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 win and for OBreaker’s win in three different games. In the first game OMaker wins if the obtained tournament has a cycle. The second game is Hamiltonicity, where OMaker wins if the obtained tournament contains a Hamilton cycle. Finally, we consider the H-creation game, where OMaker wins if the obtained tournament has a copy of some fixed digraph H.