The Inefficient Stake: Blame, Coverage, and the Welfare Case for a Cap

Abstract

When a monitor is made to bear more of the blame for what he fails to catch, he is expected to look harder. On the fold this programme has been studying he does the opposite. Raising his stake raises the share of blame that attaches to him faster than it lowers the chance there is blame to attach, so a larger exposure buys less coverage, not more — and past an upper edge it buys none at all, because the monitor’s engaged branch ceases to exist and he settles at the corner. The rate at which the stake destroys coverage is not merely negative; it is bounded away from zero by a quantity the companion papers on this fold’s geometry have already derived for a different purpose. This paper prices that fact. It puts a distribution of tasks over stakes, assigns monitors to them, and adds to the substrate the one thing it has never contained — a price for the harm a missed failure causes someone other than the monitor. Against that price a cap on the stake dominates : lower is better at every harm price the analysis covers, under every resolution of the model’s own ambiguity, and the reason it is not the fragile result a dominance over a policy grid usually is, is that it rests on the sign of a derivative rather than on the balance of two magnitudes. A higher price for harm makes coverage more valuable, and there is no cap that buys coverage. Two further things follow, and both concern what such a welfare statement can and cannot be. Because the substrate’s payoff folds, the welfare of any policy that leaves tasks in the bistable region is an interval rather than a number; the cap is therefore also an instrument of determinacy, and the policy that dominates turns out to buy determinacy for free — so the trade-off between doing well and knowing how well, which the neighbouring literature formalises, is empty here. And because the two prices the functional carries never enter a monitor’s problem, no observation of monitors can identify them; they are recoverable only from a planner’s own choices across policy regimes, and only when those regimes differ in the average stake their coverage falls on. The claim is a direction and the magnitudes are flagged. What the argument does not carry — deterrence outside coverage, participation, and the fairness of bearing what one causes — is named, because each is a way the case for a cap would stop.