# Exercises

```Selected Problems from Text to Practice
(11/09/2007, Solutions Posted)
1 Introduction 1
Exercise 5.3 (Altruistic preferences) 1
Exercise 6.1 (Alternative representations of preferences) 1
2 Nash Equilibrium 3
Exercise 16.1 (Working on a joint project) 3
Exercise 17.1 (Games equivalent to the Prisoner’s Dilemma) 3
Exercise 34.1 (Guessing two-thirds of the average) 4
Exercise 34.3 (Choosing a route) 5
Exercise 37.1 (Finding Nash equilibria using best response functions) 6
Exercise 38.2 (Dividing money) 7
Exercise 41.1 (Strict and nonstrict Nash equilibria) 7
Exercise 50.1 (Other Nash equilibria of the game modeling collective
decision-making) 8
3 Nash Equilibrium: Illustrations 11
Exercise 58.1 (Cournot’s duopoly game with linear inverse demand and different
unit costs) 11
Exercise 63.1 (Interaction among resource-users) 12
Exercise 68.1 (Bertrand’s oligopoly game) 13
Exercise 85.1 (Second-price sealed-bid auction with two bidders) 16
Exercise 90.3 (Internet pricing) 19
4 Mixed Strategy Equilibrium 23
Exercise 110.1 (Expected payoffs) 24
Exercise 114.1 (Mixed strategy equilibrium of Hawk–Dove) 25
Exercise 117.2 (Choosing numbers) 26
Exercise 120.2 (Strictly dominating mixed strategies) 26
Exercise 136.1 (Best response dynamics in Cournot’s duopoly game) 29
5 Extensive Games with Perfect Information: Theory 33
Exercise 163.1 (Nash equilibria of extensive games) 33
Exercise 164.2 (Subgames) 33
Exercise 168.1 (Checking for subgame perfect equilibria) 33
Exercise 177.3 (Comparing simultaneous and sequential games) 34
Exercise 179.3 (Three Men’s Morris, or Mill) 35
6 Extensive Games with Perfect Information: Illustrations 37
Exercise 189.1 (Stackelberg’s duopoly game with quadratic costs) 38
Exercise 196.4 (Sequential positioning by three political candidates) 38
7 Extensive Games with Perfect Information: Extensions and Discussion 43
Exercise 210.2 (Extensive game with simultaneous moves) 43
Exercise 224.1 (Exit from a declining industry) 45
8 Coalitional Games and the Core 49
Exercise 245.1 (Three-player majority game) 49
Exercise 248.1 (Core of landowner–worker game) 49
Exercise 249.1 (Unionized workers in landowner–worker game) 49
Exercise 258.1 (House assignment with identical preferences) 51
Exercise 267.2 (Empty core in roommate problem) 51
```