Rundowns: why connected four-card hands play well in PLO

A rundown is a starting hand of four cards in consecutive ranks, such as 9-8-7-6 or J-10-9-8. In Pot-Limit Omaha (PLO) these hands are worth playing far more often than the equivalent four cards in Hold'em would be, and the reason is arithmetic about straights. This post counts the hands, counts the straights, and then says where the shape stops being good enough.

How rare a rundown is

You are dealt one of C(52,4) = 270,725 different four-card hands. Count ace as both low and high and there are 11 sets of four connected ranks: A-2-3-4 up to 10-J-Q-K, and J-Q-K-A on the end. Each set can come in 4 × 4 × 4 × 4 = 256 suit combinations, so there are 11 × 256 = 2,816 rundowns. That is 1.04% of starting hands, about one deal in 96.

So the hand is rare, and you will fold most of what you are dealt. When one arrives, you want to know what it is worth.

Why connectedness makes straights

Omaha makes you use exactly two hole cards and three board cards (the exactly-two rule). A rundown gives you six two-card pairs to choose from, and every one of them is a pair of connected or near-connected ranks. Take 9-8-7-6:

• Board 10-J-Q: 9 and 8 from your hand make 8-9-10-J-Q.
• Board 10-J-8: 9 and 7 make 7-8-9-10-J.
• Board 5-4-3: 7 and 6 make 3-4-5-6-7.

Those are three quite different boards, each of which gives you a straight. A hand with scattered ranks has far fewer combinations that fit together. This is the same effect that produces the big draws in wraps, and a rundown is the hand most likely to flop one.

The numbers

I counted every possible board against three fixed hands. Suits do not matter for straights, so only the ranks are fixed. After you hold four cards, 48 remain, so there are C(48,3) = 17,296 possible flops and C(48,5) = 1,712,304 possible five-card boards.

Two things stand out. First, 9-8-7-6 is ahead of K-9-5-2 by a factor of 2.4 (27.5% against 11.4%), and on the flop alone by a factor of 3.3. Second, the high rundown A-K-Q-J does much worse than the middle one. It sits at the end of the rank line, so fewer sets of three board cards fit around it: nothing above the ace, and only the ten below the jack. A middle rundown has room on both sides, which is also why it can make a wrap.

This is a count of any straight. It says nothing about whether yours is the best straight, and that is where rundowns get into trouble.

Where rundowns go wrong

The low end of the hand makes second-best straights

With 9-8-7-6, a board of 10-J-Q gives you a straight that is not the nuts: anyone holding K-9 (a king-high straight) or A-K (the ace-high one) beats it. A straight that is not the best one can be worth a lot of chips on a board that also offers a flush or a full house. The same applies to 5-4-3-2: it makes plenty of straights, but several of them are to the low end where a higher straight is also on the board. For the general version of this trap, see why the second-best flush is a trap, which works the same way.

No pairs, so no sets

A rundown has four different ranks, so it can never flop a set. Its value is all in straights, plus whatever flush draws the suits add. A hand with a pair has a second way to win, and a rundown has one.

Suits decide how good it is

Of the 256 suit combinations of a rundown, 36 are double-suited (two suits, two cards each), which is 14.1%. Those hands also carry two flush draws, and double-suited hands are worth noticeably more than rainbow ones. A rainbow rundown has the straights and nothing else.

Practical rules

1. Prefer the middle. 9-8-7-6 and 10-9-8-7 have room on both sides. A-2-3-4 and J-Q-K-A do not.
2. Add suits where you can. A double-suited rundown is a better hand than a rainbow one with the same ranks.
3. Be careful once the board pairs or goes three to a suit. Your straight is then the weaker hand against a full house or flush.
4. Check whether it is the best straight before you put chips in. The count above is for any straight, not the nuts.

Checking it in the app

The listing for Omaha Poker: PLO Offline Game says it gives a pre-flop starting-hand grade, so you learn which four-card hands are worth playing, and live hand insight with your win chance and your outs grouped by what they make. The listing does not say how the app grades rundowns specifically, so the figures here are game maths rather than app behaviour. Dealing yourself hands and looking at the grade against the shape of the four cards is a quick way to see how the app treats connected hands for yourself.