[Buzz]

O.K. let’s revisit this topic (which is honestly not very well settled in my own mind). Something seems not quite right about equating winning half a pot twice to scooping once.

I agree that winning half a pot twice is equivalent to scooping once and losing once. But.... well, let me try to explain with an example from a passive $4/$8 game.

Let’s say your opponents will contribute a total of $80 to the pot and it will cost you $24 to play a hand all the way to the showdown.
• When you scoop, you will be awarded all $104 in the pot of which $24 is your own investment.
• When you split the pot, you will be awarded $52, of which $24 is your own investment.

Each time you split such a pot, you only actually win $28. When you win half such a pot twice, what you actually win is $56 (and you get your own investment back). If you go home after splitting two such pots and losing none, you’ll be $56 richer.

On the other hand, when you scoop once, you actually win all $80 your opponents have contributed to the pot (and you get your own investment back). If you go home after scooping one such pot and losing none, you’ll be $80 richer.

You have to scoop one pot and also lose another to end up with the same number of dollars as splitting two pots and losing none.

If instead, you scoop one pot and then get out of the next (instead of contributing and losing $24), in that case, scooping the one pot is better than splitting two pots.
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I realize that winning half the (same sized) pot twice is the same as scooping once and losing once. (You get your own $24 back plus winning a total of $80 invested by your opponents).

But if you scoop once and don’t lose at all, you’ll end up with more money than if you win half the pot twice and don’t lose at all.
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Let's consider it from the standpoint of odds after the turn.

When you’re looking at your hand and the board after the turn,
• if it will cost you $16 more to continue to see the showdown if you like the river card,
• if there will be $104 in the pot at the showdown if you continue (including the $8 you’ve already invested plus the $16 it will hencefore cost you to see the showdown), and
• if seeing the river will only cost you $8 if you miss (instead of $16 because you will fold if you miss) then at that point in the hand,
• then you are getting 88 to 8 implied pot odds to win the whole pot.

(The pot at that point contains $8 of what was formerly your own money plus at the showdown it will contain $80 of your opponent’s money).

If you only win half the pot, you’ll be awarded $52, of which $16 is what you will henceforth contribute if you call the turn bet.

When you’re only playing for half the pot, if you continue, you’ll be playing to win $36. You’ll be investing $8 to possibly win $36. You’re getting 36 to 8 implied odds for half the pot.

So when you’re playing to win this whole pot, on the third betting round your implied pot odds are 11 to 1, and when you’re playing to win half this pot,
on the third betting round your implied pot odds are 4.5 to 1.

I'm not done thinking this out yet, and I'm not sure I ever will be.

But although I agree that scooping (the same sized pot) 2000 times in 10000 plays to the showdown is the same as winning half the (same sized) pot 4000 times in 10000 plays, there still seems something amiss in equating two half wins to one scoop.

If there is a "blatant" mistake here, I'm oblivious to it (although I'm sure I'm very capable of making a mistake).

It certainly is not my intention to lead anyone astray. I am, as always, looking for the truth. Alas, sometimes I make mistakes. I will continue to correct them if I see them. (In this regard, the record speaks for itself).

Buzz