INTRODUCTION

Many problems in multiagent decision making can be addressed using tournament solutions. Examples of wellstudied tournament solutions are the Copeland set, the Banks set, or the minimal covering set [4]. A common benchmark for tournament solutions is which desirable properties they satisfy (see e.g., [4], for an overview of such properties).

In 2011, Brandt [1] proposed a new tournament solution called ME and an associated graph-theoretic conjecture. If the conjecture had held, ME would have satisfied virtually all desirable properties that are usually considered in the literature on tournament solutions. In 2013, however, the existence of a counter-example with about 10 136 alternatives was shown. The proof is non-constructive and uses the probabilistic method [3].

This left open which of the properties are actually satisfied by ME . In this paper, we resolve these open questions. Using the counter-example by Brandt et al. [3] we show that ME fails to satisfy most properties (such as monotonicity and stability) while it does satisfy a stronger version of idempotency, irregularity, and membership in the Banks set. We also prove that computing ME is NP-hard.

PRELIMINARIES

A tournament T is a pair (A, ), where A is a set of alternatives and is an asymmetric and complete (and thus irreflexive) binary relation on A, usually referred to as the dominance relation. The dominance relation can be extended to sets of alternatives by writing a B when a b for all b ∈ B. Let BT (a) denote the set of all subsets B ⊆ A such that |B is transitive and a B \ {a}.

A tournament solution is a function that maps a tournament to a nonempty subset of its alternatives. The Banks set BA chooses maximal elements of inclusion-maximal transitive subtournaments, i.e.,

BA(T ) = {a ∈ A : ∃B ∈ BT (a) such that b : b B}. A subset of alternatives B ⊆ A is called S-stable for tour- nament solution S if a / ∈ S(B ∪ {a}) for all a ∈ A \ B.

In particular, we refer to BA-stable sets as extending sets. The union of all inclusion minimal extending sets defines the tournament solution ME [1], i.e.,

ME (T ) = {B is BA-stable : ∀C B : C is not BA-stable}.

An example of a tournament where BA and ME differ is given in Figure 1

PROPERTIES OF ME

Dominance-based properties.

First, we consider two properties that are based on the dominance relation. Monotonicity prescribes that a chosen alternative should still be chosen if it is reinforced. The second property, independence of unchosen alternatives, states that the choice set should be unaffected by changes in the dominance relation between unchosen alternatives. Theorem 1. ME satisfies neither monotonicity nor independence of unchosen alternatives.

Choice-theoretic properties.

An important class of properties concern the consistency of choice and relate choices from different subtournaments of the same tournament to each other. A relatively strong property of this type is stability, which requires that a set is chosen from two different sets of alternatives if and only if it is chosen from the union of these sets [2]. S(A ∪ B)).

In Figure 2, the logical relations between the different properties are depicted. Theorem 2. ME satisfies α ⊇ but neither α ⊆ nor γ ⊇ .

As a consequence, ME is not stable but satisfies idempotency. It is still open whether ME satisfies γ ⊆ .

Relationships to other tournament solutions.

Besides the axiomatic properties of ME , we are also interested in its set-theoretic relationships to other tournament solutions.

Theorem 3. For all tournaments T , ME (T ) ⊆ BA(T ).

This also implies that the irregularity of BA [4, Theorem 7.1.3] extends to ME .

It is unknown whether the tournament equilibrium set is always contained in ME and whether ME is always contained in the minimal covering set.

Computational complexity.

An important property of every tournament solution is whether it can be computed efficiently. By a reduction from 3SAT, we can show that this is not the case for ME . Theorem 4. Deciding whether an alternative in a tournament is contained in ME is NP-hard.

Membership of the problem in NP seems rather unlikely. The best upper bound we know of is Σ p 3 .

CONCLUSION AND DISCUSSION

We have analyzed the axiomatic as well as computational properties of the tournament solution ME . Results were mixed. In conclusion, ME It is worth pointing out that ME 's violation of monotonicity, stability, and independence of unchosen alternatives crucially depends on the existence of tournaments with more than one minimal extending set. Not only is the size of known tournaments of this type enormous (about 10 136 alternatives) but, furthermore, these tournaments are very likely to be extremely rare. In effect, ME does satisfy these properties in all scenarios in which tournaments only admit a unique minimal extending set. Hence, it is fair to say that ME satisfies the considered properties for all practical purposes. This, in turn, may be interpreted as a criticism of the axiomatic method in general: For what does it mean if a tournament solution (or any other mathematical object) in principle violates some desirable properties, but no concrete example of a violation is known and will perhaps ever be known?