A non-trivial upper bound on the threshold bias of the Oriented-cycle game

In the Oriented-cycle game, the two players, called OMaker and OBreaker, alternately direct edges of the complete graph on n vertices. OMaker directs exactly one edge, whereas OBreaker is allowed to direct between one andb edges in each round. OMaker wins if the final tournament contains a directed cycle, otherwise OBreaker wins. It is not that difficult to see that OMaker has a winning strategy for this game whenever b < n/2 − 2, whereas OBreaker wins for b > n− 3. We show that OBreaker has a strategy whenever b > 5n/6. Moreover, we also show that in a variant, where OBreaker is asked to direct exactly b edges in each round, OBreaker wins whenever b > 19n/20. Joint work with Anita Liebenau.