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The Chooser-Picker 7-in-a-row game

Explore the strategy-stealing argument and variations of the 7-In-A-Row game using hypergraphs. Analyze the Chooser-Picker version and learn about winning strategies. Includes an auxiliary game on a tiled plane.

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The Chooser-Picker 7-in-a-row game

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  1. The Chooser-Picker 7-in-a-row game András Csernenszky July 3 2008 Szeged

  2. Hypergraph games (an example) • The vertices of the graphs are the fields of an infinite graph paper. • The winning sets are the horizontal, vertical or diagonal consecutive cells of length 5. • If one of the player (because of the strategy stealing argument [J.Nash], this is the first player) gets a length 5 line, then he wins otherwise the game is draw. Given an F = (V, F) hypergraph and two players: the first player and the second player. The players alternate each other choosing one of the hypergraph’s edge. We call these games as hypergraph games.

  3. Hypergraph games • If both player plays well then who wins that game? • Is it possible that no one wins? • What happens if we change the length of the winning sets?

  4. Changing the rules: the „Weak” Games (Maker–Breaker games) • Theorem: If Breaker wins the „weak” game, then the original game is draw. • ~Breaker has weaker chance to win than the second player in the original game. • Maker (~first player) wins: if he/she occupies a winning set • Breaker (~second player) wins: if he/she prevents Maker’s win. • -There are only two possible outcomes of the game. Original game „Weak” game The first player (x) need to block the 2nd player offence. Twofold threat! -Maker (x) wins

  5. Changing the rules: the Chooser-Picker games Beck’s conjecture: If Breaker wins the Maker-Breaker game, then also Picker (as a second player) wins the Chooser-Picker game. ~These games are close to each other. If we believe that Breaker wins the game, then the Chooser-Picker version can be analyzed (and also Picker win expected) • Picker picks two vertices, and Chooser chooses one of them, the other one remains to Picker. • Chooser wins by getting a full winning set, and Picker wins if he prevents this (as the in Maker-Braker games). • If there is odd number of vertexes, then the last one goes to Chooser by definition. „Weak” game: Chooser-Picker game Maker wins! Chooser wins!

  6. Using Chooser-Picker games(an example: 4x4 Tic-Tac-Toe) • We think that this game is draw. • If we could prove that Breaker wins the weak version of thisgame then we are ready. • We can check quickly whether the Chooser-Picker version of this game is a Picker win? • And now we should start the more time-consuming proof; that Breaker also wins that game… • The size of the game tree is the same, but if we know a winning strategy for Picker we can prove is somewhat easier.

  7. About the k-in-a-row games • The first player wins for k = 5 on the 19×19 or even in the 15 × 15 board [Allis] • The first player wins if k<=4, and the game is a blocking draw if k >=9 [Shannon and Pollak] • k=8 is also a draw [T. G. L. Zetters] • OPEN questions: • k=5 on infinite board? • k=6, 7?

  8. k=7 • The original game is believed to be a draw. • It would be stronger to prove that the weak version is a Breaker win. • At first we examine the Chooser-Picker version of this game.

  9. The Chooser-Picker 7-in-a-row-game:An auxiliary game • We consider a tiling of the plane, and play an auxiliary-game on each tile (sub-hypergraph). • It is easy to see, if Picker winsall of these sub-games, then Picker wins the game played on any K board which is the union of disjoint tiles.

  10. The Chooser-Picker 7-in-a-row-game:Tricks-1 • If Picker wins the Chooser-Picker game on (V,F), then Picker alsowins it on (V \ X,F(X))

  11. The Chooser-Picker 7-in-a-row-game:Tricks-2 • If in the course of the game (or just already at the beginning) there is a two element winning set {x, y} then Picker has an optimal strategy starting with {x, y}

  12. The Chooser-Picker 7-in-a-row-game:Tricks-3 • It helps Chooser’s game if we change one of Picker’s square to a free square, and it is also advantageous for Chooser if he/she gets one of the free squares (P « FREE«C).

  13. Playing on the 4x8 board…

  14. Results: • Picker wins the Chooser-Picker 7-in-a-row (case study on the 4x8 auxiliary board, few pages). • It is an additional information for the original 7-in-a-row game. • We checked M-B game on the same auxiliary board, but it is a Maker win! (brute force computer search) • We should find other auxiliary games (and check the C-P version at first).

  15. Thank you for your attention!

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