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Final Exam

Date: March 28, 2012

Instructor: Yosuke YASUDA

1. Extensive Form (16 points)

For each of the game trees (a) and (b) below, answer the following questions: (1) How many information sets are there?

(2) How many subgames (including the entire game) are there? (3) Derive the subgame perfect Nash equilibrium.

2. Duopoly Game (20 points)

Consider a duopoly game in which two firms, denoted by Firm 1 and Firm 2, simultaneously and independently select their own prices, p¹ and p², respectively. The firms’ products are differentiated. After the prices are set, consumers demand A_{− p}1₊

p2

2 units of the firm 1’s good and A − p²+ ^p1

2 units of the firm 2’s good. Assume that the firms have identical (and constant) marginal costs c(< A), and the payoff for each firm is equal to the firm’s profit, denoted by π¹ and π².

(1) Write the payoff functions π¹ and π² (as a function of p¹ and p²). (2) Derive the best response function for each player.

(3) Find the pure-strategy Nash equilibrium of this game.

(4) Derive the prices (p¹, p2) that maximize joint-profit, i.e., π¹+ π².

(5) Suppose that this game is played finitely many times, say T (≥ 2) times. De- rive the subgame perfect Nash equilibrium of such a finitely repeated game. Assume that payoff of each player is sum of each period payoff.

(6) Now suppose that the game is played infinitely many times: payoff of each player is discounted sum of each period payoff with some discount factor δ ∈ (0, 1). Assume specifically that A = 16, c = 8. Then, derive the condition under which the trigger strategy sustains the joint-profit maximizing prices you derived in (3) (as a subgame perfect Nash equilibrium).

3. Auction (14 points)

Suppose that a seller auctions one object to two buyers, = 1, 2. The buyers submit bids simultaneously, and the buyer with higher bid receives the object. The loser pays nothing while the winner pays the average of the two bids ^{b+ b}

′

2 ^{where b is} the winner’s bid, b^′ is the loser’s bid. Assume that the valuation of the object for each buyer is private and it is independently and uniformly distributed between 0 and 1.

(1) Suppose that buyer 2 takes a linear strategy, b² = αv². Then, derive the probability such that buyer 1 wins (as a function of b¹).

(2) Solve a Bayesian Nash equilibrium. You can assume (without proof) that equilibrium bidding strategy is symmetric and linear: b1 = αv1, and b2 = αv2.

Answers

1. Extensive Form (16 points) (1) (a) 1 (b) 2

(2) (a) 4 (b) 3

(3) (a) (AHJ, DF ) (b) (AIJ, DG), (BIK, DF )

2. Duopoly Game (24 points)

(1) Firms’ payoffs are described as follows: π1 _{= (p}1_{− c)}



A− p¹+^p2 2

 π2 _{= (p}1_{− c)}



A− p²+^p1 2

 .

(2) Let us first consider the Firm 1’s maximization problem: maxp1^≥0

(p¹− c)^A− p¹+^p2 2



The FOC is:

A− p¹ +^p2

2 ^{− (p1− c) = 0.}

⇒ p¹ = ^{A+ c}

2 ⁺

1 4^p2.

Therefore, the best reply function of Firm 1 is written as BR1_(p2_{) =}

A+ c

2 ⁺

1 4^p2.

By almost same calculation, we obtain firs 2’s best reply function as well: BR2_(p1_{) =}

A+ c

2 ⁺

1 4^p1.

(3) Let p^∗ = (p^∗1, p^∗2) be the Nash equilibrium prices. Since p^∗ is an intersection of the firms’ best reply functions, the following equalities must hold:







p^∗1 = ^{A+ c}

2 ⁺

1 4^p

∗ 2

p^∗_{2 =} ^{A+ c}

2 ⁺

1 4^p

∗ 1

Solving these simultaneous equations give us (p^∗1, p^∗2) = ^{ 2(A + c)}

3 ^,

2(A + c) 3

 .

(4) Let ˆp = (ˆp1,pˆ2) be the joint-profit maximizing prices. Then, ˆp is the solution of the following maximization problem:

pmax₁,p₂≥0 ^(p 1_{− c)}



A− p¹+ ^p2 2

+ (p²− c)^A− p²+^p1 2



The FOC’s are:



A− p¹+ ^p2 2

− (p¹− c) +^{p2− c}

2 ^{= 0, (1)}

p1_{− c}

2 ⁺



A− p²+^p1 2

− (p²− c) = 0. (2)

Note that these equalities can be written as (1) ⇐⇒ A − 2p¹+ p²+ ^c

2 ^{= 0,} (2) ⇐⇒ A − 2p²+ p¹+ ^c

2 ^{= 0.} Solving the above simultaneous equations, we obtain

(ˆp1,pˆ2_{) =}

 A+ ^c

2^{, A+} c 2

 .

(5) When the stage has a unique Nash equilibrium, the subgame perfect Nash equilibrium of any finitely repeated games is also unique: players choose the stage game Nash equilibrium irrespective of the history of the game. So, in our game, players choose (p^{1, p2}) = (p^∗1^{, p}

∗

2), the solution derived in (1), for all periods (independent of the history), which constitutes a unique subgame perfect Nash equilibrium.

(6) Consider the following trigger strategy: Choose p = ˆp as long as no one has deviated, and switch to p = p^∗ forever after someone deviates. Note that under this trigger strategy, each player receives ˆ^πi = 72 each period during the cooperative phase, and π_i^∗ = 64 on the punishment phase (note that p^∗_i = 16, ˆp_i = 20, i = 1, 2.) If Firm i unilaterally deviates from p = ˆp, it obtains π(pi^,20). The maximum deviation profit, denoted by π_i^′, is the solution of the following optimization problem:

maxpi

(pi− 8)(16 − pi +²⁰ 2 ⁾ FOC: 34 − 2pi = 0 ⇒ p^′_i = 17.

The associated profit is π^′_i = π(17, 20) = 81. So, the incentive constraint becomes

72 + 72δ + 72δ²+ · · · = ¹

1 − δ^{72 ≥ 81 +} δ 1 − δ⁶⁴

⇐⇒ δ ≥ ^{81 − 72} 81 − 64 ⁼

9 17^.

Thus, p = ˆp can be sustained via trigger strategy if and only if δ ≥ 17⁹.

3. Auction (14 points) (1) We can derive

Pr{b^{1 > b2}} = Pr{b^{1 > αv2}}

= Pr{^b1

α ^{> v2} =} b1

α^.

(2) Given that buyer 2 follows the equilibrium bidding strategy, solve the buyer 1’s optimization problem (E is the conditional expactation given b¹ > b2_).

maxb₁ ^E

 v1₋

b1_{+ b}2

2



Pr{b¹ > b2_}

By assumption and (1), this is equivalent to

maxb₁ ^v 1₋

b1 ₊^b₂¹

2

!b1

α

Note that E[b² | b^{1 > b2}] = ^b2¹, since b² is uniformly distributed between 0 and b1 (given that b¹ > b2) and thereby its average should be the half of b¹. Solving the first order condition, we obtain

0 = v¹− ³

2^{b1 ⇔ b1 =} 2 3^v1. So, there is a symmetric and linear equilibrium such that

b_i = ²

3^vi for i = 1, 2.