Problem Set 2 and 3: Solutions
Advanced Microeconomics II (Fall, 2nd, 2013)
1. Question 1 (6 points) See the following game tree.
(a) How many information sets (including singleton sets) does this game have? (b) How many subgames (including the entire game) does this game have?
(c) Find all (pure-strategy) subgame perfect equilibria. 2. Question 2 (3 points)
Players 1 (proposer) and 2 (receiver) are bargaining over how to split the ice-cream of size 1. In the first stage, player 1 proposes a share {x, 1 − x} to player 2 where x ∈ [0, 1] is player 1’s own share. Player 2 can decide whether accept the offer or reject it. If player 2 accepts, then the game finishes and players get their shares. If player 2 rejects, the game move to the second stage, in which the size of the ice-cream becomes δ(∈ (0, 1)) of the original size due to melting. In the second stage, by flipping a coin, the ice-cream is randomly assigned to one of the players. Suppose each player maximizes expected size of the ice-cream that she can get. Derive a subgame perfect Nash equilibrium of this game.
3. Question 3 (5 points)
Players 1 and 2 are forming a firm. The value of their relationship depends on the effort that each expends. Suppose that player i’s utility from the relationship is x2j+ xj− xixj, where xi is i’s effort and xj is the effort of the other player. Assume x1, x2 ≥ 0.
(a) Find the Nash equilibrium of this game. Is it Pareto efficient?
(b) Suppose that the players interact over time, which we model with the infinitely repeated version of the game. Let δ denote the (common) discount factor of the players. Under what conditions can the players sustain some positive effort level k = x1 = x2 > 0 over time?
4. Question 4 (5 points)
Consider a game of election with asymmetric information among voters. Whether candidate A or candidate B is elected depends on the votes of two citizens (denoted by 1 and 2). The economy may be in one of two states, α and β. The citizens agree that candidate A is best if the state is α and candidate B is best if the state is β. The payoff for each citizen is symmetric and given as follows: 1 if the best candidate wins, 0 if the other candidate wins, and 1/2 if the candidates tie. Suppose that citizen 1 is informed of the true state, whereas citizen 2 believes it is α with probability 0.9 and β with probability 0.1. Each citizen may either vote for candidate A, vote for candidate B, or not vote.
(a) Formulate this situation as a Bayesian game.
(b) Show that the game has exactly two pure strategy Nash equilibria, the one of which involves weakly dominated strategy.
5. Question 5 (6 points)
Two players, 1 and 2, each own a house. Each player i values her own house at vi
and this is private information. The value of player i’s house to the other player j(6= i) is 3
2vi. The values vi are drawn independently from the interval [0, 1] with uniform distribution. Suppose players announce simultaneously whether they want to exchange (E) their house of not (N). If both players agree to an exchange, the exchange takes place. Otherwise no exchange occurs.
(a) Find a Bayesian Nash equilibrium of the 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 the value of player i’s house to the other player j becomes 5
2vi?
6. Question 6 (6 points)
Consider the following labor market signaling game. There are two types of worker. Type 1 worker has a marginal value product of 1 and type 2 worker has a marginal value product of 2. The cost of signal z for type 1 is C1(z) = z and for type 2 is C2(z) = (1 − c)z. The worker is type 1 with probability p and 2 with probability 1 − p. There are two firms that play a Bertrand wage bidding game for the services of the worker, which simplifies wage determination: the equilibrium wage becomes the expected marginal value product of the worker.
(a) Show that there is a separating PBE in which type 1 does not signal and type 2 chooses z ∈ [1,1−c1 ].
(b) Show that the equilibrium payoff to the worker in the pooling equilibrium with no signaling is w(p) = 2 − p.
7. Question 7 (9 points)
Consider a game between two friends, Amy and Brenda. Amy wants Brenda to give her a ride to the mall. Brenda has no interest in going to the mall unless her favorite shoes are on sale (S) at the large department store there. Amy likes these shoes as well, but she wants to go to the mall even if the shoes are not on sale (N ). Only Amy subscribes to the newspaper, which carries a daily advertisement of the department store. The advertisement lists all items that are on sale, so Amy learns whether or not the shoes are on sale. Amy can prove whether or not the shoes are on sale by showing the newspaper to Brenda. But this is costly for Amy, because she will have to take the newspaper away from her sister, who will yell her later for doing so.
In this game, the nature first decides whether or not the shoes are on sale, and this information is made known to Amy. That is, Amy observes whether the nature chose S or N . The nature chooses S with probability p and N with probability 1−p. Then Amy decides whether or not to take the newspaper to Brenda (T or D). If she takes the newspaper to Brenda, then it reveals to Brenda whether the shoes are on sale. In any case, Brenda must then decide whether to take Amy to the mall (Y ) or to forget it (F ). If the shoes are on sale, then going to the mall is worth 1 unit of utility to Brenda and 3 to Amy. If the shoes are not on sale, then traveling to the mall is worth 1 to Amy and −1 to Brenda. Both players obtain 0 utility when they do not go to the mall. Amy’s personal cost of taking the newspaper to Brenda is 2 units of utility, which is subtracted from her other utility amounts.
(a) Draw the game tree of this game.
(b) Does this game have a separating perfect Bayesian Nash equilibrium? If so, fully describe it.
(c) Does this game have a pooling perfect Bayesian Nash equilibrium? If so, fully describe it.
Hint: A separating equilibrium means that Amy takes different strategies in S and N , while she chooses the same strategy in a pooling equilibrium. Your answer in (c) might depend on the value p.
Answer
1. Question 1 (6 points)
(a) There are 6 information sets. Note that a singleton set is a set which contains only one element. So, there is 1 proper information set (with more than one element) and 5 singleton information sets.
(b) There are 5 subgames.
(c) There are two subgame perfect Nash equilibria: (ADGI, LN ) and (BDGI, KN ). The resulting payoffs are (2, 0) in the former, (3, 4) in the latter equilibrium. 2. Question 2 (3 points)
x = 1 − δ
2. Player 2 accepts the offer if and only if x ≤ 1 − δ
2. Note that each player will receive δ
2 as his/her expected payoff in the second stage. Taking this into account, the proposer must give at least δ
2 to the receiver. 3. Question 3 (5 points)
(a) BRi(xj) = 0 if xj > 0 and any non-negative number if xj = 0. So, there is a unique Nash equilibrium, (0, 0). This equilibrium is not Pareto efficient. If both players choose strictly positive effort k(> 0), then the payoff of each player becomes k2+ k − k2 = k, which exceeds the Nash equilibrium payoff 0. (b) Consider the trigger strategy that use the stage game Nash equilibrium (0, 0) as its punishment. That is, playing (k, k) unless no one deviates, switching to (0, 0) forever if someone once deviates. Such a trigger strategy becomes a subgame Nash equilibrium if the following condition holds.
k + δk + δ2k + · · · ≥ (k + k2) + δ · 0 + δ2· 0 + · · ·
⇒ k
1 − δ ≥ k
2+ k ⇒ δ ≥ k
1 + k 4. Question 4 (5 points)
(a) The nature chooses A with probability 0.9 and B with probability 0.1. A realized state can be seen as a type of citizen 1 and can be observed only by her. There is no type for citizen 2. The action for each player is 1(=vote for candidate 1), 2(=vote for candidate 2), N (=not vote). The payoffs should be described according to the above situation (I skip and leave it to you).
(b) The best reply of citizen 1 for each state is
BR1A(1) = 1 and N , BR1A(2) = BR1A(N ) = 1 BR1B(2) = 2 and N , BR1A(1) = BR1A(N ) = 2.
Note that the best reply of citizen 2 against the possible equilibrium strategies of citizen 1 is
BR2(N N ) = BR2(N 2) = 1, BR2(1N ) = BR2(12) = N .
Looking for the mutual best raply, we find the following two Nash equilibria (N 2, 1) and (12, N ).
Note that the citizen 1 chooses N 2 in the first equilibrium, which is weakly dominated by the strategy 12.
5. Question 5 (6 points)
We focus on symmetric equilibria, and thereby let θi = θ. (You can show that there exists no asymmetric equilibrium.)
(a) θ = 0 is a unique BNE. That is, no exchange occurs in equilibrium. Suppose that θ > 0 constitutes a BNE. Then, the expected value of exchange is
1 2θ ×
3 2 =
3 4θ. Clearly, players with vi ∈ 3
4θ, θ
do not have incentive to exchange. Thus, we derive contradiction.
(b) θ = 0, 1 are BNE; the former involves a weakly dominated strategy while the latter does not. In this sense, θ = 1 is essentially a unique BNE. Suppose that 0 < θ < 1 constitutes a BNE. Then, the expected value of exchange is
1 2θ ×
5 2 =
5 4θ. Clearly, players with vi ∈
θ, min 5 4θ, 1
have incentive to exchange. Thus, we derive contradiction.
6. Question 6 (6 points)
(a) In a separating PBE, type i worker receives the wage equal to her productivity i (= 1, 2). When type 1 chooses z1 = 0, the payoff becomes 1 − C1(0) = 1. Since she must prefer not to signal (so as to manipulate type 2),
2 − C1(z2) ≤ 1 ⇐⇒ 1 − z2 ≤ 0 ⇐⇒ z2 ≥ 1 should be satisfied. Since type 2 must also prefer separating,
2 − C2(z2) ≥ 1 − C2(0) ⇐⇒ 2 − (1 − c)z2 ≥ 1 ⇐⇒ z2 ≤ 1 1 − c
should hold. By combining these two inequalities, z2 ∈ [1,1−c1 ] (and z1 = 0) constitutes a separating PBE.
(b) If no one signals, the expected productivity is 1 × p + 2 × (1 − p) = 2 − p. Then, the payoff of type i worker is 2 − p − Ci(0) = 2 − p, independent of i. 7. Question 7 (9 points)
See the handout: Joel Watson (2008), Strategy, 2nd ed., page 343-345.
