The Almost Sure Evolution of Hierarchy Among Similar Competitors

Abstract

While generic competitive systems exhibit mixtures of hierarchy and cycles, real-world systems are predominantly hierarchical. We demonstrate and extend a mechanism for hierarchy; systems with similar agents approach perfect hierarchy in expectation. A variety of evolutionary mechanisms plausibly select for nearly homogeneous populations, however, extant work does not explicitly link selection dynamics to hierarchy formation via population concentration. Moreover, previous work lacked numerical demonstration. This paper contributes in four ways. First, populations that converge to perfect hierarchy in expectation converge to hierarchy in probability. Second, we analyze hierarchy formation in populations subject to the continuous replicator dynamic with diffusive exploration, linking population dynamics to emergent structure. Third, we show how to predict the degree of cyclicity sustained by concentrated populations at internal equilibria. This theory can differentiate learning rules and random payout models. Finally, we provide direct numerical evidence by simulating finite populations of agents subject to a modified Moran process with Gaussian exploration. As examples, we consider three bimatrix games and an ensemble of games with random payouts. Through this analysis, we explicitly link the temporal dynamics of a population undergoing selection to the development of hierarchy.

Figures (7)

• Figure 1: The ratio of moments $\epsilon$ and correlation $\rho$ for three learning rules, with Hessian blocks drawn from symmetric and skew-symmetric Gaussian families, and diffusion drawn as the product of Gaussian matrices. The isovariant case uses $\Sigma_x \propto I$ to establish a reference. We require that the $H_{xx}$ block is negative definite with largest eigenvalue $\lambda \leq -\sqrt{T}/10$. Sampling from the conditional distribution is performed using a mixed Langevin-Gibbs sampler yamada2018hessian. The scaling ensures that the conditioning of $H_{xx}$ converges for large $T$, thus the eccentricity of the steady states converge. Median results are shown with solid lines while shaded regions represent the [5%,95%], [10%,90%],[20%,80%], and [40%,60%] intervals for 1,000 samples. Recall that $\rho = 0.5$ implies perfect transitivity
• Figure 2: Results from the replicator example. Left: The trajectory of the centroid with tangent arrows for a performance function with a peak at the origin, where all competitors are selected to start centered at a point away from the origin. Left inset: For each epoch, predicted movement of the centroid according to the replicator equation compared to actual movement. Top right: Moments of the trait distribution per epoch, per dimension. Middle right: Covariance of the trait distribution per epoch. Bottom right: Relative intransitivity, per epoch. For a video animation of the trait distribution over time, see https://github.com/ccebra/Evolution-Almost-Sure-Hierarchy/tree/main
• Figure 3: Left: Step-by-step intransitivity for the three bimatrix games under control parameters. Right: The empirical correlation $\rho$ versus concentration $\kappa$ (trait-averaged standard deviation) in the trait distribution. Colors represent the norm of the gradient of the performance function, $g$, for PD and Stag, and $H_{xx}^2/(g \text{ Cov})^2$ for chicken, all evaluated at the final centroid. The lines mark quartic ($\mathcal{O}(\kappa^4)$) and quadratic convergence ($\mathcal{O}(\kappa^2)$) from the maximum empirical $\rho$. Data is generated using repeated trials with varying genetic drift to produce clusters with varying concentration. Trials with multiple clusters at the end of evolution (all in chicken) have been removed.
• Figure 4: Trait concentration and transitivity for random performance functions in 4-dimensional trait spaces under control parameters under the softmax selection criterion. Left: Intransitivity by step according to the HHD for random performance functions with different trigonometric and linear weights. (Black): Equal weighting of trigonometric and linear. (Red): Trigonometric has double weight over linear. (Blue): Function is entirely trigonometric. Middle: Correlation versus predicted. Right: Kendall transitivity measure by step for control parameters.
• Figure 5: Trait concentration and bimatrix game performances. (a): The normalized concentration of the strategies for different bimatrix games. The vertical black line is the predicted Nash equilibrium of our performance function. (b): The performance heatmap for our Moran process. The contours indicate the final locations of the competitors, and the horizontal and vertical lines are the predicted Nash equilibrium.