When delegation works better with a gap

A manager’s local knowledge can make investment decisions better informed. But if the manager consistently favors larger projects than headquarters does, unrestricted authority also creates a predictable conflict. Restricting investment controls that conflict while potentially blocking useful responses to market conditions.

In Delegation without Priors, Wei He and six coauthors ask how to balance these costs when headquarters cannot assign reliable probabilities to the conditions it might encounter. Their answer combines a guaranteed default, a gap of unavailable actions and a higher range of choices whose boundaries are randomized.

This is a mathematical result in an arXiv preprint, not an experimental finding. The supplied record does not establish journal peer review.

Designing for an unknown distribution

The model has two decision-makers: a principal, such as headquarters, and an agent, such as a local manager. Only the agent observes the relevant circumstances, represented by a number between zero and one.

The principal’s ideal action equals that number. The agent’s ideal action is always higher by a fixed amount, called the bias. Both dislike deviations from their preferred actions according to squared distance, so large mistakes are disproportionately costly. Payments cannot be used to reconcile their interests.

Before circumstances are known, the principal commits to a rule determining which actions the agent may choose. She knows the possible range of circumstances but nothing about their probability distribution.

Instead of maximizing an average calculated from assumed probabilities, she minimizes worst-case regret: the largest expected squared error across possible circumstances, compared with what she could achieve if fully informed. For a randomized rule, the expectation averages over the rule’s lottery. It does not guarantee a small error in every individual draw.

Why fixed restrictions hit a limit

The authors first establish a stark benchmark. Among fixed sets of permitted actions, the best worst-case performance can be achieved either by allowing every action between zero and one or by fixing the action at one-half. Which is preferable depends on the bias.

More elaborate fixed menus cannot improve this benchmark. Any rule that responds to the agent’s information leaves the principal exposed somewhere to at least the full squared bias. Eliminating responsiveness avoids that exposure but sacrifices adaptation.

Randomization can hedge these competing risks. In one illustrative example, the bias is one-half. The best fixed rule has worst-case regret of one-quarter. A lottery that sometimes fixes the action at one-quarter and otherwise offers the interval from one-half to one reduces that figure to three-sixteenths.

That example demonstrates an improvement; it is not the paper’s fully optimal construction.

A default, a hurdle and room to act

The optimal construction always makes one common default action available. Actions immediately above it are excluded. Beyond that gap, the agent may receive an interval of discretion with randomly drawn lower and upper boundaries. Some draws offer only the default.

The gap matters because the agent would otherwise abandon the default for a modest upward adjustment even in the lowest state. Removing those nearby alternatives makes departure worthwhile only when circumstances justify crossing a larger hurdle from the agent’s perspective.

The interval’s floor controls when departure becomes attractive. Its ceiling controls how far the agent can go afterward. Randomizing these boundaries balances expected losses across states rather than leaving a particular permissive action permanently available.

How the authors establish optimality

The proof considers a broader class of mechanisms that assign action lotteries to reports about circumstances, while making truthful reporting optimal for the agent.

With squared-distance losses, the relevant features of each lottery are its mean and variance. Raising the mean toward the agent’s preferred action makes higher reports tempting. In the improving mechanisms studied here, responsiveness therefore requires additional variance, which is costly to both parties and helps discourage misreporting.

The authors construct an optimal balance and a least-favorable distribution that proves no admissible mechanism can do better. They then show that random delegation can reproduce the optimal means and variances. Thus, letting the agent choose from randomized menus loses nothing relative to the broader mechanism class.

What the result does—and does not—say

The same rule remains optimal when calibrated to a known upper bound on an otherwise uncertain upward bias. But when bias varies with circumstances, the structure can change: sufficiently small disagreement at low states can make a connected interval with a random ceiling optimal instead.

The conclusions depend on the specified preferences, information structure, commitment and absence of payments. The paper offers a possible explanation for investment rules that exclude intermediate project sizes, not evidence that firms actually use optimal lotteries or that such rules outperform alternatives in practice.