## 1 Introduction

Given a set of alternatives and binary non-transitive preferences over these alternatives, how can we consistently choose the “best” elements from any feasible subset of alternatives? This question has been studied in detail in the literature on tournament solutions [see, Lasl97a, Hudr09a, Mose15a, BBH15a]. The lack of transitivity is typically attributed to the independence of pairwise comparisons as they arise in sports competitions, multi-criteria decision analysis, and preference aggregation.^{1}^{1}1Due to their generality, tournament solutions have also found applications in unrelated areas such as biology [Schj22a, Land51a, Slat61a, AlLe11a].
In particular, the pairwise majority relation of a profile of transitive individual preference relations often forms the basis of the study of tournament solutions. This is justified by a theorem due to McGa53a, which shows that every tournament can be induced by some underlying preference profile. Many tournament solutions therefore correspond to well-known social choice functions such as Copeland’s rule, Slater’s rule, the Banks set, and the bipartisan set.

Over the years, many desirable properties of tournament solutions have been proposed. Some of these properties, so-called *choice consistency conditions*, make no reference to the actual tournament but only relate choices from different subtournaments to each other. An important choice consistency condition, that goes under various names, requires that the choice set is invariant under the removal of unchosen alternatives. In conjunction with a dual condition on expanded feasible sets, this property is known as *stability* [BrHa11a]. Stability implies that choices are made in a robust and coherent way. Furthermore, stable choice functions can be rationalized by a preference relation on *sets* of alternatives.

Examples of stable tournament solutions are the *top cycle*, *the minimal covering set*, and the *bipartisan set*

. The latter is elegantly defined via the support of the unique mixed maximin strategies of the zero-sum game given by the tournament’s skew-adjacency matrix. Curiously, for some tournament solutions, including the

*tournament equilibrium set*and the

*minimal extending set*, proving or disproving stability turned out to be exceedingly difficult. As a matter of fact, whether the tournament equilibrium set satisfies stability was open for more than two decades before the existence of counterexamples with about alternatives was shown using the probabilistic method.

Bran11b systematically constructed stable tournament solutions by applying a well-defined operation to existing (non-stable) tournament solutions. Bran11b’s study was restricted to a particular class of generating tournament solutions, namely tournament solutions that can be defined via qualified subsets (such as the *uncovered set* and the *Banks set*). For any such generator, Bran11b gave sufficient conditions for the resulting tournament solution to be stable.
Later, BHS15a showed that for one particular generator, the Banks set, the sufficient conditions for stability are also necessary.

In this paper, we show that *every* stable choice function is generated by a unique underlying simple choice function, which never excludes more than one alternative.
We go on to prove a general characterization of stable tournament solutions that is not restricted to generators defined via qualified subsets. As a corollary, we obtain that the sufficient conditions for generators defined via qualified subsets are also necessary. Finally, we prove a strong connection between stability and a new property of tournament solutions called *local reversal symmetry*. Local reversal symmetry requires that an alternative is chosen if and only if it is unchosen when all its incident edges are inverted. This result allows us to settle two important problems in the theory of tournament solutions.
We provide the first concrete tournament—consisting of 24 alternatives—in which the tournament equilibrium set violates stability. Secondly, we prove that there is no more discriminating stable tournament solution than the bipartisan set. We also axiomatically characterize the bipartisan set by only using properties that have been previously proposed in the literature. We believe that these results serve as a strong argument in favor of the bipartisan set if choice consistency is desired.

## 2 Stable Sets and Stable Choice Functions

Let be a universal set of alternatives. Any finite non-empty subset of will be called a *feasible set*.
Before we analyze tournament solutions in sec:tsolutions, we first consider a more general model of choice which does not impose any structure on feasible sets.
A *choice function* is a function that maps every feasible set to a non-empty subset of called the *choice set* of . For two choice functions and , we write , and say that is a *refinement* of and a *coarsening* of , if for all feasible sets .
A choice function is called *trivial* if for all feasible sets .

Bran11b proposed a general method for refining a choice function by defining minimal sets that satisfy internal and external stability criteria with respect to

, similar to von-Neumann–Morgenstern stable sets in cooperative game theory.

^{2}

^{2}2This is a generalization of earlier work by Dutt88a, who defined the minimal covering set as the unique minimal set that is internally and externally stable with respect to the uncovered set (see sec:tsolutions).

A subset of alternatives is called -*stable* within feasible set for choice function if it consists precisely of those alternatives that are chosen in the presence of all alternatives in . Formally, is -stable in if

Equivalently, is -stable if and only if

(internal stability) | |||

(external stability) |

The intuition underlying this formulation is that there should be no reason to restrict the choice set by excluding some alternative from it (internal stability) and there should be an argument against each proposal to include an outside alternative into the choice set (external stability).

An -stable set is *inclusion-minimal* (or simply *minimal*) if it does not contain another -stable set. is defined as the union of all minimal -stable sets in .
defines a choice function whenever every feasible set admits at least one -stable set. In general, however, neither the existence of -stable sets nor the uniqueness of minimal -stable sets is guaranteed. We say that is *well-defined* if every choice set admits exactly one minimal -stable set. We can now define the central concept of this paper.

A choice function is *stable* if is well-defined and .
Stability is connected to rationalizability and non-manipulability. In fact, every stable choice function can be rationalized via a preference relation on *sets* of alternatives [BrHa11a] and, in the context of social choice, stability and monotonicity imply strategyproofness with respect to Kelly’s preference extension [Bran11c].

The following example illustrates the preceding definitions. Consider universe and choice function given by the table below (choices from singleton sets are trivial and therefore omitted).

The feasible set admits exactly two -stable sets, itself and . The latter holds because (internal stability) and (external stability). All other feasible sets admit unique -stable sets, which coincide with . Hence, is well-defined and given by the entries in the rightmost column of the table. Since , fails to be stable. , on the other hand, satisfies stability.

Choice functions are usually evaluated by checking whether they satisfy choice consistency conditions that relate choices from different feasible sets to each other.
The following two properties, and , are set-based variants of Sen’s and [Sen71a].
is a rather prominent choice-theoretic condition, also known as Cher54a’s *postulate * [Cher54a], the *strong superset property* [Bord79a], *outcast* [AiAl95a], and the *attention filter axiom* [MNO12a].^{3}^{3}3We refer to Monj08a for a more thorough discussion of the origins of this condition.

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