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Econ-446 · economics · Economics

Interference and equilibrium

LLM status Partially solved unverified

Statement

Original subject group: Econometrics, causal inference and measurement

Audit classification: Published research agenda. The publication explicitly motivates interference and equilibrium-adjustment research. The revised finite-population estimand distinguishes assignment distributions from full-policy endpoints.

Definitions and precise research target

Consider a finite population of product markets, each containing sellers and buyers whose prices, entry and purchase choices adjust after a platform policy. For market $m$, let $Z_m$ be the vector of seller treatment assignments, and let $Y_m(Z_m)$ be total monetary consumer and producer surplus after a prespecified adjustment horizon, with producer surplus measured net of real costs. Arbitrary within-market interference is allowed; markets are assumed independent conditional on recorded common shocks. The target is \[ \tau=E[Y_m(\mathbf1)-Y_m(\mathbf0)], \] where $\mathbf1$ and $\mathbf0$ assign the policy to every or no eligible seller. An experiment may randomize only intermediate treatment saturations, so it directly identifies outcomes for the tested assignment distributions rather than the two endpoints. Determine which additional full-saturation market experiments, exposure restrictions or equilibrium models permit informative identification or bounds for $\tau$. Specify any exposure mapping and restrictions on price adjustment before estimation, and distinguish immediate assignment effects from outcomes after entry and prices settle. Construct uncertainty statements that reflect market-level randomization and untested saturation extrapolation. The agenda explicitly recognizes interference and equilibrium adjustment; the particular surplus measure, horizon and proposed endpoint estimand here are an adaptation rather than a theorem already stated by the source.

Definitions, assumptions and mathematical qualifications. Let the fixed market cohort have weights $\nu_m\geq0$, $\sum_m\nu_m=1$, so $\tau=\sum_m\nu_m\{Y_m(\mathbf1)-Y_m(\mathbf0)\}$; superpopulation expectation is a separate target. Each potential outcome includes the complete market assignment and a fixed response horizon, with common monetary welfare units and equilibrium selection. An exposure mapping is a given function $g_m(Z_m)$ and imposes $Y_m(Z_m)=\widetilde Y_m(g_m(Z_m))$; it is a substantive restriction. Intermediate-saturation randomization identifies averages over its tested assignment law. Endpoint identification requires positive endpoint assignment probability or specified extrapolation assumptions. Conditional independence of market shocks does not remove within-market spillovers.

Variants and assumption changes

  1. Editorial, endpoint design. Randomize entire markets to all-treated or all-control, estimate the finite-cohort full-policy contrast with market-level randomization uncertainty.
  2. Editorial, untested endpoints. Test only partial saturations, impose a stated Lipschitz or monotonicity restriction on selected equilibrium welfare as a function of saturation, and bound the two endpoints; test the restriction where possible.

References and source audit

Checked source. 2025 conference draft, Section 1.1, third research direction, PDF p. 5. Full primary text retrieved. https://conference.nber.org/confer/2025/DTs25/farronato.pdf.
  1. Chiara Farronato. 2025. Digital Platforms as Information Aggregators: Implications for their Design and Regulation. 2025 conference draft, Section 1.1, third research direction, PDF p. 5. https://conference.nber.org/confer/2025/DTs25/farronato.pdf.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-446-A1 · Partial result · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-446

Sharp saturation bounds and rare endpoint assignments

Status: Partial; the full catalogue problem remains unresolved.

Abstract

A finite collection of saturation experiments and a Lipschitz restriction yield exact bounds on the full-treatment welfare contrast. A two-endpoint feasibility problem and piecewise interpolation establish sharpness. Separate randomization calculations distinguish absent endpoint support from exponentially rare endpoint support. The mathematical results do not validate the scalar exposure model or identify welfare after unobserved equilibrium adjustment.

The experiment and the target

Fix a finite market cohort, weights $\nu_m\ge0$ summing to one, and potential total surplus $Y_m(z)$ for every seller assignment vector. Surplus is measured after a fixed adjustment horizon in a common monetary numeraire, with real costs counted once. A selected equilibrium is part of each potential outcome. The catalogue contrast is \[ \tau=\sum_m\nu_m\{Y_m(\mathbf1)-Y_m(\mathbf0)\}. \tag{1} \] Individual seller randomization may change prices, entry, and transactions throughout a market. A sum of direct seller effects therefore need not equal (1).

Two different support cases matter. A design assigning exactly $k$ of $n$ sellers, with $0<k<n$, never observes an endpoint. A design treating each seller independently with probability $p\in(0,1)$ assigns the endpoints positive probabilities. In a large market those probabilities can be tiny, but they are not zero. The first case has an identification problem without extrapolation restrictions; the second can have a severe precision problem.

For a design with no endpoint support, unrestricted potential outcomes can be changed at $\mathbf0$ or $\mathbf1$ while leaving every supported outcome fixed. Thus the data law stays unchanged while (1) changes. This is an explicit nonidentification construction. Outcome bounds constrain the possible changes but do not convert unsupported endpoints into identified outcomes.

A complete Lipschitz characterization

Impose the substantive scalar exposure restriction $Y_m(z)=f_m(s(z))$, where $s(z)$ is the treated fraction. Let $f(s)=\sum_m\nu_mf_m(s)$. Suppose repeated experiments identify $v_k=f(s_k)$ at $0<s_1<\cdots<s_K<1$. This is a population-moment statement, not a claim that a single assignment identifies each market's response curve.

Assume $0\le f(s)\le M$ and $|f(s)-f(t)|\le L|s-t|$ for known finite $M,L$. Require the observed values to satisfy the corresponding pairwise inequalities. Define \[ \begin{aligned} A_-&=\max\{0,\max_k(v_k-Ls_k)\},& A_+&=\min\{M,\min_k(v_k+Ls_k)\},\\ B_-&=\max\{0,\max_k(v_k-L(1-s_k))\},& B_+&=\min\{M,\min_k(v_k+L(1-s_k))\}. \end{aligned} \tag{2} \]

Theorem: The sharp identified interval for $f(1)-f(0)$ is \[ [\,\max\{B_--A_+,-L\},\ \min\{B_+-A_-,L\}\,]. \tag{3} \] The response class is incompatible with the data if the implied endpoint feasibility set is empty.
Proof: Write $a=f(0)$ and $b=f(1)$. Necessarily $a\in[A_-,A_+]$, $b\in[B_-,B_+]$, and $|b-a|\le L$. Conversely, for any such pair, all point pairs among $(0,a),(s_1,v_1),\ldots,(s_K,v_K),(1,b)$ obey the Lipschitz inequalities. Interpolate linearly between successive points. Each segment has slope bounded in absolute value by $L$, so the triangle inequality across segments proves the global bound. Interpolation also remains in $[0,M]$.

The set of differences $b-a$ from the endpoint rectangle is $[B_--A_+,B_+-A_-]$: a proposed $d$ is achievable exactly when $[A_-,A_+]\cap[B_--d,B_+-d]$ is nonempty. Intersect this interval with $[-L,L]$. Every resulting difference is attained by the interpolant just constructed, proving sharpness. ∎

Take $s_1=1/4$, $s_2=3/4$, $v_1=2$, $v_2=3$, $L=4$, and $M=5$. Then $A=[1,3]$, $B=[2,4]$, and $\tau\in[-1,3]$. Rising welfare between tested saturations does not force the full-policy contrast to be positive. The curve may fall outside the tested interval.

Adding monotonicity in this example gives $A=[1,2]$ and $B=[3,4]$, hence $\tau\in[1,3]$. More generally, for nondecreasing interior values one additionally imposes $a\le v_1$ and $b\ge v_K$; monotone piecewise interpolation proves sharpness of the resulting endpoint problem. Monotonicity of welfare is economically substantial because congestion and entry can reverse benefits from greater treatment.

If the $v_k$ are estimated, one may optimize jointly over simultaneous confidence intervals and all Lipschitz inequalities. Whenever those intervals cover every true $v_k$, the optimized outer interval contains the true contrast by feasible-set inclusion. This supplies a coverage argument but not narrow bounds. Pointwise intervals at a nominal level do not automatically deliver the same simultaneous coverage.

An unbiased endpoint estimator without scalar exposure

Allow arbitrary interference again. Suppose market assignments are independent across markets and endpoint probabilities $p_{m1},p_{m0}$ are positive and known. Under consistency, \[ \widehat\tau=\sum_m\nu_m\left[ \frac{{\mathbf 1}\{Z_m=\mathbf1\}Y_m(Z_m)}{p_{m1}} -\frac{{\mathbf 1}\{Z_m=\mathbf0\}Y_m(Z_m)}{p_{m0}} \right] \tag{4} \] is unbiased for (1). To prove this, condition on the full potential-outcome table. On an endpoint assignment its observed outcome equals that endpoint potential. Each indicator divided by its assignment probability has expectation one. Taking expectations term by term gives (1), independently of all intermediate outcomes.

The endpoint events are mutually exclusive, so a single market's unweighted contribution has variance \[ \frac{Y_m(\mathbf1)^2}{p_{m1}}+ \frac{Y_m(\mathbf0)^2}{p_{m0}} -\{Y_m(\mathbf1)-Y_m(\mathbf0)\}^2. \tag{5} \] Independence across markets makes total variance the sum of (5) times $\nu_m^2$. The unobserved joint endpoint pair means its exact numerical value is generally unavailable from a single realization. It is a valid design-variance formula, not a feasible variance estimator without further work.

For $n$ sellers independently treated with probability $1/2$, both endpoint probabilities equal $2^{-n}$. If both endpoint potential outcomes are one, the market contribution has mean zero and variance $2^{n+1}$. This gives exponential variance growth despite positive support. Randomizing whole markets directly to all-treated or all-control gives endpoint probabilities $1/2$ and variance $4$ in the same example. Increasing the number of seller observations inside a market is therefore not interchangeable with adding independent endpoint-assigned markets.

One can also see the practical rarity directly: with $M_0$ independent equal-sized markets, the probability of observing no all-treated market is $(1-2^{-n})^{M_0}$. When $M_0$ is much smaller than $2^n$, an unbiased estimator often receives no treated-endpoint contribution. Unbiasedness alone is a poor summary of its usefulness.

What the proof does not identify

The scalar exposure mapping is not established by randomization. Treating a large seller can alter the market differently from treating a small seller even when the fractions match. Equilibrium switching can violate a proposed Lipschitz constant, and outcome horizons can change the response curve. A one-month curve does not automatically describe welfare after years of entry.

Total transactions or revenue are also not total surplus. Demand, real costs, and the ownership boundary must justify the outcome measure. Consumer benefits may include participants who leave the platform, and ignoring them changes the population.

The exact interpolation result and randomization calculation are standard techniques supplied in full. The broad catalogue question remains unresolved because the relevant welfare measurement, exposure restrictions, equilibrium selection, and useful number of market experiments have not been established for an actual platform.

Completion Estimate

70%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt proves sharp Lipschitz saturation endpoint bounds and exact endpoint randomization and variance formulas, separating absent support from rare support. Credible exposure mappings, useful market-level uncertainty, surplus measurement, and unobserved equilibrium adjustment still require application-specific identification.

References

  • [f] C. Farronato (2025). Digital Platforms as Information Aggregators: Implications for their Design and Regulation, conference draft, section 1.1. The primary source motivates interference and equilibrium adjustment; it does not supply the response restrictions imposed here. https://conference.nber.org/confer/2025/DTs25/farronato.pdf.

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