Game Theory: Unusual Question #3 and #4

[Aisthesis]

Expanding B's raising criteria to pre-empt A makes sense to me.

But the rest of what you say conflicts with your solution to the problem: You have B value-raising exactly the same hands in #3 and #4 (5/6 through 1).

But you have B bluff-raising more often in #4 (1/9) than in #3 (1/18). That makes no sense to me if "the bluff-raise criterion is set by the value-raise criterion."

But I definitely see your point on B making some additional pre-emptive raises. It's just a little surprizing to me that you don't then expand B's value-raises at all.

[PairTheBoard]

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

[Aisthesis]

On raising frequency, B seems to me to have exactly the same pot odds. But A is risking less to win more when he raises, since he has already put $1 in blind.

That's not Bozeman's view of optimum. He agrees with JA, who has B raising top 1/6. Cf. his comment to well's post on this: "Jerrod and I got the same strategies, and you and I got the same value, Jerrod apparently making an arithmetic mistake."

So, Bozeman thinks Jerrod's criteria are correct but that the EV for B is 17/72 on that scenario rather than 1/4.

[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

[DoggSnott]

who gives a fiddler

[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?

[well]

So we agreed upon the solution of #4.

Now I have thought of a new question, but not yet about an answer.
Anyway, I'll post the question first.

Suppose A and B are playing #4 and swap the blind after each hand.
They both start off with, say, $15.
The game is over when one of them has it all.

How would the strategies change during this tournament?

Next Time.