Counterfeiting in Omaha Hi-Lo: when the board kills your low

Counterfeiting is what happens when a board card repeats the rank of one of your low hole cards. It is how a hand that looks like the nut low ends up worth much less, and it is why experienced players want three low cards in their hand rather than two. This post works through two examples and then puts exact numbers on how much a third low card helps.

It assumes you know how a low is built: five different ranks, all 8 or lower, ace low, exactly two from your hand and three from the board. How a low hand is made in Omaha Hi-Lo covers that, and the exactly-two rule explains why you cannot borrow a third card from your hand.

Two ways a board card counterfeits you

Your own low can never get worse when another board card arrives, because more board cards only add combinations. Counterfeiting is about what the card does to everyone else, and to the cards you still need.

1. It hands your key card to your rivals

You hold A-3-K-Q and an opponent holds 4-6-K-Q. The flop is 2-5-7.

• You make 7-5-3-2-A (ace and 3 from your hand, 2, 5 and 7 from the board).
• Your opponent makes 7-6-5-4-2 (4 and 6 from their hand, the whole flop).

Lows are compared from the highest card down: both have a 7, then your 5 beats their 6, so you are ahead.

The turn is another ace. Your hand is still 7-5-3-2-A, no worse than before. But the opponent now has an ace on the board to use: 4 and 6 from their hand with A, 2 and 5 from the board make 6-5-4-2-A. That beats your 7-low. The ace you held put you ahead, and it is now available to every player. You have been counterfeited.

2. It leaves the board one usable low card short

A board card with the same rank as one of your hole cards cannot be used with that card, because a low needs five different ranks.

The board is 2-3-4-K-K. Compare two hands:

• A-5-K-Q makes 5-4-3-2-A, the best low there is. Ace and 5 come from the hand, and 2, 3 and 4 from the board.
• A-2-K-Q has no low at all. With A and 2 in the hand, the board can only supply 3 and 4. There is no third card of 8 or lower that differs from your hole cards, and the 2 on the board is dead to you.

Both hands hold two low cards on the same board. A-2 repeats a rank the board already shows, so it is counterfeited, while A-5 does not. On this board the two best low cards are the worst pair to hold.

Why a third low card helps

With three low cards in your hand you have three different pairs to build from. Take A-2-3-K on the board 2-4-8-K-K.

• A-2 cannot work. The board offers only 4 and 8 once the 2 is excluded.
• A-3 can. Ace and 3 from the hand, 2, 4 and 8 from the board, make 8-4-3-2-A.

A-2-K-Q gets nothing from that board. With three low cards the board can repeat one and you still have a pair that avoids it. A-2-3-4 has six pairs to choose from.

The figures below come from counting every possible five-card board against a fixed hand, so they are exact. After you hold A-2-K-Q, 48 cards remain and C(48,5) = 1,712,304 boards are possible. On 1,057,824 of them (61.8%) the board shows three or more different ranks of 8 or lower, so a low exists for somebody. You make one on 625,344 and miss on 432,480, which is 40.9%. The same count for other hands:

The table does not depend on which low ranks you hold: 6-7-K-Q gives the same counts as A-2-K-Q. Only the number of low ranks matters for whether you make some low. The ace and 2 matter for whether it is the best one.

Why A-2-3 is the strongest protection

Three small cards do two jobs. They give you the spare pair described above, and they make the lows you can build the lowest available. A-2-3-K has both.

There is a small cost: you hold cards the board would otherwise supply, so boards showing a low fall from 61.8% of the total with A-2-K-Q to 58.3% with A-2-3-K. That 3.5-point fall is small beside the drop in missed lows.

What follows at the table

1. Count your low ranks. A third low card is worth more than a high card in the same hand.
2. Look at what the board repeats. If the board shows 2 and 3 and you hold A-2, you are short of a low card. If it shows 2 and 3 and you hold A-5, you are not.
3. Remember the quarter. Quartering shows what a tied low half is worth.

Checking it in the app

The listing for Omaha Poker: PLO Offline Game says it has Omaha Hi-Lo (Omaha 8 or better) tables, hand history in which you can replay any pot street by street, and live hand insight showing your outs grouped by what they make. The listing does not describe how the app treats counterfeited lows, so the figures here are game maths, not app behaviour. Replaying a showdown from the hand history and checking which ranks the board repeated shows the effect on your own hands.