Is there something magical about having 2 opponents?

[Jordan Olsommer]

Hi all - I was just mucking around with the ever-so-useful Gocee chart (http://www.gocee.com/poker/HE_Value.htm) of race probabilities for certain hands versus x number of random hands, to see what kind of equity edge the hands have.

Calculating the edge is simple, of course - e.g. AA has according to the chart an 85.3% probability of holding up over one random hand, and the average hand in a two-handed game has a 50% probability of winning, therefore AA has an equity edge of 35.3%, since it figures to win 35.3% more than its fair share of pots. Right so far?

If so, then a curious thing arises when you look at the data. Here are the preflop equity edges for AA-TT against 1, 2, and 3 opponents.

AA (1) 35.3%, (2) 40.07%, (3) 38.9%
KK (1) 32.4%, (2) 35.57%, (3) 33.2%
QQ (1) 29.9%, (2) 31.57%, (3) 28.5%
JJ (1) 27.5%, (2) 27.87%, (3) 24.2%
TT (1) 25.1%, (2) 24.37%, (3) 20.2%

For AA-JJ, the preflop equity edge actually goes up when you add a second opponent, but then back down when you add a third! (it goes further down with more opponents for 3+ opponents or for TT or lower). Why is this?? What's so special about having two opponents as opposed to one that makes the edge increase?

[Disclaimer: if you find some glaring incredibly boneheaded basic math mistake in my calculations, please point it out quickly before my window of opportunity to delete this post and save face erodes. Thanks /images/graemlins/smile.gif]

[AaronBrown]

You haven't done anything stupid, although your definition of "equity edge" is not the only possible way to analyze this.

Basically, strong hands with high cards or pairs have the biggest edge with the fewest opponents. Hands with straight or flush possibilities have the biggest edge with the most opponents. In between are hands with both or neither.

There are a few exceptions. One that you noticed is the top four pairs are so powerful, they maintain their edge even against two opponents. It's also true that top suited connectors or one-gap hands play so well as both strong hands and drawing hands that they share this property: AKs, AQs, KQs, KJs, QJs, QTs, JTs.

There's nothing magic about two opponents. Most hands play their best against either 2 or 9 opponents, but a few have their maximum in between.

[AaronBrown]

I should have added that this refers to playing all-in showdown against different numbers of opponents. Don't confuse this with how many players you want in the pot if you hold one of these hands. That's a different question.

If you play at a table of ten, even if only one other player is in the pot with you, you can't use statistics from a two-person game. He's likely to have a much better hand than a random player. In a statistical sense, you are playing against some of the other eight opponents, because his hand probably represents something like the best hand they had among them.

[scottgiese]

I don't think what you've defined "equity edge" has much practical value. You're saying in an N-handed pot, the average equity per hand is 100%/N, which is true. Then if I have a hand that has x equity against a collection of N-1 random hands, then my "equity edge" is x-(1/N).

Well, okay, but I don't really know what that's supposed to be telling us, even ignoring the fact that I'm rarely playing against N-1 random hands.

Perhaps more meaningful is how much better my pot equity in this situation is than that of a random hand. That can be expressed by the pot equity of my hand divided (not minus) the pot equity of a random hand, against N-1 random hands. Your table for this becomes:

AA (1) 1.7, (2) 2.2, (3) 2.6
KK (1) 1.65, (2) 2.1 ...

So the equity of AA in a heads-up matchup vs a random hand is 1.7x the equity of a random hand in the same situation. The equity of AA vs 2 random-handed opponents is 2.2x that of a random hand.

You can see that using this metric, adding random opponents always increases your "edge", if you like.

Don't confuse this with the value of this hand -- it's not. A random hand is "worth" nothing vs one, two, or 10 random opponents. AA, KK, QQ, JJ, TT are worth a lot more.

Nothing magical, just math. But I don't think we're on to anything profound here.

[scottgiese]

Bah, as soon as I posted it, I realized an error. AA has 85% pot equity vs 1 random hand. It has an equity edge vs a random hand of 85%/15% (5.6), not 85%/50% (1.7), since that latter is measuring the edge against all hands in the pot (including itself), which is silly.

Really you only care about the edge over your opponents' hands, not your own hand, since you aren't generally worried about your own hand beating you.

Far more practical, then, is to say the "equity edge" of AA vs 1 random hand is 5.6, and its edge vs 2 random hands is 73%/(27%/2) = 5.4, and its edge vs 3 random hands is 64%/(36%/3) = 5.3.

So AA is worth 5.6 times as much as your opponent's hand, if he doesn't look at his cards and pushes all-in. It's worth 5.4x as much as either of your opponent's hand if two of them do this. 5.4 is less than 5.6, but since you're now getting 2-1 implied odds, it's still a better situation.

If you know a game where people are betting like this, AA is pretty good against 1, 2, 3, 4, or more opponents who bet like this. KK-TT are no slouches either, I bet, but you can do the math.

[Jordan Olsommer]

[ QUOTE ]
So AA is worth 5.6 times as much as your opponent's hand, if he doesn't look at his cards and pushes all-in.

[/ QUOTE ]

...or AA is going to win 35.3% more often than your opponent's hand, if he doesn't look at his cards and pushes all-in. I fail to see how we're not talking about the exact same thing here.

[Siegmund]

The only thing "magical" going on is that the number you are subtracting makes its largest leap downward, from 1/2 to 1/3, at the first step, and as long as the value of a hand decays slowly with an increasing number of opps, you will see this kink at the beginning of your graph and then a gradual dropoff after that.

There is nothing wrong with your calculation; but your calculation doesn't prove that 2 opponents is a good number of opponents to have for all these hands. Remember that every hand that shows down puts the same amount of money in the pot. What you might find more useful is to weigh the loss in equity edge vs. the gain in pot size:

For instance with AA, you can invest $1 and expect to get back
$1.71 = 85% of $2 vs. one opponent all-in for $1
$2.20 = 73% of $3 vs. two opponents all-in for $1 each
$2.56 = 64% of $4 vs. three opponents
$2.80 = 56% of $5 vs. four opponents, on up to
$3.11 vs. a full table.

A hand like A2o, on the other hand, you expect to get back
$1.09 heads up
$1.05 vs. two opponents
$1.01 vs. three opponents
$.98 vs. four opponents
$.91 vs. full table

Some hands are more complicated, such as K2s, which has some high-card value that falls off quickly....
$1.06 heads up
$1.04 vs. 2 or 3 opponents
...and is then replaced by the flush-making value in multiway hands:
$1.05 vs. 4 opponents
$1.09 vs. 5 opponents
up to $1.19 vs. a full table.

[Jordan Olsommer]

That was a very easy to understand and thorough explanation of what is going on here. Thank you very much, good sir.