Unused Potential Micro Prelim Question- Summer 2011
In Sun Tzu's The Art of War, we are told: "When your army has crossed the border [into enemy territory], you should burn your boats and bridges, in order to make it clear to everybody that you have no hankering after home."
Suppose an invading army can be one of two types, Strong or Weak, and the defending army cannot distinguish between the two. The invader is committed to attacking, but can choose whether or not to burn its bridges, cutting off its own option to retreat. The defending army can choose to Fight or Yield. If the defending army fights, a Strong invading force will win with probability 80%, and a Weak invader will win with probability 50%.
An invading army receives the following payoffs:
- Y (for yield) if the defending army yields
- W (for win) if the defending army fights and the invader wins
- R (for retreat) if the defending army fights, the invader loses, and the invader still has the option to retreat
- D (for dead) if the defending army fights, the invader loses, and the invader cannot retreat because he burned his bridges
Normalize D = 0, and assume Y > W > R > D = 0. The defending army gets the following payoffs:
- 100 if it fights and wins
- 30 if it yields
- 0 if it fights and loses
Note that 20% × 100 < 30 < 50% × 100, so the defending army would prefer to fight when the invader is weak but yield when the invader is strong. Let p be the prior probability that the invading army is strong.
1. First, suppose the defending army cannot see whether the invader burned his boats and bridges, so the two armies are effectively in a simultaneous-move game.
(a) Show that burning bridges is a weakly dominated strategy.
(b) Calculate all the Bayesian Nash equilibria of the static game if p < 2/3, and if p > 2/3.
2. Now suppose instead that the defending army can see whether the invader burned his boats and bridges before deciding whether to fight or yield.
(a) Show there cannot be a fully separating perfect Bayesian equilibrium, i.e., an equilibrium where the defender learns for certain whether the invader is strong or weak.
(b) If p < 2/3, there is a semi-separating PBE in which all strong invaders and some weak invaders burn their bridges, and the defending army mixes between yielding and fighting when it sees the invader burn its bridges. Calculate the equilibrium strategies, and verify that this is indeed an equilibrium.
(c) If p < 2/3, there is also a pooling PBE where both types of invaders play the same strategy. Show the equilibrium strategies and the beliefs that support this equilibrium, and show that this equilibrium is robust to the Cho-Kreps Intuitive Criterion.