Set-Valued Demand and Coherent Counterfactuals

Conceptual map

V-11.01Printed set values
V-11.02Reference calculations
V-11.03The modularity condition
V-11.04Choice-kernel variation
V-11.05Coherent repairs

1. A bundle is a set-valued economic object

When consumption time or another shared resource is reoptimized for every bundle, the value function is defined on sets. Context-specific marginal values are not fixed item utilities.

Let $$N$$ be a finite item set and $W:2^{N}\to \mathbb{R}$ the optimized value of each available set. For two items with attention weights three and four under the quadratic allocation example, $W(\varnothing)=0$ , $$W({a})=3$$ , $$W({b})=4$$ , and $$W({a,b})=5$$ . The interaction is $$5-3-4+0=-2$$ .

Definition 1 · Modularity

A set function is modular when $W(S)=W(\varnothing)+\sum _{j\in S}w_{j}$ for fixed coefficients independent of the surrounding set.

2. Zero cross-differences characterize fixed item increments

Proposition 1 · Finite modularity criterion

A set function on a finite universe is modular if and only if every cross-difference $\Delta _{i}\Delta _{j}W(S)$ is zero for all distinct $$i,j$$ outside $$S$$ .

Proof. A modular sum has context-independent first differences, so all cross-differences vanish. Conversely, zero cross-differences make the marginal contribution of each item unchanged as other items are added. Starting from the empty set and adding items in any order then gives $W(S)=W(\varnothing)+\sum _{j\in S}[W({j})-W(\varnothing)]$ . ∎

The example fails because its only pair cross-difference is $$-2$$ . Marginal additions at the full bundle are $$W({a,b})-W({b})=1$$ and $$W({a,b})-W({a})=2$$ . Treating one and two as fixed singleton utilities predicts pair value three and empty value two when subtracted from the full anchor. Both disagree with the original set function.

3. Demand and welfare must use one coherent value map

With a zero-valued outside option and a standard logistic shock difference, purchase probability for set $$S$$ is Λ( $W(S)-p_{S}$ ). At zero prices, exact probabilities for { $$a$$ }, { $$b$$ }, and { $$a,b$$ } are approximately 0.952574, 0.982014, and 0.993307.

The modularized values one, two, and three give approximately 0.731059, 0.880797, and 0.952574. If demand uses one value function while bargaining or welfare uses another, the same counterfactual menu has inconsistent take-up and surplus. A coherent implementation calls a single set-value routine, or documents an approximation and propagates it through every downstream object.

4. Support and approximation boundaries are part of the audit

Failure case · Reference marginals transported to unseen bundles

Marginal values computed at the full bundle identify local deletions there. They determine every bundle only under modularity or another stated structural restriction.

An empty set needs a normalized value. Zero attention weights create items with no direct contribution but possible interactions under richer technologies. Time allocation at a boundary can make $$W$$ nondifferentiable in parameters. Unobserved bundles leave set values unidentified without shape restrictions. Menu-dependent shock distributions change the choice kernel even when deterministic set values agree. Capacity, prices, and bargaining alter availability and must use the same bundle identifiers. For large $$N$$ , evaluating all $2^{|N|}$ sets is infeasible and approximation error needs a domain-specific bound.

5. Implementation, exercises, and sources

Enumerate supported sets, recompute optimized values, and test cross-differences. Store set keys canonically. Compare any additive reconstruction with exact values on all available subsets, then pass the same value routine and shock law to demand, pricing, bargaining, and welfare. Report unsupported sets and approximation envelopes.

Download the volume verification script →

Exercises

Reproduce the interaction, reference marginals, and six probabilities above.
Prove that a two-item set function is modular exactly when its single cross-difference is zero.
Add a third item and list the pairwise cross-differences required by the proposition.

Partial solutions

1. Use Λ( $$z$$ )=1/(1+ $e^{-z}$ ) at exact values 3,4,5 and approximate values 1,2,3. 2. The condition rearranges to $W(\{a,b\})=W(\{a\})+W(\{b\})-W(\varnothing)$ . 3. Check each pair in every context that excludes the pair, including both empty and singleton contexts.

Satoru Fujishige, Submodular Functions and Optimization.Finite set functions and cross-differences.
Donald M. Topkis, Supermodularity and Complementarity.Lattice differences and comparative statics.
Kenneth Train, Discrete Choice Methods with Simulation.Choice kernels and menu probabilities.

6. Audit checkpoint

Universe, canonical set keys, empty-set normalization, supported bundles, optimized value routine, resource constraints, boundary allocations, first and cross-differences, modularity test, reference context, additive reconstruction error, shock law, menu dependence, price and availability maps, demand, bargaining and welfare consistency, capacity, unsupported sets, and computational scaling.

7. Scope boundary

The chapter covers finite set values and additive-reconstruction audits. General random-set demand, combinatorial auctions, and approximation algorithms require further theory.