Optimal Calling Range Based On These Conditions?

[Phoenix Wright]

Anyone else by themselves with a deck of playing cards and sometimes create poker scenarios, or conditions for you to solve? No? Just me? All right then

Recently, I invented this "game" for practice and curiosity. What would be my "optimal" range here and how would I calculate this myself?

Conditions of this "game"/scenario:

I play Heads-up No Limit but (to make it tougher on me and not just "luck") I begin the game with 25% of all starting chips in play. If the game has 1,000 chips in play, then I begin with 250 and the opponent with 750. In this "game", the opponent:

from the Big Blind: ALWAYS checks to the Flop and then shoves All-in on the Flop

from the BTN (also SB in Heads-Up) ALWAYS completes and then shoves All-in on the Flop.

Note: Any bets by us are met with All-in re-raises (including preflop open-raises if we so chose).

What strategy should we (basically calling range) use here? Also how would any blinds or antes impact the strategy? The smaller blinds are (or antes if used), then the deeper stacked we might be essentially playing, but is that relevant to this scenario and if so, then how much?

For precision-sake, assuming starting stacks are Hero 250 vs Villain 750 (remember we begin with only 25% of chips in this "game") and blinds of 10/20 with 50 ante (huge ante I know, but I didn't want this "game" to take all day - just to see how this scenario would run), then what would our optimal calling/shoving range look like? Since the format is Heads-up and our opponent plays a pure strategy (not a mixed strategy where they take actions some percentage of the time, here each action occurs 100% of the time [shoving All-in on the Flop]), then I imagine there would be a mathematical range were could come up with to optimally solve this "game."

p.s. Maybe I've been reading too much about GTO and Nash Equilibrium lately

 

[Phoenix Wright]

Any math people out there?

 

[ChrisFoxWallace]

This question can definitely be solved with Pio Solver, though I think it would require a paid membership. Normally a solver just solves against other GTO ranges and behaviors, but with a Pio membership you can "node lock" which means determining your opponents range and behavior outside of GTO ranges. In a few hours you could probably solve this partially.

By partially solved, I mean that it would be solved for each hand that started with the starting conditions as if you were only playing one hand, and also assuming you had an infinite bankroll. It would not account for anything like only being able to play this tournament once and solving for the most likely win frequency. That would be very complicated, but might be doable with some programming skills.

You could get started on a solution by looking at the ranges in a simple equity calculator like Equilab and calling with any hand that is a favorite against two random cards on any given flop. There are less than two thousand possible flops as a practical way of seeing this exercise, but it would still take some time to get all those ranges figured out and dumped into a database or a spreadsheet.

Doing some work with Equilab you could probably get to about 95% right with just a few hundred and estimating the rest based on that information. If you know that two pair is good enough to call with on a board with three completely unconnected and unsuited cards, then it covers a lot of potential hands on all those kinds of flops and saves you a ton of time.

 

[Phoenix Wright]

Thank you for the in-depth answer; this is the kind of thing I was looking for

but this sounds a bit more complicated than I thought

Originally, I just assumed you could type the scenario into a poker calculator (something more advanced than equilab) and get something closer to nash equilibrium with just the click of a button and letting the computer run a while to calculate. I imagined maybe 10 minutes to run, but this sounds more complicated than I expected due to behaviors outside of GTO such as absolute decisions.

Anyhow, thanks again for the reply

 

[maestro121920]

my brain can't fathom this, too hard and complicated for me, i will just go all in blind

 

[Hermus]

In this toy game without ICM and without rake, because we can force the opponent to play any2 by pre-flop raising, by open-shoving any2 we have an EV of 0 so as a baseline we can only lose if we play a range weaker than any2 (dropping strong hands in favour of weak hands).

Because we don't lose any value by betting/raising pre, and we have no fold equity, any hand that is good enough to play we call/check with.

If you want to calculate a pre-flop calling-range you'll just calculate the pre-flop equity of every hand against any2. Given that 32o, the lowest equity hand in headsup poker, still has 32% equity against any2, and you'll need 13|1 == 7% the equilibrium pre-flop strategy on the button is to complete any2. Removing ante's changes nothing because in that situation we still get 3|1 == 25% pot odds. The only way that the blinds will have any effect on the strategy is if the big blind is artificially large compared to the small blind to where the pot odds presented are higher than 32%.

The method for constructing a post-flop calling range is exactly the same. We gain nothing by betting, so we check every hand. The opponent shoves any2 (remember we never raise pre so villain's range still contains any2) and we call with any hand with positive EV on that particular texture. At this point, the size of the pot at the start at the hand is 140 at 180 effective = 1.8|1 == 36%. You can use equilab or flopzilla to calculate the equity of a hand against a any2 range on any particular texture. The calling range will obviously be different for every texture.

Essentially, what we did here is not a Nash equilibrium because for all intents and purposes the opponent is not a player in a non-cooperative game. The opponent is the game, and we beat the game by calling any2 pre and calling any +EV hand post. If the opponent also aimed to maximise EV then suddenly we're playing poker, and we can approach the Bayesian Nash equilibrium (Nash equilibrium in games with imperfect information) by iteratively maximising the EV equations of each player through linear programming. Doing this by hand is essentially impossible for anything more complicated than the AKQ game (in the data science and machine learning community often referred to as Kuhn poker if you're interested in some research).

 

[ChrisFoxWallace]

Hermus has it right, and those were my initial thoughts as well. We play everything preflop without raising, and then call any hand that is getting the right price against any two post flop. Assuming 10/20/50, each hand will with 90 in the pot because the button will always complete.

If we are the BB, then we start the flop with 180 because we paid the ante. So we are getting 270 to 180 pot odds. 90 in the pot already and the 180 double up we will get from our opponent. That reduces to 3/2, so if we have 40% equity against a random hand, we call, otherwise we fold. You can do the same with the button by just running the odds without paying the ante.

Equilab will find the equity for an individual hand against any two, so if you have equilab open you could play perfectly quite easily. But that is solving each hand rather than creating a whole solution set, which I figured was what you are asking for.

 

[LOKIE77]

Yes as a warm up and prior to playing I routinely deal out 6 or 9 hands and look at position, card showdown value go over the probability and odds of being dealt said hands so I have that ll memorized (the math). So I do pre-flo analysis what hand i would play form various posn and then deal out the flop, turn and river going thru the situ and if I would bet rasie or fold. Work pretty good in theory but while in live gane remember you can't control the cards or what your opponent does or plays. Good luck at the tables

 

[Evan Jarvis]

Hey Phoenix,

You're a Legend!

Just thought I'd let you know homie