Abstract
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.
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Notes
- 1.
- 2.
By symmetry, we may assume without loss of generality that \(p\le 1/2\).
- 3.
See, e.g., Cormen et al. [8] for the definitions of asymptotic notations.
- 4.
A strongly connected tournament is also said to be strong. Strong connectedness is equivalent to irreducibility and to the property of having a Hamiltonian cycle [22].
- 5.
A king is an alternative that can reach any other alternative via a directed path of length at most two [19]. Therefore, all alternatives of a tournament are kings if and only if every pair of alternatives can reach each other via a directed path of length at most two. Such a tournament has been studied in graph theory and called an all-kings tournament [25].
- 6.
Note that the set of Condorcet winners is not a tournament solution because it can be empty.
- 7.
This is known in graph theory as the set of kings (cf. Footnote 5). An alternative definition, which is also the origin of the name “uncovered set”, is based on the covering relation. An alternative \(a_i\) is said to cover another alternative \(a_j\) if (i) \(a_i\) dominates \(a_j\), and (ii) any alternative that dominates \(a_i\) also dominates \(a_j\). The uncovered set corresponds to the set of alternatives that are not covered by any other alternative.
- 8.
One way to interpret the possible intransitivity of the preferences is as a result of noise in the voters’ true preferences. Laslier [17] introduced the term Rousseauist cultures for this kind of models.
- 9.
Our setting is slightly different for the last two values of p, as we explain later in this section.
- 10.
Since the probability that \( CNL \) selects all alternatives is equal to the corresponding probability for \( COND \) for any fixed n by symmetry, and \( UC ^\infty \) selects all alternatives exactly when \( UC \) does, the results for \( CNL \) and \( UC ^\infty \) are captured by those for \( COND \) and \( UC \), respectively.
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Acknowledgments
This material is based upon work supported by the Deutsche Forschungsgemeinschaft under grant BR 2312/11-1 and by a Stanford Graduate Fellowship. The authors thank Felix Brandt, Pasin Manurangsi, and Fedor Petrov for helpful discussions and the anonymous reviewers for helpful comments.
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Saile, C., Suksompong, W. (2018). Robust Bounds on Choosing from Large Tournaments. In: Christodoulou, G., Harks, T. (eds) Web and Internet Economics. WINE 2018. Lecture Notes in Computer Science(), vol 11316. Springer, Cham. https://doi.org/10.1007/978-3-030-04612-5_26
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