[Xhad]

LEVEL 5

Now B has to defend against this tactic, lest he continue to bust his .95's every time A snows with a .60. To recap A's strategy:

ARB: Always stand pat.
ABR: Always stand pat.
RBA: Always draw.
RAB: Always draw.
BAR: Stay if A + B > 1, otherwise draw.
BRA: Stay if A + B > 1, otherwise draw.

B is not concerned about the cases where A draws anymore because play there is automatic. The concern is whether to draw when A stays, or stay hoping to catch a snow.

The first thing to do is put A on a hand range if he stands pat. His hand range for snows is 1-b to b (note that b always has a hand > .5 here, so 1-b will always be less than b). His hand range for legitimate hands is b to 1. This gives a hand range of 1-b to 1.

This would mean that the probability of a snow is [b – (1-b)]/[1-(1-b)], which simplifies to 2 – 1/b. So EV of staying pat is $100 (2-1/b) since B always wins if he stands pat against a snow.

But of course this isn't the whole story. Sometimes B will draw against a snow only to still be ahead after the draw, and sometimes B will fail to catch up when A had a legitimate hand.

So what we next need to do is figure out A's average hand. Since we already know his range is (1-b) to 1, this would mean that his average hand is [1-(1-b)]/2, or 1-(.5)b.

So the EV of drawing would be $100 (1 – A's average hand) (which is $50b), since that is the probability of winning if we draw.

All this would mean that to stay pat, the following would have to be true:

$100(2-1/b) > $50b

Which simplifies to:

-1/2 b^2 + 2b – 1 > 0

Solving the quadratic forumula yields 2 – sqrt(2) or 2 + sqrt(2), the latter is not a legal hand so 2 – sqrt(2) (or a little under .60) is the minimum snow-catching hand.