Topics in Microeconomics 2009-10 Kirori Mal College, University of ...

Topics in Microeconomics 2009-10 Kirori Mal College, University of Delhi 

HW 1 

1. Consider the following game: 

2.


HW 2 

Suppose a parent and child play the following game, first analyzed by Becker (1974). First child takes the 

action, A ≥ 0, that produces income for the child, IC(A), and income for the parent, IP(A). Second, the 

parent observes the incomes IC and IP and then chooses a bequest, B, to leave to the child. The child’s 

payoff is U(IC+B)= log(IC+B); the parent’s is U(IP-B)+kU(IC+B) = log(IP - B)+klog(IC + B), where k > 0 

reflects the parent’s concern for the child well-being. Assume: bequest B can be positive or negative, the 

income functions IC (A) and IP (A) are strictly concave and are maximized at AC >0 and AP > 0, 

respectively. Verify the “Rotten Kid” theorem: in the backwards induction outcome, the child chooses the 

action that maximizes the family’s aggregate income, IC(A)+IP(A), even though only the parent’s payoff 

exhibits altruism.


HW 3 

A. Read Chapter 5 of the text book. 

B. Refer the problem with players Karl, Rosa and Ernesto that was discussed in class. Find a 

Nash equilibrium (that is not subgame perfect) other than the one we found in class. 

C. Do the following problems: 

1. There is a selection committee with three members (1, 2 and 3) who have to choose one of four 

candidates (a, b, c and d). The preferences of the members among the candidates are as follows 

u1(d) < u1(c) < u1(b)< u1(a) 

u2(c) < u2(d) < u2(a)< u2(b) 

u3(c) < u3(a) < u3(b)< u3(d) 

where ui is the utility function of person i. The following procedure is used to choose among the 

candidates. First, Member 1 vetoes one of the candidates, thereby knocking him out of contention. 

Then Member 2 vetoes one of the three remaining candidates, thereby knocking him out of 

contention. Finally, Member 3 vetoes one of the two remaining candidates. The candidate not 

vetoed by any of the members is chosen. Each member wishes to maximize his utility. 

Who will Member 1 veto? Who will Member 2 veto? Who will Member 3 veto? Justify your 

answers. 

(Hint: From the point of view of any particular committee member, maximization of his utility 

requires him to look forward and predict the behavior of members who will exercise their veto 

after that particular committee member. In particular, you are asked to find subgame perfect 

outcome of the game) 

Also answer the following questions. 

a) Write down the set of players, set of terminal histories, player function, and players’ 

preferences for the game. 

b) How many strategies does each player have? 

c) Find a Nash equilibrium of the above game which is not Subgame perfect. 

2. Three friends take part in the following pizza splitting game. In period 1, friend 1 is asked to split 

the pizza of size 1 into at most two parts (i.e. he is free to not split the pizza at all). Let the two 

resulting fractions of pizza be p1 and 1-p1 . In period 2, friend 2 is asked to make a single split on 

any one of the existing pizza parts (again he is free to make no split at all). Once the two friends 

have made their splitting decisions, period 3 comes and friend 3 is asked to choose one of 

possible many pizza parts on the table. This is followed by friend 2 choosing one of the remaining 

parts. Whatever is left is consumed by friend 1. If you were friend 1 (and you and your friends all 

liked pizza), what would your choice of p1 be?


HW 4 

A. Do the following problems: 

1. Consider a market in which there are two firms, both producing the same good. Firm i’s cost 

of producing qi units of the good is Ci(qi) = 8qi for each i ∈ {1, 2}; the price at which output 

is sold when the total output is Q is Pd(Q) = max{24 – Q, 0}, where Q = q1 + q2. Each firm’s 

strategic variable is output and the firms make their decisions sequentially: one firm chooses 

its output, then the other firm does so, knowing the output chosen by the first firm. 

a) Model this situation as an extensive game. 

Players: 

Terminal Histories: 

Player function: 

Preferences: 

b) Find the subgame perfect equilibrium and outcome of Stackelberg’s duopoly game. 

Equilibrium: Strategy of firm 1 → 

Strategy of firm 2 → 

Outcome: 

c) Consider a variant of the situation in which firm 2 has some fixed cost of production (f > 0): 

C2(q2) = f + 8q2 for q2 > 0 

= 0 for q2 = 0 

Find the subgame perfect equilibrium and outcome of Stackelberg’s duopoly game in each of 

the following cases: 

i) f = 1 



Outcome: 

ii) f = 9 



Outcome: 

iii) f = 25 



Outcome:


2. Consider a market in which there are three firms, all producing the same good. Firm i’s cost 

of producing qi units of the good is Ci(qi) = 0 for each i ∈ {1, 2, 3}; the price at which output 

is sold when the total output is Q is Pd(Q) = max{32 – Q, 0}, where Q = q1 + q2 + q3. Each 

firm’s strategic variable is output and the firms make their decisions sequentially: one firm 

chooses its output, then the second firm does so, knowing the output chosen by the first firm, 

and finally the third firm makes its choice of output knowing the output chosen by the first 

two firms. 


Players: 



Preferences: 

b) Find the subgame perfect equilibrium and outcome of Stackelberg’s duopoly game. 




Outcome: 

c) Can you suggest a Nash Equilibrium strategy profile which results in an outcome different 

from the one which results from subgame perfect equilibrium above? 




Outcome:


3. A legislature has k members, where k is an odd number. Two rival bills, X and Y, are being 

considered. The bill that attracts the votes of a majority of legislators will pass. Interest 

group X favors bill X, whereas interest group Y favors bill Y. Each group wishes to entice a 

majority of legislators to vote for its favorite bill. First interest group X gives an amount of 

money (possibly zero) to each legislator, then interest group Y does so. Each interest group 

wishes to spend as little as possible. Group X values the passing of bill X at $VX > 0 and the 

passing of bill Y at zero, and group Y values the passing of bill Y at $VY > 0 and the passing 

of bill X at zero. (For example, group X is indifferent between an outcome in which it spends 

VX and bill X is passed and one in which it spends nothing and bill Y is passed.) Each 

legislator votes for the favored bill of the interest group that offers her the most money; a 

legislator to whom both groups offer the same amount of money votes for bill Y (an arbitrary 

assumption that simplifies the analysis without qualitatively changing the outcome). For 

example, if k = 3, the amounts offered to the legislators by group X are x = (100, 50, 0), and 

the amounts offered by group Y are y = (100, 0, 50), then legislators 1 and 3 vote for Y and 

legislator 2 votes for X, so that Y passes. (In some legislatures the inducements offered to 

legislators are more subtle than cash transfers.) 

We can model this situation as the following extensive game. 

Players The two interest groups, X and Y. 

Terminal histories The set of all sequences (x, y), where x is a list of payments to legislators 

made by interest group X and y is a list of payments to legislators made by interest group Y. 

(That is, both x and y are lists of k nonnegative integers.) 

Player function P(∅) = X and P(x) = Y for all x. 

Preferences The preferences of interest group X are represented by the payoff function 

where bill Y passes after the terminal history (x, y) if and only if the number of components of 

y that are at least equal to the corresponding components of x is at least 0.5 (k + 1) (a bare 

majority of the k legislators). The preferences of interest group Y are represented by the 

analogous function (where VY replaces VX, y replaces x, and Y replaces X). 

Find the subgame perfect equilibrium and outcome of the above game in each of the 

following cases: 

i) VX = 900, VY = 600 and k = 5 

Equilibrium: Strategy of group X → 

Strategy of group Y → 

Outcome: 

ii) VX = 1200, VY = 600 and k = 3 

Equilibrium: Strategy of group X → 

Strategy of group Y → 

Outcome:


4. (Multiple choice question) There is a pile of 17 matchsticks on a table. Players 1 and 2 take 

turns in removing matchsticks from the pile, starting with player 1. On each turn, a player 

has to remove a number of sticks that equals the square of a positive integer, such that the 

number of matchsticks that remain on the table equals some non-negative integer. The 

player, who cannot do so, when it is his /her turn, loses. Which of the following statements is 

true? 

a) If player 2 plays appropriately, he/she can win regardless of how 1 actually plays. 

b) If player 1 plays appropriately, he/she can win regardless of how 2 actually plays. 

c) Both players have a chance to win, if they play correctly. 

d) The outcome of the game cannot be predicted on the basis of the data given. 

5. Each of n ≥ 2, i = 1, ..., n can make contributions si ∈ [0, w] (w > 0) to the production of 

some public good. The contributions are made sequentially, that is, beginning with individual 

1, each individual i makes the contribution after 1, …, i - 1 have contributed and before i + 

1,…., n make their contributions. Assume that each individual i observes the contributions of 

the preceding i - 1 contributors before choosing her contribution. Their payoff functions are 

given by πi (s1, .., sn) = n(min{s1, .., sn}) − si. 


Players: 



Preferences: 

b) Find the subgame perfect equilibrium and outcome of the game in question. 

Equilibrium: Strategy of individual 1 → 

Strategy of individual 2 → 

Strategy of individual 3 → 

.. 

Strategy of individual i → 

Outcome: 

B. Read Chapter 5 and 6 of the text book and try all the problems.


HW 5 

1. Consider the following game played between players 1 and 2. Player 1 moves first and plays 

either L or R. This is observed by both the players. If L is played, players 1 and 2 simultaneously 

choose from the sets {l, r} and {t, d} respectively and the game ends with payoffs (3, 1), (0, 0), (0, 

0) and (1, 3) if (l, t), (l, d), (r, t) and (r, d) are played respectively. If Player 1 plays R, players 1 

and 2 simultaneously choose from the sets {m, n} and {q, s} respectively and the game ends with 

payoffs (2, 1), (1, 2), (1, 2) and (2, 1) if (m, q), (m, s), (n, q) and (n, s) are played respectively. 

(i) Model this game as an extensive form game with perfect information and simultaneous 

moves. 

(ii) Compute all subgame perfect Nash equilibria (SPNE) in the game. 

2. Players 1 and 2 choose an element of the set {1, ...,K}. If the players choose the same number, 

then player 2 pays 1 to player 1; otherwise no payment is made. Find all pure and mixed strategy 

Nash equilibria of this game. 

3. Define a Bayesian game. Formulate problems 4 through 8 as Bayesian games (i.e. specify 

players, states, actions, signals, beliefs, payoff functions) 

4. Consider a Cournot duopoly game with incomplete information. Suppose that demand is given by 

p = max{1 – Q, 0}, where Q is the total quantity produced in the industry. Firm 1 selects a 

quantity q1, which it produces at zero cost. Firm 2’s cost of production is private information 

(selected by nature). With probability ½, firm 2 produces at zero cost. With probability ½, firm 2 

produces with a marginal cost of ¼. Call the former type of firm 2 “L” and the latter type “H” 

(for low and high cost, respectively). Firm 2 knows its type, whereas firm 1 knows only the 

probability that L and H occur. Let q2 H and q2 L denote the quantities selected by the two types of 

firm 2. Then when firm 2’s type is L, its payoff is given by u2 L = (1 – q1 – q2 L )q2 L . When firm 2’s 

type is H, its payoff is u2 L = (1 – q1 – q2 H )q2 H – (q2 H )/4. As a function of the strategy profile (q1; q2 L , 

q2 H ), firm 1’s payoff is u1 = ((1 – q1 – q2 L )q1)/2 + ((1 – q1 – q2 H )q1)/2 

Solve for Bayesian Nash Equilibrium. (Ans. (q1; q2 L , q2 H )= (3/8; 5/16, 3/16)) 

5. Consider a simple simultaneous-bid poker game. First, nature selects numbers x1 and x2. Assume 

that these numbers are independently and uniformly distributed between 0 and 1. Player 1 

observes x1 and player 2 observes x2, but neither player observes the number given to the other 

player. Simultaneously and independently, the players choose either to fold or to bid. If both 

players fold, then they both get the payoff –1. If only one player folds, then he obtains –1 while 

the other player gets 1. If both players elected to bid, then each player receives 2 if his number is 

at least as large as the other player’s number; otherwise, he gets –2. Compute the Bayesian Nash 

equilibrium of this game. (Hint: Look for a symmetric equilibrium in which a player bids if and 

only if his number is greater than some constant α. Your analysis will reveal the equilibrium 

value of α.) 

6. Consider the following game. Nature selects A with probability ½ and B with probability ½. If 

nature selects A, then players 1 and 2 interact according to matrix “A”. If nature selects B, then


the players interact according to matrix “B”. Suppose that, before the players select their 

actions, player 1 observes nature’s choice. (That is, player 1 knows which matrix is being 

played.) Player 2 does not observe nature’s choice. These matrices are pictured here. 

Solve for Bayesian Nash equilibrium. 

A B 

1↓2 → L R 1↓2 → L R 

U 2, 1 0, 0 U 1, 2 0, 0 

D 0, 0 1, 2 D 0, 0 2, 1 

7. Suppose that a public good is provided to a group of people if at least one person is willing to 

pay the cost of the good. Assume that the people differ in their valuations of the good, and each 

person knows only her own valuation. Denote the number of individuals by n, the cost of the good 

by c > 0, and individual i’s payoff if the good is provided by vi. If the good is not provided then 

each individual’s payoff is 0. Each individual i knows her own valuation vi. She does not know 

anyone else’s valuation, but knows that all valuations are at least vL and at most vH, where 0 ≤ vL 

< c < vH. She believes that the probability that any one individual’s valuation is at most v is F(v), 

independent of all other individuals’ valuations, where F is a continuous increasing function. 

Suppose F(v) = max{0, min{(v – vL)/ (vH – vL), 1}} i.e. uniform distribution from vL to vH. 

The following mechanism determines whether the good is provided. All n individuals 

simultaneously submit envelopes; the envelope of any individual i may contain either a 

contribution of c or nothing (no intermediate contributions are allowed). If all individuals submit 

0 then the good is not provided and each individual’s payoff is 0. If at least one individual 

submits c then the good is provided, each individual i who submits c obtains the payoff vi − c, and 

each individual i who submits 0 obtains the payoff vi. Solve for the Bayesian Nash equilibrium 

when n = 2 and n = 3. 

8. Consider a Coumot duopoly operating in a market with inverse demand P(Q) = max{a – Q, 0}, 

where Q = q1+ q2 is the aggregate quantity on the market. Both firms have total costs c(qi) = cqi, 

but demand is uncertain: it is high (a = aH) with probability θ and low (a = aL) with probability 

1– θ. Furthermore, information is asymmetric: firm 1 knows whether demand is high or low, but 

firm 2 does not. All of this is common knowledge. The two firms simultaneously choose quantities. 

Suppose aH = 100, aL = 80, θ = 0.5, and c = 0. What is the Bayesian Nash equilibrium of this 

game? 

9. Consider the first-price sealed-bid auction, but each bidder i observes only his own valuation vi. 

The valuation is distributed uniformly and independently on [0, v*] for each bidder. Derive 

symmetric (pure strategy) Bayesian Nash equilibrium of this auction if there are two bidders. 

Suppose that bids can only be non-negative. (Look for an equilibrium in which bidder i’s bid is a 

linear function of his valuation.)


HW 6 

1. Consider an “all-pay auction” with two players (the bidders). Player 1’s valuation v1 for the 

object being auctioned is uniformly distributed between 0 and 1. That is, for any x∈ [0, 1], player 

1’s valuation is below x with probability x. Player 2’s valuation is also uniformly distributed 

between 0 and 1, so the game is symmetric. After nature chooses the players’ valuations, each 

player observes his/her own valuation but not that of the other player. Simultaneously and 

independently, the players submit bids. The player who bids higher wins the object, but both 

players must pay their bids. That is, if player i bids bi, then his/her payoff is – bi if he/she does not 

win the auction; his/her payoff is vi – bi if he/she wins the auction. Calculate the Bayesian Nash 

Equilibrium strategies (bidding functions). (Hint: The equilibrium bidding function for player I is 

of the form bi(vi) = kvi 2 for some k.) 

2. Consider a “common-value auction” with two players, where the value of the object being 

auctioned is the same for both players. Call this value Y and suppose that Y = y1 + y2, where y1 

and y2 are both uniformly distributed between 0 and 10. That is, for any x ∈ [0, 10], we have y1 < 

x with probability x/10. Suppose that player 1observes y1 (but does not observe y2) and that 

player 2 observes y2 (but does not observe y1). The players simultaneously submit bids, b1 and b2. 

If player i bids higher than does player j, then player I wins the auction and gets the payoff Y – b 

whereas player j gets 0. 

a) Suppose player 2 uses the bidding strategy b2(y2) = 3 + y2. What is player 1’s best-response 

bidding strategy? 

b) Suppose each player will not select a strategy that would surely give him/her a negative 

payoff conditional on winning the auction. Suppose also that this fact is common knowledge 

between the players. What can you conclude about how high the players are willing to bid?


HW 7 

In a 2 x 2 signaling game, there can be any or all of the following Weak Sequential Equilibria (WSE): 

both types of Player 1 may play pure strategies in equilibrium (if they play the same strategy, we say it is 

a pooling equilibrium; if they differ, we say it is a separating equilibrium); one type of Player 1 may play 

a pure strategy while the other plays a mixed strategy (leading to a semi-separating equilibrium); or both 

types of Player 1 may play mixed strategies. (We won’t deal with the latter two cases.) 

When looking for a WSE... 

1. Decide whether you’re looking for a separating or pooling equilibrium. 

2. Assign a strategy (a message for each type) to Player 1; make sure it is not strictly dominated. 

3. Derive beliefs for Player 2 according to Bayes’ rule at each information set reached with positive 

probability along the equilibrium path. Set arbitrary beliefs for Player 2 at information sets that are 

never reached along the equilibrium path. 

4. Determine Player 2’s best response. 

5. In view of Player 2’s response, check to see whether Player 1 has an incentive to deviate from the 

strategy you assigned her in any state of the world (in other words, for all types of Player 1). If she does 

not, you have found a WSE. If she does, this is not an equilibrium - return to step 2 and assign Player 1 a 

different strategy. 

6. Once you have exhausted all possible strategies within an equilibrium subset, return to step 1 and 

select a different type of WSE. 

Find WSE for the following game: 

A signaling game in which there are two types of player 1, strong and weak, the probabilities of these 

types are 0.9 and 0.1 respectively, the set of messages is {B, Q} (the consumption of beer or quiche for 

breakfast), and player 2 has two actions, F(ight) or N(ot). Player 1's payoff is the sum of two elements: 

she obtains two units if player 2 does not fight and one unit if she consumes her preferred breakfast (B if 

she is strong and Q if she is weak). Player 2's payoff does not depend on player 1's breakfast; it is 1 if he 

fights the weak type or if he does not fight the strong type.

Topics in Microeconomics 2009-10 Kirori Mal College, University of ...

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