Closing the Compounding Seat: Equilibrium Charge-Setting on a Destructible Seat

A monitor whose seat compounds and whose own inattention can destroy it grows more attentive as it grows more prudent: the classical penalty paradox inverts. That is a decision problem’s result — the charge is a parameter of the environment, and it leaves open the question a paradox about penalties must eventually answer, which is whether any penalty-setter would choose to impose such a charge on such a monitor. This paper closes the game. A charge-setter commits to a scalar charge and the monitor best-responds by the companion paper’s correspondence, imported unchanged; because that correspondence is genuinely multivalued on a band of charges, every equilibrium statement here names its selection. Existence is not a formality and is not assumed: attainment is generic on the destructible seat, where the band is extinct and the response continuous, and can fail on the insulated benchmark at one edge of the open band and only under a selection that abandons the engaged branch there — a dichotomy the paper characterizes rather than conventions away. The closure’s economics is then a deterrence story, because a charge-setter who values only vigilance has no reason to charge at all. Its central finding is that the seat decides which margin absorbs the monitor’s prudence. On an insulated seat the paradox survives the closure but migrates: equilibrium vigilance does not fall — the charge-setter’s own penalty does, and the deterrence it purchases with it. On a destructible seat the reversal survives and the charge-setter takes both margins. The two seats are the endpoints of one rotation, and one law crosses the whole construction unbroken — equilibrium vigilance never falls in the monitor’s prudence, for an exact reason the paper identifies and at a boundary it characterizes. Welfare inverts the moral. The monitor’s own tail, which aligns its behavior, is precisely what lets a charge-setter charge more before it disengages; the tail subsidizes the charge-setter, and a planner who counts the monitor’s welfare pays for the subsidy. In proportional terms the closure costs more on the destructible seat than on the insulated one, at every deterrence family and either court tested. And a planner who counts the monitor heavily enough does not moderate the penalty but abandons it, at a boundary fixed by the charge-setter’s own appetite for the first unit of charge and signed by the seat — earlier over the insulated monitor, later over the exposed one. The failures the programme studies require no one to collude. On the seats where the mechanism lives, they turn out to require no one to be less vigilant either