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Econ-544 · economics · Economics

Interacting platform designs

LLM status Partially solved unverified

Statement

Original subject group: Digital, platform and data economics

Audit classification: Published research agenda. The source explicitly motivates combinatorial interaction experiments. Eight fully observed cells are already identifiable; meaningful research must concern unobserved combinations, growing dimension or behavioral adjustment.

Definitions and precise research target

Consider an online retail platform partitioned into geographically separate markets where cross-market spillovers are negligible. Let $r,b,d\in\{0,1\}$ indicate respectively a price-sensitive ranking, a quality badge and standardized product disclosure. The eight triples $z=(r,b,d)$ define feasible joint interventions; $z=0$ means the all-zero baseline. Under each $z$, sellers choose prices, quality and whether to participate after observing platform rules. Let $W(z)$ be equilibrium total surplus per potential buyer, defined as consumer utility net of search and payments plus seller and platform profits net of real operating and quality costs. Transfers among these parties cancel. Define $\Delta(z)=W(z)-W(0)$ and, for ranking and badge interaction with disclosure fixed at zero, $I_{rb}=W(1,1,0)-W(1,0,0)-W(0,1,0)+W(0,0,0)$. Analogous contrasts describe the remaining pair and triple interactions. Identify these quantities using market-level factorial assignment with sufficient time for seller adjustment, or a validated structural counterfactual that permits those adjustments. Record both immediate behavior and subsequent price, quality and entry responses. Determine which sparse experimental designs recover the signs of economically material interactions under declared smoothness or sparsity restrictions. This is a joint-intervention welfare-design problem; additive individual-feature effects are an assumption to be tested, not the desired equilibrium estimand.

Definitions, assumptions and mathematical qualifications. An equilibrium-selection rule, identical shock distribution and common eligible-buyer denominator define $W(z)$ for each of eight policies. The triple interaction is $I_{rbd}=W(1,1,1)-W(1,1,0)-W(1,0,1)-W(0,1,1)+W(1,0,0)+W(0,1,0)+W(0,0,1)-W(0,0,0)$. Market randomization requires strictly positive assignment probability to each tested cell. With all eight cells observed, these finite contrasts are ordinary estimands; they are not an unsolved combinatorial theorem. In the $K$-feature extension, state $W(z)=\sum_{S\subseteq\{1,\ldots,K\}}\beta_S\prod_{k\in S}z_k$, the maximum interaction order and an explicit sparsity restriction before choosing a reduced design.

Variants and assumption changes

  1. Editorial, complete factorial. Randomize all eight cells and estimate each pair and triple interaction with market-level uncertainty; no sparsity restriction is needed.
  2. Editorial, large feature set. Let $K$ grow, allow at most $s$ nonzero interactions of order at most $d$, and seek budget-limited designs that recover material interaction signs under stated signal and noise bounds.

References and source audit

Checked source. 2025 conference draft, Section 1.1, second research direction, PDF pp. 4--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, second research direction, PDF pp. 4--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-544-A1 · Partial result · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-544

Factorial welfare interactions and restricted sparse recovery

Status: Partial; the full catalogue target remains unresolved.

Abstract

A degree-bounded welfare function can be recovered by observing all feature subsets up to that degree, using an exact finite-difference inversion. Uniform cell error gives explicit sign-certification thresholds. Missing combinations remain unidentified without those restrictions. For a sparse extension, an injectivity condition and a restricted singular-value bound yield finite-design identification and deterministic recovery guarantees.

Welfare is a joint-policy response

Let $z\in\{0,1\}^K$ specify platform features, and let $W(z)$ be expected equilibrium total surplus per baseline eligible buyer after a fixed adjustment horizon. Consumer utility, seller profits, and platform profits include the same payments with opposite signs; real operating, quality, and search costs are counted once. An equilibrium selector and shock law are fixed across interventions.

Write $W(S)$ when precisely feature subset $S$ is enabled. Every function on the Boolean cube has a unique multilinear expansion \[ W(z)=\sum_{S\subseteq[K]}\beta_S\prod_{j\in S}z_j. \tag{1} \] For ranking, badge, and disclosure, the pair and triple coefficients are the catalogue's factorial contrasts relative to other features being off. They are not necessarily the interactions at every other feature setting.

Finite-difference inversion

Theorem: For every subset $S$, \[ \beta_S=\sum_{T\subseteq S}(-1)^{|S|-|T|}W(T). \tag{2} \] Consequently, if $\beta_S=0$ whenever $|S|>d$, observing all $W(T)$ with $|T|\le d$ identifies the entire welfare function using $\sum_{j=0}^d\binom Kj$ cells.
Proof: At an indicator vector of $T$, equation (1) gives $W(T)=\sum_{U\subseteq T}\beta_U$. Substituting into (2), the coefficient on $\beta_U$ is $\sum_{T:U\subseteq T\subseteq S}(-1)^{|S|-|T|}$. It is one when $U=S$ and zero otherwise by the binomial identity. This proves inversion and uniqueness. Under degree at most $d$, all remaining coefficients are known to be zero, so the recovered low-order coefficients determine every cell. ∎
For $K=3,d=2$, baseline, three single-feature, and three pair cells identify the model. The missing all-three value is predicted as \[ W(111)=W(110)+W(101)+W(011) -W(100)-W(010)-W(001)+W(000). \tag{3} \] The formula follows from setting the triple interaction to zero. It is a testable restriction if the eighth cell is later observed, not an implication of randomization in the other seven.

Without the degree restriction, changing only $W(111)$ preserves all seven observations while changing the triple interaction by the same amount. For example, seven zero-valued cells are compatible with $W(111)=1$ or $-1$ if the declared outcome range permits both. Equivalently, one may add $\theta z_1z_2z_3$ to any response function. Even a single nonzero interaction can be invisible to all lower-order cells.

Noise and material sign recovery

Suppose each estimated tested cell satisfies $|\widehat W(T)-W(T)|\le e$ simultaneously. Formula (2) gives \[ |\widehat\beta_S-\beta_S|\le2^{|S|}e. \tag{4} \] There are $2^{|S|}$ terms, each with coefficient of absolute value one, so the triangle inequality proves the bound. Thus a displayed coefficient with absolute value greater than $2^{|S|}e$ certifies its sign; alternatively, a true coefficient exceeding twice that threshold guarantees both a detected nonzero interval and the correct sign.

If each cell has $n$ independent market observations in $[-M,M]$, its mean error exceeds $e$ with probability at most $2\exp(-ne^2/(2M^2))$. For $J$ tested cells, a union bound gives simultaneous error at most $\alpha$ when \[ e=M\sqrt{2\log(2J/\alpha)/n}. \] Independence is between experimental markets; many buyers within one market do not create that many independent equilibrium realizations. Unequal counts can be handled with cell-specific error radii and their sum in (4).

For pair interactions the worst-case cell error multiplier is four; for a triple it is eight. With $e=0.01$ surplus units, an estimated pair contrast $0.08$ has a positive interval $[0.04,0.12]$, while an estimated triple contrast $0.05$ is not sign-certified. These are numerical checks, not estimated platform effects.

What sparsity additionally requires

Restrict possible coefficients to a dictionary $\mathcal S$ of subsets with $|S|\le d$. For tested assignments $z_1,\ldots,z_J$, define matrix $A_{jS}=\prod_{k\in S}z_{jk}$. Suppose $\beta$ has at most $s$ nonzero coordinates. Observations of expected welfare satisfy $w=A\beta$.

Proposition: All $s$-sparse coefficient vectors are uniquely identified from $w$ if and only if no nonzero vector in $\ker A$ has at most $2s$ nonzero coordinates.
Proof: Two distinct $s$-sparse vectors with the same image have a nonzero difference in the kernel supported on at most $2s$ coordinates. Conversely, split the support of any such kernel vector into two sets of size at most $s$. Assign its coordinates on the first to one vector and their negatives on the second to another. Their difference is the kernel vector, so they are distinct $s$-sparse vectors with the same image. ∎
This equivalence is for unrestricted coefficient magnitudes and signs. Extra nonnegativity or welfare-range constraints can change the necessary condition by excluding one of the constructed vectors.

For quantitative stability suppose every vector $v$ supported on at most $2s$ coordinates obeys \[ \|Av\|_2/\sqrt J\ge\gamma\|v\|_2,\qquad\gamma>0. \tag{5} \] Let the true coefficient vector satisfy $\|\beta\|_2\le R$, and choose an $s$-sparse vector in that ball minimizing squared fit to $\widehat w$. A minimizer exists because the feasible set is a finite union of compact balls. If $\|\widehat w-w\|_2/\sqrt J\le e$, comparison to the true vector gives fitted residual at most $e$, and therefore \[ \|A(\widehat\beta-\beta)\|_2/\sqrt J\le2e,\qquad \|\widehat\beta-\beta\|_2\le2e/\gamma. \tag{6} \] The second inequality uses (5), since the difference is $2s$-sparse. Every coordinate error is bounded by the same quantity. Large coefficients consequently have stable signs.

This is an exhaustive sparse-search guarantee, not a claim that the optimization is computationally easy or that a proposed platform design satisfies (5). Random design results would need their own probability and sample-size proofs. A design with duplicated columns cannot satisfy even the necessary injectivity condition.

Equilibrium and identification limitations

Feature interactions can emerge from seller repricing even when immediate buyer responses are additive. Measuring cells before quality investment or entry adjusts defines another $W$. Conversely, a statistical interaction is not proof of a specific strategic mechanism. A matched price, quality, and entry model must explain it.

The finite factorial and sparse linear-algebra results are standard tools supplied with complete proofs. They solve restricted recovery tasks, while the broad catalogue question still requires defensible degree/sparsity assumptions, welfare measurement, independent market assignment, and a common equilibrium horizon. No actual interaction sign or optimal platform bundle is asserted.

Completion Estimate

65%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt proves degree-bounded factorial inversion, noise thresholds for interaction signs, and exact sparse-design injectivity and stability conditions. Full platform welfare design still requires defensible interaction restrictions, constructive efficient sparse experiments, equilibrium adjustment and surplus measurement, and empirical price, quality, and entry responses.

References

  • [f] C. Farronato (2025). Digital Platforms as Information Aggregators: Implications for their Design and Regulation, section 1.1. https://conference.nber.org/confer/2025/DTs25/farronato.pdf. The primary draft motivates combinatorial design experiments; the degree and recovery restrictions here are stated assumptions.

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