The Compounding Seat: When Prudence Pays for Vigilance
Year: 2026
Type: preprint
Abstract: An auditing programme that models recurring internal-control failures as a no-collusion equilibrium has priced the monitor’s incentives on an additive substrate, where his continuation value enters as a level — and its immediate predecessor showed that a risk-averse monitor on that substrate faces a riskless corner : because his safe action is to abandon the post, and an abandoned post draws a failure with certainty and a fixed floor of blame, risk aversion has nothing to price at the corner and can only buy an earlier exit. Prudence, there, cannot pay for vigilance. This paper replaces the level with a compounding base — a career, a licence, a partnership stake that grows while it is held and is written down when the post it sits on fails — and shows that the sign can reverse : on a base the monitor’s own inattention can destroy, and only there, prudence pays for vigilance. The mechanism is that a monitor on a compounding base ranks outcomes by the time-average growth rate of his capital, not the ensemble average of its level, and an absorption identity — a constant-absolute-risk-aversion taste over the log-growth increment is exactly a constant-relative-risk-aversion growth criterion at a curvature shifted up by one, verified to machine precision — collapses the whole family of concave growth tastes into one, so the reversal is a property of compounding entering the objective and not of any functional form. The results form a ledger over two switches, seat destructibility and the criterion: wherever the seat is insulated the corner lottery is degenerate and prudence buys exit for every criterion (the programme’s sign is preserved); the reversal requires the conjunction of a destructible seat, a criterion that prices its compounding, and a catastrophe tail heavier than a critical weight. That critical tail is characterized on a mean-preserving family that holds the seat’s expected hazard fixed and varies only the shape of the loss, so the reversal is attributable to tail composition alone (the family coincides exactly at the risk-neutral pole); it has a closed form organized by a tail-shape function $\beta(L) = -\log(1-L)/L$ that recurs throughout, its existence is Jensen’s inequality itself, and the reversal region is widest at an interior hazard budget and vanishes at both extremes. The reversal is a first-order growth-rate effect — a volatility drag the compounding forces into the objective — cleanly separated from a second-order effective-curvature drift that a level preference on a large base can mimic but the single claim does not rest on; and as the tail approaches total loss the monitor attends at any survivable charge, the penalty paradox run in reverse, with the operative survivability ceiling switching from corner-survivability under bounded adjudication lotteries to an engaged-ruin bound under unbounded ones. Geometrically the riskless corner’s trap — the hysteresis by which vigilance once lost is hard to recover — dies on the destructible seat: sufficient prudence or tail weight converts abandonment from a first-order (hysteretic) to a second-order (continuous, reversible) transition, whose normal form on a noiseless court is a boundary cusp, distinct from the interior cusp, along a decreasing extinction boundary. But the cusp lives at skin-scale charges where adjudication risk contributes at first order to the corner’s curvature even as it perturbs the levels only at second order, so under any genuine court noise the cusp relocates below the operating range, corner abandonment turns continuous, and the surviving hysteresis is an attention crash between two engaged levels detached from the corner — whose own classification is left open. A robustness analysis partitions the results by their fragility to the court’s signal and finds the partition itself informative: a spine — the risk-neutral pole, the corner value, and the entire $\beta(L)$ drag infrastructure — is exactly signal-free; the headline signs survive every signal structure with only second-order shifts; and exactly two things break, the amplification’s endpoint and the trap’s fine geometry. The analysis is confined to the monitor’s decision problem, with the charge schedule exogenous and no equilibrium claim; the single-agent objects it delivers — the best-response correspondence, its regime geometry, and the comparative statics — are stated for a companion closure. The paper is orthogonal to its predecessor on both the substrate and the criterion axes, an orthogonality the absorption identity makes exact: it is a standalone extension, not an amendment, and it locates the programme’s “prudence cannot pay for vigilance” precisely as the statement for a level monitor on any substrate, or for any monitor on an insulated seat. Results are calibration-flagged throughout; the identities and the signs are established analytically, and every threshold, location, and normal-form coefficient is flagged as calibration-dependent and dual-routed against the computational machinery.
Author: HANAEDA, MIKIO
Language: English
Subfield: Accounting
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7172352647
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