Game Theory: Unusual Question #3 and #4

[PairTheBoard]

When B raises A is getting 3-1 pot odds to call. So B should bluff raise 1 time for every 3 value raises.

When A raises B is getting 5-1 pot odds to call. So A should bluff raise 1 time for every 5 value raises.

Do I have this right? Game Theory is new to me as well.

PairTheBoard

[Aisthesis]

Well, you should probably take this with a grain of salt pending an answer from a non-newbie to this area. But it sounds to me like the proportions between bluff- and value-raises are correct GIVEN the pot odds you menion.

However, when A is making a raising decision in problem #4, B is still just getting 3:1 on the call, not 5:1. A bet $1 blind, then B put in a $1 limp, and now A has added one additional dollar with the raise. So, there is still just $3 in the pot, and it will cost B an additional $1 to show down.

[Aisthesis]

[ QUOTE ]
The answer to the second problem is:
(There are co-optimal solutions, but where possible I present undominated ones)

B:
folds on [0,19/36]
raise-bluffs on [19/36,7/12] *
limp-folds on [7/12,2/3]
limp-calls on [2/3,5/6]
value-raises on [5/6,1]

A:
calls a raise on [2/3,1]
folds to a raise on [0,2/3]
raises after B limps on [3/4,1]
checks after B limps on [1/12,3/4]
raises after B limps on [0,1/12]

The value of the game is 1/4.

To answer David's question, B should just raise the same amount of hands when the A's blind is made live. B, however, doesn't limp with his thinnest value calls, because he'll face a raise sometimes. He just folds them.

Jerrod Ankenman

[/ QUOTE ]

Ok, everyone seems to agree that JA's EV for B is actually 17/72. I haven't checked this, but I'll just accept it.

But I think I can improve on A's strategy if B plays that way. Everything is the same except the following:

A bluff-raises on [0,2/9]
A value-raises on [2/3,1]

This dramatically reduces B's EV to 179/1296 if my arithmetic is right.

(17/72 = 306/1296)

[PairTheBoard]

Yes of course. I don't know what I was thinking. 3-1 either way isn't it. oh well. nevermind.

Did we ever get it settled on what's suppose to be the correct strategies?

PairTheBoard

[Jerrod Ankenman]

[ QUOTE ]
[ QUOTE ]
The answer to the second problem is:
(There are co-optimal solutions, but where possible I present undominated ones)

B:
folds on [0,19/36]
raise-bluffs on [19/36,7/12] *
limp-folds on [7/12,2/3]
limp-calls on [2/3,5/6]
value-raises on [5/6,1]

A:
calls a raise on [2/3,1]
folds to a raise on [0,2/3]
raises after B limps on [3/4,1]
checks after B limps on [1/12,3/4]
raises after B limps on [0,1/12]

The value of the game is 1/4. (later determined to be wrong/JA)

To answer David's question, B should just raise the same amount of hands when the A's blind is made live. B, however, doesn't limp with his thinnest value calls, because he'll face a raise sometimes. He just folds them.

Jerrod Ankenman

[/ QUOTE ]

Ok, everyone seems to agree that JA's EV for B is actually 17/72. I haven't checked this, but I'll just accept it.

But I think I can improve on A's strategy if B plays that way. Everything is the same except the following:

A bluff-raises on [0,2/9]
A value-raises on [2/3,1]

This dramatically reduces B's EV to 179/1296 if my arithmetic is right.

(17/72 = 306/1296)

[/ QUOTE ]


The difference between the strategies is simply the hands [2/3,3/4] and [1/12,2/9].

When A has [2/3,3/4]:

When B has:
limp-folds on [7/12,2/3] -- same result as checking
limp-calls on [2/3,3/4] -- same result as checking
limp-calls on [3/4,5/6] -- loses one extra bet

When A has [1/12,2/9]:

When B has:
limp-folds on [7/12,2/3] -- win 2 bets extra
limp-calls on [2/3,5/6] -- lose 1 bet extra

Since 5/6 - 2/3 = 2/12 and 2/3 - 7/12 = 1/12, this is a break-even.

So your strategy loses an extra bet when A in [2/3,3/4] and B in [3/4,5/6] and doesn't gain any offsetting bets when compared to the optimal strategy.

Jerrod

[Jerrod Ankenman]

[ QUOTE ]
I thought this was the agreed upon correct solution. Posted by wells and agreed to by Bozman?

http://forumserver.twoplustwo.com/sh...;o=14&vc=1

It has B value raising from 3/4 up to 1 as opposed to the 5/6 up to 1 for problem #3.

Although the Raise by B meets similar conditions as in #3, the Calls do not. So it could make sense for the Calling Interval to change. Both on the low side to folds and the high side to raises. Assuming the Linked to Solution is Correct.

Also Aisthesis, I understand you to say that A's bluffing frequency should be greater than B's because of the pot odds. When A raises B has greater Pot Odds to call. Doesn't Game Theory say the raising frequency should be less in that case?

PairTheBoard

[/ QUOTE ]

I think the one posted by well has at least one error. The solution I posted on 6/5 (which has a lightbulb next to it) is correct except that the value of the game is wrong.

B raises the same amount in both cases, because there's no third bet to consider. If a third bet were included (ie, B could limp-reraise), then the raising amounts would be different. As it stands, A calls to prevent B from bluff-raising, and B value raises half of the times that A will call.

Jerrod

[Aisthesis]

Well, Bozeman's idea that B should raise pre-emptively to limit the option for A got me to thinking about this a bit more, and I realized that there's also a case for why B should raise LESS: His raising hands are all great trap hands if A raises him!

Moreover, in #4, I hate to have B give up on hands less than 1/2 because he probably has the better hand. Maybe as last to act, A does have some kind of edge here, but I'm not ready to admit it yet.

Anyhow, the following suggestion is far from "airtight" but it does improve on Jarred's EV, and I'm at least convinced that it's the optimal way for A to counter B's strategy:

B just limps on [1/2,1] and calls a raise at [2/3,1]

A will then bluff-raise on [0,1/9] and value-raise on [2/3,1]

My reasoning is this: B actually again has to call with the top 1/3 of his hands to prevent frivolous bluffing.

A, on the other hand, as I brought up in a previous post, is getting 4 times the "bang for his buck" on the raise. Hence, he will value-raise 4 times as often, but also now knows the range of B's hands as [1/2,1]. So, where B raised the top 1/6, A will raise the top 2/3 of B's range, hence [2/3,1].

For the same reason, it would seem that A would want to bluff-raise on 2/9 of his hands rather than just 1/18. But that assumed that B would call only 1/3 of the time. B is in fact going to call twice that often. So, A can only bluff 1/9 of his hands. Note that the function of this bluff is simply to drive B's call criteria on the raise down as far as possible. If A didn't bluff at all, B could make more profit by limp-calling only on [3/4,1], where he has pot odds given A's value-raise criteria.

Anyhow, I find it difficult to believe that B's truly optimal strategy here should be to refrain from making the raise altogether. But this strategy for him has a value of 1/4 (A's raises are all just break-even, and otherwise everything is the same as in scenario #3a, where B couldn't raise), which is better than in Jarred's and Bozeman's strategy.

Can anyone beat this strategy by letting B use his raise option?

[Aisthesis]

Yes, I can't find any mistakes in your logic there, anyway.

Interesting in your answer, however, is that it would seem that there are lots of co-optimal bluff-raising strategies for A. Hence, neither 1/12 nor 2/9 is any kind of "magic number" for when A would want to bluff-raise. Does that mean that 1/12 is just the minimum amount of bluffing to force B down to 2/3 for the limp-call?

If so, I guess it also means that if A starts bluffing more, B could make even more profit by reducing his calling criteria after the limp.

Anyhow, your comparison looks right to me. Sorry for the unjustified critique! (Please bear with me... you've obviously been playing this game longer than I have...)

[Aisthesis]

Well, JA's lightbulb strategy still seems to be the best raising strategy for B, and he obviously knows what he's doing in this game.

But I'm waiting to hear what's wrong with my (as far as I can tell) improvement in the post "stab at #4."

On that one, I basically just took as axiomatic that B was never going to raise and would limp on [1/2,1]. On what I come up with as A's best raising strategy (and B's best limp-call strategy), it seems to me that B does better than in JA's game. So, I'm just waiting to hear why I'm wrong...

[PairTheBoard]

I was going on the assumption that well's solution was correct. Jarrod seems pretty sure he is correct though and he has B raising with the same frequency as #3, on [5/6,1] plus bluffs. That would certainly do better at making the Poker Point about frequent limping since well's solution actually has B raising more than he limps. Jarrod has B limping a lot more than he raises.

PairTheBoard