[Paper Review] Stable marriage problem under Monte Carlo simulations, influence of preferrence corelation on relaxation time

Piotr Nyczka, Jerzy Cisło|arXiv (Cornell University)|May 31, 2011

Game Theory and Voting Systems1 references3 citations

TL;DR

This study uses Monte Carlo simulations to investigate how correlation in preference lists affects relaxation time in the stable marriage problem. By modeling attractiveness and personal taste as multidimensional vectors, the authors show that higher preference correlation (lower dimensionality of attractiveness/taste vectors) drastically reduces relaxation time, with a surprising minimum at n=2 for intermediate system sizes.

ABSTRACT

In this paper we consider stable marriage problem under the Monte Carlo simulations. We investigate how correlation in lists of preferrences can affect simulation results such as: relaxation time, time distribution of relaxation times etc. We took into account attractiveness of individuals and it's different types as well as personal taste.

Motivation & Objective

• To model realistic relationship formation dynamics beyond deterministic algorithms by incorporating random encounters and individual preferences.
• To investigate how correlations in preference lists—driven by shared attractiveness and taste dimensions—affect system relaxation time.
• To explore the impact of system size (N) and preference correlation strength (via vector dimension n) on convergence speed to stable states.
• To test whether natural selection might favor common tastes and individualized preferences for reproductive efficiency.

Proposed method

• Employed Monte Carlo simulations with random pairwise encounters between N men and N women at each time step.
• Defined preference rankings using a scalar product between individual taste vectors (T) and attractiveness vectors (A), both normalized and randomly initialized.
• Simulated relationship formation where agents switch partners if both prefer each other over current partners, ensuring stability via Nash equilibrium checks.
• Measured relaxation time τ as the number of Monte Carlo steps until no unstable pairs remain.
• Varied the number of attractiveness/taste dimensions (n) to control correlation strength: lower n implies higher correlation.
• Averaged results over 100 independent simulations for statistical robustness.

Experimental results

Research questions

• RQ1 How does the correlation strength in preference lists—controlled by the dimensionality n of attractiveness and taste vectors—affect the relaxation time τ in the stable marriage problem?
• RQ2 Does the relaxation time τ scale differently with system size N under correlated versus uncorrelated preference lists?
• RQ3 Is there an optimal level of preference correlation (i.e., optimal n) that minimizes relaxation time for finite system sizes?
• RQ4 Why does the relaxation time exhibit a minimum at n=2 for intermediate N (50 < N < 350), despite higher correlation (n=1) yielding faster convergence for larger N?
• RQ5 To what extent do random encounters and individualized preferences influence the speed of reaching stable matchings compared to deterministic algorithms?

Key findings

• Relaxation time τ is drastically shorter under highly correlated preferences (n=1) compared to uncorrelated ones, with a gap of approximately 10³ for N=20.
• For N > 350, relaxation time τ decreases monotonically with increasing preference correlation (i.e., decreasing n), with τ(N) following a power law for n=1 across all N.
• For 50 < N < 350, the relaxation time τ reaches a minimum at n=2, indicating that weaker correlation (higher individualization) accelerates convergence in small systems.
• Even for n > 1000, the relaxation time gap between correlated and uncorrelated cases remains substantial, indicating persistent influence of correlation strength.
• The relaxation time τ(N) for correlated preferences (n=1) follows a power law for all N, while for higher n, the power-law regime is limited to N < Nc before τ grows faster than power law.
• The results suggest that natural selection may favor shared tastes (low n) for faster pairing and higher reproductive efficiency in larger populations, but individualized preferences (n=2) may optimize matching speed in smaller groups.