Poker Combination Calculation Exercise — My Answer

Hello Ed,

Very interesting exercise, thank you for it.

Just something I think about moving all-in with a 5 cards board which is the nuts. In a theorical way, of course it is an EV+ move, even if fold equity is very low. But in a practical way we should consider what we almost always neglect : rake. In that case there is a little computation to do with rake percentage and fold equity. And not sure it is an EV+ move in high limits.

Best regards,

Zeuwan

Moving all in on a nuts board may not be the correct play when the rake is high compared the the stack size for example, online small stake tables can have very high rakes.

Moving all in on in such a table will just serve to increase the rake and result as a bigger loss when u tie. Keeping the pot small may be your best move here.

Just occassionally people wil fold when you move all in on a nut board – especially at very fishy low stakes tables. I guess its a case of observing which opponents aren’t switch on to whats going on.

All-in on nut boards?

Given that 5% rake is taken up to $3 and the pot exceeds $60 (0.05 * $60 = $3), pushing is +EV. In a smaller pot your fold equity has to exceed the cost of a push (rake). Since every fold gains you a substantial fraction of the pot, that should be easy to do, right?

Example
calculated for two different pot sizes, $10 and $40

You are on the river with 2 opponents in a $10 and $40 pot respectively, facing a nut board. It is checked to you, stacks are $50 and you push.

When checking behind
(will be compared to the scenarios when going all-in)
$10 pot – $0.5 rake / 3 players = 3.17
$40 – $2 / 3 = 12.67

When going all in
Scenario 1: Both opponents fold, you win
$10 – $0.5 = $9.5 – $3.17 = 6.33
$40 – $2 = $38 – 12.67 = 25.33
Scenario 2: One opponent folds, the other calls
($110 – $3) / 2 = $53.5 – $50 wager = $3.5 – $3.17 = 0.33
(140 – $3) / 2 = 68.5 – $50 = 18.5 – 12.67 = 5.83
Scenario 3: Both opponents call
(160 – $3) / 3 = $52.33 – 50$ = $2.33 – $3.17 = (0.84)
(190 – 3) / 3 = 62.33 – 50 = 12.33 – 12.67 = (0.34)

It turns out that, in the $10 pot, both players have to fold once for every 7.5 ($40 pot: 74.5) times the all-in fails – one player has to fold once for every 0.4 ($40 pot: 16) times that both opponents called for the play to be more profitable than checking behind.

So in a small pot the play is to check, which probably is the best option in the bigger pot as well – dependant on your estimate of how often your opponents are folding. However, checking is clearly wrong when the pot is so big that the rake is capped already and shoving is a freeroll for folds.