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.
The discriminative power of tournament solutions has also been investigated empirically by @cite_12 . Building on the observation that the distributions of real-world tournaments are typically far from uniform, these authors examined the behavior of eleven common tournament solutions on tournaments generated according to stochastic preference models and empirical data. The stochastic models that they used include the impartial culture model, the Mallows mixtures model, and the P ' o lya-Eggenberger urn model. They reported that under these more realistic models, most tournament solutions are in fact much more discriminating than the analytical results for uniform random tournaments suggest.
@cite_12
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Tournament solutions constitute an important class of social choice functions that only depend on the pairwise majority comparisons between alternatives. Recent analytical results have shown that several concepts with appealing axiomatic properties tend to not discriminate at all when the tournaments are chosen from the uniform distribution. This is in sharp contrast to empirical studies which have found that real-world preference profiles often exhibit Condorcet winners, i.e., alternatives that all tournament solutions select as the unique winner. In this work, we aim to fill the gap between these extremes by examining the distribution of the number of alternatives returned by common tournament solutions for empirical data as well as data generated according to stochastic preference models.