Replicator-Mutator Dynamics and Safe Asset Status of the United States: Dashboard

Introduction

The paper carries in one dynamical system five features of safe asset status that the literature identifies: several safe assets coexisting at a stable interior configuration, liquidity as a network benefit at low concentration and a congestion cost at high concentration, the slow erosion of institutional fundamentals, gradual and reversible transitions, and frequency-dependent competition, here among U.S. Treasuries (the incumbent, asset 1), a diversified basket of the other safe sovereigns, and gold, with institutional disruption as the mutation rate μ₁.

Theoretical Foundations

Congestion turns the coordination game of the safe asset literature into a congestion game with an interior equilibrium that is unique at the calibration, and the replicator-mutator equation carries selection and mutation together: an asset whose fitness exceeds the population mean gains share, and holders leave an asset at its mutation rate.

\dot{x}_i \;=\; \sum_j Q_{ij}\, f_j(x)\, x_j \;-\; x_i\, \phi(x)

(2)

x_i is asset i's share and f_j(x) asset j's fitness, Equation (3); Q_ij, an entry of the mutation matrix, is the probability that a holder of asset j moves to asset i in a period (Q_jj = 1 − μ_j), and φ(x) is the population mean fitness.

The Model

Each asset's fitness is its intrinsic safety plus a concave network benefit less a convex congestion cost, so it rises with the asset's own share and then falls; gold carries the largest congestion cost and the basket the smallest.

f_i(x_i) \;=\; a_i + b_i\, x_i^{\gamma} - c_i\, x_i^{\eta}, \qquad 0<\gamma<1<\eta

(3)

Figure 1 of the paper, the fitness functions of the three assets.

Analytical Results

At the calibration the interior equilibrium is unique and lies on the falling branch of the incumbent's fitness, so the superiority σ, the incumbent's fitness relative to the population mean, rises as disruption erodes the share while the incumbent holds the majority, and the equilibrium moves continuously and reversibly toward the basket, the flattest profile; a structural rise in the incumbent's own congestion cost c₁ instead lowers σ through unity.

+0.890∂σ/∂μ₁, superiority against the mutation rate, at the calibration

−0.022∂σ/∂c₁, superiority against the incumbent's congestion cost

≈1.66c₁ at which the incumbent cedes the majority, against the calibration's 1.30

≈1.75c₁ at which σ falls through unity, the incumbent's share there ≈0.48

≈2.09c₁ at which the basket overtakes the incumbent

Explorer: the equilibrium against the mutation rate

The controls set the incumbent's mutation rate μ₁, intrinsic safety a₁, network benefit b₁, and congestion cost c₁, and the benefit exponent γ, which the three assets share; the panels draw the equilibrium shares and σ along μ₁, the mark at the setting of the controls; the regime follows the kernel's rule, Regime I where Treasuries hold at least half, II where they still lead the basket, and III where the basket leads.

mutation rate μ₁intrinsic safety a₁network benefit b₁congestion cost c₁benefit exponent γ

Equilibrium shares against the mutation rate

Superiority σ against the mutation rate

The paper's counterpart: Figures 2 and 3.

Numerical Analysis and Simulation

Calibrated, the equilibrium reproduces the composition of the safe asset universe; the incumbent cedes the majority and then the plurality to the basket as the mutation rate rises, and a single basin of attraction covers the simplex; below a mutation rate of about 0.0074 a second equilibrium, with gold rather than the basket second, coexists with the dominant one.

Table B.1 of the paper, the baseline calibration; at the calibration σ(x*) = 1.009.
Parameter	U.S. Treasuries	Basket	Gold
a, intrinsic safety	0.935	0.940	0.960
b, network benefit	1.048	0.778	0.663
c, congestion cost	1.300	0.980	1.600
γ, benefit exponent	0.60	0.60	0.60
η, congestion exponent	2.00	2.00	2.00
μ, mutation rate	0.02	0.04	0.03

Table B.1 of the paper, the mutation targets τ: the probability that a mutating holder of the column's asset moves to the row's asset.
Destination	from U.S. Treasuries	from the basket	from gold
to U.S. Treasuries	—	0.46	0.60
to the basket	0.55	—	0.40
to gold	0.45	0.54	—

0.620 · 0.270 · 0.110equilibrium shares of Treasuries, the basket, and gold

1.009σ, the incumbent's fitness relative to the mean, at the calibration

≈0.055μ₁ at which the incumbent cedes the majority, against the calibration's 0.02

≈0.085μ₁ at which the basket overtakes the incumbent

Explorer: trajectories and basins on the simplex

With all three congestion costs scaled by a common multiplier, 1 at the calibration, and the other settings of the first explorer, the first panel draws trajectories from nine starting distributions and the second the basins of attraction, a numbered mark at each attractor.

common multiplier of the three congestion costs

Trajectories on the simplex

Basins of attraction

Shade: the Treasury share at the attractor a starting distribution reaches, darker for a larger share; a white line divides two basins, and a numbered mark stands at each attractor.

below a quartera quarter to a halfa half to three quartersabove three quarters

The paper's counterpart: Figures 8 and 9.

The Self-Limiting Feedback Loop

On the dominant-equilibrium branch, disruption lowers the incumbent's share, the lower share relieves its congestion and raises σ, and the higher σ opposes further erosion.

Figure 10 of the paper, superiority rising as the incumbent's share falls.

The Absence of Critical Slowing Down

The dominant eigenvalue of the linearized system stays bounded away from zero along the equilibrium curve, closest to it near the calibration, so the model predicts no early-warning signature before the transition.

Figure 12 of the paper, the dominant eigenvalue along the transition.

Empirical Calibration

The observed composition of the global safe asset universe places the system at a mutation rate μ₁ of about 0.02, inside Regime I, the stable majority; the paper states the calibration's limits, among them the crisis-time flight to safety it does not capture, in Sections 8.5 and 8.6.

≈48.7trillion dollars, the safe asset universe at the end of 2025

≈62 · 27 · 11percent, Treasuries, the basket, and gold

≈59 to 72percent, the defensible range of the Treasury share

≈71 to ≈56percent, the dollar's share of disclosed reserves, 2001 to mid-2025

Explorer: the observed position on the equilibrium curve

The control sets the observed Treasury share; the panel places it on the equilibrium curve at the calibration and reads the implied mutation rate, σ, and the slope of σ against the mutation rate there.

observed Treasury share

The observed share on the equilibrium curve at the calibration

The paper's counterpart: Figure 15.

Implications for the United States

The paper reads the projected fiscal path, below, as a rising mutation rate, which erodes the share along a smooth, reversible curve; under elevated disruption it predicts a diversified successor, with gold second, and lowering the incumbent's congestion cost acts on the one channel the restoring force leaves unopposed.

≈100 to 156percent of GDP, federal debt held by the public, 2025 to 2055, the Congressional Budget Office's baseline

3.3 to ≈7percent of GDP, net interest, 2026 to 2056

Conclusion

Erosion of the incumbent's status by disruption is self-limiting, and the transition to a basket-dominant configuration is smooth and reversible; the model predicts no critical slowing down before that transition.