Scoop or split: the pot math of three boards

In most poker games a hand either wins the pot or doesn't. In Patch-Patch the pot is cut into three equal shares, one per board, and a hand can win any number of them. That changes what "winning" means. Taking two boards out of three can be a small profit or a big one depending on how many players are in, and taking one board can lose chips or break even. This article puts numbers on it.

A third of the pot per board

The rule is simple: each board pays a third of the pot, in whole chips (when the pot doesn't divide by three, the first board takes a third rounded down, the second half of what's left, the third the rest). A tie on a board shares that board's third. Winning all three boards alone is a scoop, and takes everything.

Here is a heads-up hand at a 600-chip pot. You win boards 1 and 3 with straights; your opponent wins board 2 with a full house.

Both players put 300 into that pot. You got 400 back: a profit of 100, a third of what you risked. Your opponent lost 100. Two boards out of three, heads-up, is a modest win.

What each result is worth

Say every player in the hand put the same amount in, call it one stack's worth, S. Heads-up the pot is 2S and each board is worth two thirds of S; three-handed the pot is 3S and each board is worth exactly S.

Two lines in that table are worth remembering:

• Heads-up, break-even is one and a half boards. Win one board and you lose a third of what you put in. Win two and you gain a third. Only the scoop doubles your chips.
• Three-handed, break-even is one board. When three players have put in the same amount, winning any one board gets your chips back, two boards doubles them, and a scoop triples them.

This is why the same hand can be a good call three-handed and a poor one heads-up: a hand that expects to win one board is break-even against two opponents and a loser against one.

The one-board rule for calling

Most decisions in Patch-Patch come down to a pot-sized bet in front of you: every bet and raise is the size of the pot (pot-only betting explains the rule). Facing one heads-up, the arithmetic is the same every time. The pot is P, your opponent bets P, and calling costs you P for a share of a final pot of 3P. Each board of that final pot is worth exactly P.

So a call pays off when, from here, you expect to win more than one board on average, counting a split board as half. Less than that, and folding is cheaper over time. More betting on later streets changes the details, but the rule is a good first check, and it applies to a pot-sized raise too: calling a raise to four times a bet also buys a third of the final pot.

Some examples, heads-up, facing a pot bet on the flop:

• Strong on one board, behind on the other two: about one board. A marginal call, often a fold if the strong board isn't safe.
• Ahead on one board and drawing well on another: comfortably more than one board. A call, and sometimes a raise.
• Drawing on all three boards with nothing made: rarely more than one board, even when the draws are good. Usually a fold.

Locked boards and the freeroll

The scoop changes the picture when one side has a board that can't be lost.

If you hold a lock on two boards heads-up, you can't lose chips on the hand, whatever happens on the third: every chip that goes into the pot returns at least two thirds to you. Bet and raise; more chips in the middle is simply more profit, and if the third board comes good you scoop.

If your opponent has a lock on one board, your best result is two boards: a profit of a third of what you put in, against a loss of all of it if he wins a second board. You need to be a strong favourite on both remaining boards to want more chips in.

Three-handed, the same logic is stronger. With all three players in for the same amount, one locked board already returns your contribution, so a player with one lock and a good second board has little to lose by betting.

Splits

A tie on a board divides that board's third among the tied players. Splits come up more in Patch-Patch than in Hold'em, because with twelve cards each, two players can easily hold the same straight. Two things follow:

• A split board is worth half a board in every calculation above.
• A tie on any board ends the scoop: the celebration and the whole pot need all three boards won alone.

If a board's share doesn't divide evenly between the tied players, the odd chip goes to the first of them clockwise from the button. Over many hands it evens out.

When to play for the scoop

The scoop is the only result that doubles your chips heads-up, so it pulls at every decision. It is worth chasing when it's cheap: when you are already sure of two boards and the third is open, or when your split gives every board a real chance without leaving one hopeless. It is not worth chasing at the cost of the boards you already have. A split that turns a sure board into a coin flip for the sake of a scoop gives back more often than it takes.

The useful habit is to count boards before each decision: how many do I expect to win from here? Heads-up, more than one and a half means you're ahead in the hand; more than one is enough to call a pot bet. Three-handed, one board is break-even. Everything else is detail.

For the rules behind all this, see the Patch-Patch rules; for how to give every board a chance, see How to split 12 cards into three hands.