Defaults can have a dramatic influence on consumer decisions. We identify an overlooked but practical alternative to defaults: requiring individuals to make an explicit choice for themselves. We study such "active decisions" in the context of 401(k) saving. We find that compelling new hires to make active decisions about 401(k) enrollment raises the initial fraction that enroll by 28 percentage points relative to a standard opt-in enrollment procedure, producing a savings distribution three months after hire that would take three years to achieve under standard enrollment. We also present a model of 401(k) enrollment and characterize the optimal enrollment regime. Active decisions are optimal when consumers have a strong propensity to procrastinate and savings preferences that are highly heterogeneous. Naive beliefs about future time-inconsistency strengthen the normative appeal of the active-decision enrollment regime. However, financial illiteracy favors default enrollment over active decision enrollment.

Defaults often have a large influence on consumer decisions. We identify an overlooked but practical alternative to defaults: requiring individuals to make an explicit choice for themselves. We study such “active decisions” in the context of 401(k) saving. We find that compelling new hires to make active decisions about 401(k) enrollment raises the initial fraction that enroll by 28 percentage points relative to a standard opt-in enrollment procedure, producing a savings distribution three months after hire that would take 30 months to achieve under standard enrollment. We also present a model of 401(k) enrollment and derive conditions under which the optimal enrollment regime is automatic enrollment (i.e., default enrollment), standard enrollment (i.e., default non-enrollment), or active decisions (i.e., no default and compulsory choice). Active decisions are optimal when consumers have a strong propensity to procrastinate and savings preferences are highly heterogeneous. Financial illiteracy, however, favors default enrollment over active decision enrollment.

We consider a moral hazard problem where the principal is uncertain as to what the agent can and cannot do: she knows some actions available to the agent, but other, unknown actions may also exist. The principal demands robustness, evaluating possible contracts by their worst-case performance, over unknown actions the agent might potentially take. The model assumes risk-neutrality and limited liability, and no other functional form assumptions. Very generally, the optimal contract is linear. The model thus offers a new explanation for linear contracts in practice. It also introduces a flexible modeling approach for moral hazard under nonquantifiable uncertainty. (JEL D81, D82, D86)

This supplement contains additional materials for the main paper. Section B contains proofs of auxiliary results not included in the main paper. Section C details how the generalized virtual values coincide with traditional (ironed) virtual values in the single-good monopoly problem. Theorems, equations, and sections in the main paper are referenced using the original numbering.APPENDIX B: ADDITIONAL PROOFS PROOF OF LEMMA 4.3 (ADAPTED FROM MADARÁSZ AND PRAT (2012)): It is easy to see that statement (b) of the lemma follows from (a) by integrating over each S k in the partition; so it suffices to prove (a).As in the proof of Lemma 4.2, we can take = max x θ u(x θ) − min x θ u(x θ), and then in any mechanism, any two types' payments can differ by at most . Also, put τ = min{ε/6 1}.By Lipschitz continuity, there exists δ such that whenever θ θ are two types with d(θ θ ) < δ, then |u(x θ) − u(x θ )| < τε/6 for all x. We show this δ has the desired property.Let (x t) be any given mechanism. Let t = min θ t(θ). Let S ⊆ (X) × R be the set of values (x(θ) τt + (1 − τ)t(θ)) for θ ∈ Θ and let S be its closure, which is compact (by the above observation on payments). Then define ( x t) by simply assigning to each type θ ∈ Θ the outcome in S that maximizes its payoff, Eu(x θ) − t. This exists by compactness. This ( x t) is a mechanism: IC is satisfied by definition and IR is satisfied since the payments have only been reduced relative to those in (x t), so each type θ has the option of getting allocation x(θ) for a payment of less than t(θ), which gives nonnegative payoff. Now let d(θ θ ) < δ. We know that the outcome chosen by θ in the new mechanism can be approximated arbitrarily closely by an element of S corresponding to some type θ ; in particular, there exists θ such that Eu x θ θ − Eu x θ θ < τε 6 and t θ − τt + (1 − τ)t θ < τε 6 (B.1) Now, we know from IC for the original mechanism thatand by the definition of the new mechanism ( x t) that Eu x θ θ − t θ ≥ Eu x(θ) θ − τt + (1 − τ)t(θ)Using (twice) the fact that d(θ θ ) < δ, the latter inequality turns into Eu x θ θ − t θ ≥ Eu x(θ) θ − τt + (1 − τ)t(θ) − τε 3

We construct a combinatorial model that is described by the cube recurrence, a nonlinear recurrence relation introduced by Propp, which generates families of Laurent polynomials indexed by points in Z 3 . In the process, we prove several conjectures of Propp and of Fomin and Zelevinsky, and we obtain a combinatorial interpretation for the terms of Gale-Robinson sequences. We also indicate how the model might be used to obtain some interesting results about perfect matchings of certain bipartite planar graphs.