Betting value and bluffs with the same size (i.e. being balanced) is one of the most basic concepts of GTO poker. It's not a simplification--it's fundamental theory. With that said, I would guess you can make some pretty big deviations from GTO in a completely anonymous pool. I personally would prefer to spend my time focusing on technical aspects of strategy that will apply everywhere, but someone who plays only on Ignition might profit more from focusing on MDA-related exploits.

From a theory perspective I think bluffing with KK has to be pretty bad here. First of all, there's no way that KK has 0 EV checking back in theory or in practice. Your value betting range is going to be very narrow, so you can be much more selective with your bluffs. You're probably going to have at least KcQc, KcJc, and QcJc at some frequency. I would bet that's probably more than enough bluffs to choose from to balance your value range.

But again, that doesn't mean OP's play is bad exploitatively. The CO's range is probably going to be completely capped here, and humans tend to bluffcatch pretty suboptimally when capped.

To fact check my previous reply, I ran a sim and

Hi, I've just recently started studying poker theory, and this seems to disagree with what I read in Janda's book. From what I remember, the solver will only use mixed strategies where these have equal EV.

Can somebody clarify why this is not the case here?

% error

Yes, that's exactly what I thought. But after run a sim I found that :
.KK have no more SDV than QJ.
.we prefer bet KK and give up with QJ more often cause of the blocker effect on AQ/ AJ.

So yes, GG, it's not an "he checks, I bet blindly lol" like I thought first, it's a very well selected bluff. I'm deeply impressed.

It will mix strategies when the EV is very close. The difference doesn't have to be exactly 0, just very close to 0. If it had to be exactly 0 it would never mix.

I simplified the tree in my solver and solved it down to 0.0265% accuracy, which means the EV calculations are accurate to within 0.5% of one big blind. The solver is still mixing nearly 50/50 with combinations that have EV differences of over 5% of a big blind. (Also FWIW the strategy is mostly unchanged from my last sim which had a more complex tree and had been solved to about 0.4% accuracy.)

If someone who's more knowledgeable about the workings of solvers sees that I'm misunderstanding something then please let me know.

Your sim is wildly different from mine. In my sim KK has much more EV checking than QcJc, which has 0 EV checking. QcJc is a 100% frequency bluff, and checking would be a 1.5 bb mistake. I don't see how KK could possibly bluff at a higher frequency that QcJc. Maybe your ranges are very different from mine. If your ranges incorporate more button flatting than mine then you could have a lot less Tx and suited combos... but still KK is always going to have more SDV than QJ, so I'm really not sure what happened with your sim.

Pretty good result for calling with one out on the turn, congrats!

It has to be exactly 0, if one action makes more money you take that action

Yes, I run differents sims. How to play river depend a lot of the IP range (linear or more polar) but also of the differents size options for CO on the turn.

Overall, it's an "he checks I bet" spot solver approuved !

I know that's the general concept, but how does that make sense in the context of the fact that the solver still mixes when the EV difference isn't 0 and the accuracy of the sim is to a finer degree than the difference between two mixed actions? It seems clear to me that the solver will always judge one action to be at least some microscopic amount better than the others, yet it still mixes.

This is maybe more of a pure game theory discussion than a poker strategy discussion at this point, but is it really impossible for the solver to accept that two actions have a microscopic difference in EV and to still mix? It seems like that's clearly what it's doing unless I'm misunderstanding what the EV numbers are telling me. I understand that at a basic level the solver is never choosing to give up EV with a hand for the sake of its range, but how do I reconcile that idea with what the solver is actually doing?

If anyone has any materials that get into fine detail of what exactly the solver is doing when it calculates strategies, I would love to see them. I've looked around and read some wikipedia pages and stuff like that, but nothing I've found seems to really explain at a technical level how a solver actually works.

I think it's just a numerical issue really, when you implement algorithms you have to accept the numerical precision won't be perfect. So what in theory was "we mix if and only if both actions have same EV" becomes "we mix if and only if both actions have EVs close enough compared to some numerical tolerance parameter we chose such that our computation finishes within an acceptable amount of time".