A Remark on the Tournament Game

Efficient Simulation of a Random Knockout Tournament

We consider the problem of using simulation to efficiently estimate the win probabilities for participants in a general random knockout tournament. Both of our proposed estimators, one based on the notion of “observed survivals” and the other based on conditional expectation and post-stratification, are highly effective in terms of variance reduction when compared to the raw simulation estimato...

Biased orientation games

We study biased orientation games, in which the board is the complete graph Kn, and OMaker (oriented maker) and OBreaker (oriented breaker) take turns in directing previously undirected edges of Kn. At the end of the game, the obtained graph is a tournament. OMaker wins if the tournament has some property P and OBreaker wins otherwise. We provide bounds on the bias that is required for OMaker’s...

On the clique-game

We study Maker/Breaker games on the edges of the complete graph, as introduced by Chvátal and Erdős. We show that in the (m : b) game played on KN , the complete graph on N vertices, Maker can achieve a Kq for q = ( m log 2 (b+1) − o(1) ) · log2 N , which partially solves an open problem by Beck. Moreover, we show that in the (1 : 1) game played on KN for a sufficiently large N , Maker can achi...

A Strategy for Maker in the Clique Game which Helps to Tackle some Open Problems by Beck

We study Maker/Breaker games on the edges of the complete graph, as introduced by Chvátal and Erdős. We show that in the (m : b) clique game played on KN , the complete graph on N vertices, Maker can achieve a Kq for q = ( m log 2 (b+1) − o(1) ) · logN , which partially solves an open problem by Beck. Moreover, we show that in the (1:1) clique game played on KN for a sufficiently large N , Make...

A Remark on a Remark by Macaulay or Enhancing Lazard Structural Theorem