game theory ( 經典例題），和幾周前的“必答題”相似，提上來討論

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. 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.

If x launches an attack by sea and y prepares a defense by sea the probability of a successful invasion is 80%.   If x launches an attack by sea and y prepares a defense by land the probability of a successful invasion is 100%. If x launches an attack by land and y prepares a defense by land the probability of a successful invasion is 60%. If x launches an attack by land and y prepares a defense by sea the probability of a successful invasion is 100%.

 

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