Once in a while, we as poker players find ourselves in the enviable position of holding the stone cold nuts on the river. Ideally in these spots, we like our opponents to also have monster hands that are second best so that we can get paid off handsomely and drag enormous pots. Often, however, our opponents will have much weaker hands than we do, and we must determine what the optimal bet size should be, i.e., how to get them to put in as much money as possible.
Let me use an example from a recent hand that I played. The game is $1-2 No Limit. I have been playing tight on a table with several loose aggressive players. After one such LAG limps in, I raise to $10 in middle position with A-K offsuit. The LAG is my only caller. After the rake is deducted, we are contesting a pot of $18. His stack after the call is $125 and I have him covered.
The flop comes down A-K-2 all different suits. He checks, and I check it back to him. Against this loose player's range, top two pair is a little bit too strong to bet. I feel that I have to let him catch up a little, or maybe induce a bluff from him on the next street. Free card, the pot remains at $18.
The turn is another Ace. I have just made the nut full house and, even better, my opponent quickly bets out $15. Bingo! The way this is going, I rate to win a very nice pot on this hand. However, I elect to just call at this point. The reason? I feel it is STILL a little too early to tip him off to the strength of my hand. I am almost certain that he does not hold an Ace. There really are not too many hands that he could hold that would call a raise from me here. But if I flat call him, he may try and push me out of the pot on the river. I am giving him a chance to make a mistake. The pot is now up to $48.
The river comes -- an irrelevant 4. My opponent doesn't hesitate too long before firing out one more bet of $18. Small for the size of the pot, this actually looks like more a value bet than a bluff. Now I am putting him on a King rather than air. Clearly at this point, last to act on the river, I have to raise his bet. But for how much? He has another $92 behind. The pot, if I call, will stand at $84. Let's run some numbers and see what looks like the best bet. For simplicity's sake, I am only going to examine three possible plays. Min-raising to $36 (2x), value-raising to $64 (about 3.5x), and shoving for his last $92 (about 5x). I will have to make some assumptions, and of course this won't be the best way to play it for every single situation. But I'm curious ... and it is always good to practice thinking about these things after the fact.
Let's start with the min-raise. I will call his raise and add another $18, which will offer him about 5.6-1 pot odds. I estimate that with his range of a king, a pocket pair above 4, or nothing, he will call the min-raise about 40% of the time. So making him put in another $18 40% of the time shows an EV of $7.20.
How about the medium-sized raise? If I make him put in an additional $46 he will be getting about 2.8-1 pot odds. I estimate with that same range, he calls this bet about 20% of the time. So he is putting in an additional $46 20% of the time ... the EV on this raise gets up to $9.20.
And the shove? Now he is looking at 1.9-1 pot odds. I am guessing he only calls this maybe 12% of the time. $92 12% of the time looks like $11.04. Also with pushing all in, since his call frequency is going to be lower, you don't have to show down your cards as often which I like.
However, you can play with the numbers a lot, and get them to fit any bet you like if you manipulate how often you think the opponent will call. There is no way to prove any estimate one way or another. But that isn't the point. The point is to use your best guess, and to always consider all the options available to you. If a player will only call a value-raise slightly more frequently than he will call an all in shove, maybe we should be pushing all in on the river more often. Players sometimes fall into the habit of 3x-ing everything ... and they forget that the beauty of no limit is that you can gun for your opponent's entire stack at any time!
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I ran this hand by Billy, and he seemed to like a raise to about 45 or 50. His reasoning was that a raise this size gets called just about as often as the min-raise. But obviously they are putting more in so it becomes more profitable. I was inclined to agree with him. If a raise to just $50 gets called 33% of the time, the expected value on the raise is $16.50. This clearly dominates the three scenarios listed in my analysis.
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