Source Clarifications

Source: arXiv v3 of Supply-Side Equilibria in Recommender Systems (arXiv:2206.13489v3). This memo records only material differences between the numbered source statements and the formalized statements. A restriction is not claimed to be necessary without a counterexample in the source domain.

Here $N$ is the number of users, $P$ the number of producers, and $\beta$ the cost exponent.

Clarified regularity and notation

For the two-user C1 results below, the paper’s definition of C1 in arxiv-update.tex:680-690 is read literally: each $H_i$ is a distribution function on $\mathbb R_{\geq 0}$, hence monotone, and $S\subseteq\mathbb R_{\geq 0}^N$. These are regularity conditions already present in the source setup, not additional economic restrictions. In Lemma 12, the free angular symbol in the displayed bracket is the polar parameter $\phi$ of the score pair $(r\cos\phi,r\cos(\theta^*-\phi))$; the curve’s bound variable is its first coordinate. This resolves only notation and does not change the conclusion or its domain.

One-dimensional law and genre convention

• Corollary 1 and Lemma 3 — Typo. Original: $F(q)=(q/N)^{\beta/(P-1)}$ on $[0,N^{1/\beta}]$. Formalized: $F(q)=(q^\beta/N)^{1/(P-1)}$, clipped to zero and one outside that interval, and the resulting equilibrium construction. Reason: the printed CDF generally does not equal one at $q=N^{1/\beta}$.

• Example 1 — Exact. In the one-user, one-dimensional Euclidean model, the displayed power-law measure is the unique symmetric mixed equilibrium.

• Theorem 1 and the paper’s genre definition — Exact with a normalization convention. Normalize each nonzero support point, $$G(\mu)=\{p/\lVert p\rVert:p\in\operatorname{supp}\mu,\ p\ne0\}.$$ Zero can belong to topological support when production approaches zero; the zero vector is left unnormalized.

• Lemma 4 — Positive-score conclusion exact; span wording corrected. Original: the equilibrium genre lies in the span of the user vectors, with strictly positive user scores stated parenthetically. A checked counterexample to the span wording uses the paper’s stated arbitrary-norm domain (model definition, arxiv-update.tex:268–274): with one user $(1,2)$, two producers, $\ell_1$ cost, exponent one, and singleton genre $(0,1)$, the constructed ray law is a source symmetric mixed Nash equilibrium and has user score $2$, but $(0,1)$ is not in the span of $(1,2)$. Against the uniform ray $(0,Q)$, $Q\sim U[0,1]$, a deviation $(a,b)\ge0$ has payoff $\min(1,a/2+b)-a-b\le0$; each ray action in support has payoff zero. Thus the example satisfies the actual Nash model, not merely a relaxed optimization condition. The lemma’s introductory sentence and proof (arxiv-update.tex:956–963) establish the positive-score conclusion directly from a single-nonzero-genre symmetric equilibrium, nonzero nonnegative users, and perturbation-cost continuity.

Single-genre optimization and thresholds

• Lemmas 1 and 5–7 — Source regularity made explicit. The score-ratio optimization uses a positive score vector $y^*$ that attains the coordinate-product maximum and satisfies $$\sum_i\frac{y_i}{y_i^*}\le N$$ for every feasible score vector $y$. Reason: positivity makes every ratio defined, and attainment supplies the optimizer used in the supporting inequality.

• Lemma 8 — Exact with bounded-product regularity. The abstract display states that the sup-inf ratio equals $N$ for every nonempty $R\subset(0,\infty)^N$. The proof’s product-maximization argument needs a finite positive product supremum; the checked version uses an attained maximum. The paper’s bounded norm-ball score-set application supplies this regularity. The abstract phrase “every set” also includes $N=1$, $R=(0,\infty)$, where every inner infimum is zero. That unbounded example explains the omitted qualification in the abstract lemma; it does not refute the source market application. Attainment is sufficient and is not asserted necessary.

• Corollary 2 — Exact. Original: $\beta^*\ge1$. Formalized: the source product-condition definition of $\beta^*$ contains exponent one under the stated compact convex source-norm regularity, as well as yielding the explicit single-ray equilibrium at $\beta=1$.

• Corollary 3 — Exact with a dimension clarification. Original: $\beta^*\ge q$ for $\ell_q$ cost, with equality for standard basis users. Formalized: $q$ belongs to the source product-condition exponent set, so $\beta^*\ge q$, and the construction gives the stated singleton equilibrium for every $0<\beta\le q$. For the standard basis with at least two users, the opposite strict product gap above $q$ gives the exact equality $\beta^*=q$. The source should state that nontrivial dimension: with only one basis user, the powered score image remains an interval for every exponent and the threshold is unbounded rather than equal to finite $q$.

• Corollary 5 — Exact with regularity conditions. For the normalized finite cases, the formalization proves the source product-condition bounds $$\beta^*\le1\quad(Z=1),\qquad \beta^*\le\frac{\log N}{\log N-\log Z}\quad(1<Z<N).$$ The proof uses the source’s nonnegative users, score-one unit witnesses, aggregate bound, and compact unit sublevels to turn the strict convex-hull product improvement into a statement about the supremum. It also proves the consequent no-singleton-equilibrium exclusion above the finite interior threshold. The prose endpoint $Z=N$, described as infinity in the source, is intentionally outside the real-valued displayed quotient because its denominator is zero.

• Corollary 4 — Exact. Original: $\beta^*=2/(1-\cos\theta^*)$ for two equally sized nonzero user populations with $\ell_2^\beta$ cost. Formalized: the source’s own product-condition definition of $\beta^*$, including its supremum, is exactly the displayed finite phase threshold. The accompanying fixed-exponent equilibrium iff is stated with the zero-safe nonzero-support genre convention. Thus the threshold equality itself is exact; only the nonzero genre convention makes the normalization well-defined.

• Corollary 6 — Exact. The welfare conclusion uses the nonzero genre convention above; the zero vector is left unnormalized.

Two-user characterization, phase results, and profit

• Lemma 2 — Exact with an atomless score-law clarification. Original: a generic necessary-and-sufficient C1–C3 characterization (support maximization, marginal-CDF compatibility, and realization by nonnegative content) written with weak CDFs. Formalized: for the source’s continuous proper norm-power model, an actual symmetric Nash law supplies all C1 ingredients—null score fibres, realised-support preimages, and attained induced fibre costs—and therefore proves the source weak-CDF C1 direction. The canonical C2–C3 probability data are also derived from the source Nash law. For the converse, atomless score laws make the CDF formula equal the tie-aware winning probability, giving the checked equivalence. This is the reading used in the paper’s continuous score-law applications. Without it, the one-dimensional zero-Dirac law satisfies the displayed weak-CDF C1–C3 data but loses the tie-payoff information. With one user, two producers and linear cost, staying at zero pays $1/2$, while producing $0<\epsilon<1/2$ pays $1-\epsilon>1/2$. That example explains the qualification for atomic candidate laws; it does not challenge the atomless application.

• Theorem 2 and Propositions 9–10 — Regularity clarification. Original: the two-user phase statement is presented for arbitrary nonnegative, linearly independent equal-population user vectors. Formalized: the same arbitrary-user equal-population market, with its angle written explicitly as $$\frac{\langle u_1,u_2\rangle}{\lVert u_1\rVert_2\lVert u_2\rVert_2} = \cos\theta, \qquad 0<\theta\leq\pi/2,$$ and with $\beta_c=2/(1-\cos\theta)$. It proves both threshold implications under the paper’s score-law regularity and an explicit finite conditional radial-law representation for the upper branch. Reason: the angle and nonzero-user clauses unpack the source’s nonnegative linear independence condition; the representation makes the phrase “conditional distribution of $\lVert p\rVert$ along each genre” mathematically definite.

• Lemma 9 — Cost formula on equilibrium support. The displayed squared cost is $$\frac{z_1^2+z_2^2-2z_1z_2\cos\theta}{\sin^2\theta}.$$ It is the minimum cost for feasible scores in the canonical two-dimensional model and holds on equilibrium support in any dimension. It need not hold at every other feasible score pair: take nonnegative unit users $u_1=(3/5,4/5,0)$, $u_2=(3/5,0,4/5)$ and content $p=(0,1,0)$. The scores are $(4/5,0)$. Zero second score forces the first and third content coordinates to vanish, so $p$ is the only nonnegative content with those scores and has squared cost $1$. The displayed formula gives $25/34$. This is a counterexample to the unrestricted cost formula, not an equilibrium counterexample; the reduction below proves why Theorem 2’s equilibrium application remains valid.

• Lemma 10 — Smooth interior first-order conditions. Identify the cost derivatives with the derivatives of the maximum-score CDFs at nonzero interior equilibrium support points, using the source’s smoothness assumptions. Absolute continuity alone gives these identities almost everywhere, rather than selecting a pointwise density at every support point. Boundary optimality uses one-sided conditions. The canonical calculation is applied after Theorem 2’s dimension reduction.

• Lemma 12 — Slope inequality by support optimality. In canonical coordinates, the formalization proves the displayed inequality $$g'(z_1)\left(\frac{\beta-2}{\beta}\cos(\theta-2\phi)-\cos\theta\right)\le0, \qquad (z_1,g(z_1))=(r\cos\phi,r\cos(\theta-\phi)).$$ Comparing four nearby score pairs using equilibrium optimality (C1) determines the graph’s monotonicity from the sign of the cost’s mixed partial derivative. This proves the slope condition without the printed density-differentiation argument. The standalone lemma uses canonical coordinates; the reduction below supplies this setting for Theorem 2.

• Propositions 7–8 — Exact. Original: positive and zero expected producer profit. Formalized: the corrected source-facing endpoints prove those literal expected-profit integrals, with the population law and payoff integrability derived from source Nash. Proposition 7 retains its explicit normalized Euclidean/nonzero-user domain; Proposition 8 retains the singleton-genre and finite-dimensional norm regularity used in its source proof.

Theorem 2: reduction to the two-user plane

• Proof route: using the displayed induced-cost formula on arbitrary feasible score pairs → first reducing an equilibrium to the canonical two-dimensional model. The source uses Lemmas 9–12 in the proofs of Propositions 9–10 (Appendices D.1 and D.4). Its equilibrium condition C1 supplies the needed restriction to the cone between the two user directions.

The formalization derives that cone restriction and proves that the Gram cost formula holds on equilibrium support in any dimension. It then maps the original equilibrium to two dimensions with exactly the same two user scores and production costs. Every candidate deviation in the canonical model has an original-space counterpart with at least as high scores and no greater cost, so the mapped law is an equilibrium as well. This establishes the reduction without assuming that rotating coordinates preserves nonnegative content.

The canonical calculations then prove the two branches of Theorem 2: below $\beta_c=2/(1-\cos\theta)$ every equilibrium is single-genre, and above it no finite-genre equilibrium satisfies the stated conditional-law regularity. The reduction preserves the relevant genre conclusions. Thus Lemma 9’s failure away from equilibrium does not restrict Theorem 2 to two dimensions or change its phase threshold.

Infinite-producer limit

• Definition 1 — Typo. Original: the product exponent is indexed by user although the weights are attached to genres. Formalized: genre weights, $$U(p)=\sum_i\prod_g F_g\!\left(\frac{u_i\cdot p}{u_i\cdot v_g}\right)^{w_g}-c(p), \qquad w_g\ge0,\quad\sum_gw_g=1,$$ on the positive-denominator domain. Reason: the corrected indexing matches the genre mixture described by Definition 1.

• Theorems 3–4 — Corrected source statements. Original: Theorem 3 states the two-genre construction for any two linearly independent nonnegative user vectors, while Theorem 4 defines $$\theta^*=\arccos\!\left( \frac{\langle u_1,u_2\rangle}{\lVert u_1\rVert_2\lVert u_2\rVert_2} \right)<0.$$ Formalized: canonical users with $0<\theta<\pi/2$ and $\beta>2/(1-\cos\theta)$. Reason: the displayed arccosine is nonnegative on the source domain, so the printed strict-negative hypothesis has no instance. The checked corrected result is the strictly acute construction; it does not claim that this angle restriction is necessary for every coherent infinite-producer formulation.

• Theorems 3–4 genre witnesses — Explicit construction. The corrected canonical endpoint identifies the two genres as $[\cos\phi,\sin\phi]$ and $[\cos(\theta-\phi),\sin(\theta-\phi)]$, where $\phi$ is the selected compact angular maximizer. It also proves that these witnesses are distinct: equality would make their first cosine scores equal and hence force $C_2=1$, contradicting the candidate condition $C_2<1$. This records the source’s two-genre construction in canonical coordinates; it is not an additional source correction.

• Theorem 4 genre weights — Typo. Original: $\alpha_1=\alpha_2=2$, despite Definition 1 requiring $\alpha_1+\alpha_2=1$. Formalized: $\alpha_1=\alpha_2=1/2$. Reason: the formalized weights define the stated equal two-genre probability mixture.

• Theorem 4 quality-cap constant — Typo. With user angle $\theta$, maximizing genre angle $\phi$, and $C_2=\cos(\theta-\phi)/\cos\phi$, the CDF constant changes from $$C_1=\frac{\sin\theta\cos\phi}{\sin(\theta-\phi)} \quad\longrightarrow\quad C_1=\frac{\sin\theta}{\cos\phi\sin(\theta-\phi)}=1+C_2^\beta.$$ Reason: at the upper support quality $a=C_1^{1/\beta}$, a genre has total two-user gross payoff $1+C_2^\beta$ and pays cost $a^\beta=C_1$. Equality gives the zero payoff required by support approaching zero. The paper’s first-order identity for $\phi$ gives the displayed equality to $1+C_2^\beta$.