  Tournament solutions provide methods for selecting the "best" alternatives
from a tournament and have found applications in a wide range of areas.
Previous work has shown that several well-known tournament solutions almost
never rule out any alternative in large random tournaments. Nevertheless, all
analytical results thus far have assumed a rigid probabilistic model, in which
either a tournament is chosen uniformly at random, or there is a linear order
of alternatives and the orientation of all edges in the tournament is chosen
with the same probabilities according to the linear order. In this work, we
consider a significantly more general model where the orientation of different
edges can be chosen with different probabilities. We show that a number of
common tournament solutions, including the top cycle and the uncovered set, are
still unlikely to rule out any alternative under this model. This corresponds
to natural graph-theoretic conditions such as irreducibility of the tournament.
In addition, we provide tight asymptotic bounds on the boundary of the
probability range for which the tournament solutions select all alternatives
with high probability.
