Unused Prelim Question – Summer 2011 – Dan Quint
Suggested Solutions
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.
Burning bridges does not change the invader’s payoff when the defending army yields, and
strictly reduces it when the defending army fights – from 0.8W + 0.2R to 0.8W for a strong invader,
and from 0.5W + 0.5R to 0.5W for a weak invader. Thus, burning bridges is weakly dominated by
not burning them.
1. (b) Calculate all the Bayesian Nash equilibria of the static game if p < 32 , and if p > 23 .
Note that the invaders’ decision of whether or not to burn its bridges does not affect the
defending army’s payoffs. Given a probability p that the invader is strong, the defender gets a
payoff of (p × 0.2 + (1 − p) × 0.5)100 = 50 − 30p from fighting, and a payoff of 30 from yielding. If
p < 23 , then 50 − 30p > 30 and the defending army’s best-response in the static game is to fight;
either type of invader’s unique best-response is then not to burn bridges. So if p < 23 , the unique
equilibrium is (don’t burn, fight).
If p > 23 , then 50 − 30p < 30, and the defender’s best-response is to yield. In that case, the
decision to burn bridges is irrelevant; the set of equilibria consists of (any strategy, yield).
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.
Suppose there was a fully separating PBE. Then there would be some strategy (either burning
or not burning bridges) that led the defender to be certain the invader was strong. In that case, the
unique best-response to that strategy would be to yield. So by imitating the strong type’s strategy,
a weak invader could get a payoff of Y . By following his equilibrium strategy, a weak invader
reveals itself to be weak, leading the defender to fight, leading to a payoff of either 0.5W + 0.5R
or 0.5W , either of which is strictly less than Y . So imitating the strong type’s strategy would be
a profitable deviation for the weak type of invader, so a fully separating equilibrium cannot exist.
2. (b) If p < 23 , 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.
In this semi-separating equilibrium, only weak invaders leave bridges standing, so doing so
reveals their weakness and leads the defender to fight. The defender’s mixed strategy in response
to burning bridges must make weak invaders indifferent between burning and not burning; so if
defenders yield with probability y when bridges burn, we need
yY + (1 − y)(0.5W ) = 0.5W + 0.5R
y(Y − 0.5W ) = 0.5R
y =
0.5R
Y − 0.5W
In turn, the weak invaders must mix such that defenders are indifferent between yielding and
fighting when they see bridges burning. If we let µ be the defender’s beliefs about the invader’s
strength when a bridge burns, this requires 50 − 30µ = 30, or µ = 23 . By Bayes’ Law, if all strong
invaders and a fraction b of weak invaders burn bridges, then
2
p
= µ =
3
p + b(1 − p)
p
which requires b = 2(1−p)
. (This is less than 1 as long as p < 23 , which we are already assuming.)
The equilibrium strategies, then, are:
• Strong invaders burn bridges
• Weak invaders burn bridges with probability b =
p
2(1−p)
• Defenders fight when the invader did not burn his bridges
• Defenders yield with probability y =
0.5R
Y −0.5W
when the invader did burn his bridges
We have already verified (by the construction of b and y) that weak invaders and defenders who
see burning bridges are indifferent among their strategies, and are therefore playing best-responses.
When the invader does not burn his bridges, the defender correctly believes that he is weak, and
therefore plays a best-response by fighting. All that remains is to show that strong invaders play a
best-response by burning bridges, which is true if
yY + (1 − y)(0.8W ) ≥ 0.8W + 0.2R
y(Y − 0.8W ) ≥ 0.2R
0.5R
Y − 0.5W
≥
0.2R
Y − 0.8W
0.5W
0.2W
0.5W
0.2W
Multiplying both sides by W
R , it’s sufficient if Y −0.5W ≥ Y −0.8W , or (Y −W )+0.5W ≥ (Y −W )+0.2W ;
this is always true if Y − W ≥ 0. So this is an equilibrium.
(This last part of the proof – showing that strong invaders are playing a best-response in this
equilibrium – was trickier than I expected, and is probably what persuaded me this question was
too hard for a prelim problem.)
2. (c) If p < 32 , 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.
In a pooling equilibrium, the invader’s beliefs on the equilibrium path must match the prior
probability, which we assumed put p < 32 likelihood on a strong invader. This means in any pooling
equilibrium, the defending army must fight. This means that the invaders cannot be burning their
bridges, since if they were, not burning the bridges would be a profitable deviation (it would lead
to strictly higher payoffs regardless of how the defenders responded). Thus, the only candidate
for a pooling equilibrium is for all invaders to not burn their bridges, and for the defending army
to fight. This will be supported by any off-equilibrium-path beliefs (upon seeing burning bridges)
which make fighting a best-response for the defending army, for example, a belief that if bridges
burn, the invaders must be weak.
Cho-Kreps does not destroy the pooling equilibrium here, because either type of invading army
would benefit from bridge burning if that were followed by the defending army yielding.1
1
(Cho-Kreps would eliminate this equilibrium if weak types would never deviate to burning their bridges, even if
it led to the defending army yielding for sure. In that case, a strong invader could hypothetically burn his bridges
and then argue, “I must be strong, otherwise there is no way to rationalize what I just did.” This logic would not
work here, since either type of invader would happily burn its bridges if that led the defending army to yield.)