Comparisons of Experiments in Moral Hazard Problems

I use a novel geometric approach to compare information in moral hazard problems. I study three nested geometric orders on information, namely the column space, the conic span, and the zonotope orders. The orders are defined by the inclusion of the column space, the conic span, and the zonotope of the matrices representing the experiments. For each order, I establish four equivalent characterizations of the orders, (i) inclusion of feasible state-dependent utility sets, (ii) matrix factorizations, (iii) posterior belief distributions, and (iv) improved incentives in certain moral hazard problems. The column space order characterizes the comparisons of implementability in all moral hazard problems. The conic span order characterizes the comparisons of costs in all moral hazard problems with a risk neutral agent and limited liability. The zonotope order characterizes the comparisons of costs in all moral hazard problems when the agent can have any utility exhibiting risk aversion.

💡 Research Summary

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The paper develops a geometric framework for comparing information structures—modeled as finite Blackwell experiments—in moral‑hazard environments. An experiment is represented by an N × M row‑stochastic matrix E, whose columns correspond to signal realizations. Three nested orders are introduced: (1) the column‑space order, defined by inclusion of the linear span of the columns; (2) the conic‑span order, defined by inclusion of the non‑negative cone generated by the columns; and (3) the zonotope order, defined by inclusion of the set of all convex combinations of the columns with coefficients bounded between 0 and 1.

For each order the author proves four equivalent characterizations: (i) inclusion of feasible state‑dependent utility sets that the principal can generate; (ii) existence of a matrix factorization E = E′G with appropriate sign or bound constraints on G; (iii) dominance of the induced posterior belief distributions (linear convex order for the zonotope, convex order for the conic span, and linear inclusion for the column space); and (iv) improvement of incentives or reduction of agency costs in specific classes of moral‑hazard problems.

The column‑space order captures implementability: if the column space of E contains that of E′, then for any target state distribution the principal can implement it with E whenever it is implementable with E′. This order ignores payment constraints and therefore does not speak to costs.

The conic‑span order adds the limited‑liability restriction (payments cannot be negative). Under risk‑neutrality of the agent, the principal’s cost‑minimization problem becomes a linear program whose feasible region is exactly the conic span. Consequently, inclusion of conic spans is necessary and sufficient for one experiment to yield weakly lower agency costs than another in all risk‑neutral, limited‑liability moral‑hazard settings.

The zonotope order further restricts coefficients to the unit interval, reflecting both lower and upper bounds on payments. This order is the appropriate comparator when the agent may be risk‑averse or when the principal faces an explicit budget constraint. The author shows that inclusion of zonotopes is equivalent to dominance in the linear convex order of the induced posterior distributions, which is weaker than Blackwell’s classic convex order but strong enough to guarantee cost dominance for any risk‑averse utility.

The three orders form a strict hierarchy:
 Column‑space ⊆ Conic‑span ⊆ Zonotope ⊆ Blackwell.
In binary state spaces the zonotope coincides with the Blackwell order; when experiments have full column rank (no redundant signals) the conic‑span, zonotope, and Blackwell orders all collapse to the same relation. Thus the classic Blackwell order remains appropriate only in non‑redundant settings.

The paper also provides easy‑to‑check matrix‑factorization criteria (e.g., E = E′G with G ≥ 0 for the conic span) and interprets the orders in terms of posterior belief distributions, linking the geometric approach to existing literature on information economics, likelihood‑ratio orders, and the informativeness principle.

Applications are illustrated through a principal‑agent production problem where the principal can only contract on noisy performance indicators. The analysis shows how the choice of indicator (experiment) determines whether the principal can implement a desired output distribution and at what cost, depending on the agent’s risk attitude and liability constraints.

Overall, the paper contributes a unified, geometry‑driven taxonomy of information comparisons that bridges implementability and cost considerations across a broad spectrum of moral‑hazard models, extending and refining the classic Blackwell ordering. Future work could explore dynamic settings, continuous state spaces, or multiple agents.