GE2256: Game Theory Applications to Business
Sem A 2024-2025
Homework 1
Due: Wednesday, 2 October
IMPORTANT NOTE: Please submit your assignment online through Canvas by the due date or deadline. You may write your answers in MS Word or use any other processor (e.g., LaTeX). However, the final file you upload on Canvas must be a pdf document. It is also fine if you write your answers on a piece of paper and then scan it or take a high-quality image using your camera/phone and create a PDF document to upload on Canvas.
Other Notes: Please note that the questions are based on the material covered in Lectures 1-2. There are five questions in this assignment. While your answers will not be graded, you must attempt to write your answers to all five questions to get full credit for this assignment.
Question 1: Determine which of the following situations describe strategic interactions (or games) and which describe individual decisions. In each case, indicate what specific features of the situation caused you to classify it as you did.
(a) China chooses a level of tariffs to apply to American imports.
(b) A party nominee for president of the United States must choose whether to use private financing or public financing for her campaign.
(c) You receive a $100 gift card for downloadable music and must choose whether to purchase individual songs or whole albums.
Question 2: Each of two players has two possible actions, Cooperate and Defect; each action pair results in the players’ receiving amounts of money equal to the numbers corresponding to that action pair in the following game matrix:
Cooperate Defect
Cooperate 18, 18 0, 27
Defect 27, 0 9, 9
For example, if player 1 chooses Cooperate and player 2 chooses Defect, then player 1 receives nothing, whereas player 2 receives $27.
The players are not “selfish”; rather, the preferences of each player i are represented by the payoff function mi (a) + αmj (a), where mi (a) is the amount of money received by player i when the action profile is a, j is the other player, and α is a given non-negative number. Player 1’s payoff to the action pair (Cooperate, Cooperate), for example, is 18 + 18α .
(a) Formulate a strategic game that models this situation in the case α = 1. Is this game the Prisoner’s Dilemma?
(b) Find the range of values of α for which the resulting game is the Prisoner’s Dilemma.
Question 3: An old lady is looking for help crossing the street. Only one person is needed to help her; if more people help her, this is no better. You and I are the two people in the vicinity who can help; we have to choose simultaneously whether to do so. Each of us will get pleasure worth a 3 from her success (no matter who helps her). But each one who goes to help will bear a cost of 1, this being the value of our time taken up in helping. If neither player helps, the payoff for each player is zero. Set this up as a game. Write the payoff table, and find all pure-strategy Nash equilibria (PSNE).
Question 4: An employer hires an employee and promises him a wage w. The employee can work (W) or shirk (S). Working is associated with the cost of effort, e, where w > e > 0. If the employee works, the employer obtains revenue r otherwise 2/r.
The employer cannot tell whether the employee is working or shirking unless she chooses to run an inspection (I) at the cost of c > 0. Inspection reveals whether the employee is working or not, and in the latter case, the wage is withheld. The decisions of the two players (W/S and I/NI) are taken simultaneously. The payoff matrix is as follows:
I NI
W w-e, r-c-w w-e,r-w
S 0,r/2-c w,r/2-w
The employee chooses a row (row player), and the employer chooses a column (column player).
Under what condition(s) will this game have a pure strategy Nash equilibrium?
Question 5: An airline loses two suitcases belonging to two different travelers. Both suit- cases happen to be identical and therefore have equal replacement value. An airline manager tasked to settle the claims of both travelers explains that the airline is liable for a maximum of $100 per suitcase, and in order to determine an honest appraised value of the suitcases, the manager separates both travelers so they can’t confer, and asks them to write down the amount of their value at no less than $2 and no larger than $100. He also tells them that if both write down the same number, 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: $2 extra will be paid to the traveler who wrote down the lower value, and a $2 deduction will be taken from the person who wrote down the higher amount. Describe the set of strategies of both players and find all the pure strategy Nash equilibria of the game. Restrict yourself to integer numbers only.