Name
Abstract | ||
|---|---|---|
In the tournament game two players, called Maker and Breaker, alternately take turns in claiming an unclaimed edge of the complete graph K-n and selecting one of the two possible orientations. Before the game starts, Breaker fixes an arbitrary tournament T-k on k vertices. Maker wins if, at the end of the game, her digraph contains a copy of T-k; otherwise Breaker wins. In our main result, we show that Maker has a winning strategy for k = (2 - o(1)) log(2) n, improving the constant factor in previous results of Beck and the second author. This is asymptotically tight since it is known that for k = (2 - o(1)) log(2) n Breaker can prevent the underlying graph of Maker's digraph from containing a k-clique. Moreover, the precise value of our lower bound differs from the upper bound only by an additive constant of 12. We also discuss the question of whether the random graph intuition, which suggests that the threshold for k is asymptotically the same for the game played by two 'clever' players and the game played by two 'random' players, is supported by the tournament game. It will turn out that, while a straightforward application of this intuition fails, a more subtle version of it is still valid. Finally, we consider the orientation game version of the tournament game, where Maker wins the game if the final digraph - also containing the edges directed by Breaker - possesses a copy of T-k. We prove that in that game Breaker has a winning strategy for k = (4 + o(1)) log(2) n. |
Year | DOI | Venue |
|---|---|---|
2016 | 10.1017/S096354831500019X | COMBINATORICS PROBABILITY & COMPUTING |
Field | DocType | Volume |
Complete graph,Discrete mathematics,Combinatorics,Tournament,Random graph,Upper and lower bounds,Repeated game,Sequential game,Game tree,Example of a game without a value,Mathematics | Journal | 25 |
Issue | ISSN | Citations |
SP1 | 0963-5483 | 1 |
PageRank | References | Authors |
0.39 | 12 | 3 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Clemens Dennis | 1 | 18 | 8.25 |
| Heidi Gebauer | 2 | 83 | 11.07 |
| Anita Liebenau | 3 | 25 | 5.19 |