[dirtylobster]

A friend just asked me about this and I tried to think it through but I'm not sure if I have it right so please correct me where/if I'm wrong.

NL 0.5/1, $100 stacks.

Hero is dealt JJ UTG, opens for $4

folded to SB who raises to $12
BB folds

Is it profitable to 4-bet to $40 here?

The problem being.... let's say he only 3-bets premium hands (QQ/KK/AA/AK). Then it's obviously a mistake to 4-bet as it prices us in to call a shove (36% equity), right? But... if we assume his 3-betting range also includes JJ/TT/99/AQ. Now our equity is about 50%.

Let's assume he always folds the second range to a 4-bet and always shoves the first range.

10 times.

5 times he'll fold, giving us 5 x $17 ($1 blind + $4 open + $12 raise) = $85

5 times he'll shove and we're priced to call. 5 x our $100 to win 5 $200 pots w/ 36% equity.... 0.36 x $1000 gives us $360 right. So $500 invested, $360 won. That shows a negative profit of -$140.

Totally this gives us $85 - $140 = -$55 over 10 hands which would lead us to the fact that 4-betting is not profitable assuming this specific range.

Am I on the right track so far?