The restoring force
The incumbent's equilibrium share lies on the falling, congestion-dominated branch of its fitness profile, so a loss of share relieves congestion and raises its fitness relative to the population mean. Above a low disruption rate the equilibrium is unique, moves smoothly and reversibly with institutional disruption, and carries no tipping point and no critical slowing down. The force opposes disruption; it does not oppose a structural rise in the incumbent's own congestion cost.
Survival of the flattest
Displaced demand flows to the asset that can absorb reliance with the smallest loss of its safety. The diversified basket, whose reliance is spread across decorrelated markets, has the flattest fitness profile and takes the plurality as disruption rises; gold, the most congested asset, takes the second share.
The line of sight
the section belongs to the restoring forcethe section belongs to survival of the flattestthe theme passes through
1
Introduction
Restoring forceSurvival of the flattest- Establishes
- Safe asset status carries five features that the literature identifies and no one framework has integrated: several safe assets coexisting at a stable interior configuration, liquidity that is 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 among assets. Carried in one dynamical system, they make the erosion of the incumbent's status self-limiting and its successor the diversified basket.
- Leads to
- What do the existing accounts of safe asset status hold, and where does this model depart from them?
2
Theoretical Foundations
Restoring force- Establishes
- A safe asset is debt about which no one profits from producing information, and Treasuries earn a convenience yield for it. In the account of Brunnermeier, Merkel, and Sannikov the status is a rational bubble that fails when belief is revised; in the coordination game of He, Krishnamurthy, and Milbradt the payoff to holding an asset rises with the number of co-holders, which yields multiple equilibria and a tipping structure. A congestion cost turns the coordination game into a congestion game with one interior equilibrium and locates failure in use, as in March 2020, rather than in belief. The replicator-mutator equation carries selection and mutation together; the model keeps its scaffolding and replaces Eigen's error catastrophe with a stable interior equilibrium.
- Exhibits
- Equation 2, the replicator-mutator equation
- Leads to
- How is the congestion cost written into the fitness of each asset?
3
The Model
Restoring forceSurvival of the flattest- Establishes
- Three assets: U.S. Treasuries, the incumbent; a diversified basket of the other safe sovereigns, the portfolio reserve managers already hold rather than an engineered security; and gold. Each asset's fitness is its intrinsic safety plus a concave network benefit less a convex congestion cost, so it rises in own share and then falls. The congestion coefficient is an inverse measure of absorption capacity and orders the assets: gold the most congested, the basket the least, the incumbent between. Holders leave the incumbent at the mutation rate μ1, the frequency of institutional disruptions such as debt-ceiling confrontations, encroachments on central bank independence, and the weaponization of payments.
- Exhibits
- Equation 3, fitness as network benefit net of congestion · Figure 1, the fitness functions of the three assets
- Leads to
- What equilibrium does the system reach, and how does it answer disruption?
4
Analytical Results
Restoring forceSurvival of the flattest- Establishes
- At the calibration an interior equilibrium exists at which Treasuries hold the majority, their fitness exceeds the population mean, and congestion has bid their marginal fitness just below the basket's; it lies on the falling branch of the incumbent's fitness, and that is the source of its stability (Proposition 1). Along it the superiority σ rises as disruption erodes the share, the restoring force of the Minority Principle (Proposition 2). Above a disruption rate μ1 of about 0.0074, below the calibration, the equilibrium is unique and moves continuously and reversibly with the disruption rate, with no error threshold, no tipping point, and no hysteresis (Proposition 3). As disruption rises the basket, the flattest profile, overtakes the incumbent and gold, at higher rates, the second share (Proposition 4). The restoring force belongs to the disruption channel: a structural rise in the incumbent's congestion cost lowers σ and carries it through unity (Proposition 5).
+0.890∂σ/∂μ1, superiority against the disruption 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 a calibrated 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
- Exhibits
- Equation 4, self-limiting erosion · Figure 2, superiority against the mutation rate · Figure 3, equilibrium shares against the mutation rate · Figure 4, survival of the flattest · Figure 5, the two routes of erosion
- Leads to
- Where on the equilibrium curve does the calibrated system rest, and how does it move as disruption accelerates?
5
Numerical Analysis and Simulation
Restoring forceSurvival of the flattest- Establishes
- The equilibrium curve divides into three regimes, a stable majority (Regime I), the erosion of the majority (Regime II), and the diversified successor with the basket holding the plurality (Regime III), regions of one smooth curve rather than states separated by a bifurcation. Calibrated, the equilibrium reproduces the composition of the safe asset universe with σ just above unity. Under an accelerating disruption rate the incumbent cedes the majority and then the plurality to the basket, continuously and reversibly. A stronger intrinsic fitness or network benefit keeps the majority to a higher disruption rate, a larger congestion cost lowers the share, and no parameter introduces a discontinuity. A single basin of attraction covers the simplex at the calibration; at baseline turnover a second appears only once congestion weakens to about 0.29 of its calibrated relative strength, and at the calibrated congestion a second equilibrium, with gold rather than the basket second, coexists with the dominant one only below a disruption rate μ1 of about 0.0074.
0.620 · 0.270 · 0.110equilibrium shares of Treasuries, the basket, and gold
1.009σ at the calibrated equilibrium
0.055μ1 at which the incumbent cedes the majority, against a baseline 0.02
0.085μ1 at which the basket overtakes the incumbent
- Exhibits
- Figure 6, the transition under accelerating erosion · Figure 7, sensitivity of the Treasury share · Figure 8, phase portraits on the simplex · Figure 9, basins of attraction under weak and calibrated congestion
- Leads to
- Why does the feedback between share and relative fitness stabilize rather than compound?
6
The Self-Limiting Feedback Loop
Restoring force- Establishes
- Disruption reduces the incumbent's share, which relieves congestion, which raises its relative fitness, which opposes further erosion: the safe asset paradox stated as a stabilizing principle. In the Eigen model σ is fixed and in a network-effect model it falls with share; here it rises as share falls, so the dominant configuration is held in place by the congestion that caps it. Safe asset choice is a congestion game with a single attracting configuration, in which too many holders and too few are both self-correcting. The recovery follows the recession of disruption, not the reversal of a structural rise in congestion.
- Exhibits
- Figure 10, superiority rising as the incumbent's share falls · Figure 11, the congestion game and its interior equilibrium
- Leads to
- What does a restoring force that never vanishes imply for the statistics of the share?
7
The Absence of Critical Slowing Down
Restoring force- Establishes
- The dynamics have no bifurcation to approach: the slowest eigenvalue of the linearized system stays bounded away from zero, closest to it near the baseline and, though not monotone, no closer at any higher disruption rate. The early warnings of a regime shift, rising variance and autocorrelation in Treasury spreads, repo haircuts, or recovery times after political shocks, should not diverge as the share erodes, and their absence separates the model from a tipping-point account.
−0.055slowest eigenvalue at the baseline, μ1 of 0.02
−0.30slowest eigenvalue at μ1 of 0.30
- Exhibits
- Figure 12, the dominant eigenvalue along the transition
- Leads to
- Does the calibrated model reproduce the observed safe asset universe, and where does it place the United States?
8
Empirical Calibration
Restoring forceSurvival of the flattest- Establishes
- The global safe asset universe, marketable safe sovereign debt with monetary gold, is matched by the equilibrium shares, the convenience yield standing for σ and the Economic Policy Uncertainty index for μ1. The observed share places the system at a low disruption rate, inside the majority regime: the majority would be ceded only if that rate roughly trebled. A compressed or negative convenience yield is the signature of a congested interior equilibrium, not of erosion past a threshold, and the partial recoveries of the dollar's reserve share are the restoring force at work. The gradual transition Eichengreen described is the whole character of the process, not a prelude to a break. The calibration states its limits 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
0.02μ1 implied by the observed share
71 to 56percent, the dollar's share of disclosed reserves, 2001 to mid-2025
- Exhibits
- Figure 13, measures of the Treasury convenience yield · Figure 14, empirical proxies for the parameters · Figure 15, the observed position on the equilibrium curve · Equation 5, state-dependent fitness, an extension
- Leads to
- What follows for the United States?
9
Implications for the United States
Restoring forceSurvival of the flattest- Establishes
- The fiscal path the Congressional Budget Office projects raises the disruption rate and erodes the share along a smooth, reversible curve, not toward a cliff. The successor is a diversified instrument rather than another single sovereign, which would carry its own congestion cost, with gold second. Stronger fiscal and institutional fundamentals, a deep Treasury market, and fewer discrete disruptions raise the share; a more diverse holder base and wider intermediation capacity lower the congestion cost and act on the one channel of erosion the restoring force leaves unopposed. The British transition was a gradual migration over decades, and the model implies that no rising variance in British bond spreads should be found before it.
100 to 156percent of GDP, federal debt held by the public, 2025 to 2055
3.3 to 7percent of GDP, net interest, 2026 to 2056
- Leads to
- What, then, does the paper conclude?
10
Conclusion
Restoring forceSurvival of the flattest- Establishes
- The feedback between share, congestion, and relative fitness runs in the stabilizing direction. Erosion of the incumbent's status is self-limiting, the transition to a basket-dominant configuration is smooth and reversible, and no critical slowing down precedes it. The calibrated system rests at a congestion-stabilized majority, inside the range in which the restoring force operates. The absence of historical recoveries reflects disruptions that persist and a successor that congestion stabilizes in its turn, not an endogenous trap.
- Appendices
- A, Proofs of the Main Propositions · B, Baseline Parameter Values, Table B.1