Country x is planning to attack country y, and country y is anticipating the attack.

Country x can either attack by land or by sea and country y can either prepare for a land defense or a sea defense.

Both countries must choose either an all land or all sea strategy, they may not divide their forces.

The following are the probabilities of a successful invasion according to both strategies used:

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      x attacks by     y defends by     prob. of success
     ---------------------------------------------------
          sea             sea                 80%
          sea             land               100%
          land            land                60%
          land            sea                100% 
     ---------------------------------------------------

1) What should the strategy of country x be, assuming the goal is to maximize the probability of a successful invasion? Assume the goal of country y to be to minimize the probability of a successful invasion.

2)What is the final probability of a successful invasion assuming both utilize an optimal strategy?

lets say that x goes by sea with probability p and land with probability 1-p.  Also y defends by sea with probability q and by land with probability 1-q.   Then the odds of x winning is

w=0.8pq+p(1-q)+0.6(1-p)(1-q)+q(1-p)

w=0.6+0.4p+0.4q-0.6pq

now y winds to choose q such that it makes the maximum w minimal and this is when dw/dp=0

dw/dp=0.4-0.6q

0.4=0.6q

q=0.4/0.6=2/3

thus y should defend by sea with probability 2/3

and by similar arguments x should attack by sea with probability 2/3.  and this gives an overall probability of invasion of 13/15

Posted by Daniel on 2009-02-06 13:59:28