# 战略决策理论代写 Strategic Decision Theory ## Strategic Decision TheoryProblem Set 9

### Problem 1  战略决策理论代写

A crime is observed by a group of n ≥ 2 people. Everybody would like the police to be informed about the crime but everybody prefers that someone else makes a call. They choose simultaneously whether to call the police or not. When nobody calls, everybody’s payoff is 0. If anybody calls the police, those who call the police receive v − c and those who do not receive v , where v > c > 0.

(a) Find all pure strategy Nash equilibria.
(b) Using the following steps find a symmetric mixed strategy Nash equilibrium.

(I) Suppose that each player other than player i calls the police with probability p. What is the expected payoff player i gets if he calls the police and what is his expected payoff if he does not?
(II) Find a symmetric mixed strategy Nash equilibrium in which each player calls with probability p ∗ . Is it unique?
(III) In the Nash equilibrium you found in (ii), is the probability that a particular player reports the crime increasing or decreasing in n? Is the probability that the police is informed about the crime (i.e. at least one player calls) increasing or decreasing in n? What happens as n approaches infinity? Interpret the result.

### Hint: You can use the fact that Pr(no one calls) = Pr(i does not call) · Pr(no one else calls).

Now, consider a model where player i’s payoff when he reports is now v − ci , instead of v − c and ci is only known to himself. Everybody believes that each ci is identically and
independently distributed over [c , c ]. Let F be the distribution function of ci and f is the associated density function with f > 0 for all ci ∈ [c , c ].

(c) Give a realistic story(s) that would justify an uncertain ci .
(d) Show that for any strategy profile of all others, a player’s best response is given by the following cutoff strategy si
(c ) with some c ∈ [c , c ] such that he calls the police if ci < c  and does not if ci > c .
(e) Find a symmetric pure strategy Bayesian Nash equilibrium in which every player plays si (c ∗ (n)). (i.e. find a value of such c ∗ (n)) What is the probability that a player does not call the police? What is the probability that nobody calls the police?

### Problem 2  战略决策理论代写

Consider a Cournot duopoly operating in a market with inverse demand P = a −Q, where Q = q1 + q2 is the aggregate quantity on the market. Throughout this question, you can ignore second-order conditions and also non-negativity constraints (i.e. a is always sufficiently high relative to c ).

(a) (From Gibbons 3.2) Assume that both firms have the same unit cost c but demand is uncertain: it is high (a = aH ) with probability θ and low (a = aL ) with probability 1−θ , where aH > aL . 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 firms simultaneously choose quantities. What is the Bayesian Nash equilibrium of this game?

(b) Now, assume that there is no uncertainty in the demand: a is fixed. However, the unit cost of each firm differs: firm i’s unit cost is ci for i = 1, 2. Suppose both c1 and c2 are commonly known and they choose their outputs simultaneously. What is the Nash equilibrium? Give equilibrium price, levels of output, and profits.

(c) Now, suppose that while the two firms are described as in part (b), firm 2 is uncertain about the cost function of firm 1. Specifically, c1 takes on one of a finite number of values: c11 < c12 < ··· < c1n , with probabilities p1 ,...,pn . Firm 2’s cost is always c2 . Firm 2 chooses its level of output x2 knowing only the distribution of c1 ; firm 1 chooses its output q1 knowing the value of c1 (therefore it is allowed to condition its level of q1 on c1 ). The two engage in Cournot competition. What is the equilibrium? Give equilibrium price, levels of output, and profits.

### Hint: An equilibrium is a quantity x ∗ 2 output by firm 2 and, for each level of firm 1’s costs, a quantity q ∗ 1 (c1 ), such that firm 2, believing that firm 1’s output will be given by the function q ∗1(c1), maximizes its expected profits with respect to its output level x∗2, and firm 1, believing that firm 2 will have output x∗2and knowingthat its own costs are given by c1, would choose to produce q∗1(c1).  战略决策理论代写

(d) Suppose the setting is as in (c), except that firm 1 can, after observing the realization of c1 but prior to the selection of x1 and x2 , provide irrefutable evidence to firm 2 concerning its cost structure if it wishes to do so. What do you think will happen in this case? (Hint: start with n = 2, in which c1 is equal to either c11 or c12.)

(e) What do you think will happen if firm 1 can make a false announcement about its cost? 