[Paper Review] Coevolutionary intransitivity in games: A landscape analysis

Hendrik Richter|arXiv (Cornell University)|Jan 16, 2015

Evolutionary Game Theory and Cooperation13 references4 citations

TL;DR

This paper introduces a landscape-based framework linking rating- and ranking-based intransitivity measures to coevolutionary fitness dynamics, demonstrating that collective ranking difference (crd) is a more robust indicator of coevolutionary intransitivity than temporal mismatch (ptm), particularly under varying randomness and player counts in a tunable random game model.

ABSTRACT

Intransitivity is supposed to be a main reason for deficits in coevolutionary progress and inheritable superiority. Besides, coevolutionary dynamics is characterized by interactions yielding subjective fitness, but aiming at solutions that are superior with respect to an objective measurement. Such an approximation of objective fitness may be, for instance, generalization performance. In the paper a link between rating-- and ranking--based measures of intransitivity and fitness landscapes that can address the dichotomy between subjective and objective fitness is explored. The approach is illustrated by numerical experiments involving a simple random game with continuously tunable degree of randomness.

Motivation & Objective

• To address the dichotomy between subjective fitness (from coevolutionary interactions) and objective fitness (generalization performance) in coevolutionary algorithms.
• To investigate how intransitivity—cyclic superiority relations—impedes coevolutionary progress despite apparent gains in subjective fitness.
• To develop and validate dynamic intransitivity measures that capture temporal inconsistencies in fitness rankings across generations.
• To evaluate the effectiveness of different intransitivity metrics (e.g., itx, kld, ptm, crd) in relation to both static and dynamic fitness evolution.
• To establish a scalable experimental framework using a random game with tunable randomness to study intransitivity effects systematically.

Proposed method

• Proposes a codynamic fitness landscape framework integrating objective fitness (f_obj) and subjective fitness (f_sub) over a shared search space S with neighborhood structure n(s).
• Introduces a simple random game model where outcomes depend on player ratings and a tunable randomness parameter p_rand, enabling continuous control over intransitivity.
• Employs rating-based intransitivity measures (e.g., itx, kld) and ranking-based dynamic measures (e.g., crd, ptm) to quantify static and dynamic intransitivity.
• Uses time-averaged measures (e.g., ⟨itx⟩, ⟨crd⟩) to assess intransitivity across generations, normalized per number of players.
• Applies statistical analysis to compare relationships between intransitivity measures and fitness evolution (e.g., max score, max generalization performance).
• Validates metric interchangeability and robustness by comparing crd and ptm across varying N (players) and p_rand (randomness levels).

Experimental results

Research questions

• RQ1 How do static intransitivity measures (e.g., itx, kld) correlate with the degree of randomness in a coevolutionary game?
• RQ2 To what extent does static intransitivity influence subjective or objective fitness in coevolutionary dynamics?
• RQ3 How do dynamic intransitivity measures (e.g., ptm, crd) reflect temporal inconsistencies in fitness rankings across generations?
• RQ4 Which intransitivity measure—crd or ptm—better captures the true coevolutionary intransitivity in relation to fitness evolution?
• RQ5 Can ranking-based measures like crd serve as a more reliable proxy for coevolutionary intransitivity than rating-based or temporal mismatch measures?

Key findings

• Static intransitivity (measured by itx) increases monotonically with the randomness parameter p_rand, while subjective and objective fitness (score, generalization performance) remain largely unaffected.
• There is a proportional relationship between the rating-based intransitivity measure itx and the probabilistic measure kld, confirming their interchangeability in the studied game model.
• The collective ranking difference (crd) scales linearly with the maximum observed score (max sc) and generalization performance (max gp), particularly at low p_rand, indicating it tracks fitness evolution.
• The temporal mismatch (ptm) measure scales less strongly than crd, especially for small N and high p_rand, suggesting crd is more sensitive to dynamic intransitivity.
• crd based on both rating and generalization performance as objective fitness show piecewise linear scaling, confirming consistency across different objective fitness definitions.
• The crd measure demonstrates superior robustness and interpretability compared to ptm, indicating it is a more meaningful indicator of coevolutionary intransitivity in dynamic settings.