## 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 |

| | Left | Right |

Row | Up |
2 , 4 |
1 , 0 |

Down |
6 , 5 |
4 , 2 |

### 2b

| | Column |

| | Left | Right |

Row | Up |
1 , 1 |
0 , 1 |

Down |
1 , 0 |
1 , 1 |

### 2c

| | Column |

| | Left | Middle | Right |

Row | Up |
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.