perplexus.info :: Probability : Attack and defense

Attack and defense (Posted on 2009-02-06)

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:

     ---------------------------------------------------
      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?

Submitted by pcbouhid

Rating: 5.0000 (1 votes)

Solution: (Hide)

Denote the country names with capital letters. 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) = 0.8xy + x(1-y) + (1-x)y + 0.6(1-x)(1-y) =
= 0.8xy + x - xy + y - xy + 0.6 - 0.6x - 0.6y + 0.6xy =
= - 0.6xy + 0.4x + 0.4y + 0.6

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:

- 0.6x + 0.4 = 0
x=2/3

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:

- 0.6*(2/3)*(2/3) + 0.4*(2/3) + 0.4*(2/3) + 0.6 = 13/15

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
solution	Charlie	2009-02-07 13:20:55
my strategy	Daniel	2009-02-06 13:59:28

Please log in:

Login:
Password:
Remember me:

Sign up! \| Forgot password

Search:

Search body:

Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (16)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:


perplexus dot info