ADVANCED ECONOMIC THEORY
Attempt two questions
1. (a) Show that a strategy for a player that is part of a Nash equilibrium
cannot be strictly dominated by another strategy.
(b) Can strategies which form a Nash equilibrium be weakly dominated for
all players? If so give an example to illustrate your answer.
(c) Brie‡y discuss the assumptions needed to justify playing the following:
(i) strategies that are not strictly dominated;
(ii) strategies that survive iterative deletion of strictly dominated strategies;
(iii) strategies that are not weakly dominated.
(d) Consider any 2-player normal form game. Show that any mixed Nash
equilibrium can be justi…ed in terms of beliefs in a world in which players
cannot randomize? Does your reasoning extend to extends to games with
more than two players? Explain your answer.
(e) Discuss brie‡y how players in a game may come to have a common belief
on how the game is to be played.
2. (a) Consider the following 2-player game
1n2 F
C
F
-4, -4 8,0
C
0,8
4,4
(i) Compute all the Nash equilibria (both pure and mixed) of this game
and the corresponding payo¤s.
(ii) Suppose that the payo¤ accruing to player 1 when this player uses
F and its rival 2 uses C is increased by a strictly positive amount ;
to (8 + ), with all other payo¤s remaining as before. In the nondegenerate mixed strategy equilibrium, does player 2 now play C more
or less frequently? Does player 1’s equilibrium payo¤ rise or fall?
Comment on your answer.
(iii) Is there a correlated equilibrium in which each players obtain a payo¤
of atleast 4? Explain your answer.
(b) Consider a game in which each player i = 1; ::; n chooses a real number si in the interval [0; 1] once and simultaneously: Let i (s1 ; ::; sn ) be the
payo¤ of player i if the players chosse a strategy pro…le (s1 ; :::; sn ). Suppose i (s1 ; ::; sn ) is di¤erentiable. Show in general any non-interior Nash
equilibrium of this game is Pareto ine¢ cient.
(c) Does a game with a …nite number of pure strategies always have a pure
Nash equilibrium? If not provide an example. Does your answer change if
the game has an in…nite number of strategies?
(d) Show that any game with a …nite number of pure strategies has a mixed
Nash equilibrium. (You may assume the following result: any normal form
game has a Nash equilibrium if the set of strategies for every player is convex
and compact and the payo¤ function for each player i is quasi-concave in
i0 s strategy and continuous in the strategies of all players). Hence, explain
brie‡y why every …nite extensive form game has a mixed subgame perfect
equilibrium strategy..
3. (a) Consider a …rst price sealed-bid auction of an object between two bidders
with incomplete information. The value of the object to each bidder i is
vi = ti + where ti is a random variable that is independently drawn from
a uniform distribution over the interval [0; 1] and
0 is some constant:
Before the auction each bidder i = 1; 2 observes only the value of ti : The
auction rules are such that each player submits a bid in a sealed envelope.
The envelopes are then opened, and the bidder who has the highest bid gets
the object and pays the auctioneer the amount of his bid. If the bidders
submit the same bid, each gets the object with equal probability. All aspects
of the structure of this game are common knowledge with the exception that
each i does not know the value of tj for j 6= i:
(i) Derive a (pure) Bayesian Nash equilibrium in which each bidder i uses
a linear strategy of the form b(ti ) = ti + for some constants and :
In this equilibrium, what is the expected payo¤ of bidder i if his type
is ti ?
(ii) What if there are n bidders? What happens to each bidder’s bid
function as n increases?
(b) Describe the one shot deviation property in dynamic games.
(c) Consider the in…nite horizon alternating bargaining game (Rubinstein
game). Here the rules are in odd periods player 1 makes an o¤er of a split
x 2 [0; 1] to player 2, which player 2 may accept or reject and in even
periods player 2 makes an o¤er of a split to player 1, which player 1 may
accept or reject . If at any period a player accepts an o¤er, the proposed
split is immediately implemented and the game ends. If she rejects, nothing
happens until the next period. Assume that each player i discounts future
payo¤s by a common factor i < 1. Show that there exists a subgame perfect
equilibrium that results in an agreement at period 1 with the following split:
player 1 receives 11 1 2 2 and player 2 receives 12 (1 1 12 ) : (You may restrict
yourself to showing that there are no pro…table one-period deviations). Is
the equilibrium you …nd history-independent?
4. (a) Suppose that a game G = (Si ; i )ni=1 ; where Si and i refer to the payo¤
function and the strategy set of player i; is played repeatedly a …nite number
of times T . Denote this …nitely repeated game by GT :
(i) Show that if G has a unique Nash equilibrium and if players do not
discount the future then GT has a unique history-independent subgame
perfect equilibrium.
(ii) Show also that GT might have a history-dependent subgame perfect
equilibrium if G has more one equilibrium (it is su¢ cient to construct
an example of a game G that has multiple equilibria and is played twice
repeatedly).
(b) Consider a repeated random state oligopoly market with n …rms who
produce a homogenous commodity at zero cost at each date. In each period
there is a random demand shock (state) s that is …rst drawn independently
with the probability q(s) from the …nite set S and revealed to the …rms.
After observing the states, the …rms simultaneously choose non-negative
prices for that period. The consumers buy from the cheapest …rm (ties are
split equally) and the aggregate demand of the consumers at price p at each
period is given by s p.
(i) Compute the Nash equilibrium outcome and the monopoly price in any
stage game if the state is s.
(ii) Suppose that along an equilibrium path of the repeated game, at every
date every …rm sets a price p(s) whenever the state is s. Characterise
the (expected) subgame perfect equilibrium payo¤ for each …rm from
such a symmetric stationary equilibrium strategy.
(iii) Show that for su¢ ciently large , setting the monopoly price at every
date and for every state s can be sustained as an equilibrium.
(iv) Fix any ; and consider any strongly (most collusive) symmetric stationary equilibrium. Show that on the equilibrium path the …rms set
prices countercyclically in high states. Provide an intuition for this
result. What happens to the equilibrium prices in low states?
5. (a) Suppose that in game a player has I information sets indexed i =
1; 2; ::; I and Ki possible actions at information set i 2 I: How many strategies does the player have?
(b) Explain why every …nite extensive form game of perfect information has
a unique subgame perfect equilibrium if no player has the same payo¤s at
any two terminal nodes (no “ties”in the payo¤s).
(c) De…ne the concept of perfect Bayesian equilibrium for dynamic games
of imperfect information.
(d) Consider the extensive form game in …gure 1.
Suppose that a > 0. Show that the game does not have a perfect Bayesian
equilibrium in pure strategies. Does it have a mixed perfect Bayesian equilibrium? If so compute the set of mixed perfect Bayesian equilibria of this
game for di¤erent values of a.
(e) Brie‡y discuss the limitations of the concept of perfect Bayesian equilibrium in dynamic games of imperfect information. Does the notion of forward
inductions help to alleviate some of these limitations? It is su¢ cient to use
examples to illustrate your to this part of the question.
Answers
1. (a) If s.dominated it can’t be a best response and hence not NE.
(b)
2; 2 0; 2
2; 0 3; 3
(2,2) is a NE and is weakly dominated.
(c) De…ne concepts and explain why the following are enough to justify the
solution concepts.
(i) Assume rationality.
(ii) Assume common knowledge of rationality.
(iii) Assume rationality and uncertainty/noise.
(d) Consider any 2-player …nite normal form game. Suppose players choose
pure strategies. Let 1 2 S1 be player 2’s belief about what player 1 does.
Symmetrically assume 2 2 S2 is player 1’s belief about what player 2
does. Suppose furthermore that both players are rational and rationality
is commonly known. Assume also that the beliefs are commonly known.
Then one can show that ( 1 ; 2 ) is a mixed NE of the game as follows.
Consider any i = 1; 2: If i (si ) > 0 ) j 6= i thinks i chooses si with positive
probability. Since j knows that i is a RAT it must be that si is a best
response to i’s belief. But j knows i’s belief - namely j . Therefore,
i (si )
> 0 ) si is a best response to
But this implies that ( 1 ;
2)
j
(1)
is a NE.
With more than 2 players …x any player i. Then the players other than
i may have di¤erent beliefs about how i behaves:Common Belief cannot
ensure that all other players have the same belief about i and therefore, it is
not su¢ cient to justify NE in terms of belief. However, if there is common
knowledge of beliefs the others cannot disagree and all must have the same
belief about i and therefore NE can be justi…ed in terms of beliefs.
(e) The anwers must brie‡y discuss how the following might bring about
common beliefs: eductive/mental reasoning, pre-play communication, focal
points, learning and evolution in a world with boundedly rational players.
2. (a)
(i) (F,C), (C,F) and a mixed strategy (1/2,1/2) and payo¤ of 2 for both
players
4+
(ii) ( 8+
; 8+4 ) for 2 and (1/2,1/2) for 1: Therefore, 2 plays C less frequently
16
(to make 1 indi¤erent) and the payo¤ of 1 declines from 2 to 8+
:
(iii) the only correlated equilibrium is to play (F; C) with prob 1/2 and
(C; F ) with prob 1=2. Extra mark explaining this is a public correlated eq
and there is no other corr eq that gives at least 4 to each player.
(b) If a Nash equilibrium happens at an interior point then a necessary
condition for NE is
d i
= 08i
dsi
P
whereas a Pareto e¢ cient point is a solution to maxs1 ;::;sn i i i (s1 ; ::; sn )
for some i 0: Thus any interior Pareto e¢ cient allocation satis…es
d i
+ 1=
dsi
i
X
j
j6=i
Clearly because of the externality 1=
i
d j
= 0 8i
dsi
X
d j
j dsi
the two necessary condi-
j6=i
tions are not the same.
(c) Standard: Irrespective of the no of strategies a game may not have a
NE in pure strategies. As an example they could give matching pennies and
Bertrand with capacity constraint as examples.
(d) Standard. For any …nite normal form game the set of mixed strategies
(the set probability distributions over a …nite set) is convex and compact.
Also the expected payo¤ for any player is linear in mixed strategies. Therefore, expected payo¤ for any i is concave (and thus quasi-concave) in i0 s
mixed strategies and continuous in mixed strategies of all players. Therefore, since the game where mixed strategies are allowed is in turn a normal
form, it follows from the result stated in the question that the mixed game
has a NE.
Finite extensive form game: working backwards the last set of subgames
must have a Nash equilibrium by the previous steps; working backwards
establishes the results.
3. (a) Suppose that the strategy of j is given by b(tj ) = tj + : Then a best
action for i that receives information ti is a bid b that maximizes
(ti +
= (ti +
b):prob [b > b(tj )]
b
b):prob
> tj
+(ti +
b):prob
+(ti +
b):prob
= (ti +
b):
b
= tj
b
< tj
b
maximising with respect to b, the foc for this problem is:
b = 1=2(ti +
+ )
Thus a symmetric an equilibrium consists of a b(ti ) = 1=2ti +
Also, the expected payo¤ to ti is t2i =2:
for all i:
When there are n bidders, do the same thing: a best action for i that receives
information ti is a bid b that maximizes
(ti +
(ti +
b):prob[b > b(tj ) for all j 6= i] =
b):prob[
b
(ti +
> tj for all j 6= i] =
b
b)
n 1
FOC for this problem is (product rule!!)
(ti +
b)(n
1) = b
Therefore,
b = 1=nf(n
1)(ti + ) + g
Thus a symmetric an equilibrium consists of a b(ti ) = nn 1 ti + for all i: As
n ! 1 the equilibrium bids converge to each type’s valuation of ti + :
(b) Standard.
(c) Standard. Consider the following payo¤ vectors y = (y1 ; y2 ) 2
z = (z1 ; z2 ) 2 s.t.
y1 =
and
z1 =
(1
1
2)
2 1
1 (1
1
and y2 =
2)
and z2 =
2 1
2 (1
1
(1
1
and
1)
2 1
1)
2 1
De…ne strategy pro…le
s1 =
o¤er y at every odd period irrespective of the past
accept an o¤er x = (x1 ; x2 ) i¤ x1 z1 at every even period irrespective of the past
s2 =
o¤er z at every even period irrespective of the past
accept an o¤er x = (x1 ; x2 ) i¤ x2 y2 at every odd period irrespective of the past
Clearly, the above induces agreement at y in the …rst period. Also, note
that
y2 = 2 z2
(2)
z1 =
1 y1
(3)
Then using one-period deviation property and the last two conditions, the
candidate must show that s is a SPE by showing that at each history si is
a best reply in both the proposer and responder role.
The above SPE is clearly history-independent.
4. (a)
(i) Let s be the unique NE of G: Then in any SPE of GT in the last round
players must play s irrespective of the past. Therefore at period T
1
they also play s irrespective of the past. Hence by backward inductions the
players play s irrespective of the past at every period.
(ii) Consider the following simultaneous-move game is played twice:
Player 1 a1
a2
a3
Player 2
b1
b2
b3
10,10 2,12 0,13
12,2 5,5 0,0
13,0 0,0 1,1
Pure strategies Nash equilibria of the one-shot game are
(a2 ; b2 )
and
(a3 ; b3 )
Now for the 2-period game consider the following strategies:
(i) Player 1: play a1 in period 1. Play a2 in period 2 if (a1 ; a2 ) was played in
period 1, otherwise play a3 . Player 2: play b1 in period 1. Play b2 in period
2 if player 1 played a1 . Otherwise play b3 :
(ii) Player 2: play b1 in period 1. Play b2 in period 2 if (a1 ; a2 ) was played
in period 1, otherwise play b3 .
Each is a best response to the other. This is because the cost of deviating in
period 1 is a punishment of 4 in the second period. But the gain from deviating in period 1 is at most 3. Also, note that that these strategies are credible
in the second period because they involve playing a Nasd equilibrium of the
one-shot game.
(b)
(i) Both NE and mutual minmax are zero prices and the monopoly is given
by pm = s=2, for a total payo¤ of (s=2)2 .
P
(ii) Let v = n1
p (^
s) (^
s p (^
s)) q (^
s) be the expected payo¤ from such a
s^2S
strategy pro…le. Necessary and su¢ cient conditions for the strategy pro…le
to be an equilibrium are, for each state s,
1
(1
n
)p(s)(s
where
v =
p(s)) + v
1X
p (^
s) (^
s
n s^2S
(1
)p(s)(s
p(s));
(4)
(5)
p (^
s)) q (^
s) :
Condition (4) ensures that the …rm would prefer to set the prescribed price
p(s) and receive the continuation value v rather than deviate to a slightly
smaller price (where the right side is the supremum over such prices) followed
by subsequent minmaxing. Condition (5) gives the continuation value v ,
which is independent of the current state. If (4) holds then a Grim type
strategy sustains v as an equilibrium. Furthermore if the equilibrium path
involves playing the stationary outcome playing p(s) for each s then
1
(1
n
)p(s)(s
p(s)) + v
(1
)p(s)(s
p(s)) + v;
(6)
for some v
0 and this implies (4)
(iii) For su¢ ciently large , setting p(s) = s=2, the myopic monopoly pro…t
maximizing price, for every s; satis…es (4) . This re‡ects the fact that the
current state of demand fades into insigni…cance for high discount factors.
(iv) In a strongly symmetric equilibrium, p(s) maximizes v , s.t. (4) and
(5). Also note that (4) can be rewritten as
(n
n v
1) (1
)
p (s) (s
p (s)) :
Hence, there exists s < max S such that for all s > s, p(s) is the smaller of
the two roots solving, from (6.1.3),
p(s)(s
and if s 2 S, then for all s
p(s)) =
(n
n v
1) (1
)
;
(7)
s; p(s) is the myopic monopoly price
s
p(s) = :
2
It is immediate from (7) that for s > s, p(s) is decreasing in s, giving
countercyclical collusion in high states.
Intuition: The most pro…table deviation open to a …rm is to undercut the
equilibrium price by a minuscule margin, jumping from a 1=n share of the
market to the entire market, at essentially the current price. The higher
the state, the more tempting this deviation. Thus, the higher the state, and
hence the higher the quantity demanded at any given price, the lower must
the equilibrium price be to render this deviation unpro…table. The function
p(s) is thus decreasing for s > s. In periods of relatively high demand,
colluding …rms set lower prices.
5. (a) I1
I2
::::
Ik :
(b) Standard
(c) Standard
(d)
1’s strategy: ( 0 ;
2’s strategy: (p; 1
1;
2
=1
0
1)
denoting probabilities Out, L and R.
p) denoting probabilities L’and R’.
2’s beliefs: ( 1 ; 1
1 ) denoting conjecture of 1 having played L and R
respectively (conditional on 2’s move.
1
= 1=2 is the critical beliefs for 2.
1
= 1=2 indi¤erent
Assume pure strategies by 1:
If
1
1
1=2 => best response is R’) R )
< 1=2 => best response is to L’) L )
1
= 0; a contradiction
1
= 1; a contradiction
Therefore no pure PBE.
Now suppose that 2 is randomising.
Expected payo¤ of 1:
L ) ap
R) a(1
p)
out) 2
Case 1: Prob Out is less than one; then 1 = 1=2 and ap = a(1 p) 2:
Therefore, possible if a 4 and then p = 1=2 and the equilibrium strategy
of 1 is ( 0 ; 1 2 0 ; 1 2 0 ) for andy 0 .
Case 2: Prob out is one; ap 2 and a(1 p) 2 which is possible if a
in which case the equilibrium is {out, (a 2)=a p 2=a; 1 = 1=2):
4;
(e) Need to discuss that beliefs in a PBE o¤-the-equilibrium is arbitrary and
one could reason in many di¤erent ways on how to set the probabilities when
the counterfactual happens. Forward induction is one way of doing so and
re…ning the equilibrium concept. It assumes that players reason rationally
about how they could have reached a counterfactual. Forward induction can
be based on domination argument or on an equilibrium argument (the latter
may be too strong).
Weakness of FI arguments:
Excludes the possibility that the past may be result of irrationality; if
we do allow it need a theory of irrationality or mistakes (proper Equilibrium)
Together with BI, FI may be too strong: results in existence problems.
The candidate needs to illustrate these with examples.