Game Theory: Unusual Question #3 and #4

[PairTheBoard]

Thanks Jerrod. I appreciate that explanation.

PairTheBoard

[Aisthesis]

Forgive me, guys, but I'm still struggling a bit with the proper approach to these problems. I think if I can completely "get it" on just one, then I'll be good to go. The more I look at it, the more I think my difficulty in grasping the simple approach may have had to do with a certain mental "fuzziness" on how pot odds work here.

So, anyhow, first I'll try walking myself through problem 3 step-by-step, spelling out a few things that are no doubt self-evident to many of you. The main thing (actually on both problems) seems to me to be working backwards from the last decisions on the part of each player. So, here goes:

1) B clearly limps on the top 1/2 of his hands here since he is getting 1:1 pot odds. At 1/2 he has exactly the odds to call, but at anything below that he doesn't.

2) When B raises, A will be getting 3:1 pot odds. By bluffing some (as yet unknown) amount, B will be able to force A to call the raise with the top 1/3 of his hands, hence [2/3,1]. Note: I am thinking that we have to know this value before B can decide how many hands he is going to value-raise (??).

3) When B value-raises and A calls, B is again getting 1:1 pot odds. So, at 5/6 (the upper half of A's calling hands), he will be getting proper odds on the value-raise.

4) So, the last remaining question is: How many bluffs will it take to get A to call at 2/3. Since A is getting 3:1 odds on the call, B will need to bluff once for every 3 times he value-raises. (1/6)*(1/3) = 1/18.

I'm hoping that this is just a paraphrase of what David was saying--spelling everything out just a little more at every step of the way. Please let me know if I'm still thinking about this incorrectly.

[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.

[Aisthesis]

Looking through Jerrod's explanation, I already see a few mistakes in my thinking.
[ QUOTE ]
Forgive me, guys, but I'm still struggling a bit with the proper approach to these problems. I think if I can completely "get it" on just one, then I'll be good to go. The more I look at it, the more I think my difficulty in grasping the simple approach may have had to do with a certain mental "fuzziness" on how pot odds work here.

So, anyhow, first I'll try walking myself through problem 3 step-by-step, spelling out a few things that are no doubt self-evident to many of you. The main thing (actually on both problems) seems to me to be working backwards from the last decisions on the part of each player. So, here goes:

1) B clearly limps on the top 1/2 of his hands here since he is getting 1:1 pot odds. At 1/2 he has exactly the odds to call, but at anything below that he doesn't.

2) When B raises, A will be getting 3:1 pot odds. By bluffing some (as yet unknown) amount, B will be able to force A to call the raise with the top 1/3 of his hands, hence [2/3,1]. Note: I am thinking that we have to know this value before B can decide how many hands he is going to value-raise (??).

[/ QUOTE ]

Here, Jerrod's explanation makes more sense: It isn't the odds the pot is laying here for A that is relevant but rather making B indifferent to bluffing. As long as A calls 1/3 of the time, then B's bluffs just break even (of course, if B bluffs too much, A can capitalize on that by calling more; similarly if B doesn't bluff enough, then A can play tighter and lose less against the value-raises).

[ QUOTE ]
3) When B value-raises and A calls, B is again getting 1:1 pot odds. So, at 5/6 (the upper half of A's calling hands), he will be getting proper odds on the value-raise.

[/ QUOTE ]

Here, pot odds aren't the issue at all. B just wants to be favorite against A's hand. If he's not a favorite when A calls, why should he put in any extra money?

[ QUOTE ]
4) So, the last remaining question is: How many bluffs will it take to get A to call at 2/3. Since A is getting 3:1 odds on the call, B will need to bluff once for every 3 times he value-raises. (1/6)*(1/3) = 1/18.

I'm hoping that this is just a paraphrase of what David was saying--spelling everything out just a little more at every step of the way. Please let me know if I'm still thinking about this incorrectly.

[/ QUOTE ]

[PairTheBoard]

Bozeman: "If this answer is correct, B would rather have the no raising game than the one raise game, even though he acts last when he raises."

Actually B is acting first. A has the blind and the button. What this really shows is that the Power of the Raise is worth most to the player in last position.

PairTheBoard