Assignment
Exercise 1
For each game, identify
a) All strategy profiles which survive iterated elimination of strictly dominated strategies,
b) All strategy profiles which survive iterated elimination of weakly dominated strategies,
c) All Nash equilibria (pure strategies).
In each game, payoffs are in alphabetical order. That is, in a given cell, the first number is Ann's utility and the second number is Bob's utility. You can gain up to 10 points per each game.
Exercise 2
Ann and Bob just went through a nasty divorce and are not on speaking terms. They have one son, Chris, who is a away in college. Next weekend, their kid is coming back home. Chris does not want to get involved in his parents' battle; he stays at his friends for the weekend.
However, knowing that Chris is in home town, Ann and Bob want to invite him for a dinner. Of course, each of them wants to have a dinner just with Chris, without the ex-spouse.
There are only two possible party days: Saturday and Sunday. If both invite Chris for the same day, then each dinner will be a disaster. This is because, in order to avoid the problem of choosing between Ann and Bob, Chris just will not show up.
If one invites Chris to the dinner for Saturday and the other chooses Sunday, then each dinner will be a success. However, the Saturday dinner will be a bigger success (comparing to the Sunday dinner). In other words, each player strictly prefers the Saturday dinner over the Sunday dinner; however, having a dinner on Sunday is still strictly better than a disaster of Chris not showing up.
Hint: this is a static game in which each player has two strategies.
a) Represent this situation in a normal form. You need to assume some specific utilities; make sure that your choice of utlities is consistent with the story.
b) Given the normal form you prepared in (a), represent this situation in a matrix form. Clearly specify who chooses among rows and who chooses among columns.
c) In the matrix you created in (b), find all strategy profiles which survive iterated elimination of strictly dominated strategies. (Do not use the iterated elimination of weakly dominated strategies.)
d) In the matrix you created in (b), find all Nash equilibria (pure strategies).
Exercise 3
An airline loses two suitcases belonging to two different travelers, Ann and Bob. Both suitcases happen to be identical and contain identical items (antiques). An airline manager tasked to settle the claims of both travelers explains that the airline is liable for a maximum of $50 per suitcase. (The manager is unable to find out directly the price of the items.) In order to determine an honest appraised value of the antiques the manager separates both travelers so they can't confer, and asks them to write down the amount of their value. For simplicity, assume that the value can be one of the following numbers: $10, $20, $30, $40, or $50.
The manager explains that if both write down the same number, then he will treat that number as the true dollar value of both suitcases and reimburse both travelers that amount. However, if one writes down a smaller number than the other, this smaller number will be taken as the true dollar value, and both travelers will receive that amount along with a bonus/malus: $10 extra will be paid to the traveler who wrote down the lower value and a $10 deduction will be taken from the person who wrote down the higher amount.
Assume that a player's utility is the amount of money s/he receives. Hint: this is a static game in which each player has five strategies.
a) Represent this situation in a matrix form. (Do not prepare the normal form.) Clearly specify who chooses among rows and who chooses among columns. You are provided will all the necessary information to compute the utilities; do not assume your own utilities.
b) In the matrix you created in (a), find all strategy profiles which survive iterated elimination of strictly dominated strategies.
c) In the matrix you created in (a), find all strategy profiles which survive iterated elimination of weakly dominated strategies.
d) In the matrix you created in (a), find all Nash equilibria (pure strategies).
Exercise 4
In this exercise, your job is to invent and solve a game. Think about some interactive problem (politics, economics, sports, your personal life, etc.). Then, express that problem in the language of game theory (matrix), and solve it (Nash equilibria).
It is important that what you propose is original. That is, what you analyze must be your own invention. Your task is to put on the hat of applied game theorist: you analyze some problems using the game-theoretic tools. You can think of your job as consisting of three steps.
a) Start with a story. Explain, in English, a problem you want to analyze. Your story should not be longer than 1 page. You can pick whatever topic you want. The only restriction is that (i) the game must have exactly two players, (ii) each player has a finite number of strategies (at least two), and (iii) the game is your own invention. If you copy or just modify a game that you found on the Internet or in a book or saw in class, then your grade for Exercise 4 will be zero. Be creative.
b) Create that matrix that depicts the story you wrote in (a). In particular, explain how you constructed the payoffs.
c) Find all (pure-strategy) Nash equilibria in the matrix you draw in (b).