Small Stakes Hold Em question

[maxsick]

In the section on "Pot Odds" (pgs. 27-31), Ed Miller describes a scenario where you are playing heads up in a $2/$4 game. Your opponent has four outs to the winning hand against your top pair. The pot contains $44 on the turn. The bet size is $4. If you bet, your opponent will be correct to call with 12-to-1 pot odds. If you don't bet, you are offering your opponent infinite odds (by allowing him to check behind for free). Is there are way to profit in this scenario?

[MCS]

Betting is still profitable for you. When you bet and he calls, you are making money on his call because you win his bet most of the time.

He has to call because of the money already in the pot, but he would be better off if you did not bet.

For you, the most profitable result is betting and having him fold. Next is betting and getting called. Next is checking and allowing a free card. Worst is folding. [img]/images/graemlins/smile.gif[/img]

[Holden97]

As you can see, this situation is profitable for both players. Clearly it is more profitable for you to bet because of your positive expectation (without seeing the last card) that you will win the hand.

[Kurn, son of Mogh]

Well, first of all, it's profitable because you win the $44 pot over 90% of the time. But being a winning poker player is about extracting maximum vale from your winning hands (and giving up minimum downside on your losing hands)

So, lets look at your options and calculate the marginal value of each option to you.

If you check the turn and he folds to your river bet when he doesn't improve, you win $0 extra the 12 times you win, and lose $8 the one time you lose (we assume you bet the river and call his raise). The marginal EV of not betting the turn is -$0.62.

If you bet the turn and then bet the river regardless of what falls, you win $4 the 12 times you win the hand (we'll assume he folds when he misses his draw) [$48], and lose $12 the one time he improves (we assume you bet the turn, bet the river and call his raise) [$12] for a marginal EV of +$2.77 when you bet the turn.

Clearly, betting the turn with the intention of betting the river and not folding is worth slightly more than 1/2 bb more than checking the turn.

Note: this analysis does not consider $ already in the pot, it's simply an analysis of the marginal profit you make by betting the turn.

Why do you call his raise on the river? Because when he raises you're getting a minimum of 13:1 on the call and he's probably bluffing more than 7% of the time.

[AKQJ10]

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As you can see, this situation is profitable for both players.

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Incidentally, the reason this is possible and not a violation of the Fundamental Theorem of Poker is that so many players have put dead money into the pot with hands they shouldn't have played in the first place.