Similarity Suppresses Cyclicity: Why Similar Competitors Form Hierarchies

TL;DR

This paper explains why most real-world competitive systems tend to form hierarchies rather than cycles, showing that similarity and smooth performance functions promote transitivity, supported by theoretical proofs and evolution experiments.

Contribution

The paper introduces a general mechanism demonstrating how similarity and smoothness in performance functions promote transitivity in competitive systems, explaining the prevalence of hierarchies.

Findings

Similar competitors tend to compete transitively.

Evolutionary training promotes trait convergence and transitivity.

The mechanism is robust across various game types and performance functions.

Abstract

Competitive systems can exhibit both hierarchical (transitive) and cyclic (intransitive) structures. Despite theoretical interest in cyclic competition, which offers richer dynamics, and occupies a larger subset of the space of possible competitive systems, most real-world systems are predominantly transitive. Why? Here, we introduce a generic mechanism which promotes transitivity, even when there is ample room for cyclicity. Consider a competitive system where outcomes are mediated by competitor attributes via a performance function. We demonstrate that, if competitive outcomes depend smoothly on competitor attributes, then similar competitors compete transitively. We quantify the rate of convergence to transitivity given the similarity of the competitors and the smoothness of the performance function. Thus, we prove the adage regarding apples and oranges. Similar objects admit well ordered comparisons. Diverse objects may not. To test that theory, we run a series of evolution experiments designed to mimic genetic training algorithms. We consider a series of canonical bimatrix games and an ensemble of random performance functions that demonstrate the generality of our mechanism, even when faced with highly cyclic games. We vary the training parameters controlling the evolution process, and the shape parameters controlling the performance function, to evaluate the robustness of our results. These experiments illustrate that, if competitors evolve to optimize performance, then their traits may converge, leading to transitivity.