Bluffing with weak hands

ChuckTs made a post on the private Stoxpoker forums a few days back where he asked about bluffing for stacks with an ace-high type hand and if this was a good time to try it. This is what the hand looked like (I hope he doesn't mind me writing about it here):

Poker Stars, $0.50/$1 NL Hold'em Cash Game, 7 Players
LeggoPoker.com - Hand History Converter

UTG+1: $99.20
MP: $101.50
CO: $132.55
BTN: $138.35
SB: $99
Hero (BB): $149.95
UTG: $122.10

Pre-Flop: A 8 dealt to Hero (BB)
5 folds, SB raises to $4, Hero calls $3

Flop: ($8) T T 9 (2 Players)
SB bets $6, Hero calls $6

Turn: ($20) 2 (2 Players)
SB bets $12, Hero raises to $27, SB calls $15

River: ($74) 5 (2 Players)
SB checks, Hero bets $63

Now, my reply in the thread got no responses. I'm going to guess it was for one of three reasons:

1. Nobody understood what I tried to say.
2. I'm completely wrong and everybody wants to save me the embarrassment by ignoring the post.
3. I'm so completely right that no more posts are needed.

Which is it? Well, who knows. But #3 seems unlikely, so instead of just reposting my reply I'll try to rephrase it a bit and make the logic a bit easier to follow. I'm also going to use a much simpler example to illustrate the reasoning. In fact, I'm going to invent a game just for this purpose.

The game works like this: Both players ante up (let's say $.50 for an initial pot of $1), and then both players draw a card from their own 13-card (deuce to ace) deck. Since they have individual decks, both players can have the same card. Once the deal is done, the person starting (let's call him the "button") can either bet or check. If the button bets, his opponent may fold. If the button bets and his opponent calls, they go to showdown. If the button checks, they go to showdown. Whoever has the highest card will take the money in the pot. No raising is allowed.

So in an example deal, your opponent draws a trey, you - with the blind - draw a card, you bet and he will almost certainly fold. The only card he can beat is if you drew a deuce and bluffed, so he doesn't stand much of a chance of winning. Easy, right?

So how should you attack this game? Clearly you should always bet your aces. They're the nuts and you can't lose. Which other cards should you bet? There's a game theory optimal answer to that, but instead of taking the long way to get there, let's instead stipulate that our opponent will only call a bet with a nine or better. Which hands should you bet?

Clearly you should bet your aces. There's no question about betting the nuts.

But should you bet your eights?

No, most certainly not! You stand nothing to gain from this. Any better hand will call, and any worse hand will fold. When you have the best hand, you will win the exact same amount of money as if you had checked, but half the time you will instead lose an extra bet instead of saving it.

Should you bet a deuce? Yes!

It comes down to this: Your opponent will fold more than half of his possible hands when you bet, so regardless of what you have, you will take the pot a little more than half of the times that you bet. A deuce can never win, so if you check it you've lost. If you bet it, more than 50% of the time he will fold and you will win back your blind. If you check it, you will always lose. The key here is that you gain from making this bet because he will fold the winning hand more than half the time and you're about even money on the bet.

You shouldn't bet the best of your bad hands, but you should bet the worst of your bad hands. This is because the value of the bluff comes from your opponent folding the best hand. This doesn't happen if you bet an eight, but it can certainly happen if you bet the deuce.

Now we return to the original hand, and hopefully you can see that the same reasoning applies. Here, Chuck has an ace-high hand. We can separate his opponent's hands into these three categories:

1. Busted draws,
2. Trips-or-better,
3. Mediocre hands who floated the turn.

Since he's getting about even money on the river bet (a little better, but play along) he needs his opponent to fold a better hand about half the time. His opponent is probably never folding trips-or-better. He's certainly folding his busted draws, but we could already beat the busted draws. And he would fold his mediocre hands if he's somewhat tight.

We can beat each hand in group #1 (the busted draws). So for Chuck's bluff to be profitable, we need group #3 (mediocre hands) to be bigger than group #2 (monsters). It's a somewhat simple exercise with PokerStove to find out if this is the case. But what I tried to explain in the original thread, and that which is the point of this post, is that if our hand had been weaker - say 87o - then a river bet would have gone from probably -EV, to definitely +EV, because suddenly we no longer beat a busted draw! Instead of needing just group #3 to be bigger than group #2, we make money if the total sum of both group #1 and #3 is greater than his monsters.

So, perhaps counterintuitively, betting A8 is a losing play, but betting 87 isn't. Weird, huh.

Did I make sense this time? I hope so. Comment, please.

/FP

Oh, and an addendum: Playing limit, this is a lesson that is learned pretty early on. Against everyone but the worst of calling stations (who might show down king-high) good limit players will check this river every time. Raising the turn and then checking the river with a mediocre hand (but one that has some chance of winning) is sometimes called a "free showdown play" and is pretty common in limit.