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Problem Set 4

ECO 444

Konrad Grabiszewski

Exercise 1

The following game is called the war of attrition. There are two players fighting over a prize. Each

player chooses the time at which to give up.

Let time t be a continuous variable that starts at 0 and runs indefinitely. A strategy of player i is

denoted by ti and specifies when the player gives up. We assume that the value player i attaches

to the prize is vi > 0 and that value does not change with time.

If player i gives up first at time ti, then player i gains zero revenue but incurs cost of ti. That is,

payoff of player i is −ti. If player j gives up first at time tj, then player i gains the prize (revenue

is vi) and incurs cost tj. Note that there is no typo: the cost of player i is tj because player i

obtains the prize after tj amount of time. If both players give up at the same time ti = tj, then

the payoff of player i is 1 2 vi − ti.

Normal-form representation.

• N = {A, B} is the set of players;

• Si = [0,∞) is the set of player i’s strategies; players choose when to give up;

• ui is the utility function of player i defined as

ui(ti, tj) =

  −ti if ti < tj 1 2 vi − ti if ti = tj

vi − tj if ti > tj

(1)

Find all (pure-strategy) Nash equilibria.

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Exercise 2

There are two firms – Firm A and Firm B – involved in a synergistic relationship. For exam-

ple, an R&D cooperation. The players simultaneously choose effort levels without any form of

communication. (That is, we are dealing with a static game.) Efforts are costly.

Let si ≥ 0 denote the effort chosen by player i. The utility function of player i is,

ui(si, sj) = (k + sj)si −s2i , (2)

where (k + sj)si is the revenue, s 2 i is the cost, and k > 0.

a) Depict this game in a normal form.

b) Find the best-response correspondences of players A and B.

c) Find all (pure strategy) Nash equilibria.

d) Comparative statics analysis: Determine the impact of parameter k on (i) equilibrium strate-

gies of A and B, and equilibrium utilities of A and B.

We enrich our model and add two exogenous parameters, cA > 0 and cB > 0. Parameter ci depicts

quality/ability of player i. This parameter affects total cost of effort of player i. Hence, we say that

ci is a cost parameter. Higher ci means lower quality/ability. Our normal-form representations

remains almost the same. We only change utility functions.

ui(si, sj) = (k + sj)si − cis2i (3)

We impose the following “technical” assumption.

4cAcB > 1 (4)

e) Find the best-response correspondences of players A and B.

f) Find all (pure strategy) Nash equilibria.

g) Comparative statics analysis: Determine the impact of parameters k, cA, and cB on (i)

equilibrium strategies of A and B, and equilibrium utilities of A and B.

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Exercise 3

In this exercise, you will develop and analyze the Cournot duopoly with two different firms (see

Handout 9). In the market, there are 2 firms – Firm A and Firm B – firms producing a single

good. Let i denote a generic firm. Each firm decides how much to produce (quantity competition).

Let qi denote the quantity produced by firm i. Let Q denote the aggregate quantity; Q = qA + qB.

Firms incur costs, and we assume a linear cost function for each firm; Ci(qi) = ciqi, where ci is

marginal cost. We impose two assumptions.

a > 2cA − cB (5)

cA > cB > 0 (6)

We assume a linear demand function with a > 0.

P (Q) =

  a−Q if Q < a

0 otherwise

(7)

a) Show that assumptions (5) and (6) imply that the following is true: a > cA and a > cB.

b) Depict this game in a normal form.

c) Find the best-response correspondences of players A and B.

d) Find all (pure strategy) Nash equilibria.

e) Comparative statics analysis: Determine the impact of parameters a, cA, and cB on (i)

equilibrium strategies of A and B, (ii) equilibrium aggregate quantity, (iii) equilibrium price,

and (iv) equilibrium utilities of A and B.

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Exercise 4

In this exercise, you will develop and analyze the Tullock contest with two different players and

common value (see Handout 11). There are two players – Ann and Bob – competing for one prize.

Ann and Bob assign the same value to the prize, v > 0 (common value). Not winning the contest

results with utility zero.

Players compete in efforts and these efforts determine the probability of winning the contest.

Effort of Ann is denoted by sA and effort of Bob is denoted by sB. We assume that sA, sB ≥ 0.

The probability of Ann winning the contest is sA sA+sB

. The probability of Bob winning the contest

is sB sA+sB

.

Efforts are costly. We assume linear cost functions. That is, CA(sA) = cAsA is Ann’s cost and

CB(sB) = cBsB is Bob’s cost. We assume the following.

cA > cB > 0 (8)

Each player maximizes her/his expected utility. Hence, we obtain utility of Ann uA and utility of

Bob uB.

uA(sA, sB) = v sA

sA + sB − cAsA (9)

uB(sA, sB) = v sB

sA + sB − cBsB (10)

Remark: There is no need to depict this game in a normal form.

a) Find the best-response correspondences of Ann and Bob.

b) Find all (pure strategy) Nash equilibria.

c) Comparative statics analysis: Determine the impact of parameters cA and cB on (i) equi-

librium strategies of Ann and Bob, (ii) equilibrium probability of Ann wining, and (iii)

equilibrium probability of Bob wining.

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