A Medium-Strenght Hand - Part II

In my last column (available at CardPlayer.com), I started an analysis of a specific no-limit hold'em hand that I hope will provide useful insights into a more general no-limit hold'em question - the question of what to do when facing a bet with a medium-strength holding. In my example, our hero faced a $70 bet into an $85 pot, with a board of Q-9-4 rainbow, from a middle-position player who had raised preflop. Our hero held two black tens.

Let's say our hero decides to raise. He has $500 in front of him, which just might be enough money to raise to something like $220, and still have a meaningful decision afterward with the chips remaining in his stack. The best thing about the raise is that our opponent often folds. Let's say he calls (or sometimes reraises) with K-Q or better (for example, he also calls with A-Q, overpairs, and sets), and half the time with two jacks. Based on the range of hands I gave him in my last column, our opponent folds 58 percent of the time in this range, and plays on the other 42 percent. When our opponent plays on, he either reraises, and we fold, or calls, and we stop putting money in the pot unless a 10 hits. I am therefore giving us an extremely small amount of equity if our raise gets called. I'll say that 4 percent of the time, we'll win the pot in that case - although when we do happen to win the pot, we'll double up.

With this information, we can calculate the expected value (EV) of raising:
EV(Raise) = .58 x ($155) + .42 x [(.04 x $585) - $220] = $89.90 - $82.57 = $7.33

My instincts upon hearing this situation were that raising is more profitable than folding, and it appears this analysis supports my instincts. What's interesting is that in my last column, I made an initial estimate for EV(Call) of $49.70. That analysis took a few more paragraphs, and it actually failed to include the river action (for example, I gave an estimate for our equity assuming the action ended after the turn). Basically, for the purposes of the EV(Call) analysis, our hero called on the flop, and then called again on the turn if a safe turn card hit (although sometimes the turn action went check, check - but we'll deal with that in a minute).

Having an additional street to deal with definitely lowers our equity, because we're always in a tough spot if our opponent bets the river. I guessed in my last column that the river wouldn't lower our equity much. Let's see if analysis bears that out, as well.
First, let's say our opponent bet the turn, we called, and now he's setting us all in on the river. Is he firing the third barrel as a bluff, or is he value-betting with a made hand? Let's give our opponent a ton of credit and say that he bluffs optimally on the river, meaning that it won't matter if we fold or call when he bets, because he won't be bluffing often enough for us to show a profit by calling. By the river, there will be $525 in the pot, and our hero will have $280 left in his stack. That means that if our opponent is playing optimally, he will be bluffing 280 ÷ (525+280+280) = 26% of the time. Using the range of hands for turn betting that I gave our opponent in my last column, we can now come up with a guess for the river action.

If our opponent value-bets K-Q or better on the river, he ends up betting a safe river 41 percent of the time after he bets the turn. He also bets any ace or king, about 17 percent of river cards. So, he fires the third barrel 17% + [(1-17%) x 41%] = 51% of the time overall. When our opponent fires the third barrel on the river, we might as well fold.

Going back to the last column, the action on the turn went check, check 35 percent of the time. Let's say after that happens, our opponent will bet $150 on the river with the strong hands he checked - top set, middle set, A-Q, and K-K, and also with some bluffs. I'll say he bluffs with A-K.

With all of these estimates, we can start to put this hand together. After our opponent bets $70 on the flop and we call:

• 17 percent of the time, the turn is an ace or king, our opponent bets, and we fold. EV= -$70

• 54 percent of the time, there is a safe turn card, our opponent bets, and we call. After that, our opponent follows through on the river with another bet 51 percent of the time (including always when a king or ace hits) and we have no equity in the pot - meaning we've lost $220 overall. About 6 percent of the time, it goes check, check and we lose, again costing us $220; 43 percent of the time, it goes check, check and we win, netting us $305.

• 29 percent of the time, the turn action goes check, check. Again, we will call on the river if no ace or king comes. Checking on the turn is actually very bad news for us, as it turns out that using my assumptions above, given that our opponent checks the turn, there is only a 16 percent chance that we have the best hand. We'd definitely prefer it if our opponent bet the turn, given the profile I've credited him with. After check, check on the turn, 17 percent of the time, our opponent bets the river and we fold, losing $70. Our EV for the other 83 percent of the time is -$120.

Finally, overall, we have:

EV(Call) = [.17 x (-$70)] + .54 x [(.57 x -$220) +(.43 x $305)] + .29 x [(.17 x -$70) + (.83 x -$120)] = -$11.90 + $3.11 - $32.34 = -$41.13

Wow, that's a bad result. I'm running out of space, but let me just quickly show you the results again, this time modifying the strategy a little.

If, after a check-check on the turn, we fold to a bet on the river:

EV(Call) = -$28.21

In addition to the above adjustment, I also tried having us call one additional bet when a king hits on the turn, but not the river.
It didn't help much.

So, after all of this analysis, the results were EV(Call, with the best turn and river strategy) = -$28.21, EV(Fold) = $0, and EV(Raise) = +$7.33. We've shown that raising is the best play then, right? Not so fast. Remember, everything is opponent-specific.
The math worked out this way in large part because we were up against a tough and smart opponent who didn't continue firing the second or third bullets with nothing very often, and who had a reasonable opening range to begin with. But even this math isn't perfect. I didn't figure the equity of hitting a 10 into the EV(Call) analysis. I ignored it because its overall impact should be pretty small, but since the results turned out so close, it's worth noting that the real EV(Call) is probably a few bucks higher. Also, this opponent always bet when an ace or king hit. If he ever checked, it would add to our EV. Finally, the opponent in this example never bluff-reraised on the flop. Against an opponent who might do that, the whole calculation has to be redone.

And to piggyback on that last sentence, these last two columns weren't proof of anything. They were calculations based on guesses, estimates, and a crude model of how a decent, solid-aggressive Internet player might play his hand. Any of my assumptions could've been off, and I might've even messed up a calculation somewhere (believe it or not, I didn't hire a team of poker math guys to go over all of it). So, it's important when doing an analysis like this to think about what we've actually learned. To me, this particular analysis showed that even when you're not value-betting or bluffing, and even when your opponent's hand is not all that likely to improve, it's sometimes a good play to raise - especially if your opponent is a strong, non-maniacal player.

Matt Matros is the author of The Making of a Poker Player, which is available online at www.CardPlayer.com. He thanks James Davis for providing feedback on this column.