The oriented-cycle game is a two-player game on the edges of a complete graph 
. OMaker moves first and directs one unused edge
per turn. In the monotone 
-biased version, OBreaker then directs between one and 
unused edges.
OMaker wins if the final tournament contains a directed
cycle, and OBreaker wins otherwise (Liebenau et al. 2026).
Let 
be the largest integer bias for which OMaker has a winning
strategy. Allowing OBreaker fewer than 
edges makes the game monotone
in 
.
The strict version, in which OBreaker must direct exactly 
edges when available, is a different
game.
The bounds
 |
combine the OMaker strategy of Ben-Eliezer et al. (2012) with the improved OBreaker strategy of Liebenau et al. (2026). The latter improves the earlier
coefficient 
for the monotone version.
