Both X and Y should choose their strategy randomly. Let x be the probability that country X attacks by sea. Let y be the probability that country Y defends by sea.
The probability of a successful invasion is :
f(x,y)=.8xy + x(1-y) + (1-x)y + .6(1-x)(1-y) =
.8xy + x - xy + y - xy + .6 - .6x - .6y + .6xy =
-.6xy + .4x + .4y + .6 .
Y is obviously going to try to minimize the probability of a successful attack. Taking the derivative of f(x,y) with respect to y yields:
-.6x + .4 = 0.
Thus X should attack by sea with probability 2/3 and by land with probability 1/3. Y should also defend by sea with probability 2/3 and by land with probability 1/3. The probability of a successful invation is -.6*(2/3)*(2/3) + .4*(2/3) + .4*(2/3) + .6 = 13/15.
Michael Shackleford, A.S.A.