[nate1729]

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Let's not lose perspective. If Nate is 75% likely to be ahead then this is a tremendous value raise versus three opponents and increasing our winning chances is merely a very nice bonus.

I suspect 75% is a rather high estimate however.

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When we're ahead we prefer to gain value, of course. The problem is that just as effective odds differ from pot odds, the weighted payoffs when we're ahead and behind require a greater degree of certainty than the immediate numbers would indicate. If we're behind, betting can be very costly, and lead to a precarious pot size. Meanwhile, if we're ahead, we're not ahead by too much, in the weighted average case. Even if we're winning currently, we do not have the same equity that top pair normally has.

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The pot is not so small when you consider implied odds. There are currently 2.5 BB in the pot. This will increase to 3.5 BB after Hero and (assume) one more player call the flop. Hero intends to bet the turn which if called in two spots will increase the pot to 6.5 BB. Toss in 2 BB for a river bet and one caller and we have 8.5 BB. Obviously it could be less but it could easily also be more.

This is going to be a 7-10 BB pot when someone wins it. That's why it's not a big deal to invest an extra 0.5 BB on the flop to improve our chances.

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I'm glad that you look at the big picture of a hand, but I fear you might be assuming some things you shouldn't. Part of the reason you predict a big pot is that you're planning to raise the flop. Bets beget bets; once the money's in there the pot's more worth fighting for. That the pot might get big is bad news, not good news, for a weak made hand suffering from a severe information defecit. And, of course, some of the predicted 7-10 BB is invested by us, and (more to the point) if the pot gets that big and we stay in it we're unusually likely to have gotten some money in badly. T9o on a 9-high monochrome board is not a hand that can confidently claim a good equity share of the entire projected size of a pot.

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It's also important to realize that "invest" is not synonymous with "spend" or "lose". If everyone were expected to call then we would be getting 3-1 odds on our raise. We would only need to win 25% of the time for our raise to be free and anything beyond 25% is immediate profit.

I think our pot equity at least meets this 25% threshold. I'm not concerned about the high cost of raising because there is no cost. This is a value raise.

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I like this thought; this is one consideration that makes this decision difficult. As I've said, however, I think the real weighted equity we're facing, and the stark contrast between immediate and effective odds here, makes it misguided to project enough of an immediate equity gain to swing the decision.

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The obvious objection is that we may not get three callers. But our likely best hand is so precarious that virtually any fold is more valuable to us than a caller. [...]

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The same precariousness is what makes us hesitant about raising for immediate equity. Remember also that the range of hands to which we can denying a profitable call is smaller than it might seem: bad spades and overcards in Daniel's hand would be our targets, but this is not likely enough to support the risk of raising. (Also note that the difference between one and two bets is less likely to cause Daniel to make a Sklansky-incorrect fold than other players. He's a hand-reader, not an odds man.)

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Another perspective on checkraising the flop is that from a purely protection view it is by far the best. The pot is as small as it is ever going to be. Ed is still live behind the potential coldcallers and will also be acting last after our threatened follow-up turn bet. If the middlemen won't fold for two bets now, then they can never be made to fold. This checkraise is the biggest bomb at our disposal and if it won't work then our hand is beyond protection.

Remember that any turn protection is inherently only half as good because it only works against one card instead of two.

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I think that your argument applies very well to many common hold'em situations, but one thing that makes this hand interesting to me is that the turn bet is comparatively more powerful than it often is. Many hands have proper odds to call a double bet on the flop, but if the turn blanks off, the turn bet will be more powerful if we're confronting opponents with 9:2 odds instead of something more attractive. (Remember that we're squeezing Daniel between our bet and Ed on the turn, so 9:2 might be less attractive than it seems.)

Also, we're not betting every turn. Just calling makes it easier to get away from a hand we're pretty sure is beaten if the turn comes ugly.

Even though flop protection might prevent an opponent from seeing two cards instead of one, turn protection is often superior. In this case, we can exploit the increased betting limit and the fact that our hand might be strong enough to protect on the turn (and not on the flop.) And, of course, Daniel and Ed aren't the only ones dodging those cards; we are too.

Thanks again for taking the time to think about my article. Your comments have helped me consider new aspects of the situation. If you continue to disagree with me (and even if you don't) I hope you share any more thoughts you might have about the hand.

--Nate