This model is easy to analyze for 2 players. If player 1 chooses x≤1/2, player 2 chooses x+2ε, and the scores are x+ε and 1–x–ε respectively. If player 1 chooses x≥1/2, player 2 chooses x–2ε, and the scores will be 1–x+ε and x–ε respectively. Thus player 1 chooses 1/2 and player 2 chooses 1/2±2ε, and the scores will be 1/2+ε and 1/2–ε.

The 3 player case is more complicated. If player 1 chooses x≤1/4, player 2 chooses (1–x)/3, and player 3 chooses any number between them in the interval (x, (1–x)/3). In this case the average scores will be (5x+1)/6, (1–x)/3, and (1–x)/2 respectively. If player 1 chooses 1/4<x≤1/2, then player 2 chooses 1–x+2ε and player 3 chooses x–2ε. In this case the scores will be 1/2–x+2ε, 1/2-ε, and x–ε respectively. The cases where player 1 chooses x≥1/2 are similar by symmetry. Thus player 1 chooses 1/4, player 2 chooses 3/4, and player 3 chooses something between in the interval (1/4,3/4), giving average scores of 3/8, 3/8, and 1/4. What are the best strategies for more than 3 players?

This game is also interesting where players get multiple turns in a specified order. For example, for the order 112, player 1 chooses 1/4 and 3/4, and player 2 chooses any number between them in the interval (1/4, 3/4), and the scores will be 3/4 and 1/4 respectively. What are the best strategies for any order of players?

Order | Optimal Plays | Optimal Scores | Author |
---|---|---|---|

1 | [0,1] | 1 | |

12 | 1/2 1/2+2ε | 1/2+ε 1/2–ε | |

123 | 1/4 3/4 (1/4,3/4) | 3/8 3/8 1/4 | |

1234 | 1/6 5/6 1/2 (1/6,5/6) | 7/24 7/24 1/4 1/6 | Joe DeVincentis |

12345 | 1/8 7/8 3/8 5/8 (1/8,7/8) | 11/48 11/48 5/24 5/24 1/8 | Dan Dima |

In fact, Dan Dima gives a convincing argument that for larger number of players n, the first (n–1) player will pick odd multiples of 1/(2n–2), with the first two players picking 1/(2n–2) and (2n–1)/(2n–2), leaving the last player to choose any number between them. This leads to scores (4n–9)/4(n–1)(n–2) for the first two players, (2n–5)/2(n–1)(n–2) for most of the players, and 1/2(n–1) for the last player.

Order | Optimal Plays | Optimal Scores |
---|---|---|

112 | 1/4 3/4 (1/4,3/4) | 3/4 1/4 |

121 | 1/2 5/6 5/6–2ε | 5/6–ε 1/6+ε |

122 | 1/2 1/2–ε 1/2+ε | ε 1–ε |

Order | Optimal Plays | Optimal Scores |
---|---|---|

1112 | 1/6 1/2 5/6 (1/2,5/6) | 5/6 1/6 |

1121 | 3/10 7/10 9/10 9/10–2ε | 9/10–ε 1/10+ε |

1122 | 1/4 3/4 1/4+ε 3/4–ε | 1/2+2ε 1/2–ε |

1211 | 1/2 x x–ε x+ε | 1–ε ε |

1212 | 1/4 3/4 3/4+3ε 1/4–ε | 1/2–ε 1/2+ε |

1221 | 1/4 3/4 1/4–3ε 3/4+ε | 1/2+ε 1/2–ε |

1222 | x x–ε x+ε y | ε 1–ε |

Order | Optimal Plays | Optimal Scores | Author |
---|---|---|---|

1123 | 1/6 5/6 1/2 (1/6,5/6) | 7/12 1/4 1/6 | |

1213 | ? | ? | ? |

1223 | ? | ? | ? |

1231 | ? | ? | ? |

1232 | ? | ? | ? |

1233 | 1/4–2ε 3/4+2ε 1/4 3/4 | 1/4–ε 1/4–ε 1/2+2ε | Bryce Herdt |

Order | Optimal Plays | Optimal Scores | Author |
---|---|---|---|

11112 | 1/8 3/8 5/8 7/8 (5/8,7/8) | 7/8 1/8 | |

11121 | 3/14 1/2 11/14 13/14 13/14–2ε | 13/14–ε 1/14+ε | |

11122 | 1/6 1/2 5/6 (1/6,1/2) (1/2,5/6) | 2/3 1/3 | |

11211 | x y z z–ε z+ε | ε 1–ε | |

11212 | ? | ? | ? |

11221 | ? | ? | ? |

11222 | 1/4 3/4 1/4–ε 3/4+ε (1/4,3/4) | 1/4+2ε 3/4–2ε | |

12111 | x y y–ε y+ε z | ε 1–ε | |

12112 | ? | ? | ? |

12121 | ? | ? | ? |

12122 | ? | ? | ? |

12211 | 5/6 1/6 1/2 1/6–2ε (1/6,1/2) | 5/6–ε 1/6+ε | Joe DeVincentis |

12212 | ? | ? | ? |

12221 | 1/2 1/6 1/2+3ε 5/6 1/6–ε | 2/3–ε 1/3+ε | |

12222 | x x–ε x+ε y z | ε 1–ε |

Order | Optimal Plays | Optimal Scores |
---|---|---|

11123 | ? | ? |

11213 | ? | ? |

11223 | ? | ? |

11231 | ? | ? |

11232 | ? | ? |

11233 | ? | ? |

12113 | ? | ? |

12123 | ? | ? |

12131 | ? | ? |

12132 | ? | ? |

12133 | ? | ? |

12213 | ? | ? |

12223 | ? | ? |

12231 | ? | ? |

12232 | ? | ? |

12233 | ? | ? |

12311 | ? | ? |

12312 | ? | ? |

12313 | ? | ? |

12321 | ? | ? |

12322 | ? | ? |

12323 | ? | ? |

12331 | ? | ? |

12332 | ? | ? |

12333 | ? | ? |

Order | Optimal Plays | Optimal Scores |
---|---|---|

12234 | ? | ? |

12134 | ? | ? |

12234 | ? | ? |

12314 | ? | ? |

12324 | ? | ? |

12334 | ? | ? |

12341 | ? | ? |

12342 | ? | ? |

12343 | ? | ? |

12344 | ? | ? |

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 12/1/17.