's your play? I want to first answer the question, "What play earns me the most chips in the long run?" Usually, that's the only question we need to answer, but in this case the payout structure is something we absolutely need to consider. But I'm going to ignore it for a second. OK, first let's assign a range of hands to our opponent. I think he'd make that bet with any flush draw, any pair on the board, some gutshots, and sometimes with absolutely nothing. So I'll give him any suited one-pair ace (30 combinations), and offsuit one pair aces down to A5o (84 combos). KQ-K8 (57), QJ-Q9 (36), JT (16), T9s (4), T8s (4), Kx of diamonds (6), Jd9d-Jd7d (3), Td7d (1), 9d8d-9d6d (3), 8d7d-8d5d (3), 7d6d-7d5d (2), 6d5d-6d4d (2), and some complete "bluffing" hands like 44-77 (24). Now, if we call this opponent's bet, I believe he will bet the same T300 with his entire range, except maybe he'll drop some of the stone bluffs. And if we call him again, I think the same T300 bet is coming on the river with a similar range he bet the turn with. If we move all-in on the flop, I think he'll call with any ace, KQ, KJ, KT, QJ, QT, JT, Kx of diamonds, Jd9d-Jd7d, Td9d-Td7d and, say, half of his remaining flush draws. OK, calculating the equity from moving in is easier, so let's do that first. Of his 275 possible hand combinations, he folds 71 of them, and calls with the other 204. The pot after our opponent's bet is T1700, and there will be T5000 total in the pot if we move in and our opponent calls. Our equity against the range of hands our opponent calls with is 46.9%, and it costs us T1800 to move our opponent in. So our equity is: (71/275)*T1700 + (204/275)*(.469*T5000 - T1800) = T439 + (204/275)*T545 = T843 Not bad. Now what happens if we call down. Let's assume we call down on any board. The bets will be small, and our opponent is more than capable of bluffing, so I think this assumption is reasonable even if the board brings a four-flush or four-straight. Let's say our opponent will bet his entire range minus 44 on the turn, and then he'll lose the 55 on the river but still bet everything else. And he'll bet T300 every time. So what's our equity? Well, (263/275) of the time, we'll win as often as we win against his whole range minus 44 and 55. (Yes this ignores the times he turns a set of fours, or turns or rivers a set of fives. But this doesn't matter much, because the idea is just that he'll be bluffing with slightly less frequency. I'm mostly using the pairs as placeholders for that idea.) Against that range, our equity is 55.8%. So our overall equity for calling down is: (263/275)*(.558*T3200 - T900) + (6/275)*(T1700) + (6/275)*(T2000) = (263/275)*(T885.6) + (6/275)*(T1700) + (6/275)*(T2000) = T928 So letting our opponent continue to bet earns us more chips (assuming I am right about how our opponent will play this hand). Indeed, while watching this pot go down, my suggested course of action was to call. But Chris moved in. He justified that there were too many draws, and he couldn't risk free cards that would lower his chip count. Thinking about it myself, I also thought it might be too important to avoid busting out in third place. This brings us back to the pay structure issue. Sit and Gos are known for their flat pay structure, with third place out of ten earning 20% of the prize pool. The funny thing is, when they get three-handed, the prize structure is 3 for first, and 1 for second. This is pretty similar to a typical tournament's three-handed prize structure, and if anything is on the steep side. Now that we "know" the chip equities for our two plays here, let's see if we can do the tournament equities. If we move in, we double up our opponent .531(204/275) = 39.4% of the time. When that happens, we are left with 5% of the chips, and I estimate our tournament equity is about .2 (where the prizes are three units for first, and one for second). (71/275) = 25.8% of the time our opponent folds and we have T3900. I estimate our tournament equity in that spot to be 1.8. And .469(204/275) = 34.8% of the time we are heads-up with (T5200/T8000) = 65% of the chips. So our tournament equity for moving in is: .394(.2) + .258(1.8) + .348(.65*3 + .35*1) = 1.34 units If we call down, we lose the pot (263/275)*44.2% = 42.3% of the time, and we're left with (T1300/T8000) = 16.3% of the chips. I estimate our tournament equity in that position to be about .65. We win the pot (12/275) + (263/275)*55.8% = 57.7% of the time and we have a weighted average of close to T4500 chips in those cases. Our tournament equity in that position is about 2. So our tournament equity for calling down is: .423(.65) + .577(2) = 1.43 units What happened? Well, we win the pot almost as often when we move in (60.6% of the time) as when we call down (57.7% of the time). This is because our opponent hardly folds any of the hands that end up beating us in the end. Also, sliding into second isn't that important at this stage of the game. First is now worth three times as much as second is. And now, just to make things really fun, what if this had been four-handed? At that stage, we have the Sit and Go flat structure of 5-3-2. Now sliding up a place really matters. Just take my word for these numbers. Tournament equity moving in: .394(.75) + .258(3.35) +.348(4.125) = 2.60 Tournament equity calling down: .423(3.3) + .577(3.45) = 3.39 Huh? Protecting ourselves against the draw still doesn't help. In fact, it's a worse mistake at this stage than it would've been three-handed! Does that make sense? Well, yes it does. Our biggest chance of busting comes when we move in, not when we fail to move in. If we call down, our worst case scenario leaves us with 1/6 of the chips. In the three-handed case, that still only puts us at about a 60 percent chance of finishing third, in my estimation. If we move in and lose, we only have 1/20 of the chips. In that case, I give us close to a 90 percent chance of finishing third. And this is, of course, worse for us in the four-handed case, when sliding up a spot really matters. Here's what I take from this. 1) Against habitual bluffers, letting them bluff is the best play even on a scary board, especially if they're going to (probably correctly) call with their draws anyway. 2) Don't let the pay structure get in the way too much. The best chip-EV play is still usually the best play. 3) If you're interested in sliding up a spot, make sure you know how to do it (hint: moving in to fight off draws is not necessarily wise). 4) This entire analysis was based on how I thought this specific opponent would continue with his hand. We'd have an entirely different essay if this guy was going to shut down with all but his monsters on the turn, and/or fold most of his draws to an all-in on the flop. As I said, Chris went on to win the Sit and Go. And incidentally, his opponent in this hand folded to Chris's all-in. Pretty boring hand, huh? Hard to imagine anyone would spend any more time thinking about it. September 2004 | October 2004 | January 2005 | |
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© 2004-2005 by Matt Matros | All Rights Reserved |
