Hand matchup calculation - multiway pots

[gulebjorn]

X-posted from Poker Theory because I couldn't get any answers there...

I was just in a NLHE tourney where KQo pushed all-in after three limpers. I was sitting in the small blind with AQs and folded. Two limpers called, holding 44 and 33. Now let's say I called as well. What would you guess that the different probabilities of winning are for all four hands?

To get you started, here are some head-up matchups:

cards %win
4s 4c 50.67
Ah Qh 48.85

3s 3c 50.02
Ah Qh 49.44

Qs Kd 23.75
Ah Qh 75.15

4s 4c 78.48
3d 3h 17.83

So far, this is nothing special and most of you know these % by heart.

Now, what happens if we put these hands in a 4-way pot? I'm sure everyone sees that AQs and 44 are the big favorites, and that KQ and 33 don't stand much of a chance.

Before I let twodimes calculate it, I would have guessed that the probabilties of these hands in a 4-way pot would look something like this:

AQs 38%
KQo 13%
44 42%
33 8%

This distribution would mean that AQ and 44 remain a coinflip, with the pair as a slight favorite, AQ vs KQ is still 3:1, and 44 vs 33 remains 5:1.

But the twodimes result says:

cards %win
4d 4h 32.02
3s 3c 17.09
Kd Qh 17.90
Ac Qc 31.84

As you can see, suddenly AQs no longer dominates KQo as badly as it does headsup, and the probability of 33 winning in a 4-way pot remains almost exactly the same as heads-up. Also, I would expect AQ to do worse, because one Q is gone, and the K interferes with some of the straight possibilities.

So, anyone have an explanation for these differences?

And more importantly, does anyone know of an easy way to guesstimate these probabilities preflop? Lets say that in a tournament, there are two all-ins before you. You are holding AK and you assume that you are up against a small pair and some suited big cards. Is there an easy way of calculating the chance of you winning the race, if we know the numbers for the matchup of the individual hands?

Thanks a lot.

[bobman0330]

Hey, just saw your reply in the Theory forum. (Couldn't get any answers indeed... [img]/images/graemlins/mad.gif[/img]). Anyways, here's inexhorable proof that all my analyses were absolutely correct.

Point 1: AQ suffers mainly from losing a Q as an out.
Result of AQ vs. 44 vs. 33
pokenum -h ah qd - 4c 4h - 3c 3d / qh
Holdem Hi: 1221759 enumerated boards
cards win %win lose %lose tie %tie EV
Qd Ah 447055 36.59 767602 62.83 7102 0.58 0.368
4c 4h 564442 46.20 650215 53.22 7102 0.58 0.464
3c 3d 203160 16.63 1011497 82.79 7102 0.58 0.168
The extra 5% or so it loses in the 4-way matchup is due to the parlay of a K hitting without an accompanying A, 3, or 4.

Point 2: 33 loses little equity in a multiway pot vs. HU against a bigger pair.
33 hitting a set and losing is a very big parlay. The probability of losing to 44 is about 24-1. I'm not really ready to crank through the math of how often it will lose to AQ/KQ, but it's a fairly small amount.

There are 2 effects here, mostly having to do with the PPs. 44 has a ton of equity against 33 HU because it can usually win without improving, and 33 has a hard time improving. Even in one of the coinflip situations, the vast of 44's equity comes from the times it holds up unimproved. Now add 8 overcards it has to dodge in addition to the 2 for 33, and its win percentage drops a lot, because winning unimproved happens much less often. 33 has to improve anyways, and when it does so it usually wins against 1 player or against 4.

KQ is in a similar spot as 33. When it's HU with AQ, it has to hit a K to win. When it's in the 4-way pot... it has to hit a K to win. A lot (though slightly less) of AQ's equity comes from holding up unimproved, which is not a possibility when against the small PPs.