[Paper Review] Recognizing Members of the Tournament Equilibrium Set is NP-hard

Felix Brandt, Felix Fischer+1 more|arXiv (Cornell University)|Nov 19, 2007

Game Theory and Voting Systems8 references19 citations

TL;DR

This paper proves that determining whether a given alternative is in the Tournament Equilibrium Set (TEQ) of a tournament is NP-hard, establishing a fundamental computational barrier for this prominent social choice solution concept. The result is shown via a reduction from 3-SAT, demonstrating that even deciding membership in TEQ is computationally intractable unless P = NP, despite TEQ's theoretical appeal as a refinement of both the Banks set and minimal covering set.

ABSTRACT

A recurring theme in the mathematical social sciences is how to select the "most desirable" elements given a binary dominance relation on a set of alternatives. Schwartz's tournament equilibrium set (TEQ) ranks among the most intriguing, but also among the most enigmatic, tournament solutions that have been proposed so far in this context. Due to its unwieldy recursive definition, little is known about TEQ. In particular, its monotonicity remains an open problem up to date. Yet, if TEQ were to satisfy monotonicity, it would be a very attractive tournament solution concept refining both the Banks set and Dutta's minimal covering set. We show that the problem of deciding whether a given alternative is contained in TEQ is NP-hard.

Motivation & Objective

• To resolve the computational complexity of determining membership in Schwartz's Tournament Equilibrium Set (TEQ), a prominent but poorly understood tournament solution.
• To close a longstanding open problem regarding whether TEQ satisfies monotonicity, by showing that the underlying decision problem is NP-hard.
• To demonstrate that, unlike the Banks set and minimal covering set, TEQ membership cannot be computed in polynomial time unless P = NP.
• To provide a formal reduction from 3-SAT to the TEQ membership problem, establishing its computational hardness.

Proposed method

• Construct a tournament T_TEQ_ϕ from a given 3-DNF formula ϕ, encoding variable assignments and clause satisfaction through dominance relations.
• Define the TEQ relation recursively using self-reference: x →_X y iff x ∈ TEQ(D_X(y)), where D_X(y) is the set of dominators of y in subset X.
• Use structural induction on subsets B ⊇ {d} to prove that d ∈ TEQ(D_k) for all relevant k, establishing the core inductive step.
• Leverage properties of Condorcet winners and transitive subtournaments to show that certain alternatives dominate others in the TEQ relation.
• Prove that if ϕ is satisfiable, then d is in TEQ(T_TEQ_ϕ), and if not, d is not in TEQ(T_TEQ_ϕ), via a reduction from 3-SAT.
• Utilize auxiliary lemmas to establish that in certain subtournaments, a specific alternative ci dominates all others via the TEQ relation, enabling the inductive argument.

Experimental results

Research questions

• RQ1 Is the problem of deciding whether a given alternative belongs to the Tournament Equilibrium Set (TEQ) of a tournament computationally tractable?
• RQ2 Does the TEQ solution concept satisfy monotonicity, a desirable property in social choice theory?
• RQ3 Can the TEQ membership problem be solved in polynomial time, or is it inherently hard?
• RQ4 Is there a reduction from a known NP-complete problem to the TEQ membership problem?
• RQ5 What is the computational complexity of computing TEQ compared to other tournament solutions like the Banks set or minimal covering set?

Key findings

• The problem of deciding whether a given alternative is in the Tournament Equilibrium Set (TEQ) of a tournament is NP-hard.
• The NP-hardness result is established via a reduction from 3-SAT, showing that solving TEQ membership is at least as hard as solving 3-SAT.
• The proof demonstrates that if a 3-DNF formula ϕ is satisfiable, then the designated alternative d is in TEQ(T_TEQ_ϕ), and if not, d is not in TEQ(T_TEQ_ϕ).
• The recursive definition of TEQ, which relies on self-reference (TEQ is used to define itself), is shown to lead to computational intractability.
• The result implies that no polynomial-time algorithm for computing TEQ exists unless P = NP.
• The hardness result holds even though the Banks set, a refinement of TEQ, can be computed in polynomial time, highlighting a key computational distinction between the two concepts.