Hi Everyone:

Here's an interesting article written by one of our authors Philip Newall:

https://www.frontiersin.org/articles...018.00210/full

Best wishes,
Mason

Fantastic paper, thanks for the link Mason. This paper and its referenced sources make for some great reading. At first glance, it is interesting how the GTO approximate solution to limit heads up holdem is even more simple with less mixed strategy and more pure strategy, such as call a reraise over 99 percent of the time, versus the weaker AI from 2008 that only called reraises 83 percent. Thus, any reraises that are bluffs should expect to always be called by a balanced opponent, and almost any folds are pure profit.

Hi Mason and Phil,

I think the problem with raise or fold strategy is that it ignores other thresholds where ev is neutral between call and raise.

The hands that earn exactly 0ev when raised will use a mixed strategy. I think I remember that Cepheus raises or folds T4o at frequency, which is indicative of this threshold

The hands that are unprofitable when called share a threshold with the hands that are profitable when called. I haven't studied the ev of individual preflop hands much, but I think that at this threshold, we will see a group of hands that earn the same fraction of the pot whether the opponent folds or calls, and this group of hands will limp occasionally. I heard through the grapevine that there are some suited connector combos that get limped occasionally which likely fall directly on this threshold.

The hands that want the big blind to call share a threshold with the hands that want the big blind to 3 bet. Both actions will result in a very high ev of at least (1.5 small bets*bb call freq) + (1.5 small bets*bb fold frequency)/hand = ev / hand. Cepheus limped AA on the button vs me and won a big pot unimproved.

There may be more thresholds. I'm not sure.

Now I think I was wrong. Oh well. Here's what I think now:

Consider the ev of my very worst big blind raising hand, which will be equal to or slightly more positive than the ev of checking that hand. I'm thinking about JTo, Q9o, J9s, T8s, 98s, etc. I don't know what Cepheus raises there in the big blind, but I think that those hands are marginal.

Now consider how this ev is affected when Cepheus limps AA on the button. Naturally, this ev goes down. Seems that if AA was never limped, this ev would go up slightly, which would indicate that other hands near the margin such as J9o, Q8o, J8s, 97s, etc, might then change to raises.

In this sense, limping strong hands on the button has a metering effect on the big blind's raising frequency vs a limp.

Since Cepheus does in fact limp AA sometimes on the button, the margin described above could be described as static, unchanging; the margin, if maintained by the metering effect caused by the buttons strong hands, will not change.

If the big blind chooses to raise and check the marginal hands perfectly, this will be self imposed indifference; this self imposed indifference is why the button will have the same ev when choosing to raise or call AA on the button.

I think this may be key in creating a profitable limp range on the button because the weaker hands that want to limp on the button have less fear of a raise, which meters the ev of the weaker limping hands and maintains this ev at a minimum of zero.

Hi Bob! So from what I've heard from the Cepheus guys is that the non-zero limping range is driven by the bot only being run for a finite amount of time. They observed that the limping range gradually reduced as the bot continued to learn.

In general I think it's very hard for a bot to drive a frequency down to exactly zero.

So why not just node lock a solver to never limp, while playing the Cepheus approved preflop raising range, and check out the difference in ev?

A solver is based on equations, which in turn are based on GTO approximations. The results of the solver would be lost within this noise. However, I like your idea. I would like to see this done with Cepheus itself, but it is likely that Cepheus can not be node locked, or, they have already done this and shown that Cepheus+nodelock>Cepheus, but have withheld the results.

Also, if Cepheus folds to a reraise with frequency 0.009, then the question of noise arrises within Cepheus itself. Is 0.009 close enough to zero to dismiss as noise? Or, if those 0.009 hands are correct, we could learn quite a bit from those hands, imo.

I beat anyone who open raises less than 100% of the time very easily. 4-betting makes c-betting or not c-betting interesting (that much is true here).

The relatively low opening raise % is perhaps the most "unusual" part of the optimal strategy. Many humans fold too much in the BB (compared to the frequencies in the article), and have assorted postflop leaks, which make a high open-raise high flop c-bet strategy a good exploitative strategy from the SB.

But I personally moved away from the 100% open raise default when I started coming up against people who were "accidentally" correct in their counter-strategy... mostly a lot of preflop 3-bets and flop check-raises.