2025-05-30

Course on Game Theory

These are problem solutions to a TTC/TGC course on game theory by Jay R. Corrigan. The solutions are organized by lecture, of which there are twelve in total.

Lecture One

Q1: Find this game’s two pure-strategy Nash equilibrium outcomes. Assume a higher payoff is better than a lower one. (Colin bolded, Rose the other).

A1: One outcome is Colin going right, while Rose goes up. This results in 10 units of payoff to each player, and there is no incentive to change since no other outcome rewards a greater outcome to either player.

The second outcome is Colin going left, and Rose going down. This results in a lower payoff.

Q2: Explain why this game is an example of a prisoner’s dilemma. Assume a higher payoff is better than a lower one. (Colin bolded, Rose the other).

A2: This game is an example of prisoner’s dilemma because there is an outcome where both participants can benefit equally, and for a greater total payoff (Colin goes right, Rose goes down). However, each participant can gain a greater individual payoff if they go another route (at the cost of the other), which each has to account for in their decision-making. If Colin and Rose do not trust each other, therefore, their dominant strategy is to go left and up respectively – losing out on a greater total and individual payoff.

Lecture Two

Q1: Rose and Colin are playing a repeated version of the prisoner’s dilemma game. If Rose is playing tit-for-tat and Colin always defects, what will each player’s payoff be in the first round? What will their payoffs be in subsequent rounds? (Colin is bolded)

A1: In the first round, Rose will cooperate and Colin will defect (0,3). In all subsequent rounds, they will both defect (1,1).

Q2: Rose and Colin are again playing a repeated version of the same prisoner’s dilemma game. If both Rose and Colin now play tit-for-tat, what will each player’s payoff be in the first round? What will their payoffs be in subsequent rounds?

A2: Both will co-operate in the first round (2,2), and co-operate in each subsequent round as well (2,2).

Lecture Three

Q1: True or false: When playing chicken, your dominant strategy is to drive straight.

A1: False.

Q2: Fill in the blanks in the following payoff matrix so it’s clear this is a game of chicken. Assume a higher payoff is preferred to a lower one.

Lecture Four

1. This is an assurance game. There are two pure nash equilibrium outcomes (B, B and R, R) and one of these (B,B) is superior in payoff to the other (payoff dominance).

2. 	TV	Dinner + Movie
   TV	1,1	1,0
   Dinner + Movie	0,1	3,3

Lecture Five

1. Assuming a probability $p$ of Rose going straight, and a probability $(1-p)$ of Rose swerving:
   1. Colin’s expected value of going straight is $-1*p + 2(1-p)$.
   2. Colin’s EV of swerving is: $0*p + 1(1-p)$.
   3. Equate these and simplify to get: $-3p + 2 = 1 - p$ -> $p = 1/2$.
   4. This tells us Colin should go straight fifty percent of the time, and swerve fifty percent of the time.
2. This is a zero-sum game. Assuming the batter thinks the pitcher will throw a fastball, he will prepare for a fastball; if he thinks he’s getting a changeup, he’ll prepare for a changeup. There is no dominant strategy here.
   1. Assuming a probability $p$ of the pitcher throwing a fastball; and a probability $(1-p)$ of the pitcher throwing a changeup.
      1. The batter’s expected value function for readying for a fastball is $p + -1*(1-p)$.
      2. His expected value function for readying for a changeup is $-p + 2*(1-p)$.
      3. If we equate these and simplify, we get $p = 3/5$.
      4. This tells us: 60% of the time, the batter must assume a fastball, and the other 40% of the time he must assume a changeup.