Another NL Triple Draw 2-7 question?

[manku]

Same game as posted before...Stacks are fairly deep, about 600 each after second round bet of $150 three ways (pot about 700 at this point).

I have 2-3-4-5-9 on button and called the $150 because bettor is aggressive/tricky. UTG, tight player, called.

UTG draws one (I have a hunch he has 2-3-4-5 too). Bettor then breaks and draws one. I stand pat (twodimes says I'm have about 76% ev).

Question: If I draw, my ev drops to about 35% (I figure the better doesn't also have a wheel draw). So I'm giving up about $280 (40%) to draw. However, I'm gonna have a difficult time betting if they both check, and I may have to make a crying call against the initial better I he fires away. Also, if I hit a seven, I can probably get my $280 back or more.

Maybe I'm just being results oriented...utg paired, better drew a 7, and I had I drawn, would have drawn a wheel!

Manku

[MarkGritter]

You generally don't want to break to draw to 2345.

Your estimate is that you have 76% equity in the $700 pot by patting, or $532, and that you have $246 equity by drawing. Is there going to be enough last-round action to close the gap?

Suppose it were heads-up, your opponent has a wheel draw (7432), and knew exactly what you had. Your opponent can bet any 8, 5, or 6 for value, and can also bluff some percentage of his losing hands. (This doesn't exactly match the situation you give above.)

If he adopts a game-theoretical strategy then he will bluff often enough to make you indifferent between calling and folding. That is, EV(fold) = EV(call). I'll work this in showdown equities so EV(fold) = 0, though Chen and Ankenman like to use ex-showdown equities.

Let's make it easy and assume there's a full bet left, and that your opponent will make a pot-sized bet either way.

EV(call) = (bluff percent)*(pot + bet size) + (1-bluff)*(- bet size)
0 = bluff*(1400) + (1-bluff)*(-700)
0 = 1400*bluff -700 + 700*bluff
700 = 2100*bluff
bluff = 1/3

That is, when he bets he should be bluffing 1/3 of the time and value-betting the other 2/3. Let's say you always fold: He wins the pot honestly about 11/42 of the time (26%) and bluffs another 13%, so he wins the pot 39% of the time, or $273 of the original pot. That reduces your share for standing pat to $427, from $518 if there was no betting.

Against a sole player, then, you shouldn't break because you fear a bet. You can work out a game-theoretical optimal calling frequency that will ensure you at least $427.

Against two players it is pretty hard to work out, because they could collude. If player A goes all-in and player B calls, you have a hard time overcalling. I don't think that it will be much lower, though.

So, could you make more by drawing? Suppose you only put your money in with a 7 or 8. Even if you get paid off every time that is still no more than 17% (probably much less since your opponents are likely to hold these cards) or an extra $119. About $350 total, say. If you really have 76% equity then this is not worth breaking for, assuming you play a game-theoretic mix of calls and folds on the last round (instead of always calling or always folding.)

Informally: Your opponent can't bluff or value-bet enough to punish you for patting, even if he knows exactly what you have. You can't value-bet enough on the end to make it profitable to draw. But, this is because there is just one bet left. If stacks were deeper then the correct play might be different.