I develop a simplicity criterion for mechanism design, based on belief precision, the number of moments of a type distribution required to compute equilibrium strategies. Focusing on a class of multidimensional procurement auctions called first-score auctions, I show a sharp dichotomy: these mechanisms either have belief precision of two or require full distributional knowledge. This distinction is governed by whether the auction satisfies the fixed-order property, which ensures that the ex post allocation ranking of types is invariant across distributions. I then provide a simple, equilibrium-free test to determine whether a first-score auction has belief precision two. Finally, I microfound the concept by introducing a prior-free game with information acquisition: when agents learn from realistic signal structures about their opponents’ types, first-score auctions with low belief precision are exactly those that admit robust equilibria. The results offer a framework to compare mechanisms along a dimension of strategic simplicity that is closely tied to the informational demands on agents and provides guidance for the design of scoring rules in practice.

We consider the design of optimal auctions when buyers may have asymmetric and/or interdependent values. All that is known to the seller is each buyer’s ex ante expected value and an upper bound on the values. We describe a new class of mech anisms which we term compound proportional auctions: Each buyer submits a bid, which is a non-negative real number. The auction then clears in a series of rounds. Within each round, a proportional auction (Brooks and Du, 2021b) is run to allocate the remaining supply that is left over from previous rounds, among a set of active buyers. At the end of the round, those active buyers with the lowest expected value become inactive. Our main result is that compound proportional auctions maximize the revenue guarantee: minimum expected revenue across all information structures and Bayes Nash equilibria. 

This paper studies when strategic understanding acquired in one mechanism can be transferred to another. We introduce a framework in which agents’ knowledge is represented as a set of payoff comparisons they can make, and use it to formalize what it means to understand that a strategy profile is an equilibrium. We first apply this framework to mechanisms that are strategically equivalent—that is, share the same game form up to relabeling of actions—and show that agents’ understanding of equilibrium transfers across such mechanisms once the relevant action correspondences are explained to them. We then define strategic analogy, a weaker notion that allows not only actions but also types to be remapped, and show that understanding of equilibrium transfers across strategically analogous mechanisms once agents recognize how actions and types correspond. Applications include single-item auctions, scoring auctions, and nonlinear pricing with capacity constraints.