Pop quiz

[damaniac]

I think you are right, but I'm not sure...

Take for example this situation...3-way.

Opponent one bets. Hero, next to act, calls getting 5-1 on a 4-1 shot. Villain 2 raises, first villain reraises. Hero calls again, getting 10-2 (5-1) on a 4-1 shot. Last villain calls. Yet hero has put in 3 bets to win 10 on this rounds on a 4-1 shot, a losing prop in the long run, yet never made a -EV call.

[mack848]

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Eric said that it was a close call between folding the turn and calling the turn/folding the river unimproved.

He did not say that it was close between either of these options and calling down - when you only get ~ half the pot odds.

Would the difference between ~7:1 and ~4:1 not have to be made up by the chance of winning unimproved?

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#1
We can call the turn and make a small profit.

#2
We can call the river and make a small profit.

How could 2 profitable calls ever become a -EV action?

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#1 We can only call and make a small profit on the turn if we fold the river unimproved. If we are obliging ourselves to call the river UI, we no longer have the odds to call the turn (we are now only getting ~4:1 on an ~8:1 shot) - unless we have the best hand a reasonable % of the time (#10%, say).

[Dagger78]

Effective odds. You must call 2 BB to win the current pot +1BB. You may be getting the odds to call on the turn and then the odds to call on the river but that doesn't make your plan (calling the turn and calling the river) +EV when taken as a whole.

We need to know how often he is bluffing.

Take the example again need 2 pair to win getting 7:1 and 10% chance of a bluff. There will be 9BBs in the pot when you call on the river. So your effective odds are 4.5 - 1.

To make the math easier let's play the same hand 80 times.

8 times he's bluffing and you win 10BB (this includes your call on the river)
10 times you improve to the best hand and win 10BB (this includes your call on the river)

You win 180 bets, and it costs you 160 bets to call the turn and the river all 80 times. So you're plan of calling down UI is worth .25BBs if he's bluffing 10% of the time.

Therefore at 7.5% chance of him bluffing it's break even, anything less than that chance and you're losing money by calling down.

Edit : I just realized we're only getting 6-1 or so on the turn as the poster said we had the odds to call INCLUDING implied odds. Also, I didn't factor in the times our opponent is bluffing and we improved when we didn't have to.

[Dagger78]

Here's the fixed numbers.

Let's assume again villian is bluffing 10% of the time, we'll win if we improve to 2 pair, we'll raise the river and he'll call if he's not bluffing, and we're getting 6- currently from the pot.

Let's run the hand 800 times.

We improve to the best hand 100 times and raise
90 of those times he calls the raise and we win 11BBs at the cost of 3BB (the call on the turn and the river raise)
10 of those times we raise, he folds and we win 10BBs, ne didn't call, at the cost of 3BB

Of the remaining 700 times he's bluffing 70 times and we win 9BB at the cost of 2BB.

The remaining 630 times he wins and costs us 2BB.

Net win 1780 bets
Cost to call down 1600 bets
Total net win 180 BB, or +.225BB per hand.

These number include exactly how many bets went into the pot, including our bets and raises, and then how many bets came back into our stack. I did this just for simplicity.

Would someone please correct me if my math is wrong?

[Octopus]

I believe (on 45 seconds thought) that your numbers are right, but you are making this more complicated that it is.

The OP has stated that call/call is better than call/fold because of pot size and the (unstated) probability that he is bluffing. He also stated that call/fold is at least as good as folding the turn. Therefore call/call must be better than fold.

[Nick Royale]

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I think you are right, but I'm not sure...

Take for example this situation...3-way.

Opponent one bets. Hero, next to act, calls getting 5-1 on a 4-1 shot. Villain 2 raises, first villain reraises. Hero calls again, getting 10-2 (5-1) on a 4-1 shot. Last villain calls. Yet hero has put in 3 bets to win 10 on this rounds on a 4-1 shot, a losing prop in the long run, yet never made a -EV call.

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Yeah, you have to look at one street separately. The first call isn't profitable since you should have estimated the risk of getting raised.

[Nick Royale]

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#1 We can only call and make a small profit on the turn if we fold the river unimproved. If we are obliging ourselves to call the river UI, we no longer have the odds to call the turn (we are now only getting ~4:1 on an ~8:1 shot) - unless we have the best hand a reasonable % of the time (#10%, say).

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According to OP we have the odds to call the river...

[W. Deranged]

Time for some algebra, kids:

Let the pot size be P, including villain's turn bet.

Let our probability of improving be i.

Let the probability we win unimproved on the end be u.

Elindaur has told us:

1 / (P + 1) < i

1 / (P + 2) < u

What we want to know is whether we have the odds to call down, namely our chance of improving plus our chance of holding up unimproved is worth the two more bets.

We know that our odds of winning are:

i + (1-i)u

We want to know whether:

2/ (P+3)

which represents the odds that the pot is laying us (the pot size going in + 1 bet from villain on the river + the 2 BB we put in divided by the 2 BB we put in), is smaller than our odds of improving:

[i + (1-i)u] = i + u - iu

Now, notice that:

2 / [P+3] < 2 / [P+2] < (1 / [P+2]) + (1 / [P+1]) < i + u

The very small random correction term, (-ui), doesn't make that much of a different, really. I could do a more full proof but it's not really that interesting.

The point is that if you have odds to call on the turn strictly to improve and the odds to call on the river strictly to snap off a bluff you have collective odds to call down.

So calling is better than folding. (Which is to say nothing of whether raising is better, which it often may be.)

[W. Deranged]

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Eric said that it was a close call between folding the turn and calling the turn/folding the river unimproved.

He did not say that it was close between either of these options and calling down - when you only get ~ half the pot odds.

Would the difference between ~7:1 and ~4:1 not have to be made up by the chance of winning unimproved?

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#1
We can call the turn and make a small profit.

#2
We can call the river and make a small profit.

How could 2 profitable calls ever become a -EV action?

[/ QUOTE ]

This is some pretty good reasoning...

[elindauer]

This post can be answered with very little math. The answer is call, and the correct logic, pointed out by a few different posters, is that if calling the turn and folding unimproved is profitable, than calling the turn and then making a profitable river call is ever more profitable.

Almost half the field got this wrong though, and I believe this is due to a misunderstanding of reverse implied odds. Don't feel bad, I've seen this concept applied incorrectly by poker authors too!

The misconception is that when you have to call 2 bets to see a showdown, you have to adjust your odds lower. It turns out that this is very simplistic logic and is often wrong, as shown in the example. Here's a better description:

If you have a draw that you think will be good enough to win should you complete it, your odds of improving are the maximum odds you need to justify a call. You will never need more money in the pot than this amount. If you have a hand that is worth a river call as well, you adjust your draw odds down. The profit from the river call means you need less money in the pot to make the call correct. The more profitable your river call (ie, the more closer it is to a call down on for it's own sake), the lower the odds you should demand to justify your draw.

Reverse implied odds rarely apply, and even then it's often just a marginal mistake to call. Reverse implied odds apply only when your draw, if you are behind, is slim or non-existent, and the only way you win is if you call down unimproved and win. In this case, you must pay attention to the fact that a pure calldown costs 2 bets, not 1. If your draw even comes close to being worth a call and you have showdown value, you probably have a call.

An example where the river bet might turn a turn call into a fold is something like holding a pocket pair against an overcard on the board. If you are behind, you need 22:1 to continue. You probably aren't even close to calling based on your draw. So the question is, can you call 2 with the hand you have now, unimproved? Sometimes you could call 1 but not 2. That is reverse implied odds in action.

The reverse implied odds concept has been explained very poorly in the past. Hopefully this will help clear things up and save you some money. If you are really folding hands like this... stop!

Good luck.
Eric