I move in blind preflop. No antes or blinds. You call when favored which I believe is Q7 or better. On average what are the chances you win?

Same question except I expose my cards.

Isn't it Q5o that is the worst Qx hand with >50% equity vs ATC?

I believe this is the range that is profitable vs a blind shove: 22+,A2s+,K2s+,Q2s+,J6s+,T7s+,98s,A2o+,K2o+,Q5o+,J8 o+,T9o

---- Equity Win Tie
ATC 41.98% 40.47% 1.51% { random }
Call 58.02% 56.51% 1.51% { 22+, A2s+, K2s+, Q2s+, J6s+, T7s+, 98s, A2o+, K2o+, Q5o+, J8o+, T9o }

I don't have a calculator for working out the numbers for when your cards are exposed, but maybe someone else has already computed it.

Equilab can do the first calculation. For some reason I get the same range, but slightly different numbers than Arty.

Equity Win Tie
UTG 41.87% 40.38% 1.49% { random }
UTG+1 58.13% 56.64% 1.49% { 22+, A2s+, K2s+, Q2s+, J6s+, T7s+, 98s, A2o+, K2o+, Q5o+, J8o+, T9o }

Weird. I just ran it again for a hundred million sims and it came out 58.1% vs 41.9%. I'm not sure why my first calc gave ATC slightly more.

I did the open hand case analytically. Assuming the hand vs hand probabilities and the analysis are right, hero calling only when showdown equity > 50% yields a 67% win probability after weighting by the combo counts, where I accounted for card removal (somewhat messy).

Note- I only considered suited or off-suit, not individual suit interactions. If same hands are included (e.g. 85o vs 85o) and you give hero half the 50% win probability, the overall win probability reduces to 66%.

It would be good if someone confirmed these results by simulation or analysis.

Were you alluding to the fact that offsuit hand ties are sometimes folds? 8s5h is a dog to 8h5c for instance.

I would have guessed a result below 66% given that most of the time the favorite isn't this strong. Did you have the player calling more than half the time (if you include ties)?

A file with the complete 1326 x 1225 heads-up hand equities was available and floating around the internet several years ago. I just tried to locate the file but came up empty.

I imagine that some 2+2'ers have access to it. (I had it on my old computer but for whatever reason it was not ported over to my current computer.)

Of course, with that file the "open hand" calculation would be straightforward.

I'm going to have to recalculate as I don't think I included all the hand vs hand possibilities. I think it will be less than 66% as it appears that those I left out mostly favor villain.

Don't know if I can get to it today.

I updated the open hand analysis to include the hand vs hand combos I inadvertently left out. Excluding ties, hero has a 64.1% win probability, which reduces to 64.0% if same hands are included (e.g. AKo vs AKo = tie).

The analysis, especially counting combos to include card removal, was a bit tricky (at least for me) so a second analysis or maybe a simulation would lend support to this result or possibly refute it.

Anybody?

statmanhal,

Based upon your analysis, what pct of time does Hero call?

Thanks much.

I evaluated all possible matchups of the 169 hand types (excluding ties =same hands), matchups such as AA vs 85o and 85o vs AA, about 28,400 matchups that generate some 810,000 combos. If one hand win prob. is > 50%, the other is obviously < 50%. Because of this symmetry, 50% of the matchups are called, plus about 1% more if same hands are included (call if win prob >=50%).

That makes sense. After thinking about it for a second or two, I guess my question was pretty silly.

I'm in the middle of a large project, but possibly I'll have some time to devote to this later in the week or next week. (I have no idea if there is any time urgency to this investigation.)

Sorry, perhaps another silly question ....

David's original question stated that Hero calls when favored (equity > 50%).

If one includes only those hands for which Hero is favored (equity > 50%), then Hero's overall average win pct will be greater than if one includes hands for which Hero has 50% equity. I realize that this is obvious.

I think it would be best to exclude any hands for which Hero has 50% equity. Perhaps this is what you are doing. And maybe this issue is not large enough to have a significant effect on Hero's win pct when calling.

The flip side is that if one does not take care in handling suits (as David points out above) and excludes all the hands that initially look to be 50/50, then the overall win pct will be estimated too high. Since one would be excluding a whole bunch of calling hands that have something like 50.3%.

This is an interesting question. At a minimum coming up with an accurate answer is interesting.

So assuming 56% if you know nothing and 64% if you know everything, does that logically mean that you are 60% if one card is randomly exposed?

I assumed hero knows exactly what the win probability is and calls only if it is greater than 50.000...%. Suits can matter if for a given hand type vs hand type match-up (e.g. T8o vs 33 – 50.07%) a particular suit combination results in exactly a 50% probability. I think this would be very rare and thus doing a suit analysis would come to almost the same result.

I bit the bullet and attempted to derive Hero's overall win probability in the scenario in which Villain goes all-in blind pre-flop every hand and exposes his cards. Hero calls whenever his equity vs. Villain's exact hand exceeds 50%.

I set up a Rube Goldberg like process where I connected two different heads-up equity calculators, a program to tally the counts (accounting for card removal, etc.), and a spreadsheet to record the information. One equity calculator was used to determine Hero's calling range vs. a specific hand for Villain, and the other was used to calculate his equity. (Don't ask me why I used two; it made sense at the time.) I performed these calculations for every possible hand for Villain.

I attempted to completely and accurately handle the exact cards (suits) since there are several instances in which Hero's call decision is influenced by how his cards' suits interact with Villain's suits.

Anyway, to make a long story short, here is what I found. I believe these results to be consistent with the results previously posted above by statmanhal.

Hero calls 49.87% of the time. It is slightly less than 50% since there are deals which are pure 50/50 and by rule Hero does not call in those cases.

Hero's average win probability is 64.18% when he calls.

Wow, that’s impressive – connecting to equity calculators and including suits. I used a table of hand vs hand equity that was only to 3 decimals and got 64.14%. Suit inclusion and rounding probably accounts for the 0.04% difference in results but the closeness of the two results should be convincing.

You guys are damn impressive with the maths. Now all I have to do is put this into a video poker "all in face up" machine that only pays the player 63% and open a casino.

Okay, but if you do I (and I assume whosnext) want a cut of the profit. If you lose money because our number is wrong or whatever, you’re off the hook.

Using a similar approach to the one I used to calculate Hero's win probability (and call pct) when Villain exposes two cards, tonight I calculated Hero's win probability (and call pct) when Villain exposes a single card chosen at random.

I find the following:

- Hero calls 56.25% of the time

- Hero's average win probability is 59.19% when he calls.

At first reading, I am somewhat perplexed by Hero's call pct being greater than 50%. It seems that Villain revealing partial information is causing Hero systematically to make calling "mistakes" (over-calling). I double-checked my procedures and am fairly confident in the above results. Perhaps my brain is swimming in all the numbers and I am not seeing the forest for the trees, or something like that.

I'd be interested in hearing anybody's thoughts on this "over-calling" phenomenon (or if you think it signifies an error somewhere).

Calling more than 50% makes sense to me.

I would imagine you end up in more accidental tie and losing scenarios as you make your calling decision based on the partial information you have, but expose yourself to tying and domination scenarios in the process.

Think of situations like QJ and a 9 is exposed. In the perfect information example I am folding to A9, K9, 99, etc but here I am going to probably call every time because there are far more hidden cards that I dominate than dominate me.

I would be very interested to see the calling percent with hands that share cards especially starting at 9/T.

I would just assume in that scenario we begin calling right at or below the median hand in the unknown cards distribution because we block some of the tying hands (i.e. if I have T9 and a T is exposed I am calling but not for T8).

In the "be careful what you ask for", the table below presents Hero's complete calling strategy when Villain exposes a single card at random.

For ease of exposition, suppose that Villain always exposes a spade (it helps clarify how Hero's calling strategy is influenced by the suit interactions).

In the table below, "s" refers to spades (meaning, of course, the suit of Villain's exposed card). When the hand is suited, the word "suited" is used.

Villain Exposes	Hero's Call Range	Hero's Call Pct	Hero's Equity When Calls
A	AA-22, AK-A9	10.59%	60.51%
K	AA-22, AK-A2, KQ-K9	24.39%	59.71%
Q	AA-33, AK-A2, KQ-K2, QJ-Q9	36.47%	58.69%
J	AA-33, AK-A2, KQ-K2, QJ-Q2, JT-J9	47.76%	57.69%
T	AA-33, AK-A2, KQ-K2, QJ-Q3, Qs2, Q2suited, JT-J7, Js6, Js5, J6-J2suited, T9, T8suited	53.10%	57.47%
9	AA-33, AK-A2, KQ-K2, QJ-Q3, Q2 (one spade), Q2suited, JT-J5, J4-J2suited, T9-T7, T6-T4suited, T3suited (excl spades), 98suited	58.12%	57.81%
8	AA-33, AK-A2, KQ-K2, QJ-Q2, JT-J5, Js4, J4-J2suited, T9-T7, T6-T4suited, 98, 9s7, 97-96suited, 95suited (excl spades)	60.71%	58.32%
7	AA-33, AK-A2, KQ-K2, QJ-Q2, JT-J5, Js4, J4-J2suited, T9-T7, Ts6, T6-T3suited, 98-97, 96suited, 95suited (excl spades), 87-86suited	62.51%	58.96%
6	AA-33, AK-A2, KQ-K2, QJ-Q2, JT-J4, Js3, J3-J2suited, T9-T6, T5-T3suited, T2suited (excl spades), 98-97, 9s6, 96-95suited, 87	65.65%	59.34%
5	AA-33, AK-A2, KQ-K2, QJ-Q2, JT-J4, Js3, J3-J2suited, T9-T5, T4-T2suited, 98-96, 95suited, 87-86, 76	69.88%	59.49%
4	AA-22, AK-A2, KQ-K2, QJ-Q2, JT-J3, Js2, J2suited, T9-T4, T3-T2suited, 98-95, 94suited, 87-85, 76, 75 (one spade), 75suited, 65suited	75.84%	59.86%
3	AA-22, AK-A2, KQ-K2, QJ-Q2, JT-J2, T9-T3, T2suited, 98-94, 93suited, 87-84, 76-75, 74suited, 65-64suited, 54suited (excl spades)	81.33%	60.22%
2	AA-22, AK-A2, KQ-K2, QJ-Q2, JT-J2, T9-T2, 98-93, 92suited, 87-84, 83suited, 76-75, 74 (one spade), 74-73suited, 65 (one spade), 65-64 suited, 54suited	85.18%	60.79%

In a nutshell, a larger number of decent hands that make correct folds when seeing two cards would start making bad calls if one of the up cards was turned face down (eg pairs, bad aces and bad kings will often trap with the hidden downcard) than the number of poorer hands that make correct calls when seeing two cards would turn into folds if one of the cards was down. (That only happens when a hand like 82 which would call when seeing 72 stops calling when seeing only the seven.)