Problem Set 2

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Problem Set 2

The goal of the problem set is to give you practice in mastering the course material.

Try to give short, but precise answers, and always justify your solution.

You are expected to solve these exercises together with the other students of the group that you have been assigned to. It is sufficient to hand in one solution for each group.

In any case, you are asked to identify all your collaborators on the front page of your solution. If you obtain a solution through research (e.g., on Internet), you must acknowledge your source (on page 2 of this PDF) and write up the solution in your own words. It is a violation of this policy to submit a problem solution that you, i.e. any member of your group, cannot orally explain to the professor.

Submit your solution by July 16 (Saturday) 8pm (JST) electronically on Waseda Moodle in a single PDF file.

Try to write your solutions directly in this PDF, but you can use additional pages if necessary.

REFERENCES

Note that doing some research and collaborating with others are permitted, but you must list here all the sources (e.g., books, articles, websites, friends, professors, family members) that you have consulted for solving this problem set.

If you have solved the problem set without consulting any sources, write“none”.

There are two official languages in Catalonia: Catalan and Spanish. We can safely assume

that in Catalonia everybody speaks and understands both languages, but the two languages are not perfect substitutes. People care about how communication flows.

The following payoff matrix represents a situation in which two people have to (simul- taneously) decide which language to use when greeting the other before engaging in a conversation.

Person 2

Catalan Spanish

Person 1

Catalan Spanish

10 10

0 5

5 0

5 5

Using Spanish is the safe option, as it gives 5 units of payoff no matter which language the other chooses. Choosing Catalan is riskier. If both persons choose Catalan they will be very happy and enjoy 10 units of payoff. Using Catalan when the other speaks Spanish, however, is considered to be very awkward and that situation gives 0 units of payoff.

(a) Is Spanish a dominant strategy Why

1 point

(b) Find (and write down) the Nash equilibria of this two-person game in pure strategies.

1 point

(c) Find (and write down) the Nash equilibria of this two-person game in mixed strate- gies.

3 points

(d) Now assume that everybody in Catalonia chooses a preferred language and uses it all the time when meeting other people. Assume that p proportion of the population chooses and uses Catalan.

i. Write a mathematical expression and represent graphically (in the graph below) a person’s expected payoff

when s/he chooses Catalan, and

when s/he chooses Spanish.

Note that these expected payoffs might depend on p.

2 points

10

9

8

7

6

5

4

3

2

1

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p

ii. Is there an equilibrium in Catalonia in which is it stable

everybody prefers Catalan If yes,

2 points

iii. Is there an equilibrium in Catalonia in which is it stable

everybody prefers Spanish If yes,

2 points

iv. Is there an equilibrium in Catalonia in which some people prefer Catalan, some prefer Spanish If yes, what is the equilibrium value of p. Is this equilibrium stable

2 points

Imagine that a large number of consumers are uniformly distributed along a boardwalk that

is 1 mile long.

Hint: “Uniformly distributed” is a mathematical expression which refers to the uniform distribution of a continuous random variable. In this exercise, it essentially describes a situation such that

half of the consumers are located along the first half mile of the boardwalk, and the other half of the consumers are located along the second half of the boardwalk; and

in general, x% of the consumers are located along the first x/100 miles of the board- walk, where x can take any value between 0 and 100.

Ice-cream prices are regulated, so consumers go to the nearest vendor because they dislike walking (assume that at the regulated prices all consumers will purchase an ice cream even if they have to walk a full mile). If more than one vendor is at the same location, the vendors split the business evenly.

Consider a game in which two ice-cream vendors pick their location simultaneously. Show that there exists a unique pure-strategy Nash equilibrium and that it involves both vendors locating at the midpoint of the boardwalk.

2 points

Colonel Spicces has four companies that he can distribute among two locations in three

different ways: (3, 1), (2, 2) or (1, 3). His opponent, Count Vicces, has three companies the he can distribute among the same two locations in two different ways: (2, 1) or (1, 2). In oder to describe the possible outcomes of the game, let si denote the companies that Colonel Spicces sends to location i and let vi denote the companies that Count Vicces sends to location i. Note that i can be either 1 or 2, given that there are two locations in this game.

If si = vi , the result is a standoff, and each commander gets a payoff of zero for location i. If si vi , the larger force overwhelms the smaller force without loss to itself.

If si > vi , Colonel Spicces gets a payoff vi , and Count Vicces gets a payoff of -vi for location i.

If si < vi , Colonel Spicces gets a payoff -si , and Count Vicces gets a payoff of si for location i. Each player’s total payoff is the sum of his payoffs at both locations. (a) Find the strategic-form representation of this simultaneous-move game, and show that the game has no Nash equilibrium in pure strategies. 1 point (b) Find the mixed-strategy Nash equilibria of the game. Hint: This is a rather difficult task, because Colonel Spicces has three pure strategies in this game. You have to follow the usual logic in order to find the mixed-strategy Nash equilibrium, but in this game Colonel Spicces will have to assign two prob- abilities: p1 to (3, 1), p2 to (2, 2); therefore (1, 3) will be chosen with probability 1 - p1 - p2 . 2 point . Show that the 3-times-repeated Prisoners’ Dilemma has 240 × 241 × 242 = 221 = 2097152 pure strategies. 2 points