overcards v. two pair

[bobman0330]

So, I've been pondering the question of who has a playing advantage in a HU situation, two overcards, or a medium-small PP. Obviously, this question varies a lot depending on limit or big-bet, position, etc. the PP wins a small amount from set value, which I'm more or less discounting for the moment.

Anyways, here's the hypo:

One blind limit holdem. Folded to the button, who raises. Both players are aware of the other's general holding (if i'm the button holding 66, i know the opponents has 87 or better. If the opponent has 87, he knows I have 66 or worse). Now, suppose the blind can pay X amount of money to have no future betting on the hand. What would that amount be in various situations, when would it be 0, etc.? Assume variance isn't an issue. I chose limit because it's simpler to analyze, and i think it's a closer issue, due to the greater availability of the call-down as a play.

In any case, I have no idea of any of the answers to this question. Thoughts?

[blaze666]

52% for the small pair to 48% to the over cards, unless the over cards are suited and connected. use this hand probability tool: http://www.pokerlistings.com/


short stacked, i'd push with any pocket pair in a tournament. and in a ring game, i'll usually call the big blind with a pocket pair, simply for set value, nothing els.

[bobman0330]

I'm familiar with the hot and cold equities. What I'm interested in is how HU playing dynamics affect things. For example, a small PP has an advantage in being able to force overcards to fold if they don't hit the flop. (5:1 pot odds on a 7:1 shot.) Small PPs lose the opportunity to turn or river a set, but this is much less important.

The key equalizer is that overcards have frequent bluffing opportunities, especially if they are aware (as stipulated) that the opponent has a small pair. I'm fairly confident that it should be possible to construct a complete game theoretical analysis of this situation and stipulate a proper Nash strategy. I'll give it a shot later, but I'm almost certainly not skilled enough to succeed.

[nate1729]

Hey, I'm nowhere near being able to figure this out, but make life a little easier on yourself to begin with and imagine that there are two only two further rounds of betting (i.e., the turn and river are dealt at the same time.)

--Nate

[bobman0330]

I don't like this, because it materially affects the drawing odds of calling with two overcards.

I see two better ways of dissecting this problem. First, we should separate out each of the 5 different flops (flushes and straights discounted, at least for now. That's another key abstraction):
1) no one improves.
2) PP improves, OC doesn't (OC is dead)
3) OC improves, PP doesn't.
4) both improve. (OC is dead)
5) OC flops a hand that PP can't beat. (PP is dead)

Note than unless OC flops something like 887 to 87, it doesn't matter (much) how much he improves. It's a reasonable assumption to clump all his other improving hands together, and call him drawing dead (i think).

Another simplification is to work out the equity of the river subgame separately and just add it in to the calling equity of the turn.

[nate1729]

OK, OK.

Simplifying:

-If the pocket pair hits a set, he wins.
-If the pocket pair does not hit a set, he wins if and only if the overcards hit.

The pocket pair has 5 chances to hit 2 of 48 cards, so is almost exactly 20% to hit his set by the end. If he does not hit his set, the overcards have 5 chances to hit 6 of 46 cards, so are 52% to hit. So the overcards only win 41.6% of the time. Thoughts on how to handle this?

[nate1729]

In case anyone out there is curious/working on this: the pair is 51.8% to win in the ranges we specified. If we neglect the times it hits a set on the turn or river, and only pay attention to sets/pairs/high card rankings, the pair would be a 54.3% favorite.

[bobman0330]

If you assume the PP never turns or rivers a set, i think the equities come out about right.

So the flop equities are:
1. OC hits, PP doesn't. (OC = 1)
2. PP hits, OC doesn't. (OC = 0)
3. Both hit. (OC = 0)
4. Neither hit. (
This simplification eliminates the usefulness of distinguishing case 5.

[nate1729]

If the PP never turns or rivers a set, it's like 54.5/45.5... PokerStove says the true equities are 50.5/49.5.