Consider the following (not so unrealistic) scenario for a conflict between Iraq and the United States in the Persian Gulf area.
Iraq moves first and decides whether or not to invade Kuwait.
If Iraq does not invade Kuwait, the game is over and Iraq receives a payoff of 0, while the United States receives a payoff of 1,000.
If Iraq invades Kuwait, the United States must decide whether or not to send troops to Saudi Arabia.
If the United States does not send troops to Saudi Arabia, then the game is over and the payoff is 1,000 for Iraq and 100 for the United States.
If the United States sends troops to Saudi Arabia, Iraq must decide whether or not to leave Kuwait.
If Iraq leaves Kuwait, the game is over and the payoff is −1,000 for Iraq (which is humiliated) and 500 for the United States.
If Iraq decides to stay in Kuwait, the United States must decide whether or not to attack Iraq.
If the United States does not attack Iraq, the game is over. The presence of U.S. troops in Saudi Arabia is viewed as a farce and the United States suffers a great loss of prestige, while Iraq claims to have conquered “the evil intruder.” Iraq therefore receives a payoff of 1,000 and the United States receives a payoff of −700.
If the United States attacks Iraq and wins the resulting war, the game is over. However, because the United States wins with great casualties, the payoffs are U* = −500 for the United States and I* = −900 for Iraq.
(a) Present this story as a two-person game in extensive form; that is, draw the game tree. What is the outcome of the game?
(b) If I* = −500 and U* = −900, what is the sub-game perfect equilibrium for the game?
