When we are playing explo we assume our opponent is not playing clairvoyantly at 1 or more nodes. Resolving equilibrium after a single node lock controls for villain clairvoyancy following a single node, which means we need make a read only at one decision point. The approach I'm introducing in this video isn't explicitly measuring the ev loss vs clairvoyant at multiple nodes, so it's vulnerable to counter exploitation.

I think 1-2% of the pot is too much but if it was <1% of the pot I'd do this.

Ben, Can you justify the 1-2% comment as too much quantitatively. Intuitively, I lean toward the idea that executing a strategy with much better precision will at least lead to predictably good results. No complex strategy can be executed with much precision at all. At least in some cases this leads to error that have very bad consequences for EV (For example in: http://chemistry.usf.edu/faculty/data/space/papers/OverBettingPolarizedRangesintheShortRunHowBigtoBet-byBrian_Space.pdf . In this case precision in frequency is required for the play to make sense at all. I have no feeling for how unusual this is).

It may be that inaccurately executing a complex strategy leads to a lot of cancelation of error and an overall better strategy than the easier to execute strategy that is 1-2% worse on the flop. However, the equilibrium nature of it suggests deviations all hurt the EV (although its a minimum and first order deviations might vanish).

Note, the simple strategy also does not propagate the loss because we can play a well constructed strategy on the turn.

I ask because you seem to be able to confidently answer that less than 1% is a good target and I have no way of understanding your insight.

I think your content is remarkable and I am grateful that you chose the share with the community.

Sincerely,
Brian Space

Brian,

When you say "deviations all hurt the EV," I take it to mean that if villain is clairvoyant, all deviations from optimal hurt hero's EV. I was trying to be careful in this video to say that I'm not assuming my opponent will maximally (or even very precisely) exploit deviations from optimal frequencies. If I don't think my opponent is likely to counterexploit my frequencies over small samples, then I'm free to explore some less traveled parts of the game tree where they may be more likely to make mistakes.

I agree with your claim under conditions of villain clairvoyance. Suppose we were given a random hand at a random "brittle" (brittle means that small freq deviations are counterexploited for a lot of EV) node in the game tree. Your overbet spot would be a good example of this. Because there are so many spots with different equilibria, it's likely our human level execution of equilibrium strategy would be quite imprecise and we'd be better off playing some simpler strategy we were comfortable with. The equilibrium difference in EV between the simple and complex strat might be as high as 1-2% in these cases, but we still might be better off ceding 1-2% of the pot to the computer.

^You can test this second claim if you like. Click around Pio until you see a spot you classify as suitably brittle such that your intuition is that you'd be better off playing a simpler strategy, and run a sim with a simple solution vs a complex solution where the complexity yields >= some potshare threshold (in our case 1-2%). Give yourself some time interval to study the brittle strategy. Next, node lock your human level memory of the brittle strategy and compute its EV vs MES as a % of pot. Do the same operation with a simple strategy. Finally, see which one of the remembered strategies performed better. Did you make more EV "playing" the simple or the complex strat?

Empirically, we can also see there are variations in framework between top regulars. Some are going full on explo, not bothering to Pio too much and simply computing EV against what they think villain's range is. Others are faithfully executing their best guess at GTO ranges using as complex a model as their computing power allows. And some (like me) are doing a bit of both. There's probably no right answer here, and I encourage everyone to switch between these frameworks and develop a method of thinking which works in their games and with their skills.

Ben,
That is a terrific answer, thank you very much.

It seems to me that the benefit of these kinds of analysis is much more likely to come from finding better parts of the game tree to visit than anything else. I am unconvinced humans can even approximate an approximation to PIO strategies in even moderately complex situations. The mathematics of the situation suggest this to me strongly. It is also my intuition from logging many hours playing vs poker snowie that humans just do not play as randomly -- they rarely can bring themselves to take low frequency actions that just feel wrong. Snowie has no such inhibitions.

Another issue is that the idea of fragility applies to particular combos and actions. I think of the problem as an EV landscape that has wells and peaks and steepness but in many dimensions. So while a particular action may be a small part of the strategy EV as a whole, for any given combo the frequencies may or may not dramatically effect the EV, and this is not easy to assess intuitively with accuracy.

Indeed, the problem with the simple strategies that give up some EV may be that those are precisely the strategies that human do respond well to. If we always bet a small amount with our entire range and the opponent knows this they can both prepare for these spots and think through them effectively in the moment.

I suspect that much of what people actually achieve is information hiding allowing them to make better vacuum EV plays without serious repercussion. This itself is an admirable goal that led me to think about bet size distributions (smearing) for a given strategy.

Again, thank you for taking your valuable time and sharing your thoughts with us. It is very much appreciated.

Brian Space

Some hands will have better EV call EV because of beating some combos of villain's value range, good removal effects, or both. Calling with hands that are +EV vs optimal ranges might be near neutral EV vs tight ranges.

We'll have many Qs, Ks, As combos that arrive at river. The <2PR but >=TP ones should mix XC and XR (and perhaps XF, but my intuition is that we have plenty of Jx and other pairs that XF riv). If we XR every one of these combos we'll be bluffing too often, but you're correct to point out that my hand should be mixing XR fairly frequently, perhaps 30%.

I think the point was that certain nodes of the game tree involve complicated strategies that include mixing to many different sizings and lines. In these hard nodes villains are more likely to make mistakes.