Soft Tournament Equilibrium

TL;DR

This paper introduces Soft Tournament Equilibrium (STE), a differentiable framework for evaluating AI agents using set-valued solutions from pairwise comparisons, addressing non-transitive interactions.

Contribution

STE is a novel, differentiable method that computes set-valued tournament solutions directly from data, with theoretical guarantees and practical evaluation.

Findings

STE accurately recovers classical solutions in the zero-temperature limit.

It demonstrates stability and consistency in cyclic and preference-based benchmarks.

The method provides calibrated membership scores for core agents.

Abstract

The evaluation of general-purpose artificial agents, particularly those based on LLMs, presents a significant challenge due to the non-transitive nature of their interactions. When agent A defeats B, B defeats C, and C defeats A, traditional ranking methods that force a linear ordering can be misleading and unstable. We argue that for such cyclic domains, the fundamental object of evaluation should not be a ranking alone but a set-valued core, as conceptualized in classical tournament theory. This paper introduces Soft Tournament Equilibrium (STE), a differentiable framework for learning and computing set-valued tournament solutions directly from pairwise comparison data. STE first learns a probabilistic tournament model, potentially conditioned on rich contextual information. It then employs differentiable operators for soft reachability and soft covering to compute continuous analogues of two seminal tournament solutions: the Top Cycle and the Uncovered Set. The output is a set of core agents, each with a continuous membership score that can be calibrated when suitable validation labels or repeated-sampling evidence are available. We develop the theoretical foundation for STE by proving consistency with classical solutions in the zero-temperature limit, establishing Condorcet-inclusion properties, and analyzing stability and sample complexity. We evaluate the method on a planted cyclic core benchmark and on real preference/execution diagnostics. This work provides a self-contained account that re-centers general-agent evaluation on a robust tournament-theoretic foundation, moving from unstable rankings toward stable, set-valued equilibria.