I disagree, I think the odds are the same in both cases. I think the distinction "fresh money" is specious.

Another example - pot is $100 , players have $100 each, and my hand has 40% equity.

Scenario 1: I bet $100 and opponent calls. The pot is $300 and I put in $100 of it.
Scenario 2: Opponent bets $100 and I call. The pot is $300 and I put in $100 of it.

In both scenarios I put in $100 and my expected share of the final pot is $300 multiplied by my hand's percentage equity, or $120 in this case. So in both scenarios, my play's EV is +$20.

[Note - in scenario 1 my EV is actually even higher as the opponent may have folded, but for the purposes of this discussion we can assume the opponent is always calling]

Now there's a word you don't see much (specious).
"superficially plausible but actually wrong"

I don't know how to respond.

I'm laughing at myself. I told you what I know is the ice cold truth, something you should definitely get straight because misunderstanding how to apply the math is a huge leak. Your response is you disagree and what I know is the truth is "specious."

Meh. Believe what you want to believe.

When the pot is $300 and you put $100 into it, doesn't that make the pot $400?

Or was the pot $100 before either you or your opponent put $100 more into it? (in which case the pot becomes $300).

I'll figure it both ways for you.

• Way 1.
  If the pot was originally $200 and became $400 when both you and your opponent added $100 more, and if your pot equity is 40%, then your fair share if you split at that point would be $400*0.40=$160.
  
  But if neither of you had bet the last $100, the pot would still be $200 and if your pot equity is 40%, then your fair share if you split at that point would be $200*0.40=$80.
  
  That $80 plus the $100 you'd save by not making the last bet would amount to $180.
  
  $180 is $20 more than $160. By not making the last bet, you'd save $20.

• Way 2.
  If the pot was originally $100 and became $300 when both you and your opponent added $100 more, and if your pot equity is 40%, then your fair share if you split at that point would be $300*0.40=$120.
  
  But if neither of you had bet the last $100, the pot would still be $100 and if your pot equity is 40%, then your fair share if you split at that point would be $100*0.40=$40.
  
  That $40 plus the $100 you'd save by not making the last bet would amount to $140.
  
  $140 is $20 more than $120. By not making the last bet, you'd save $20.

In either case, the money you'd save by not making the last bet is the same, $20. That's because your equity is only had 40% of that last bet. (I'll explain more thoroughly in the next paragraph).

If you put $100 fresh money into the pot and your opponent does likewise, then $200 fresh money goes into the pot. When your pot equity is 40%, only $80 of the $200 is your fair share. (And $120 is your opponent's fair share). You thus lose $20 on that last bet while your opponent makes $20.

The only reason to bet when you only have 40% equity is to falsely convince your opponent you have a better hand than he does, so that he will fold. If you can't convince him that you have a better hand or if you're not setting him up to call when you bet your winners in the future, then it's purely stupid to bet.

Buzz

It looks like we can't really get any further, I've presented my logic, which nobody has found a problem with, but you just disagree with the conclusion.

I said "Another example - pot is $100 , players have $100 each, and my hand has 40% equity." $100 (the existing pot) + $100 (my bet) + $100 (opponent's bet) = $300 final pot size.

That doesn't change the fact that betting shows a direct profit of $20. Your arguments only show that there may be a way to show an even bigger profit (which I'm not disputing of course). We seem to be arguing over whether the profit should be $20 or $40. They are both profits though. You don't say that you made a loss because you didn't make the maximum possible profit.

Those two reasons almost always apply, i.e.

• By betting you give the opponent a chance to make a mistake right now
• By betting you make the opponent more likely to make a mistake on future hands

There are more reasons to bet too:

• By betting the current street, you remove the chance for the opponent to put you to a difficult decision on a later street, where you might make a mistake (i.e. you close the action now when you know you are making a profit, removing the uncertainty that you may make a loss on a later street)
• You may have under-estimated your equity and may in fact be ahead
• You are out of position, and the opponent is likely to bet if you check
  

The last one is really important. The list of reasons to check is a lot shorter than the above list:

• Hoping opponent checks behind and also checks the hand down to showdown

Playing against aggressive players shorthanded, this is relatively rare compared to all the situations in the "reasons to bet" list. And hoping your opponent doesn't do what you don't want him to do, often doesn't seem to work out

Clarification for anyone skim-reading this post, we are talking about a specific scenario here -- in other scenarios the reasons to bet or check may differ, e.g. there are cases where you check to induce a bluff; this is not one of them

Consider the situation where we're first to act in my original scenario "pot is $100 , players have $100 each, and my hand has 40% equity." We judge that the opponent is extremely likely to bet if we check. So the action will either go: we bet and he calls or folds; or we check, he bets, and we call.

We call because it's profitable to call by $20 as we already agreed on. However, if we bet first then our profit is at least $20 (which you posted math suggesting that you agreed with !). Our profit (both immediate and future) may be more than $20 though, for all the reasons in the bullet points above: he might fold; he might actually be behind and call; he might play wrong on a future hand; etc.

So the profit is at least $20 in either case, but only by betting are we likely to make a bigger profit.

I would only check and go for the $40 profit if I thought that the chance the opponent checks it down is greater than the chance of all the reasons I just mentioned to bet happening. I.e., the opponent was very likely to also check it down and he is even more likely to call if I bet and my hand was such that the future cards would make my future decisions easy, e.g. if my opponent bets the next street I would be nearly 100% sure that the turn card worsened my hand and it is now safe to fold. Because it's certainly an easy decision to go all in when you know it is showing a profit

In a way. I'm saying there's a greater net without the final bet. That is, if you only get back 40% of your and your opponents final bet, you only get back 80% of the final bet. You lose 20% of your final bet.

If there is no last bet, you have no loss on the last bet.

Wow. You lost me somewhere in there. When you responded to str8 or better, you seemed to think the odds you're getting for calling are the same as the odds you're getting for initiating money into the pot. I tried to explain why they're not. That's all. I'll iterate it one more time for you, and I'll put it in a pretty color for you.

The odds you're getting for calling are not the same as the odds you're getting for initiating money into the pot.

Buzz

You have a loss on the last bet.

If there is no last bet, you have no loss on the last bet.

I'll try again to explain.

If you're heads-up and you only get 40% of a pot, you show a net loss. But because we consider money you put in the pot on the first three betting rounds no longer yours, you may show a loss on the last bet bet, but recover enough of what you and your opponent put in the pot on the first three betting rounds to more than cover your loss on the last bet. This is almost always the case in a fixed-limit game.

If the last betting round was heads-up, but you had more opponents earlier, and if you get 40% of the pot, you may show a net profit overall while still showing a loss on the last betting round.

Either way, without any fresh money going into the pot on the last betting round, you show a greater profit.

True. And that's a good reason to bet. His only mistake would be folding. If he's absolutely not going to make that mistake then you have no reason to initiate fresh money into the pot.

If he's highly unlikely to make the mistake of folding a winning hand, then it becomes a question of how often he would fold the winning hand. (That's another issue).

Yes. That's another possibility.

However, we're talking about odds here. You seem to be under the false impression that you win the same amount whether you initiate money into the pot or not. That's only true if your opponent will bet if you don't. And your opponent won't always bet if you don't. Your opponent can make the rather common mistake of not betting when he should. By betting yourself, you take away the opportunity for your opponent to make this mistake when he's ahead.

Think of it this way: Aren't you betting because you don't want to make the mistake of missing a bet you should make? Isn't it a mistake for you to end the betting with a check when you have the winning hand? Well... if you get the chance to do so, give your opponent the opportunity to make that mistake when he has the winning hand.

See it?

Off the top of my head, that seems true if one of you gets all-in.

Always a possibility.

That's not a good reason to bet.

??????

Probably. I think you're missing the point.

Correct to here.

Whoa. If we bet first and Villain raises, our profit may be 0, or in some cases we may show a loss. And that's not what the math I posted suggested. Pas de tout.

That's true.

Wrong.

Wrong.

Buzz

This is interesting! Why aren't we afraid that by raising we are building a pot for a better flush given that we have fourth nut draw?

Going a bit further Jeff Hwang in his book Pot-Limit Omaha says we should only draw at nut flushes so couldn't we also say that applies to betting/raising with non nut flushes.

I realize his book is intended for players starting out in Omaha but am interested in your responses.

No.

Assuming no full house or better, the probability of winning with a flush depends on (1) whether an opponent has a better flush or not.

Assuming no full house or better, the probability of winning with a flush draw depends on (1) whether an opponent has a better flush or not (2) IF the flush draw is positive. When you multiply those two probabilities together, your chance with a flush draw is substantially lower than your chance with a flush.

Buzz

Buzz once again please excuse my ignorance but what do you mean by a "positive" flush draw.

As I re-read what I wrote, I can understand your confusion.

I meant, if the flush becomes enabled.

Someone with a non-nut flush already (assuming no full house or straight flush is possible) only has to worry about one thing: losing to a higher flush.

Someone who doesn't have a flush yet has to worry about two things: first the same thing someone with the flush already has to worry about, and in addition making the flush.

I'll use numbers and maybe it will become clearer.

When Hero has two hearts and the board on the turn also has two hearts, the probability of making a heart flush from Hero's perspective is 9/44=~0.20.

Six handed, when Hero already has the second nut (usually king) flush on the turn, assuming all opponents have seen the flop and anyone with the suited ace of hearts, a set, or two pairs would be continuing, the probability of losing with that king heart flush is roughly 0.50. (Actually a slightly greater danger than bumping into an ace high flush is the possibility of the board pairing and losing to a full house or quads).

Thus the probability of winning in a six handed game with a king heart flush when the ace is not on an unpaired board is ~0.50.

In a six handed game when Hero has two hearts and the board on the turn has two hearts, the probability of (1) making a flush on the river and then (2) winning with it is
~0.20*0.50=~0.10.

In other words,
• when Hero already has the second nut flush on the turn, his probability of winning is 50%.
• when Hero is drawing to he second nut flush on the turn, his probability of winning is only 10%.

I hope those numbers make it crystal clear to you.

Jeff gave good advice when he advised not to draw to a non-nut flush. (It's, of course, more complicated because usually Hero would have multiple draws with a non-nut flush draw only being one of them).

But if you already have a non-nut flush, at least if it's the second nut flush, you probably should play it, rather than fold it. With a baby flush, or a ten high flush you just have to use your judgement.

Hero, by the way, is not drawing for the spade flush after the turn. Hero already has made a straight (Broadway) on the turn.

After the flop, Hero should not draw for the ten flush alone, but I don't think he is. Instead, after the flop Hero has multiple draws: a straight, the non-nut flush, the back door low.

As stated in my first post in this thread, I like simply calling Villain's flop bet. Hero has improved with the flop, but is still drawing.

Buzz

Buzz, what is your opinion about the starting hand in the OP, and the position it was played from? I've decided to learn PLO8, and I bought Dan Deppen's book. In his section on starting hands he said that hands like the one in the OP (two big, two wheel, no Ace) were "sucker" hands, and the best they usually do is to flop marginally in both directions. I've been folding these hands, even at late as the cutoff. How playable are they, in your opinion?

when we're first to act on the river and expect villian to check behind hands we beat that would call a bet and bet hands that beat us with which he'd flat call our bet (rather than raise), then betting is likely better than checking even when behind (like in the 40% equity scenario) and have zero fold equity.

however, when we're last to act on the river, behind and with no fold equity, betting just costs us money.

I think Dan still reads this forum. He can answer questions about his book better than I can.

Here's how ProPokerTools rates (K2)(T3):If you're playing six handed pot-limit, do you want to play a hand from the cut-off position that is probably the second best starting hand at the table or not?

I think "it depends."

If you're learning the game (pot-limit Omaha-8), I think following Dan's advice about these hands is a good approach.

Buzz

Since it was my hand, I guess I'm the best one to explain my thinking here.
When I looked at this 23 hand, I liked it for the fact that I was one off the button, was double-suited with two Broadway cards, AND, most importantly, saw 4 folds in front of me and no raises. If nearly everyone is limping along, I fold this hand. If the pot is raised, I fold this hand. Since neither event happened, I feel as though my outs to the nut low (namely, those that include an ace) are very live. Moreover, the high cards and suits give my outs to scoop. So I'm hoping to see a cheap flop and getting away from the hand if I miss the flop, where I'm initially defining "missing the flop" as a flop with no ace. That didn't happen. Instead, I flopped an OESD and a FD, which caused me to reassess my hand. When villain pots the flop, I figure he has a set, two pair, or a huge draw. Even with the first two, I have outs to pass him, and I have position.

Not sure what difference putting it in colour makes. It's obvious to me that the odds are the same. It's obvious to you that the odds are different. It seems we are at an impasse on this point.

Villain cannot raise. Maybe you have misread the scenario we are discussing? I'll restate it for clarity.

The flop has been dealt. There is $100 currently in the pot. Hero has $100 in chips left in his stack. Villain has $100 in chips left in his stack. The other players have all folded. There's no rake. Hero is first to act, and believes that his hand has 40% equity.

To keep the discussion simpler, Hero can either bet $100 or check, and villain can either bet $100 or check or call. (See below for discussion of other bet amounts)

So why is this wrong? The possibilities are:
* We fold: win $0.
* We bet : wins $20 if villain calls or $100 if villain folds.
* We check-fold: wins $0 if villain bets, $40 if villain checks.
* We check-call: wins $20 if villain bets, $40 if villain checks.

We could allow other bet sizes. But hopefully it's clear (assuming we don't do something stupid like raise and then fold) that the profit is bounded by the $20 - $40 range. For example if we bet $30 and villain calls, we profit $34. Or if we bet $30 and villain goes all in, we call, and win $20 ( as reckoned from the flop decision, not the automatic call)

It's a fundamental reason to bet. We show the same profit when villain bets after our check, as we do when we bet and he calls. So, according to the Dominance Principle, we just need to compare the other cases (i.e. we bet and villain folds, vs. we check and villain checks).

Check my table of possible outcomes above. If Villain folds to our bet just 25% of the time, then we make a profit of $40 by betting. This already matches the best case scenario of check-calling. Even if we knew villain checks behind every time we check, it's still more profitable to bet if he is going to fold more than 25% of the time

Or if villain only folds 10% of the time, but he bets 60% of the time when we check; then betting wins $28, and check-calling wins $24.

In reality you don't go through all this math every time, you just remember that the fold equity has a much bigger 'weight' in this calculation than the "villain lets us off the hook equity".

If my logic isn't convincing, maybe David Sklansky's would be; check page 213 of 'The Theory of Poker', where he discusses when you should bet first on the river even though you know you are an underdog, taking into consideration fold equity and how likely it is for villain to check behind with a better hand and so on. Although the scenario is different, the principles are similar.