I've been playing a lot of TD lately, and I'm having some difficulty with marginal situations. A lot of my TD play is short handed, so this question is probably more geared towards sh play, but is still relevant to a full game.

I know that this question is partly opponent/game dependent, but I want to see what kind of debate we can get.

I always think it is tricky when I have position and my opponent has check-called a bet before the last draw and drawn one while I stand pat. (Heads up pot)When should I pay off my opponent if he bets into me after my last draw?

My range for standing pat here is relatively large, and assume that my opponent knows that. Also, the pot is usually big enough that it would seem I could always call here to catch a bluff. Should I? In reality, people probably aren't bluffing too much here.

I am a favorite to win the pot if I have a jack or better and my opponent is drawing. But it's not correct to call any bet if I never win any additional bets when my opponent misses and I always lose a bet when my opponent makes a hand.

Is it correct to fold my worst hands here? Im talking about a hand like J8732 and my opponent bets into me after the last draw. How often should I be calling/folding here?

i play mostly HU when i play TD.
i'm probably calling with this hand in HU, well i'd say most of the time. especially depending on the previous rounds of betting.
It very much depends on the opponent your playing, and if you think they understand the game well enough.
lots of variables with this

I'm going to start becoming a more active member of twosevenTDthree, but I've been busy lately... within the next two weeks.

Let's build this part of strat. I love this game--draw games in general.

Here's one way of looking at it:

Lets say the pot is X units, and your opponent bets 1 unit into your pat hand. If you call at least X/(X+1) times, the bluff play will be unprofitable to your opponent in the long run (at the breakeven X/(X+1), he gains X units 1 time for each 1 unit he loses X times). So you can safely fold randomly at least 1/X of the time without losing any EV over the straight strategy of calling every time to bluffs. You also gain EV from saving bets to players who DO make their hands. 1/(X+1) is an initial lower bound on your optimal "fold ratio"

But if you fold MORE than 1/(X+1) of the time, and your opponent picks it up, it will be best for him to start betting blind (since his bluffs have +EV). Unfortunately for him, he cannot gain EV from you this way every hand, because he will sometimes hit his draw! You can put an better lower bound on the fold ratio by assuming your opponent bets blind, and solving for the fold ratio that maximizes your return. As you raise your fold ratio, eventually the losses from the -EV to bluffs will overtake the +EV from saving bets to real hands - the point at which this happens is a better lower bound to use. Since few opponents will bluff every time (even if they realize this logic, their counterstrategy will be too obvious to you), it is actually optimal to fold MORE OFTEN than this lower bound even to word class players.

Note that this type of logic also applies to calling on the river when you are both drawing one, you get an A or a K or something and your opponent bets to you. You can't call every time.

Another point - I don't believe it is a good play to stand pat with J-high in position against someone who is drawing one if the pot is small. You are a slight favorite after the flop, but you obviously have -EV from calling your opponents bets when he makes a hand and chooses to call, or folding to your opponents bluffs when you choose to fold. Since J-high is only about 55% favorite to win straight up according to Daniel's book (a little better if your 4 low cards are smooth), when you subtract out the -EV, you might not even get 50% equity! By drawing, you pretty much guarantee yourself 50% equity by virtue of your position if you reckon to be no rougher than your opponent. If the pot is large, you have to like the jack, since the small relative bet size will detract only very slightly from your EV, and position is worth little since bluffing will rarely work.

WOW. GREAT POST. This reply answered my question perfectly! Post in strat more, this is some quality stuff.

thanks for the compliment. I thought a bit more about this problem, and found some more interesting stuff.

As I argued earlier, a safe folding % for J-high is 1/(X+1) since it is the highest folding % that doesn't allow bluffs to make money. But what is the marginal amount of money that can be made by raising this fold% further?

By folding more often, you save money from the value bets, but lose money to the bluffs. Lets say your opponent makes his hand and bets out M% of the time. Also, lets say he bluffs B% of the time. So by lowering the fold%, you will gain M% from the real hands to every (X+1)B% you lose to the bluffs. This way we can calculate a counterstrategy for the drawing hand - how often does he need to bluff to make it poor strategy to fold more often than 1/(X+1)? Solve for the break even M = (X+1)B to bet B = M/(X+1).

Think about what this means. Lets say that your opponent has a typical draw like 2358 or 2347. He has seen 8 card total (his 5 cards + maybe 3 discards), and he will probably bet out if he makes 8-6 or better. So he has 8 outs from 44 unseen cards., and will be betting out 18% of the time. The pot has to be big enough to justify his call, lets say X = 6. So B = 18%/(6+1) = 2.6% Yes, thats right. He only needs to bluff 2.6% of the time to blow your aggressive folding strategy (i.e. more than 1/(X+1)% of the time). Assuming he only bluffs his worst hands, he need only bluff when he pairs his 8, and even then only 1/2 the time!

He could bet out with weaker hands like 8-7 if he suspects you've rapped pat with rough (which makes M a little larger) but even so, the necessary bluff % is quite small. In otherwords, you should very rarely fold more often than 1/(X+1), only if your opponent is virtually guaranteed to never bluff.

The profit slope at 1/(X+1) goes in both directions of course. My experience has been that most "river bluffers" bluff more often than 2.6%. You should punish these people by folding less often than 1/(X+1)% to capture more EV. Fold at the regular rate to the "non-bluffers" to give the bluffers hope, but anyone who EVER bluffs should probably be called almost every time, unless they are selective enough to bluff less than 2.5% of the time. You will save enough X+1s to make up for all the 1s you lose to the real hands.