The Chooser-Picker 7-in-a-row game András Csernenszky July Szeged

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.

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?

Changing the rules: the „Weak” Games (Maker–Breaker games)  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 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.

Changing the rules: the Chooser-Picker games   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! 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)

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 this game 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.

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 =9 [Shannon and Pollak]   k=8 is also a draw [T. G. L. Zetters] OPEN questions: k=5 on infinite board? k=6, 7?

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.

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 wins all of these sub- games, then Picker wins the game played on any K board which is the union of disjoint tiles.

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

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}

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).

Playing on the 4x8 board…

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).

Thank you for your attention!