# Note 10 - Back to Zero Sum Games

## Table of Contents

## Review: Proof of Arrow’s Impossibility Theorem

Deferred to textbook, page 224.

## Zero Sum Games

Recall the **payoff diagram** from the first few notes. This will be useful later on:

Player 1 \ Player 2 | A | B | C |

A | -1 | 0 | 2 |

B | -2 | 3 | 4 |

C | 1 | 2 | 3 |

A zero-sum game has the property that the sum of the outcomes will be 0.

Looking at outcomes, we have a distribution of values for the game. If the expected value of the game is 0, it is called **fair**. If the value is positive, it **favors Player 1**. If the value is negative, it **favors Player 2**.

### Matching Pennies

Player 1 and 2 choose heads or tails, independently. If choices match, Player 2 gets 1 cent from Player 1, otherwise Player 1 gets 1 cent from Player 2.

Payoff diagram (aka **strategic description**):

P1 \ P2 | H | T |

H | -1 | 1 |

T | 1 | -1 |

### Odd-Even

Both players call out 1 or 2. If the sum is even, P2 wins. Otherwise, P1 wins.

P1 \ P2 | 1 | 2 |

1 | 2 | -3 |

2 | -3 | 4 |

If Player 2 calls 1, then on average, Player 1 wins 1/2. If Player 2 calls 2, then on average, Player 1 wins -1/2.

Suppose instead Player 1 calls 1 3/5 of the time.

- If P2 calls 1, the expected value is 0.
- If P2 calls 2, the expected value is 0.2.

Based on our payoff diagram,

- Player 1 wants to maximize the minimum of \(\{-2p -3 (1-p), -3 - 4(1-p)\}\)
- Player 2 wants to minimize the maximum of \(\{-2p -3 (1-p), -3 - 4(1-p)\}\)