The Chooser-Picker The Chooser-Picker

7-in-a-row game7-in-a-row gameAndrás CsernenszkyAndrás Csernenszky

July 3 2008July 3 2008

SzegedSzeged

Hypergraph games Hypergraph games (an example)(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 callthese games as hypergraph games.

Hypergraph gamesHypergraph games

If both player plays well then who wins If both player plays well then who wins that game?that game?

Is it possible that no one wins?Is it possible that no one wins?

What happens iWhat happens iff we change the length of we change the length of the winning sets?the winning sets?

Changing the rules: Changing the rules: the „Weak” Games the „Weak” Games

(Maker–Breaker games)(Maker–Breaker games)

Maker (~first player) wins: ifMaker (~first player) wins: if he/she he/she occupies a winning set occupies a winning set Breaker (~second player) wins: if Breaker (~second player) wins: if he/she preventhe/she prevents s Maker’s winMaker’s win.. -There are only two possible outcome-There are only two possible outcomess of the game. 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 Changing the rules: the Chooser-Picker gamesChooser-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-Breakergame, 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 gamesUsing Chooser-Picker games(an example: 4x4 Tic-Tac-Toe)(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 About the k-in-a-row gamesgames

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?

k=7k=7 TThe original game is he original game is believed to be a believed to be a

draw.draw.

It It would be strongerwould be stronger to prove that the to prove that the weak version is a Breaker win.weak version is a Breaker win.

At first we examine the Chooser-Picker At first we examine the Chooser-Picker version of this game.version of this game.

The Chooser-Picker 7-in-a-row-game:The Chooser-Picker 7-in-a-row-game:An auxiliary gameAn 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:The Chooser-Picker 7-in-a-row-game:TricksTricks-1-1

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:The Chooser-Picker 7-in-a-row-game:TricksTricks-2-2

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:The Chooser-Picker 7-in-a-row-game:TricksTricks-3-3

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 Playing on the 4x8 4x8 board…board…

Results:Results:

Picker wins the Chooser-Picker 7-in-a-row (case study Picker wins the Chooser-Picker 7-in-a-row (case study on the 4x8 auxiliary board, few pages)on the 4x8 auxiliary board, few pages)..

It is an additional information for the original 7-in-a-row It is an additional information for the original 7-in-a-row gamegame..

We checked M-B game on the same auxiliary board, We checked M-B game on the same auxiliary board, but it is a Maker winbut it is a Maker win!! (brute force computer search) (brute force computer search)

We should find other auxiliary games (and check the We should find other auxiliary games (and check the C-P version at first).C-P version at first).

Thank you for your Thank you for your attention!attention!