Question 1

The new food delivery company DoorDash is thinking of entering New Zealand. Its rivalry is the existing UBER Eats. Both these companies simultaneously choose whether to operate in January, February, March, or April or not to operate at all. Thus the strategy set for both these companies is. Assume that the profit received by a monopolist in month m is 10 x m – 15, whereas each duopolist (so both are operating) would earn 4 x m – 15. A company makes zero profit for any month that is not operating. Also, the value of m while calculating the profits for a particular month in the above equations is as follows {January = 1, February = 2, March = 3, April = 4}.

The payoffs of the strategic form of this game are the sum of the profit of all the months. For example, if DoorDash operates in February and UBER Eats in March, then DoorDash earns zero profit in January; 5 (=10 x 2 – 15) in February; -3 (=4 x 3 – 15) in March; and 1 (=4 x 4 – 15) in April, for a total profit of $3. UBER Eats earns zero profit in January and February, -3 in March, and 1 in April, for a total profit of $-2. Then the payoffs (DoorDash, UBER Eats) = (February, March; 3, -2).

• Derive the strategic form.
• Find the strategies that survive the IDSDS.
• Find the Nash Equilibria.

Question 2

Assume that you are at the Christchurch airport waiting to board a flight to visit your family for Christmas. Let us say 190 passengers want to board the flight. Assume that flight attendants are absent to guide you properly. So, everyone is rushing for no reason. There are two doors to enter the flight, and each passenger decides which one to take.

The satisfaction/ utility of entering the front door is 10 units (the way to the front door has a footpath bridge that protects you from the rain). In contrast, the back door is the least preferred and has 0 satisfaction/ utility (because it takes more time to reach the seat, you must go out of the terminal, and it is raining heavily). In deciding which door to take, each passenger cares only about net satisfaction/ utility. Passengers’ payoff is assumed to be their net utility.

Net Utility of a Passenger = Satisfaction of Taking a Door – Time Cost to Reach the Seat

If m passengers are taking the front door, the time cost for a passenger to reach the seat is assumed to be m units. In contrast, if m passengers take the back door, the time cost is 2m units to reach the seat. Passengers make simultaneous decisions as to whether to take the front or back door.

• Derive each passenger’s payoff function.
• Find a Nash Equilibrium

Question 3

Players are getting ready for the FIFA world cup. Two cloth brands are thinking to sponsor the jersey for teams Croatia and Moldova: Alleson Athletic and Boxercraft. Croatia’s and Moldova’s team managers have asked their players to choose one among the two. Let us now assume the intrinsic superiority of Alleson Athletic is not as great, and that network effects are stronger for Boxercraft.

These modifications are reflected in different payoffs. Now, the payoff from wearing Boxercraft is 100b and from wearing Alleson Athletic is 15 + 5a; n number of soccer players are The Banshees of Inisherin Row Player Alex Alex simultaneously deciding between Alleson Athletic and Boxercraft. Hint: Lecture 8

• Find all Nash Equilibria.
• Let us now assume that a third option exists, which is to wear the brand Augusta; it has a payoff of 200. Players simultaneously decide among Alleson Athletic, Boxercraft, and Augusta. Find all Nash equilibria.

Question 4

Return to the internship game from Lecture 8. Suppose a student’s beliefs as to what the other 9 students will do are as follows: with a probability of 0.2, three of them will apply to JPM and six of them to LM; with a probability of 0.5, four of them will apply to JPM and five of them to LM; and with a probability of 0.3, five of them will apply to JPM and four of them to LM. Assume a student chooses so as to maximize his expected utility, where will he apply?

Question 5

Three flatmates are standing in line to get a bungy jumping ticket. They decide between the pure strategies, jump and don’t jump. The payoff from jumping is 180/m, where m is the number of flatmates that choose to jump, and the payoff from not jumping is 65. Write the answers related to the following questions only. Do not think beyond the given three below questions.

• There are three asymmetric pure-strategy Nash Equilibria. What are they?
• One player chooses the pure strategy, and the other two players symmetrically randomize. What are the three Nash Equilibria in this situation?
• There is a symmetric mixed-strategy Nash Equilibrium in which each player randomizes in choosing the pure strategy. What is that one solution?