I was trying to figure out if flop underbets (< 50% of the pot) can be more profitable than a "normal" cbet in heads-up situations.

f = opponent folding frequency
P = pot size before we bet
b = a normal cbet size (>= 50% of the pot)
b' = our under bet, so b' < b for a given normal cbet size b

The EV of a normal cbet is:
(1) EV(cbet flop) = f(p+b) - (1-f)b

The EV of an underbet is:
(2) EV(undercbet flop) = f(p+b') - (1-f)b'

(1) - (2) = (b-b')(2f-1)

Since b > b' the EV of the normal cbet will be smaller than the underbet EV if 2f-1 < 0 <=> f < 0.5, that is if the opponent folds less than half of the time.

Since most opponents fold more than half of the time (given that a random hand misses the flop 2/3 of the time) there are no benefits in underbetting the pot on the flop readless.

Is this obvious or am I leveling myself? :-)

looks like it should be (1) EV(cbet flop) = f(p) - (1-f)b and similarly for the underbet case

Actually I started by writing the equation as you wrote but then I realized the reference point is after you have bet the flop.

For example, if you bet 100 into 200 you are laying your opponent 3:1 so you need him to fold at least 1/4th of the time. The EV equation for the breakeven situation is 0 = f*(200+100) - (1-f)*100 <=> f = 1/4, not 0 = f*200 - (1-f)*100 which would give f = 1/3.

if you mean what i think you mean by "reference point", then you can use whatever reference point you want as long as it's consistent. if you get confused, i recommend you just work w/ the expectation of your total stack size at the end of the hand.

if you bet 100 into 200 (and just give up and lose whenever villain doesnt fold), you break even if hes folding 1/3 the time.

If someone will fold 1/3 of the time no matter what, the most profitable bluff bet size would be a min bet, right? But in real life, calling frequencies generally change with bet size. But if you are in a specific situation where a villain's range is polarized to hands that will always either fold or call/raise regardless of bet size, you can and should bet small.

There are several other things you might consider:

1. You assume that you can only win with a fold. I did not see where you say you are betting with air, so the EV equation should include a win without a fold using a card equity factor, or, at minimum, you should recognize the result is a 'worst case' analysis.

2. Fold frequency is generally not independent of bet size, and your equation does not account for it directly. After finding the break-even fold frequency, you would then have to compare it to the likelihood villain will fold with that frequency for the bet size used.

Edit: I see that the above post has a similar thought on bet size and fold equity.

OFC I am wrong. It is the opponent that needs to call 1/4th of the time to breakeven. ty

Yes this is the scenario I was thinking of. Say a paired flop where even a 40% calling range of a preflop raise heads-up misses the flop approximately 80% of the time (propokertools). On this flops opponent's ranges are polarized and even if they bluff c/r at a given (small) frequency they will probably fold a significant portion of the time. In this case a small bet is more profitable than a bigger one.

I was thinking about betting with air so this is a worst case scenario where I would have (hypothetically ofc) zero equity when called.
The fact that the folding frequency is not static makes it tricky to model this with a simple EV equation...

Also, for balance's sake i don't know if i'd want to start underbetting my bluffs because then you would have to underbet your value hands in the same spots to avoid being transparent. However, against a non-thinking opponent this point is moot.

Lower c-bet, better risk/return ratio, lower variance, just lots of win.

Gets me to thinking: What kinds of flops/opponents' hands would call a half-pot bet but fold to a pot size bet?

Bottom pair with an over card?
Two overs and a gut shot?
Weak flush draw?
Bottom/idiot end of an open ender?
Under-pair when top is not A or K? i.e. 99 on flop Q-7-5

Would typical opponents with these holdings call for half pot but fold to a Pot size C-bet?

How big a difference would it make in the analysis?

Also, I think there is a huge difference in cash play versus tournament play when trying to figure out the calling frequencies, but if we assume all have 100+BB stacks, we can reduce that difference.

Well, where i play...

i fold to most pot bets if i can't raise or chase a monster draw vs a full stack

i call 1/2 pot bets on the flop/turn/river all the time, occasionally stealing later, but usually just snapping off bluffs and bluff inducing wimps who need to maintain momentum but don't have the balls to pot or bet 75-80%

i can't remember the last time i folded a mediocre hand to a 1/2 pot bet on the river when the board is a busted flush

i also can't remember the last time i called a pot bet on the river with the flush hitting without some crazy reason to

facts about people in poker (and in general)
-they're wimps
-they're greedy
-they're emotional
-they take their ego too seriously

when people make wimpy bets, especially minimum, call them just to show them they're going to have to man up. Unless they're going to at least bet strength on the river, folding to underbets is the biggest hole in a nit player whose terrified of a big pot without the nuts

Going by what you just said your going to go broke once you move past small stakes.. Thinking players bet small for pot control. By calling little bets every street your just building a pot thats ultimately going to be taken from you. Little bets also induce an oppenant to spazz with air or overvalue a medium hand that going to get valued. Betting big you'll get trapped and even worse trap yourself.