Problem 126 Answer
Strategy of z
After x and y bid the $1000 range will be divided into three intervals. Call
A the interval from $1000 to the lesser of the two prior bids, B the interval
between the two prior bids, and C the interval between $2000 and the greater of the
two prior bids. If max(B,C)/(B+C) > (A/$1000) then bid $.01 more than x or y, depending on
which will give you the greater interval. If max(B,C)/(B+C) < (A/$1000) then bid
Strategy of y
Call x's bid X.
If x bids more than $1750 then y should bid X - X*(sqr((2000-X)/1000)-(2000-x)/2000)+ $.01 .
If x bids less than $1750 then y should bid 2X-$1999.99 .
Strategy of x
x should bid $1750.01 .
Michael Shackleford, A.S.A., 10/20/1998