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266

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Chapter 12

Bayesian Games

do not randomize but rather have strict best responses. The interested reader should
refer to Govindan (2003) for a short and elegant presentation of Harsanyi’s approach.

12.6

Summary
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12.7

In most real-world situations players will not know how much their opponents
value different outcomes of the game, but they may have a good idea about
the range of their valuations.
It is possible to model uncertainty over other players’ payoffs by introducing
types that represent the different possible preferences of each player. Adding
this together with Nature’s distribution over the possible types defines a
Bayesian game of incomplete information.
Using the common prior assumption on the distribution of players’ types, it is
possible to adopt the Nash equilibrium concept to Bayesian games, renamed
a Bayesian Nash equilibrium.
Markets with asymmetric information can be modeled as games of incomplete
information, resulting in Bayesian Nash equilibrium outcomes with inefficient
Harsanyi’s purification theorem suggests that mixed-strategy equilibria in
games of complete information can be thought of as representing pure-strategy
Bayesian Nash equilibria of games with heterogeneous players.

Exercises
12.1

12.2

12.3

Chicken Revisited: Consider the game of chicken in Section 12.2.1 with the
parameters R = 8, H = 16, and L = 0 as described there. A preacher, who
knows some game theory, decides to use this model to claim that moving to
a society in which all parents are lenient will have detrimental effects on the
behavior of teenagers. Does equilibrium analysis support this claim? What if
R = 8, H = 0, and L = 16?
Cournot Revisited: Consider the Cournot duopoly model in which two firms,
1 and 2, simultaneously choose the quantities they supply, q1 and q2. The
price each will face is determined by the market demand function p(q1, q2) =
a − b(q1 + q2). Each firm has a probability μ of having a marginal unit cost
of cL and a probability 1 − μ of having a marginal unit cost of cH . These
probabilities are common knowledge, but the true type is revealed only to
each firm individually. Solve for the Bayesian Nash equilibrium.
Armed Conflict: Consider the following strategic situation: Two rival armies
plan to seize a disputed territory. Each army’s general can choose either to
attack (A) or to not attack (N ). In addition, each army is either strong (S)
or weak (W ) with equal probability, and the realizations for each army are
independent. Furthermore the type of each army is known only to that army’s
general. An army can capture the territory if either (i) it attacks and its rival
does not or (ii) it and its rival attack, but it is strong and the rival is weak. If
both attack and are of equal strength then neither captures the territory. As
for payoffs, the territory is worth m if captured and each army has a cost of
fighting equal to s if it is strong and w if it is weak, where s &lt; w. If an army
attacks but its rival does not, no costs are borne by either side. Identify all

12.7 Exercises

12.4

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267

the pure-strategy Bayesian Nash equilibria of this game for the following two
cases, and briefly describe the intuition for your results:
a. m = 3, w = 2, s = 1.
b. m = 3, w = 4, s = 2.
Grade Gambles: Two students, 1 and 2, took a course with a professor who
decided to allocate grades as follows: Two envelopes will each include a grade
gi ∈ {A, B, C, D, F }, where each of the five options is chosen with equal
probability and the draws for each student i ∈ {1, 2} are independent. The
payoffs of each grade are 4, 3, 2, 1, and 0, respectively. Assume that the game
is played as follows: Each student receives his envelope, opens it, and observes
his grade. Then each student simultaneously decides if he wants to hold on to
his grade (H ) or exchange it with the other student (X). Exchange happens
if and only if both choose to exchange. If an exchange does not happen then
each student gets his assigned grade. If an exchange does happen then the
grades are bumped up by one. That is, if student 1 had an initial grade of C
and student 2 had an initial grade of D, then after the exchange student 1 will
get a C (which was student 2’s D) and student 2 will get a B (which was
student 1’s C). A grade of A is bumped up to an A+, which is worth 5.
a.
b.

Assume that student 2 plays the following strategy: “I offer to exchange for every grade I get.” What is the best response of student 1?
Define a weak exchange Bayesian Nash equilibrium (WEBNE) as a
Bayesian Nash equilibrium in which each student i chooses si (gi ) = X
whenever
E[vi (X, s−i (g−i ), gi |gi )] ≥ E[vi (H, s−i (g−i ), gi |gi )].

12.5

That is, given his grade gi and his (correct belief about his) opponent’s
strategy s−i , choosing X is as good as or better than H . In particular
a WEBNE is a pair of strategies (s1, s2) such that given s2 student 1
offers to exchange grades if exchange gives him at least as much as
holding, and vice versa. Find all the symmetric (both students use the
same strategy) WEBNE of this game. Are they Pareto ranked?
c. Now assume that the professor suggests modifying the game: everything works as before, except that the students must decide if they
want to exchange before opening their envelopes. Using equilibrium
analysis, would the students prefer this game or the original one?
d. From your conclusion in (c), what can you say about the statement
Not All That Glitters: A prospector owns a gold mine where he can dig to
recover gold. His output depends on the amount of gold in the mine, denoted
by x. The prospector knows the value of x, but the rest of the world knows
only that the amount of gold is uniformly distributed on the interval [0, 1].
Before deciding to mine, the prospector can try to sell his mine to a large
mining company, which is much more efficient in its extraction methods. The
prospector can ask the company owner for any price p ≥ 0, and the owner
can reject (R) or accept (A) the offer. If the owner rejects the offer then the
prospector is left to mine himself, and his payoff from self-mining is equal to
3x. If the owner accepts the offer then the prospector’s payoff is the price p,
while the owner’s payoff is given by the net value 4x − p, and this is common
knowledge.

268

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Chapter 12

Bayesian Games

Show that for a given price p ≥ 0 there is a threshold type x(p) ∈ [0, 1]
of prospector, such that types below x(p) will prefer to sell the mine,
while types above x(p) will prefer to self-mine.
b. Find the pure-strategy Bayesian Nash equilibrium of this game, and
show that it is unique. What is the expected payoff of each type of
prospector and of the company owner in the equilibrium you derived?
Reap and Weep: A farmer owns some land that he can farm to produce crops.
Farming output depends on the talent of the farmer. The farmer knows his
talent, but the rest of the world knows only that a farmer’s talent is uniformly
distributed: θ ∈ [0, 1]. The farmer’s payoff from farming his land is equal
to his talent θ . Before setting up his farm, the farmer approaches the local
manufacturing plant and offers to work on the production line. The farmer
can ask the plant owner for any wage w ≥ 0, and the owner can reject (R)
or accept (A) the offer. If the owner rejects the offer then the farmer must
return home and settle into his farming. If the owner accepts the offer then the
farmer’s payoff is the wage w, while the owner’s payoff is given by the net
value 23 θ − w, and this is common knowledge.
a. Define the set of pure strategies for each player and find the purestrategy Bayesian Nash equilibria of this game.
b. Averaging over the type of farmer, what are the possible levels of social
surplus (sum of expected payoffs of the farmer and the owner in their
potential relationship) from the equilibrium you derived in (a)?
c. A local policy maker who is advocating for the increase of social
surplus is proposing to cut water subsidies to the farmers, which would
imply that a farmer of type θ would get a payoff of 21 θ from farming his
land. This policy has no effect on the productivity of manufacturing.
Using the criterion of social surplus, can you advocate for this policy
maker using equilibrium analysis?
Trading Places: Two players, 1 and 2, each own a house. Each player i values
his own house at vi . The value of player i’s house to the other player, i.e., to
player j = i, is 23 vi . Each player i knows the value vi of his own house to
himself, but not the value of the other player’s house. The values vi are drawn
independently from the interval [0, 1] with uniform distribution.
a. Suppose players announce simultaneously whether they want to exchange their houses. If both players agree to an exchange, the exchange
takes place. Otherwise no exchange takes place. Find a Bayesian Nash
equilibrium of this game in pure strategies in which each player i accepts an exchange if and only if the value vi does not exceed some
threshold θi .
b. How would your answer to (a) change if player j ’s valuation of player
i’s house were 25 vi ?
c. Try to explain why any Bayesian Nash equilibrium of the game described in (a) must involve threshold strategies of the type postulated
in (a).
Jury Voting: Consider the jury voting game in Section 12.4.
a. Show that both players choosing CC is a Bayesian Nash equilibrium
of the two-player game.
b. Would it be better for the two players if only one of them decided on
the fate of the defendant? Why or why not?
a.

12.6

12.7

12.8