Complementary Inputs, Option Production, and Supply Incentives

Conceptual map

IV-12.01Options and preferences
IV-12.02Alternative enabling paths
IV-12.03Time and care constraints
IV-12.04Package experiments
IV-12.05Supply incentives

1. Opportunity production precedes choice

Complementary inputs determine which options are feasible. Preferences and equilibrium determine which feasible options are chosen and supplied.

Let input universe $$N$$ be finite. An option has a monotone feasibility function $f:2^{N}\to\{0,1\}$ : if $$f(S)=1$$ and $S\subseteq T$ , then $$f(T)=1$$ . Minimal enabling sets form an antichain $\mathcal{P}$ , and $$f(S)=1$$ exactly when some $P\in \mathcal{P}$ is contained in $$S$$ .

Proposition 1 · Möbius representation

Every set function has the unique expansion $f(S)=\sum _{T\subseteq S}m(T)$ , where

\[m(T)=\sum _{R\subseteq T}(-1)^{|T\setminus R|}f(R).\](1)

Proof. Substitute (1) into the expansion and interchange sums. For each $R\subseteq S$ , the remaining coefficient is $\sum _{R\subseteq T\subseteq S}(-1)^{|T\setminus R|}$ , which equals one when $$R=S$$ and zero otherwise. ∎

2. Alternative enabling paths create interaction coefficients

Let $N=\{a,b,c\}$ and let an option be feasible when either both $$a$$ and $$b$$ are present or $$c$$ is present. Minimal paths are $\{a,b\}$ and $\{c\}$ . Nonzero Möbius coefficients are $$m({a,b})=1$$ , $$m({c})=1$$ , and $$m({a,b,c})=-1$$ .

The negative top coefficient removes double counting when both routes are supplied. Monotonicity alone does not imply nonnegative Möbius coefficients.

3. A minimum-enablement program has a price certificate

Suppose inputs $$a,b,c$$ cost 2,3,6. Enabling the option through package { $$a,b$$ } costs five; route { $$c$$ } costs six. A package-level relaxation is

\[\begin{aligned}\text{Primal:}\quad & \min_{y\ge0}\ 5y_{ab}+6y_c && \text{subject to }y_{ab}+y_c\ge1, \\ \text{Dual:}\quad & \max_{\lambda\ge0}\ \lambda && \text{subject to }\lambda\le5,\ \lambda\le6.\end{aligned}\](2)

Both optima equal five. The dual value $\lambda=5$ certifies that no mixture of the declared packages can cost less. Individual-input sharing across several options requires a larger integer or linear program and can create an integrality gap.

4. Feasibility, take-up, and welfare remain distinct

Failure case · Produced option counted as chosen outcome

Supplying an enabling path makes an option available. Choice also depends on preferences and competing options; realized service depends on capacity and equilibrium supply.

An option with no enabling path is infeasible under every package. The empty set can enable an outside option and then gives a nonzero constant coefficient. Zero-cost inputs and overlapping packages create multiple optimal implementations. Incomplete experimental support identifies only combinations of set-function values and can leave Möbius terms partially identified.

5. Implementation, exercises, and sources

Enumerate observed packages, test monotonicity, recover Möbius coefficients only on supported subsets, list minimal paths, and solve both primal and dual cost problems. Report support gaps, integrality status, feasibility, duality gap, and which later model maps opportunity into choice.

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Exercises

Compute all eight feasibility values and Möbius coefficients in Section 2.
Verify both sides of (2).
Add a second option sharing input $$a$$ and compare package-level and individual-input formulations.

Partial solutions

1. Feasible sets are { $$c$$ }, { $$a,c$$ }, { $$b,c$$ }, { $$a,b$$ }, and { $$a,b,c$$ }; inclusion-exclusion gives the three stated coefficients. 2. Choose $y_{ab}=1$ and $\lambda=5$ . 3. Shared procurement can lower joint cost relative to summing option-specific packages.

Satoru Fujishige, Submodular Functions and Optimization.Set functions and subset-lattice structure.
Alexander Schrijver, Theory of Linear and Integer Programming.Covering programs and dual certificates.
Chae-Yeon Xon (2026), “Subject to Availability: Option Production under Complementary Inputs.”Related childcare and work-option application.

6. Audit checkpoint

State inputs, package support, monotonicity, outside option, minimal paths, Möbius convention, unsupported subsets, costs, shared inputs, integer or fractional variables, primal and dual residuals, feasibility, take-up model, capacity, and welfare target.

7. Scope boundary

The chapter covers finite complementary-input feasibility and elementary cost certificates. Preference, provider equilibrium, and general mechanism design require separate models.