# D2: Game Theory

Imagine a game, a bit like *paper, scissors, stone* where each player holds up 1, 2 or 3 fingers instead.
Depending on what your opponent does, you win or lose money. Let's imagine that if you both hold up one finger, you win £1 from
your opponent. If you hold a 1 and your enemy holds up 2 fingers, no-on wins anything. If you hold a 1 and ... I'll
tell you what, I'll put this into a table.

## Payoff Matrices

He plays 1 | He plays 2 | He plays 3 | |
---|---|---|---|

You play 1 | You win £1 | Draw | You win £1 |

You play 2 | He wins £4 | He wins £1 | You win £1 |

You play 3 | You win £2 | Draw | He wins £1 |

The idea of a *game* where all money is conserved (ie all the money you win comes from your opponent and all money
he wins, you lose) is called a *zero-sum* game. All D2 games are zero-sum games. Can you see what each player would
do in this game?

If we look at the worst-case scenario for us we see that if we play 1, the worst that can happen is that we break even.
If we play 2, we could lose £4 and if we play 3 we could lose £1. By playing 1 all the time we can never lose - the worst
that could happen is that we break even. This is an example of a *playsafe strategy* where we play the strategy that
stops us losing money. It's a *maximin* strategy as we are trying to maximise the minimum we could win (losses count
as negative).

From now on, I will just put numbers in the table for player 1. Positives mean he wins that amount, negatives mean he loses - ie player 2 wins. The game we just talked about will look like this...

He plays 1 | He plays 2 | He plays 3 | |
---|---|---|---|

You play 1 | 1 | 0 | 1 |

You play 2 | -4 | -1 | 1 |

You play 3 | 2 | 0 | -1 |

To find the playsafe strategy for each player we add an extra column and row.

He plays 1 | He plays 2 | He plays 3 | Worst | |
---|---|---|---|---|

You play 1 | 1 | 0 | 1 | 0 |

You play 2 | -4 | -1 | 1 | -4 |

You play 3 | 2 | 0 | -1 | -1 |

Worst | 2 | 0 | 1 |

### Saddle Points

Because both players have reached the same conclusion (player 1 can guarantee the score is *at least* 0,
player 2 can guarantee the score is *at most* 0), the score will always be 0.

When both players have the same worst-case scenario (like the zero here) the game is said to have a *saddle point.
*

If player 2 *always* played 2 then player 1 should never deviate from his strategy of playing 1. The score should always
be 0 and zero is called the *value* of the game.

### Dominance

Study this game from player 2's point of view. We are now pretending to be the "He" in the table rather than the "You". Our aim is for negative numbers.

He plays 1 | He plays 2 | He plays 3 | |
---|---|---|---|

You play 1 | 1 | 0 | 1 |

You play 2 | -4 | -1 | 1 |

You play 3 | 2 | -2 | -1 |

Notice anything?

Strategy 2 is said to *dominate* strategy 3. Player 2 will never use 3, so we might as well remove the whole column.

Also, for player 1, strategy 1 dominates strategy 2 as you never do better using 2 than you do using 1.

## Mixed Strategy Games

Consider this game

B plays 1 | B plays 2 | |
---|---|---|

A plays 1 | -4 | 5 |

A plays 2 | 2 | -3 |

Who would you rather be: player A who wins with positive numbers, or player B who wins with negatives?

The playsafe strategy for player A is strategy 2 as the worst that could happen if he uses 1 is a loss of 4 whereas the worst that could happen if he plays 2 is a loss of 3. So 2 is slightly safer. Player B's playsafe strategy is 1 (check for yourself that this is correct). If both players used their playsafe strategy then player A would win 2 (so player B would lose 2) each time.

But if player B notices this, he should change his tactics to use strategy 2. Now he wins 3. But player A notices that he should now play 1 to win 5... and we end up going round in circles.

So what should each player do and who is favourite to win if they both play a good strategy?

Here's a clip where I answer these questions.