Consider a strategic interaction between two players, an attacker and a defender. The defender is defending a location that has two access roads: an easy path and a hard path. Therefore, the defender can defend either the easy path or the hard path. Likewise, the attacker can attack from either the easy path or the hard path. The four possible outcomes of the interaction are as follows. • If the attacker attacks via the easy path and the defender defends the easy path, then they meet and fight. On the easy path, there is a 50-50 chance that a given army wins. Let the payoffs to this outcome be (1,1). • If the attacker attacks via the easy path and the defender defends the hard path, then the attacker can capture the location without serious confrontation. Let the payoffs to this outcome be 2 to the attacker and 0 to the defender. • If the attacker attacks via the hard path and the defender defends the easy path, then they do not meet. In this scenario, however, the attacker suffers serious losses due to moving through the hard path and is not able to capture the location. Let the payoffs to this outcome be (1, 1). • If the attacker attacks via the hard path and the defender defends the hard path, then they meet. However, due to the serious losses that the attacker endures on the hard path, the attacker can be easily defeated by the defender. Let the payoffs to this outcome be 0 to the attacker and 2 to the defender. (a) Assume that players move sequentially: the attacker moves first, the defender sees the attacker coming and chooses to defend a certain path. i. Find the Nash equilibria (in pure strategies) of this game. ii. Which of the Nash equilibria is subgame perfect?
(b) Assume that players move sequentially: the defender moves first, the attacker sees the defender's preparations and chooses to attack via a certain path. i. Find the Nash equilibria (in pure strategies) of this game. ii. Which of the Nash equilibria is subgame perfect? (c) Assume that players move simultaneously: the attacker does not know where the defender is going to defend, and the defender does not know where the attacker is going to attack. i. Find the Nash equilibria (in pure strategies) of this game.