Problems

Chapter 4 (First Edition)

Problems 2 and 3

These problems provide practice for solving games and locating equilibria.

Problem 2

Find all Nash equilibria in pure strategies in the following non-zero-sum games.

2a

  Column
  LeftRight
RowUp 2 , 4 1 , 0
Down 6 , 5 4 , 2

2b

  Column
  LeftRight
RowUp 1 , 1 0 , 1
Down 1 , 0 1 , 1

2c

  Column
  LeftMiddleRight
RowUp 0 , 1 9 , 0 2 , 3
Straight 5 , 9 7 , 3 1 , 7
Down 7 , 5 10 , 10 3 , 5

Problem 3

"If a player has a dominant strategy in a simultaneous-move game, then she is sure to get her best outcome." True or false? Explain and give an example of a game that illustrates your answer.

Solutions

4.2

(a) Down is dominant for Row and Left is dominant for Column. Equilibrium: (Down, Left) with payoffs of (6, 5).

(b) Down and Right are weakly dominant for Row and Column, respectively, leading to a Nash equilibrium at (Down, Right). Brute force also shows another Nash equilibrium at (Up, Left).

(c) Down is dominant for Row; Column will then play Middle. Equilibrium is (Down, Middle).


4.3

False. A dominant strategy yields you the highest payoff available to you against each of your opponent's strategies. Playing a dominant strategy does not guarantee that you end up with the highest of all possible payoffs. For example, in a prisoner's dilemma game, both players have dominant strategies but neither gets the highest possible payoff in the equilibrium of the game.