A counterexample to a conjecture of Schwartz

Brandt, Felix; Chudnovsky, Maria; Kim, Ilhee; Liu, Gaku; Norin, Sergey; Scott, Alex; Seymour, Paul; Thomassé, Stéphan

doi:10.1007/s00355-011-0638-y

Cited by 19 publications

(18 citation statements)

References 10 publications

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“…8 Another prominent Condorcet extension-the tournament equilibrium set (Schwartz, 1990)-was conjectured to satisfy SSP and monotonicity for almost 20 years. This conjecture was recently disproved by Brandt et al (2013). In fact, it can be shown that the tournament equilibrium set as well as the related minimal extending set (Brandt, 2011) can be For generalized strategyproofness as defined by Nehring (2000) (see Remark 6), the second condition is not required and every coarsening of a strategyproof SCF is strategyproof.…”

Section: The Resultsmentioning

confidence: 99%

Set-monotonicity implies Kelly-strategyproofness

Brandt

2015

Soc Choice Welf

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This paper studies the strategic manipulation of set-valued social choice functions according to Kelly's preference extension, which prescribes that one set of alternatives is preferred to another if and only if all elements of the former are preferred to all elements of the latter. It is shown that set-monotonicity-a new variant of Maskin-monotonicity-implies Kellystrategyproofness in comprehensive subdomains of the linear domain. Interestingly, there are a handful of appealing Condorcet extensions-such as the top cycle, the minimal covering set, and the bipartisan set-that satisfy set-monotonicity even in the unrestricted linear domain, thereby answering questions raised independently by Barberà (1977a) and Kelly (1977).

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Section: The Resultsmentioning

confidence: 99%

Set-monotonicity implies Kelly-strategyproofness

Brandt

2015

Soc Choice Welf

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“…In the tournament of Figure 1, TEQ coincides withT C. Schwartz conjectured that every tournament admits a unique minimal TEQ-retentive set. This conjecture was recently disproved by a non-constructive argument using the probabilistic method (Brandt et al, 2012). While this proof showed the existence of a counter-example, no concrete counterexample (or even the exact size of one) is known.…”

Section: Retentive Setsmentioning

confidence: 98%

“…As was shown by Laffond et al (1993a) and Houy (2009a,b), TEQ satisfies any one of a number of important properties such as monotonicity if and only if Schwartz's conjecture holds. Brandt et al (2012) recently disproved Schwartz's conjecture by showing the existence of a counter-example of astronomic proportions. The interest in TEQ and retentiveness in general, however, is hardly diminished as concrete counter-examples to Schwartz's conjecture have never been encountered, even when resorting to extensive computer experiments (Brandt et al, 2010).…”

Section: Introductionmentioning

confidence: 99%

Minimal retentive sets in tournaments

et al. 2013

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Tournament solutions, i.e., functions that associate with each complete and asymmetric relation on a set of alternatives a non-empty subset of the alternatives, play an important role in the mathematical social sciences at large. For any given tournament solution S , there is another tournament solutionS which returns the union of all inclusion-minimal sets that satisfy S -retentiveness, a natural stability criterion with respect to S . Schwartz's tournament equilibrium set (TEQ) is defined recursively as TEQ =TEQ. In this article, we study under which circumstances a number of important and desirable properties are inherited from S toS . We thus obtain a hierarchy of attractive and efficiently computable tournament solutions that "approximate" TEQ, which itself is computationally intractable. We further prove a weaker version of a recently disproved conjecture surrounding TEQ, which establishesT Ca refinement of the top cycle-as an interesting new tournament solution.

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“…8 Schwartz (1990) showed that TEQ ⊆ BA and conjectured that every tournament contains a unique inclusion-minimal TEQ-retentive set, which was later shown to be equivalent to TEQ satisfying any one of a number of desirable properties for tournament solutions including stability and monotonicity (Laffond et al, 1993a;Houy, 2009a,b;Brandt et al, 2014;Brandt, 2011;Brandt and Harrenstein, 2011;Brandt, 2015). This conjecture was disproved by Brandt et al (2013), who have non-constructively shown the existence of a counterexample with about 10 136 alternatives using the probabilistic method. Since it was shown that TEQ satisfies the above mentioned desirable properties for all tournaments that are smaller than the smallest counterexample to Schwartz's conjecture, the search for smaller counterexamples remains an important problem.…”

Section: The Bipartisan Set and The Tournament Equilibrium Setmentioning

confidence: 99%

On the structure of stable tournament solutions

et al. 2016

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A fundamental property of choice functions is stability, which, loosely speaking, prescribes that choice sets are invariant under adding and removing unchosen alternatives. We provide several structural insights that improve our understanding of stable choice functions. In particular, (i) we show that every stable choice function is generated by a unique simple choice function, which never excludes more than one alternative, (ii) we completely characterize which simple choice functions give rise to stable choice functions, and (iii) we prove a strong relationship between stability and a new property of tournament solutions called local reversal symmetry. Based on these findings, we provide the first concrete tournament-consisting of 24 alternatives-in which the tournament equilibrium set fails to be stable. Furthermore, we prove that there is no more discriminating stable tournament solution than the bipartisan set and that the bipartisan set is the unique most discriminating tournament solution which satisfies standard properties proposed in the literature.

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A counterexample to a conjecture of Schwartz

Cited by 19 publications

References 10 publications

Set-monotonicity implies Kelly-strategyproofness

Set-monotonicity implies Kelly-strategyproofness

Minimal retentive sets in tournaments

On the structure of stable tournament solutions

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