Hello,

A friend and I were working what the best possible range is when playing HU against a guy who tilts and shove 100% of hands 100bb deep.

We worked out what range you should call.

For sake of simplicity we rounded it up to 60% equity, and we call 49% of the time. 51% of the time we fold our small blind/big blind.

Our winrate came to an astonishing 940bb/100. Now we were trying to figure out what the standard deviation for this example could be.

Please assume that every hand we get reset to 100bb (i really dont want to have to calculate having >100bb stacks)

Thanks

So effectively we lose 1 blind 25.5% of the time, lose 0.5 blinds 25.5% of the time, lose 100 blinds 19.6% of the time, and win 100 blinds 29.4% of the time, with an EV of +9.4175 bb/hand (matches the amount you gave).

Then we can just use the formula for standard deviation of a distribution, s.d. = sqrt(sum[probability of each result(result - average)])

That gives a s.d. of 69.36 bb/hand, or 693.6 bb/100 hands if you prefer that measure. So even after just 100 hands, being behind would be a 1.35 s.d. event.

I'm surprised we call 49% and fold 51%, it seems like it should be the other way around, but won't make much difference and can't get poker stove to work for some reason.

With the villain as the small blind, the proper calling range is 22+,A2s+,K2s+,Q2s+,J5s+,T7s+,98s,A2o+,K2o+,Q5o+,J7 o+,T8o+ for 51.4%. The average BB per hand for this strategy is 7.523, not 9.4.

Ok. I must have made a mistake.
Thanks for the correction

Thanks

This really helps me!

Can you break down the formula and show me your working out so that I can do this in the future.

As someone elese pointed out I made a mistake in the calling range and winrate, so need to redo it.

Yeah, I get the same answers for this. Where Kar gets this range is simply where the equity is greater than 49.5% (which is better than folding the bb). This range of hands gives an average equity of 57.786% when calling.

OP, your simplifying assumption of rounding up to 60% equity is unnecessary, and gives bad results because having an average 60% equity when calling significantly increases the max win rate, and would change the answer you are trying to ask.

When Hero is SB, his range needs to change slightly in this maniacal scenario, because he only has a small blind in preflop, so the equity needed to gamble for stacks is now 49.75%. This actually makes us throw out J7o and T8o in our stack-off range.

So when hero is SB, the range is now:
22+,A2s+,K2s+,Q2s+,J5s+,T7s+,98s,A2o+,K2o+,Q5o+,J8 o+,T9o+ for 49.623% calling hands. This yields an average equity of 58.081%, and this will yield a profit for Hero of 7.768 bb/hand on the button.

So since we know the optimal results for a toy game like this, we can appropriately calculate it's variance and standard deviation.

When Hero is on the button, there are 3 scenarios to consider, folding, winning a shove, losing a shove, so the variance for this is as follows:

(0.50377)*(-0.5 - 7.768)^2 + (0.49623)*(0.58081)*(100 - 7.768)^2 + (0.49623)*(0.41919)*(-100 - 7.768)^2 = 4902 bb^2/hand^2

The standard deviation when on the button is just the square root of the variance which is 70.0bb/hand, and since SD scales with the square root of hands, the SD becomes 700bb/100.

When in the BB, the variance becomes:
(0.48567)*(-1 - 7.523)^2 + (0.51433)*(0.57786)*(100 - 7.523)^2 + (0.51433)*(0.42214)*(-100 - 7.523)^2 = 5087 bb^2/hand^2.

SD on the BB becomes 71.3bb/hand, or 713bb/100.

The average SD can be found by adding the two variances and dividing by 2 since variance scales linearly with hands.

Average variance: 4994.5 bb^2/hand^2. Average SD 70.67 bb/hand, or 706.7bb/100.

The average win rate is 765bb/100.

The chances of being behind after 100 hands is a negative 1.08SD event and will happen 14% of the time.

tl;dr

Cheers for the response. Appreciate the time you took there

What Im going to try and work out is if calling with a tigher range (30% or so) would reduced the standard deviation, and whether the likelihood of being behind after 100 be less (though are winrate would be less).

Im going to post my results here, and if I make any mistakes Id like to here it.

Thanks again for you help.

EDIT: or is there a way to find out what the highest % chance of ahead, if we pick a certain range mathematically. Or do we have to do trial and error for each range till we find the least likely % we will be behind are a certain percentage. Im sure there is some higher level formula out there that I dont know.
As you are probably aware im trying to find the most optimal range against a fish who is titling and shoving hands (with experiencing the smallest possible chance of being behind after 100 hands). I know Ive asked alot^^, and I do appreciate it all :P

Finding the range that minimizes the chance of being down after 100 hands is interesting, I doubt there is a better way to do it than trial and error. I suggest you make a spreadsheet that will calculate the s.d. and expectation for you when you input the calling frequency and equity, to speed things up.

Of course, if your only goal was to minimize the chance of being down after 100 hands, you would never call again if you won one flip, if you hadn't called so far you would always call on the 100th hand etc, but that isn't as interesting a question as seeing which static range is best over 100 hands.

Over a suitably long period, the highest EV play will emerge as the strategy that minimizes the chance of being down, though this may well be 1000s of hands.

There is one problem with the general idea though, the central limit theorem might not really apply here over just 100 hands, since we are dealing with a rather weird shaped distribution. But ignore that and find the answer anyway

I would assume this would be a trial and error type of calculation. Things like ProPokerTools and PokerStove have default ranges for hand distributions, but those distributions aren't quite optimal for a villain shoving 100% (They cost about 0.2bb/hand in win rate). Ideally you rank all 169 hand combinations vs. a random shove and add/delete the lowest equity hands you would call with and calculate win rates/SDs for each case.

Actually, I'm bored/curious about this...so I looked into it. The optimal solution also creates the least probability of trailing after 100 hands. Although folding more often would reduce your standard deviation, it reduces your win rate even more. So the optimal strategies discussed earlier still look to yield the least probability of being behind after 100 hands, although I'm not a big enough math/poker genius to proclaim this would always be the case for an arbitrary number of hands.

Here is a table for the optimal strategies for BB and SB respectively as a reminder (Note: I used the same strategy for SB and BB for these calcs)

Min eq to call	Call freq	Av. Eq. when calling	Av. WR/hand	Av. SD/hand	Prob. of losing after 100 hands
0.495	0.5143	0.5779	7.6446	70.3204	0.1385
0.4975	0.4962	0.5808	7.6418	70.2910	0.1385

Here is a table of results if Hero calls all hands with a minimum equity vs. a random hand from 60% to 48%. Folding too much devastates win rates fairly fast.

Min eq to call	Call freq	Av. Eq. when calling	Av. WR/hand	Av. SD/hand	Prob. of losing after 100 hands
0.6	0.1418	0.6617	3.9421	63.4128	0.2671
0.59	0.1750	0.6495	4.6134	64.9789	0.2389
0.58	0.1991	0.6416	5.0385	65.9212	0.2223
0.57	0.2519	0.6278	5.8750	67.4935	0.1920
0.56	0.2760	0.6222	6.2003	68.0528	0.1811
0.55	0.3183	0.6132	6.6927	68.8382	0.1655
0.54	0.3484	0.6073	6.9851	69.2718	0.1566
0.53	0.3891	0.5997	7.2985	69.7167	0.1476
0.52	0.4042	0.5968	7.3827	69.8468	0.1453
0.51	0.4585	0.5871	7.5858	70.1742	0.1398
0.5	0.4857	0.5825	7.6261	70.2670	0.1389
0.49	0.5249	0.5760	7.6228	70.3321	0.1392
0.48	0.5490	0.5719	7.5571	70.3408	0.1413

Pretty sick that the best we can do vs. a full blown tilt shove monkey is only 7.67% of his 100bb stack per hand. I would have guessed it would have been more than this.

Hey tringlomane, you sure that you calculated the s.ds correctly?

I got about 37 / hand for the 0.6 equity to call case, I just did a rough and ready calculation rather than factoring in the two different positions, as follows:

85.82% of the time, lose 0.75 bb (average over the two positions), 9,38% of the time, win 100 bb, 4.79% of the time lose 100 bb. Average equity 3.948 / hand (close enough to your value). But the s.d of this strategy is 37.44 / hand.

It looks like somehow you altered the equity per hand when calculating s.ds (making them decrease slowly as equity -> 100%), but didn't change p(call) and p(fold) which also makes the s.d. decrease rapidly? Not really sure, and perhaps the error is mine somewhere?

Dammit. You're right. I made an error in the variance formula of my spreadsheet and carried it all the way through. Thanks for noting that.

Here is the data corrected.

Optimal strategies: BB first, SB second.

Min eq to call	Call freq	Av. Eq. when calling	Av. WR/hand	Av. SD/hand	Prob. of losing after 100 hands
0.495	0.5143	0.5779	7.6446	71.3101	0.1419
0.4975	0.4962	0.5808	7.6418	70.0300	0.1376

Here are the stats for alternative strategies:

Min eq to call	Call freq	Av. Eq. when calling	Av. WR/hand	Av. SD/hand	Prob. of losing after 100 hands
0.6	0.1418	0.6617	3.9421	37.4533	0.1463
0.59	0.1750	0.6495	4.6134	41.5790	0.1336
0.58	0.1991	0.6416	5.0385	44.3399	0.1279
0.57	0.2519	0.6278	5.8750	49.8475	0.1193
0.56	0.2760	0.6222	6.2003	52.1743	0.1173
0.55	0.3183	0.6132	6.6927	56.0188	0.1161
0.54	0.3484	0.6073	6.9851	58.6153	0.1167
0.53	0.3891	0.5997	7.2985	61.9556	0.1194
0.52	0.4042	0.5968	7.3827	63.1512	0.1212
0.51	0.4585	0.5871	7.5858	67.2904	0.1298
0.5	0.4857	0.5825	7.6261	69.2738	0.1355
0.49	0.5249	0.5760	7.6228	72.0489	0.1450
0.48	0.5490	0.5719	7.5571	73.7113	0.1526

To minimize your chances of losing after 100 hands to about 11.6%, you should call off hands with roughly 55% minimum equity. The win rate is 87.5% of optimal and the SD is 79.3% of optimal. With this strategy you call off 31.8% of the time which leads to the following range:

(> 55% minimum equity)
44+, A2s+, A3o+, K5s+, K7o+, Q8s+, Q9o+, J9s+, JTo

bump for detail and "Poker Theory" is asking this exact question right now.