Difficult (I think) Q I posted over in the poker theory forum

[eastbay]

I'm not even sure I have the right answer, but I do have an answer.

NL, HU game theory Q (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=3613166&page=0&view=c ollapsed&sb=5&o=14&fpart=1)

[lorinda]

My first instinct is that I feel like the house has the edge in this game as they have position ever hand. I want a decent size stack compared to the house and small compared to the blinds.

Lori

[lorinda]

Argh.

I think that if the blinds are small enough that the house has no other strategy than to call, then we can have an edge as we are the only ones who have a decision to make.

I'm going to guess that with equal stacks of around 500 chips each (or whatever the correct stack is to make folding 23o correct) then we can have an edge.

Lori

[lorinda]

My range of stacks where I believe I could have an edge is therefore <Whatever range is required so that the house always have to call if we play, but that we sometimes are able to fold>

Lori

[eastbay]

[ QUOTE ]
My range of stacks where I believe I could have an edge is therefore <Whatever range is required so that the house always have to call if we play, but that we sometimes are able to fold>

Lori

[/ QUOTE ]

Your intuition about wanting a stack which is some smallish multiple of the blinds is correct according to my solution.

However, in my solution, at the point where the game is a wash, both players have hands that they will play and pass on.

eastbay

[lorinda]

Please excuse the thinking out loud.

I believe it's possible to have bigger stacks than my initial range, but I'm not sure how to quantify that and suspect I will now be slowing down my post rate /images/graemlins/wink.gif

Lori

[bawcerelli]

if the house is playing "ICM" perfect, how can you have an edge? seems like stack sizes and blinds don't matter other than affecting what's pushed. then again i'm probably missing something.

[lorinda]

That was my first thought, but if we have (and I'm not sure of the exact numbers) a 32% equity of winning with 23o, then with stacks of 550, 550 we are better off saving our 350 (to return 1100 which is only 31.4% I think).

However, even if BB knows we are only ever folding 23o, he should still call with everything.

We therefore have more control over the game than the BB, so logically we have an edge at that point.

At some point, the BB has more correct options than we do because he has enough chips and enough information to be able to do better things than us. There is a range however where we can remove all options from him (my above range) and there is another range where we can remove enough options from him that we still have an edge... that's the tough bit /images/graemlins/frown.gif

Edit: Sigh... those numbers are backwards, you would need a hand that has 31.4% equity to push there, and you have 32% , so you would push. Please assume the stacks were 600 each there, so to push you would need a hand with 33% equity (400 to return 1200)

Lori

[eastbay]

[ QUOTE ]
if the house is playing "ICM" perfect, how can you have an edge? seems like stack sizes and blinds don't matter other than affecting what's pushed. then again i'm probably missing something.

[/ QUOTE ]

Note that the house doesn't know what strategy we're going to play, so the house chooses the strategy that minimizes our maximum edge we can get over all the strategies we could possibly play.

This is far different from guessing what strategy we might play, and trying to maximize its edge against us.

In other words, the house plays as defensively as possible.

eastbay

[Jman28]

This would take a lot of work to figure out. My intuition though is that there definitely is a stack size where the player has an edge.

[eastbay]

[ QUOTE ]
This would take a lot of work to figure out. My intuition though is that there definitely is a stack size where the player has an edge.

[/ QUOTE ]

Confession: this took several evenings with a computer and the latest papers on computational AI to get the answer I arrived at.

I'm wondering if someone smarter than me can arrive at a similar answer in a more intuitive way.

I'm also not 100% sure my answer is correct, so if some other computer geek wants to duplicate it using the full blown analysis I did, that would be cool too. /images/graemlins/wink.gif

eastbay

[Jman28]

[ QUOTE ]

I'm wondering if someone smarter than me can arrive at a similar answer in a more intuitive way.

[/ QUOTE ]

Man, I gotta wake up tomorrow and go to a wedding. This sounds like a challenge though. Damnit. I may or may not try and figure this out tonight.

EDIT: 2k posts even. Hmm. Yay?

[eastbay]

[ QUOTE ]
[ QUOTE ]

I'm wondering if someone smarter than me can arrive at a similar answer in a more intuitive way.

[/ QUOTE ]

Man, I gotta wake up tomorrow and go to a wedding. This sounds like a challenge though. Damnit. I may or may not try and figure this out tonight.

[/ QUOTE ]

Nothing personal. I just wonder if there's a hand-wavy way to get at the basic answer. There may not be, it may be intrinsically in the details. I'm not sure.

eastbay

[Deuce2High]

Eastbay, you are needed over here!

http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=3613384&page=0&view=c ollapsed&sb=5&o=14&vc=1

Consider your time taken to answer that question my only customer service request as a proud customer of SnG-Analyzer!

[lorinda]

Please note that in my assumptions every % chance of winning a pot from 0.001 to 99.999 are possible. The fact that this is not the case WILL have a bearing on the results, but I have windows calculator as my only ally, so I am basically trying to just use logic here.

Fact 1: There is a point where the stacks are so small that we have an edge because we have more control over the game than our opponent.

I am happy that I have shown the above to be true.

Now I'm going to try to create another fact. (Assuming my logic in the first fact makes sense)

If we push less than half of the time, we cannot be favorite against someone who copies our strategy

We start with x chips each.
We push 50% of the time
Our opponent calls 50% of the time
Every time we fold, we lose 200 chips.
Every time we push and our opponent folds, we win 400 chips
Every time we push and our opponent calls, we break even.

Half of the time we lose 200 (-100 EV)
1/4 of the time we win 400 (+100 EV)
The rest of the time we break even (0 EV)

At this point, the game is even, so our opponent definitely has an effective counter to our play if we push 50% of the time or less.

Therefore, our next fact is "We must push over half of the time"

If we push 30% of the time, the house again just copies us.

We fold 70% and lose 200 (-140 EV)
We push and steal 21% and win 400 (+84 EV)
We push and get called 9% and break level (0 EV)

If we push 70% of the time and the house copies us..

We fold 30% and lose 200 (-60 EV)
We push and steal 21% and win 400 (+84EV)
We push and chop 49% and break level (0EV)

Clearly at this point, the house can no longer copy our play.

Lori

[lorinda]

If we push less than half of the time, we cannot be favorite against someone who copies our strategy

I believe that the above means that the bank can get away with calling at least 50% of the time.

Edit: I also feel that there are some dodgy concepts in what I have tried to show so far. I think I've got as far as I dare, and I'm not even sure these simple facts are right, and that's before even factoring in chips properly.

Maybe leaving up this route of thinking will help someone else attack the problem without computing it.


Lori

[Gigabet]

Intuition tells me 6000. I will give my reasoning after I find out if I am close to your answer or not.

[eastbay]

[ QUOTE ]
Intuition tells me 6000. I will give my reasoning after I find out if I am close to your answer or not.

[/ QUOTE ]

Each player has 6k chips? Not terribly close.

eastbay

[microbet]

I'll just guess. 10.

If the stacks are too deep, your push or fold only strategy will be bad. If the stacks are very short, you will be pushing any two and they will be calling any two. (So, if there were no profitable stack size 1 BB would be optimal)

Edit: I mean 10 Big blinds.

[lorinda]

[ QUOTE ]
Edit: I mean 10 Big blinds.

[/ QUOTE ]

I think this is still too large.

Edit: I think we're pushing a lot here and the BB is calling a lot. I can see that it's over the original 1100 or so that I thought in my first case, but I think we get to the heart of the main strategy pretty quickly when we get past the 23o=fold point.

I'm going for 2400 chips each.

Lori

[microbet]

Actually, just based on feel, in push/fold mode I like having 7 BBs HU.

Maybe 8.

[eastbay]

[ QUOTE ]
[ QUOTE ]
Edit: I mean 10 Big blinds.

[/ QUOTE ]

I think this is still too large.

Edit: I think we're pushing a lot here and the BB is calling a lot. I can see that it's over the original 1100 or so that I thought in my first case, but I think we get to the heart of the main strategy pretty quickly when we get past the 23o=fold point.

I'm going for 2400 chips each.

Lori

[/ QUOTE ]

I guess there's two questions in here. One is what stack size gives us the max edge, and then what is the critical stack size where we slip into being a dog.

I don't know how long I should drag this out, but I will say you guys are getting pretty close to one or both of those answers.

eastbay

[eastbay]

[ QUOTE ]
Actually, just based on feel, in push/fold mode I like having 7 BBs HU.

Maybe 8.

[/ QUOTE ]

A man of fairness. If you like breaking even, you're stunningly close (you've bracketed the answer).

For bastards like Lorinda who want max edges, that's too deep.

eastbay

[the shadow]

Thanks for posing the problem. Here're some random thoughts.

For the moment, I assume that the house's stack in the BB is greater than the SB's stack.

When the SB has a large stack (in terms of numbers of BBs), then the game is unfair to the SB. My speculation is that the BB's edge increases as to SB's stack increases, although not necessarily proportionately.

At some point, the game is fair to both players. Based on microbet's suggestion and eastbay's clue, that point seems to be when the SB's stack is between 7-8 BBs.

As the SB's stack decreases, the game becomes unfair to the BB.

However, at some point, as the SB's stack decreases further, the SB's edge decreases and ultimately the game becomes fair again to both players. For example, if the SB's stack is less than 0.5 BB, the game is a coin flip fair to both players.

I need to think further about the point at which the SB's edge is greatest, but in the meantime I've got a question back to eastbay.

Let's assume for the moment that the SB's stack is greater than the BB's stack. Does your simulation show that the game maps out the same way? If so, that would support the view that the optimal push-call strategy for both players depends solely on the small stack's size in terms of BBs.

The Shadow

[eastbay]

I'm not matching up with someone else's optimality solver, so I want to resolve that before making any more assertions about my solution.

eastbay

[eastbay]

[ QUOTE ]
I'm not matching up with someone else's optimality solver, so I want to resolve that before making any more assertions about my solution.

eastbay

[/ QUOTE ]

Ok, looks like the other guy had a bug and he's reproducing my results exactly now, so I'm confident in my answer.

eastbay

[eastbay]

[ QUOTE ]
Thanks for posing the problem. Here're some random thoughts.

For the moment, I assume that the house's stack in the BB is greater than the SB's stack.


[/ QUOTE ]

Right, you're always covered, basically.

[ QUOTE ]

When the SB has a large stack (in terms of numbers of BBs), then the game is unfair to the SB.


[/ QUOTE ]

Correct.

[ QUOTE ]

My speculation is that the BB's edge increases as to SB's stack increases, although not necessarily proportionately.


[/ QUOTE ]

Yes.

[ QUOTE ]

At some point, the game is fair to both players. Based on microbet's suggestion and eastbay's clue, that point seems to be when the SB's stack is between 7-8 BBs.


[/ QUOTE ]

Correct. Very near 7.8 BB, the game is fair.

[ QUOTE ]

As the SB's stack decreases, the game becomes unfair to the BB.


[/ QUOTE ]

True.

[ QUOTE ]

However, at some point, as the SB's stack decreases further, the SB's edge decreases and ultimately the game becomes fair again to both players. For example, if the SB's stack is less than 0.5 BB, the game is a coin flip fair to both players.


[/ QUOTE ]

Also true. Which means that there must be a maximum somewhere in between.

[ QUOTE ]

I need to think further about the point at which the SB's edge is greatest, but in the meantime I've got a question back to eastbay.

Let's assume for the moment that the SB's stack is greater than the BB's stack. Does your simulation show that the game maps out the same way?


[/ QUOTE ]

The minimum stack defines the value of the game, since this is all that can get into the pot. If I have 4k chips and BB has 6k, this is no different than if I have 6k and he has 4k.

[ QUOTE ]

If so, that would support the view that the optimal push-call strategy for both players depends solely on the small stack's size in terms of BBs.

The Shadow

[/ QUOTE ]

Note that this calculation does not factor in any tournament-esque considerations. That is, we are viewing each hand of this game independently. So there may be issues relating to "survival value", etc., that aren't considered here that are relevant to SnG play. I am simply valuing the game by the expectation of each hand by itself.

I have a way brewing to address that problem pretty rigorously, though.

eastbay

[eastbay]

[ QUOTE ]
I'm not even sure I have the right answer, but I do have an answer.

NL, HU game theory Q (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=3613166&page=0&view=c ollapsed&sb=5&o=14&fpart=1)

[/ QUOTE ]

For deep money, this game is unfair to the SB. Below a critical stack size, the advantage shifts to the SB. This critical stack size is about 7.8 BB. The edge for the SB then increases to a maximum at about 4 BB, and then sharply decreases again.

#BB +/-chips for SB
12 -33.0
11 -25.9
10 -18.1
9 -10.1
8 -1.6
7 +6.7
6 +15.3
5 +22.5
4 +26.4
3 +20.6
2.5 +11.8
2 +3.8

At the breakeven point, the optimal strategies are:

sb push hands: 22+,A2+,K2+,Q4o+,Q2s+,J7o+,J2s+,T7o+,T4s+,97o+,95s +,87o,84s+,76o,74s+,64s+,53s+,43s (0.631976)

bb call hands: 22+,A2+,K2+,Q7o+,Q3s+,J9o+,J7s+,T9o,T8s+,98s (0.457014)

There are no mixed stratgies (hands that you play some of the time but not always) at this point, interestingly enough.

At the point of maximum edge for the small blind, the optimal strategies are:

sb push hands: 22+,A2+,K2+,Q2+,J2+,T5o+,T2s+,96o+,93s+,86o+,84s+, 76o,74s+,64s+,54s (0.737557)

bb call hands: 22+,A2+,K2+,Q2+,J2+,T5o+,T2s+,96o+,93s+,86o+,84s+, 76o,74s+,64s+,53s+ (0.740573)

Again, there are no mixed strategies, and moreover, within one hand they are playing the same strategy! I suspect this may point to a deeper principle of some kind, but I'm not sure what it might be.

So, my answer to the original question is: "I would insist that my stack be less than 3120 chips, and if I get to choose my stack, I choose 1600 chips."

eastbay

[the shadow]

eastbay, thanks to you and Marv (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=3616890&page=&view=&s b=5&o=&vc=1) for your work on these simulations.

The Shadow

[the shadow]

[ QUOTE ]
At the point of maximum edge for the small blind, the optimal strategies are:

sb push hands: 22+,A2+,K2+,Q2+,J2+,T5o+,T2s+,96o+,93s+,86o+,84s+, 76o,74s+,64s+,54s (0.737557)

bb call hands: 22+,A2+,K2+,Q2+,J2+,T5o+,T2s+,96o+,93s+,86o+,84s+, 76o,74s+,64s+,53s+ (0.740573)


[/ QUOTE ]

It's interesting that the gap disappears (and is even just fractionally negative) at this point.

[ QUOTE ]
So, my answer to the original question is: "I would insist that my stack be less than 3120 chips, and if I get to choose my stack, I choose 1600 chips."


[/ QUOTE ]

If you push and lose, the game's over. Even if you push and win, it's time to cash out (at least half your chips) since the game's now unfair to you.

The Shadow

[eastbay]

[ QUOTE ]

[ QUOTE ]
So, my answer to the original question is: "I would insist that my stack be less than 3120 chips, and if I get to choose my stack, I choose 1600 chips."


[/ QUOTE ]

If you push and lose, the game's over. Even if you push and win, it's time to cash out (at least half your chips) since the game's now unfair to you.

The Shadow

[/ QUOTE ]

No, it's not a tournament. Each hand has a specific stack size that you play with.

eastbay

[eastbay]

[ QUOTE ]
eastbay, thanks to you and Marv (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=3616890&page=&view=&s b=5&o=&vc=1) for your work on these simulations.

The Shadow

[/ QUOTE ]

Thanks!

eastbay