[Jerrod Ankenman]

[ QUOTE ]
Indeed, my solution contained an error.
[ QUOTE ]
Player A
[0,1/12] : raise
[1/12,3/4] : check
[3/4,1] : raise, when called

[0,2/3] : fold
[2/3,1] : call, when raised

Player B
[0,19/36] : raise
[19/36,7/12] : check
[7/12,2/3] : call, fold when raised
[2/3,3/4] : call, call when raised
[3/4,1] : raise

With this, the value of the game for player B is 17/72

[/ QUOTE ]

For the honest raise I actually found [5/6,1] (just as in the other posted solution).
And of course, instead of check it should read fold by player B...

[/ QUOTE ]

B is bluff-raising way too many hands in this solution - more than half!

[ QUOTE ]
So I like it I got it right, but what I am interested in,
is how others found this solution.

[/ QUOTE ]

Here's how I solved it.

B has the option of raising. So A's calling strategy will just be to make B indifferent to bluff-raising. Since B is putting in 2 to win 1, A needs to call (1)/(1+2) or 1/3 of the time.

Then B can value raise 1/2 of the time that A calls and make profit. So B's value raise is his 1/6 of best hands.

B's ratio of bluffs to bets is 1/3, so he bluffs with 1/18 of the hands he would fold with.

Ok, now in addition to raising and folding, B's going to call sometimes. When he calls, A has the option of raising. If this occurs, A will be betting 1 unit to win 2. So of the times that B calls, B has to call a raise at least 2/(2+1) or 2/3 of the time.

A's raises in the top 1/6 of hands are always profitable because B never has a hand in that range; A can then value raise hands above the midpoint of B's calling range.

So now you can get two variables going and write some kind of equation for A's limping threshold and B's raising threshold and solve it if you wish.

However, I resorted to a little trickery.
We know that when we're done, B will be indifferent to raising at the point 5/6. When B is at 5/6, two bets are going in all the time when A has a hand between 5/6 and 1. A will call B's raise an additional 1/6 of the time and lose (because he calls 1/3 of the time). So we know that if B limps at 5/6, A will raise with 1/6 of hands that B beats, so that B will be indifferent to raising at that point.

Ok, so now A raises 2/6 of the time total (1/6 that win and 1/6 that lose) and his bluffs make up 1/4 of the total hands he raises.

So he bluffs 1/12 of the time, and value raises with hands worse than 5/6 the remaining 2/6 - 1/12 - 1/6 = 1/12.

So his value raising threshold is 5/6-1/12 or 3/4.

This point is the midpoint of A's limp-call area, so he limp calls with 8/12 - 10/12, and since his limp-fold area is half as wide as his limp-call area, he limp-folds with 7/12 and above, and fill in his bluff-raise range to the right of that, etc.

Jerrod