Shall we make sample size contractable?

Statistical protocols not only evaluate evidence but also shape the incentives of strategic agents who decide whether and how to experiment. Building on the principal--agent hypothesis-testing framework of Bates et al [2022], we study a setting in which the agent chooses the sample size of a costly experiment. This raises a natural design question: should sample size be contractable? We compare a common payoff rule that does not depend on the sample size with a sample-size-dependent contract. We show that when sample size is not contractable, null deterrence forces the contract to be calibrated to the cheapest feasible experiment; when it is contractable, the payoff rule can instead be calibrated to the cost of the chosen experiment. These constraints admit a natural e-value interpretation. We show that, for any fixed sample size, the optimal incentive-aligned contract is an all-or-nothing threshold rule. Allowing the contract to condition on sample size weakly increases statistical power at every fixed sample size and, after the agent optimizes its sample size, weakly increases the optimal expected payoff of non-null agents. We illustrate these effects in a Gaussian model.