Another Fourth St. Decision

[David Sklansky]

The pot is 3 handed. The board is Ad Kd 7d 2c. The game is 10-20 and the guy to your right bets $20. You somehow know that the player to your left has the lone ten of diamonds and an irrelevant rag. You have two pocket sevens. A raise will definitely fold the Td and a call will definitely keep him in. If you raise we will stipulate the first guy will only call. Also for simplicity sake we will not consider the river bet. Assume it doesn't exist.

The first bettor will make his bet with any AQ, AJ with the jack of diamonds, KQ with the queen of diamonds, KJ with the jack of diamonds, QJ both diamonds, Q9 both diamonds, or J9 both diamonds. Given he will play and bet all those hands but no others, with what size pot should you raise vs. call his $20 bet?

[deadbart]

Well, I was the first person to correctly answer the last Fourth Street Decision problem, but you gave all the credit to pudley4. I've learned my lesson - I'm just going to estimate this one. /forums/images/icons/wink.gif

The bettor can have 9 hands that don't have a diamond (AQ) and 15 that do. If he does have any diamonds, you just want to call, because it only costs you one bet and you get the same number of bets into the pot. The weighted average of diamonds left in the deck is (9*9+12*8+3*7)/24 = 8.25. 8.25/42 is pretty darn close to 20%, so the diamond will hit 20% of the time. So, raising will save you the pot 20% of the times that the bettor doesn't have a diamond, or 0.2*9/24 = 7.5% of the time. So you want to raise when 7.5% of the pot is greater than the amount it "costs" you to raise.

Raising costs you about 20% of a bet when the bettor holds a diamond and one comes to beat you on the river (12 of 14 hands) and around 76% of a bet when the bettor has a made flush (3 of 24 hands). In the hands where the bettor is drawing dead, raising costs you nothing. So raising costs you about (12*.2+3*.76)/24 = 0.195, or about 20% of a bet.

So you want to raise when 7.5% of the pot is greater than 19.5% of a bet. So if the pot is 2.6 big bets or bigger, you want to raise. I still claim this is an estimate, and again I probably messed it up somewhere.

David

[HiatusOver]

$260(maybe a bit more?)...this is just a quick estimate. There is a chance that I am missing something and am way off, but I think I am probably in the ballpark.

[mikelow]

I would raise as the pot will always have at least 3BB at this point. You want to drive out the player with ten of diamonds
from beating you whenever the bettor doesn't have a big diamond.

[msk]

David,

I count 12 ways to have AQ, and 12 ways the bettor might have a diamond or two the way you presented this. So by raising, you lose half as many hands, but each time you lose you lose twice as much. Thus raising protects twice
as many pots for about the same cost. Always raise.

Mark

[deadbart]

There are 12 ways to have an AQ, but only 9 ways to have AQ with no diamond. That said, you will realistically nearly always raise.

[Poker blog]

David,

Perhaps I see this wrong, but I don't think the following sentence is what you meant:
"Also, for simplicity sake we will not consider the river bet. Assume it doesn't exist."

If the river bet doesn't exist, then why would you ever raise? If the river bet doesn't exist, then this is the last round of betting, and that's it. If you call, you get two big bets from your opponents for just one big bet of your own. But if you raise, you get two big bets from your opponent for two big bets of your own. Clearly, you would never raise.

I don't think this was how you intended your question to be, so could you clarify what you meant when you said the river bet doesn't exist? Or am I just on crack?

[cybertilt]

The third player is a rue, of the hands given our first player either has flush or has one card to getting one. thus don't raise since you will get the same return for your bet as raising with the same results. You have 7 outs to bet a flush, the pot size needs to be $120 (six to one you will improve).

[msk]

deadbart,

Right, I wondered why I was getting 12 /forums/images/icons/confused.gif since I didn't think 12 and 12 was exactly right for another reason...

There are 9 ways for AQ, 12 ways for a diamond, and of those 3 are already ahead. So in fact, there are 9 ways that getting the player with Td out matters and 9 ways where it doesn't matter. So I still think that if there is a penny in the pot it is better to raise.

Mark

[deadbart]

I think there are 9 ways to have AQ without a diamond, 12 ways to have exactly one diamond (including AxQd) and 3 ways to have two diamonds.

[pudley4]

9 ways to have AQ without Qd.
3 ways to have AQ with Qd.
3 ways to have AJ with Jd.
3 ways to have KQ with Qd.
3 ways to have KJ with Jd.
1 way each for QdJd, Qd9d, Jd9d.

So 9 ways where you want to get the 3rd player out, and 15 ways where you want him in.

My calculations show you should always raise, regardless of the pot size, but I need to double-check.

[pudley4]

1-Your opponent could have 24 possible hands.
9 AQ w/out Q /forums/images/icons/diamond.gif (Call these Hand 1)
3 each AQ w/Q /forums/images/icons/diamond.gif , AJ /forums/images/icons/diamond.gif , KQ /forums/images/icons/diamond.gif , KJ /forums/images/icons/diamond.gif (call these Hand 2)
1 each Q /forums/images/icons/diamond.gif J /forums/images/icons/diamond.gif , Q /forums/images/icons/diamond.gif 9 /forums/images/icons/diamond.gif , J /forums/images/icons/diamond.gif 9 /forums/images/icons/diamond.gif (call these Hand 3)

2-There are 42 remaining cards in the deck.

3-The pot size after your opponent bets is X

(For simplicity's sake, I will not be dividing by 24 for the total number of hands, or by 42 for the total number of river cards. I will also be referring to bets instead of dollars)

Hand 1
If your opp has Hand 1, he is drawing dead if you raise. If you just call, your other opp can draw out on you 8 times. So your EV for a raise is 42(X+1) (42 cards on the river give you a win, so you win the pot, X, plus 1 bet). Your EV for a call is 34(X+1)-8(1) (34 cards give you the win, 8 cards cost you the 1 bet you called.)

Hand 2
Notice that in Hands 2 and 3, your other opponent is drawing dead, so you win or lose the same number of pots regardless of whether he is in the hand or not. Obviously you'll want him to stay in, because in these cases, you'll be winning the same amount of money while risking less.

In Hand 2, you'll win 35 times and lose 7 times no matter what you do. Your EV for a raise is 35(X+1)-7(2). Your EV for a call is 35(X+1)-7(1).

Hand 3
In Hand 3, you'll win 10 times and lose 32 times no matter what you do. Your EV for a raise is 10(X+1)-32(2). Your EV for a call is 10(X+1)-32(1).

Now, calculate the EV of a raise:

Hand 1 + Hand 2 + Hand 3=9(42X+42)+12(35X+21)+3(10X-54)=828X+468

Calculate the EV of a call:

Hand 1 + Hand 2 + Hand 3=9(34x+26)+12(35x+28)+3(10x-22)
=756x+504

We want EV(raise)>EV(call), so 828x+468>756x+504. This is true for all X>1. Remember that X=pot size after your opponent's bet, so regardless of the pot size before the turn, as soon as he bets, you raise.

(I hope my math is right) /forums/images/icons/laugh.gif

[HiatusOver]

I disagree with those who say you should always raise, I think most people here are off-base. You are investing an extra $20 dollars to fold the 10d...more than half of the time the 10d will be drawing dead, so you obviously will have wasted money in these situations...on top of that, the 10d will only hit a non-board pairing diamond 8 out of 44 times WHEN IT IS DRAWING LIVE (which is only about 40% of the time). Because of these two facts, it is CLEARLY not worth it for you to invest the extra 20 dollars here in a small pot of let's say $80 because the 10d will not beat you anywhere close to 1 out of every 4 times.

[Poker blog]

I'm gonna reply to my own post. I am on crack. Actually, the real problem was that it was 6am in the morning and I still hadn't gone to bed yet. Now that exams are almost over, I'll get a chance to think about it.

I actually thought I had a caveat about how I hadn't slept and couldn't think straight in my original post, but I didn't. It's the crack, I guess.

[cero_z]

By my gorilla math, calling makes about $24/play, and raising makes $35/play, so I think you should raise regardless of pot size. Given the wording of the question, this seems wrong, so I look forward to a human helping out where apes have failed.

[HiatusOver]

This will be my last post under this topic until we get an answer, but I think I found the error in my previous reasoning. Because you are ahead in most on the scenarios, investing the extra $20 dollars does not actually "COST" you $20 dollars and in fact it costs you much less...therefore my previous posts were way off...i think

[woodman]

according to the question call & raise only options. Also, the only time the twit comes into play is when the first guy is drawing dead, hence:

(9/24)*(36/43)*pot size should be greater than the extra $20 to nock out the twit

Therefore pot above $63.70 raise, below, let the twit play.

[deadbart]

It doesn't cost you $20 to knock out the twit, since you win most of the time. Thus, the pot can be much smaller than your result.

[woodman]

correction, twit has 8 flush outs when no. 1 is drawing dead:
9/24*35/43*pot size>20
hence for a pot size less than $65.52 let the twit play otherwise raise

[woodman]

its a matter of giving the 2nd guy the correct odds to call or not, hence his odds of winning determine the correct pot size. so i'll stand by my answer......for now

[pudley4]

</font><blockquote><font class="small">In reply to:</font><hr />
its a matter of giving the 2nd guy the correct odds to call or not, hence his odds of winning determine the correct pot size

[/ QUOTE ]

I disagree. If the first guy is drawing dead, you don't want to give the 2nd guy any chance to call, regardless of whether he has pot odds or not. It doesn't matter who you collect the 1 extra bet from - the player behind you or the player who's drawing dead - in this scenario you'll still get only one more bet.

However, which is better - getting 1 bet from a guy who has no chance to beat you, or getting 1 bet from a guy who has a slim chance to beat you? All else being the same (which it is) you'll always take the sure money from the first bettor.

[woodman]

after careful re-consideration i will word my answer as follows:
scenario #1 you raise, EV = 0.82*(pot size + $40) - 0.18*$40
scenario #2 you call, EV = 0.75*(pot size + $40) - 0.25*$20
when is #1 &gt; #2

0.82(x+40)-0.18*40 &gt; 0.75(x+40)-0.25*20

when x &gt; -$8.57 raise ---- in other words raise any amount possible

i wish the teacher would tell us the answer so i could put the nails in this one and move on to something more constructive. /forums/images/icons/laugh.gif