Pith. sign in

REVIEW 2 major objections 6 minor 1 cited by

Competition and Collusion in Two-Sided Markets with an Outside Option

T0 review · 2 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read With small cross-side network effects, competitive platforms beat colluding ones for users on both sides, giving higher utility, higher participation, and lower prices.

desk verdict A carefully executed but local extension of Tan-Zhou with an outside option; the central collusion comparison is plausible, but the uniqueness claim overreaches and the epsilon regime is never sized. read the letter →

arxiv 2505.06109 v1 pith:W2Q4YKVA submitted 2025-05-09 econ.GN q-fin.EC

Cristian Chica , Yinglong Guo , Gilad Lerman This is my paper · ORCID
classification econ.GNq-fin.EC
keywords two-sidedmarketsplatformcompetitioncollusionoutsideoptionnetworkexternalitieslogitdemandmarketparticipationpricingformulas
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish how the outside option—users' ability to stay out of the market altogether—changes the economics of platform competition and collusion. It derives pricing formulas for $N$ symmetric platforms under logit (Gumbel) preferences and linear network externalities, expressed in terms of the normalized net deterministic utility $z_k$ that each side receives relative to the outside option. The central comparative claim is that when cross-side network externalities are sufficiently small, the competitive outcome dominates collusion on both sides: users get higher normalized net utility, market participation is higher, and prices are lower. The paper also establishes that ignoring the outside option can lead a model to overestimate or underestimate equilibrium prices depending on taste heterogeneity versus within-side externalities, and that raising the number of platforms always raises market participation while its effects on prices, consumer surplus, and profits depend on parameter regimes.

What carries the argument

The load-bearing object is the normalized net deterministic utility $z_k=(u_k-u_{0,k})/\beta_k$, the per-platform deterministic utility advantage over the outside option divided by taste dispersion. Under the logit assumption, market share on side $k$ is $\omega(z_k)=1/(e^{-z_k}+N)$, so all equilibrium quantities can be re-expressed through $z$. The argument runs through two first-order conditions written in utility space: $\beta z=(\Phi-H(z))\Omega(z)-u_0$ for competition and $\beta z=(\Phi-H^C(z))\Omega(z)-u_0$ for collusion; the pricing formulas $p=H(z)\Omega(z)$ and $p^C=H^C(z)\Omega(z)$ follow. The comparison of regimes is carried by the price-gap identity $p^C-p^*=\Phi(x^C-x^*)+\beta(z^*-z^C)$, which decomposes the collusion premium into a network-effect loss from reduced participation and a direct utility loss. Many proof steps reduce to checking that certain polynomial coefficients are positive or negative under condition (19) and its variants.

What would settle it

For a fixed parameter set satisfying condition (19)—say $N=2$, $\beta_b=\beta_s=1$, $\varphi_{bb}=\varphi_{ss}=-1$, $u_{0,b}=u_{0,s}=0$—numerically solve the two first-order conditions (13) and (20) along the ray $(\varphi_{bs},\varphi_{sb})=(t,t)$ increasing $t$ from $0$. The largest $t$ before $z_b^*\leq z_b^C$ or $p_b^*\geq p_b^C$ gives the effective radius of the ball; if that radius is smaller than empirically estimated cross-side effects for dating apps or ride-hailing, the paper's "small externalities" condition fails to cover the motivating markets.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Proposition 4.11: for any number of platforms $N\geq 2$ and within-side externalities satisfying condition (19) (a bound on within-side effects relative to taste heterogeneity), there exists an $\varepsilon>0$ such that for all sufficiently small cross-side externalities $(\varphi_{bs},\varphi_{sb})$ inside $B_\varepsilon(0)$, the symmetric competitive Nash equilibrium has $z_k^* > z_k^C$, $Nx_k^* > Nx_k^C$, and $p_k^* < p_k^C$ on both sides $k\in\{b,s\}$. In words, with small cross-side network effects, competing platforms deliver more utility to users, attract more users onto the market, and charge lower prices than a colluding cartel would. The paper further claims that the sign of the effect of a better outside option on equilibrium prices and consumer surplus is not fixed: it depends on whether user taste heterogeneity $\beta_k$ is large or small relative to the within-side externality $\varphi_{kk}$, with explicit threshold functions such as $g_{p,u}(N)$ and $f_{p,u}(N)$ separating the regions. It also claims that as $N\to\infty$, platforms charge the efficient price $\beta_k$ and market participation becomes complete, and that finite increases in competition always raise participation while prices, consumer surplus, and profits can go either way depending on the same heterogeneity-and-externality comparison.

Load-bearing premise

All of the paper's main conclusions are proved only for cross-side network effects close enough to zero, with no number given for "close enough," so a reader cannot tell whether real markets with measurable cross-side effects qualify.

Editorial extensions

If this is right

  • If Proposition 4.11 is right, a cartel of platforms hurts both sides of the market in all three measurable ways—higher price, lower participation, lower net utility—provided cross-side effects are small, so antitrust scrutiny of platform mergers does not need to choose a side to protect.
  • Any empirical or policy model that assumes full market coverage is misspecified even in direction: in high-heterogeneity regimes the no-outside-option price $p_{k,u}$ overstates the true price, while in low-heterogeneity, positive-within-side-externality regimes it understates it.
  • Competition policy that increases the number of platforms will always expand total participation when within-side externalities are not too strong, but it can raise prices and lower consumer surplus in homogeneous-taste markets with positive within-side effects.
  • In the perfect-competition limit, platforms earn a price equal to taste dispersion ($p_k=\beta_k$), full participation is restored, and a positive outside option leaves users with negative net utility; the model thus recovers standard Bertrand-like benchmarks as $N\to\infty$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the paper's epsilon is existential and unquantified, a natural next step is to compute the maximal radius of the ball $B_\varepsilon(0)$ for calibrated parameter values; if that radius is smaller than estimated cross-side effects for dating or ride-hailing markets, the policy conclusions would not be operational.
  • The price-gap decomposition suggests an empirical test: estimate market shares and user utilities before and after a platform merger, then check whether the observed price increase is roughly $\Phi(x^C-x^*)$ (participation-driven network loss) plus $\beta(z^*-z^C)$ (utility-driven loss); this would distinguish collusion from efficiency explanations.
  • The sign-clustering results for $\partial p_k^*/\partial N$ imply that entry can be anti-competitive in consumer terms when tastes are homogeneous; testing this would require measuring taste dispersion and within-side externalities separately, for example from app-level churn and same-side engagement data.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper studies a symmetric N-platform, two-sided market with single-homing users and an outside option. Users have Gumbel idiosyncratic preferences and linear network externalities, so market shares follow a multinomial logit model. The authors derive implicit pricing formulas for the competitive Nash equilibrium and the collusive optimum in terms of normalized net deterministic utilities z, prove existence and uniqueness of symmetric equilibria when cross-side externalities are sufficiently small, and establish comparative statics in the outside option utility and the number of platforms. The headline result is Proposition 4.11: under condition (19) and sufficiently small cross-side externalities, competition yields higher normalized net deterministic utility and higher market participation on both sides, and lower prices, than collusion. Several auxiliary results characterize when a better outside option or more platforms raise or lower prices, consumer surplus, and profits, and the paper applies these to dating-app markets in Section 6.

Significance. The Appendix A derivations are careful: the FOC systems (13) and (20) are properly derived from the logit demand structure via Lemmas A.1-A.4, and the limits as u0_k tends to -infinity recover the no-outside-option prices in (27), which is a useful consistency check with the existing literature. The paper also ships a Mathematica notebook for the polynomial sign checks underlying several propositions, which is a concrete reproducibility strength. If the local uniqueness issue identified below is resolved, the comparison of competition and collusion with an outside option is a worthwhile contribution to the two-sided-markets literature, and the comparative statics in Sections 4 and 5 provide falsifiable predictions expressed in terms of primitive parameters. The main limitations are that every central result is local in an unquantified neighborhood of zero cross-side externalities, and that the uniqueness proofs currently establish only a continuity branch rather than global uniqueness.

major comments (2)
  1. [Section 3, Proposition 3.3 (and Proposition 3.5)] The implicit function theorem is applied at (φ_bs, φ_sb) = 0 to the FOC system (91), yielding a local branch z(φ1) of solutions. This proves uniqueness of the branch that passes through the zero-cross-side solution, but it does not rule out additional solutions of (91) for a fixed small nonzero φ1; the same gap affects the collusive FOC (20) in Proposition 3.5. Consequently, the statements 'the unique symmetric CNE', 'the unique CE', and Proposition 4.11's comparisons 'in equilibrium' are currently proven only for the branch selected by continuity. The paper needs either a global uniqueness argument, such as uniform negative diagonal dominance of the Jacobian of (91) over all z in R^2 for small φ1, or a reformulation that restricts all comparative statements to that branch and shows that every equilibrium of the game lies on it.
  2. [Propositions 3.3, 3.5, 4.1, 4.4, 4.6, 4.7, 4.8, 4.11, 5.1-5.4] All of the paper's main results hold for (φ_bs, φ_sb) in an unspecified ball B_ε(0), and the proofs extend sign conditions from the exactly-zero cross-side case by continuity without providing any lower bound on ε. A reader therefore cannot determine whether a market with, say, cross-side externalities that are one-tenth of the within-side terms lies inside the regime. Since Section 6 applies the results to dating apps with presumably nonzero cross-side effects, this is a substantive limitation of the policy conclusions. Please either provide explicit bounds on ε in terms of the primitives, or restrict the policy discussion to the zero-cross-side limit and describe the finite-ε results as qualitative robustness statements.
minor comments (6)
  1. [Section 4, after Eq. (32)] The sentence following equation (32) contains a stray fragment, 'competition.', which should be completed or removed.
  2. [Throughout] The symbol ε denotes a different constant in every proposition; since no quantitative value is attached, please state explicitly that the radii are not uniform across propositions and that no lower bound is asserted.
  3. [Proof of Proposition 3.2] The claim that Proposition 3.1 plus the implicit function theorem gives a locally one-to-one price-to-share map should cite the determinant computation in (78) and the hypotheses of Lemma A.1; Proposition 3.1's contraction condition M_T M_φ < 1 is not imposed here.
  4. [Proposition 5.3, part (ii)] The bound z*_k < (1/5) ln 2 appears without economic interpretation; the authors should explain where this constant comes from and whether it is ever binding in the parameter regions discussed in Section 6.
  5. [Figure 7] The caption says the red region is drawn 'while excluding the condition involving z*' and then describes it as restricted; please clarify exactly which inequalities are plotted and which are omitted.
  6. [Section 6] The dating-app illustration is explicitly speculative; if it is retained, it should be clearly labeled as a qualitative illustration and not presented as empirical support for Proposition 4.11.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: pricing formulas and competition-vs-collusion ordering are derived from primitives, not from fitted inputs or self-citation chains.

full rationale

The paper's central derivation is self-contained. The pricing formulas (17) and (22) are obtained by differentiating the primitives: the user utility specification (1), the outside option (2), the Gumbel logit assumption (6), and the linear externality matrix (7). The first-order conditions (13) and (20) are intermediate equations, not assumed conclusions. Proposition 3.3 and Proposition 3.5 establish existence and uniqueness by showing monotonicity of the relevant maps at zero cross-side externalities and then invoking the implicit function theorem; Proposition 4.11 compares the competitive and collusive equilibria through the explicit difference M_k - M^C_k in equation (162), and derives the participation and price orderings from monotonicity of the share function Omega and the price decomposition (32). None of these steps reduces to its own input. The self-citations, namely Chica et al. (2021) and Chica et al. (2024), appear only as background remarks on tractability and as an illustrative discussion of AI pricing, and they do not carry the existence, uniqueness, or comparison results. The proof's reliance on a local implicit-function branch for nonzero cross-side externalities is a potential proof gap about global uniqueness, and the unquantified radius epsilon is a limitation, but these are correctness or scope concerns rather than circularity. The paper is not circular in any of the seven enumerated senses.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters appear; all quantities are exogenous model primitives. The outside option is a standard component of discrete choice models, not a new entity, and no new mechanisms are introduced.

assumptions (5)
  • domain assumption Idiosyncratic preferences are i.i.d. Gumbel with scale beta_k (Assumption I, equation (6)).
    Gives logit choice probabilities and is needed for the explicit form of T_k and the first-order conditions in utility space.
  • domain assumption Network benefits are linear with constant matrix Phi (Assumption II, equation (7)).
    Used to write the first-order conditions (13) and (20); a general phi_k is treated only for the existence of market shares.
  • domain assumption Each user single-homes and may choose one outside option; platforms have zero marginal costs.
    Fixes the participation margin and the objective function; multi-homing is listed as future work.
  • domain assumption Sufficiently small cross-side externalities, meaning (phi_bs, phi_sb) lies in an unspecified epsilon ball around zero.
    Required by all main propositions, and no quantitative epsilon is provided.
  • domain assumption Restriction to symmetric equilibria with a unique symmetric solution under condition (19).
    The model solves for symmetric CNE and CE; asymmetric deviations are considered only in the second-order condition proof.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Competition and Collusion in Two-Sided Markets with an Outside Option." pith.science (2026). https://pith.science/paper/W2Q4YKVA

@misc{pith2026250506109,
  author       = {Pith},
  title        = {Pith review of: Competition and Collusion in Two-Sided Markets with an Outside Option},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W2Q4YKVA}},
  note         = {Machine review of arXiv:2505.06109}
}
read the original abstract

We introduce pricing formulas for competition and collusion models of two-sided markets with an outside option. For the competition model, we find conditions under which prices and consumer surplus may increase or decrease if the outside option utility increases. Therefore, neglecting the outside option can lead to either overestimation or underestimation of these equilibrium outputs. Comparing collusion to competition, we find that in cases of small cross-side externalities, collusion results in decreased normalized net deterministic utilities, reduced market participation and increased price, on both sides of the market. Additionally, we observe that as the number of platforms increases in the competition model, market participation rises. Profits, however, decrease when the net normalized deterministic utility is sufficiently low but increase when it is high. Furthermore, we identify specific conditions that quantify the change of price and consumer surplus when the competition increases.

Figures

Figures reproduced from arXiv: 2505.06109 by the authors.

Figure 1
Figure 1. below shows the region described by (19) when N = 4 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Classification of the sign of z ∗ k based on (φkk,βk), k ∈ {b,s}, according to Proposition 4.1, where N = 4 and u 0 k = −1 (left) or u 0 k = 0.5 (right). The red and blue regions correspond to negative and positive z ∗ k , respectively. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 4
Figure 4. Classification of the sign of z C k based on (φkk,βk), k ∈ {b,s}, according to Proposition 4.8, where N = 4 and u 0 k = −1 (left) or u 0 k = 0.5 (right). The red and blue regions correspond to negative and positive z C k , respectively. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Learning to Charge More: A Theoretical Study of Collusion by Q-Learning Agents

    econ.GN 2025-05 reject novelty 7.0 of 10

    Q-learning firms converge to a supracompetitive price forever if the Q-function at the end of experimentation favors that price in the relevant states.

Reference graph

Works this paper leans on

41 extracted references · 40 canonical work pages · cited by 1 Pith paper

  1. [1]

    Anderson, S. P. and De Palma, A. (1992). The logit as a model of product differentiation.Oxford Economic Papers, 44(1):51–67. (Cited in page 6, 13, 18,

    work page 1992

  2. [2]

    Gilbert, E. M. (2019). Antitrust and commitment issues: Monopolization of the dating app indus- try.NYUL Rev., 94:862. (Cited in page

    work page 2019

  3. [3]

    Chica, C., Chuk, K., and Tamayo, J. A. (2021).Exclusive Dealing and Entry by Competing Two- sided Platforms. Harvard Business School. (Cited in page 3, 8, 10, 11, 13,

    work page 2021

  4. [4]

    Dewenter, R., Haucap, J., and Wenzel, T. (2011). Semi-collusion in media markets.International Review of Law and Economics, 31(2):92–98. (Cited in page 4,

    work page 2011

  5. [5]

    and Vickers, J

    Armstrong, M. and Vickers, J. (2001). Competitive price discrimination.The RAND Journal of Economics, pages 579–605. (Cited in page 2,

    work page 2001

  6. [6]

    Bardey, D., Cremer, H., and Lozachmeur, J.-M. (2014). Competition in two-sided markets with common network externalities.Review of Industrial Organization, 44:327–345. (Cited in page

    work page 2014

  7. [7]

    Berry, S. T. (1994). Estimating discrete-choice models of product differentiation.The RAND Journal of Economics, pages 242–262. (Cited in page

    work page 1994

  8. [8]

    Besanko, D., Gupta, S., and Jain, D. (1998). Logit demand estimation under competitive pricing behavior: An equilibrium framework.Management Science, 44(11-part-1):1533–1547. (Cited in page

    work page 1998

Show all 41 references
  1. [9]

    Bishop, R. L. (1960). Duopoly: Collusion or warfare?The American Economic Review, 50(5):933–961. (Cited in page

    1960

  2. [10]

    Armstrong, M. (1996). Multiproduct nonlinear pricing.Econometrica: Journal of the Econometric Society, pages 51–75. (Cited in page

    1996

  3. [11]

    Armstrong, M. (2006). Competition in two-sided markets.The RAND Journal of Economics, 37(3):668–691. (Cited in page 3, 4,

    2006

  4. [12]

    and Jullien, B

    Caillaud, B. and Jullien, B. (2003). Chicken & egg: Competition among intermediation service providers.The RAND journal of Economics, pages 309–328. (Cited in page 2,

    2003

  5. [13]

    Teh, T.-H., Liu, C., Wright, J., and Zhou, J. (2023). Multihoming and oligopolistic platform competition.American Economic Journal: Microeconomics, 15(4):68–113. (Cited in page 4,

    2023

  6. [14]

    Cohen, M. C. and Zhang, R. (2022). Competition and coopetition for two-sided platforms.Pro- duction and Operations Management, 31(5):1997–2014. (Cited in page 4,

    2022

  7. [16]

    62 Correia-da Silva, J., Jullien, B., Lefouili, Y ., and Pinho, J. (2019). Horizontal mergers between multisided platforms: Insights from cournot competition.Journal of Economics & Management Strategy, 28(1):109–124. (Cited in page

    2019

  8. [17]

    Brander, J. A. and Spencer, B. J. (1985). Tacit collusion, free entry and welfare.The Journal of Industrial Economics, pages 277–294. (Cited in page

    1985

  9. [18]

    and Gortmaker, J

    Conlon, C. and Gortmaker, J. (2020). Best practices for differentiated products demand estimation with pyblp.The RAND Journal of Economics, 51(4):1108–1161. (Cited in page

    2020

  10. [19]

    Gal-Or, E. (2020). Market segmentation on dating platforms.International Journal of Industrial Organization, 68:102558. (Cited in page

    2020

  11. [20]

    Jeitschko, T. D. and Tremblay, M. J. (2020). Platform competition with endogenous homing. International Economic Review, 61(3):1281–1305. (Cited in page

    2020

  12. [21]

    and Wang, X

    Hsu, J. and Wang, X. H. (2005). On welfare under cournot and bertrand competition in differenti- ated oligopolies.Review of Industrial Organization, 27:185–191. (Cited in page

    2005

  13. [22]

    P., De Palma, A., and Thisse, J.-F

    Anderson, S. P., De Palma, A., and Thisse, J.-F. (1992).Discrete choice theory of product differ- entiation. MIT press. (Cited in page

    1992

  14. [23]

    Halaburda, H., Jan Piskorski, M., and Yıldırım, P. (2018). Competing by restricting choice: The case of matching platforms.Management Science, 64(8):3574–3594. (Cited in page

    2018

  15. [24]

    Chica, C., Guo, Y ., and Lerman, G. (2024). Artificial intelligence and algorithmic price collusion in two-sided markets.arXiv preprint arXiv:2407.04088. (Cited in page 23,

    2024 arXiv

  16. [25]

    and Pavan, A

    Jullien, B. and Pavan, A. (2019). Information management and pricing in platform markets.The Review of Economic Studies, 86(4):1666–1703. (Cited in page

    2019

  17. [26]

    and Pinho, J

    Lefouili, Y . and Pinho, J. (2020). Collusion between two-sided platforms.International Journal of Industrial Organization, 72:102656. (Cited in page

    2020

  18. [27]

    and Sato, S

    Peitz, M. and Sato, S. (2023). Asymmetric platform oligopoly. Technical report, University of Bonn and University of Mannheim, Germany. (Cited in page

    2023

  19. [28]

    Perloff, J. M. and Salop, S. C. (1985). Equilibrium with product differentiation.The Review of Economic Studies, 52(1):107–120. (Cited in page

    1985

  20. [29]

    and Chon ´e, P

    Rochet, J.-C. and Chon ´e, P. (1998). Ironing, sweeping, and multidimensional screening.Econo- metrica, pages 783–826. (Cited in page

    1998

  21. [30]

    and Tirole, J

    Rochet, J.-C. and Tirole, J. (2003). Platform competition in two-sided markets.Journal of the European Economic Association, 1(4):990–1029. (Cited in page 1,

    2003

  22. [31]

    and Tirole, J

    Rochet, J.-C. and Tirole, J. (2006). Two-sided markets: a progress report.The RAND Journal of Economics, 37(3):645–667. (Cited in page

    2006

  23. [32]

    Ryan, C. (2017). Computer and internet use in the United States: 2016.American Community Survey Reports, ACS-39, U.S. Census Bureau, Washington, DC.(Cited in page

    2017

  24. [33]

    The public and online dating in

    SSRS (2024). The public and online dating in

    2024

  25. [34]

    and Zhou, J

    Tan, G. and Zhou, J. (2021). The effects of competition and entry in multi-sided markets.The Review of Economic Studies, 88(2):1002–1030. (Cited in page 2, 3, 4, 5, 8, 10, 11, 15, 18, 20,

    2021

  26. [35]

    and Wright, J

    63 Tan, H. and Wright, J. (2021). Pricing distortions in multi-sided platforms.International Journal of Industrial Organization, 79:102732. (Cited in page

    2021

  27. [36]

    (1988).The theory of industrial organization

    Tirole, J. (1988).The theory of industrial organization. MIT Press. (Cited in page

    1988

  28. [37]

    J., Adachi, T., and Sato, S

    Tremblay, M. J., Adachi, T., and Sato, S. (2023). Cournot platform competition with mixed- homing.International Journal of Industrial Organization, 91:103002. (Cited in page

    2023

  29. [38]

    Varian, H. R. (1989). Price discrimination.Handbook of industrial organization, 1:597–654. (Cited in page 11,

    1989

  30. [39]

    R., Danskin, G., and Ellsworth, G

    Wells, J. R., Danskin, G., and Ellsworth, G. (2015; revised in 2021). Amazon.com, 2021.Harvard Business School Case Study 716-402. (Cited in page

    2015

  31. [40]

    J., Froeb, L

    Werden, G. J., Froeb, L. M., and Tardiff, T. J. (1996). The use of the logit model in applied industrial organization.International Journal of the Economics of Business, 3(1):83–105. (Cited in page

    1996

  32. [41]

    Weyl, E. G. (2010). A price theory of multi-sided platforms.American Economic Review, 100(4):1642–1672. (Cited in page

    2010

  33. [42]

    and Weyl, E

    White, A. and Weyl, E. G. (2016). Insulated platform competition.Available at SSRN 1694317. (Cited in page 3,

    2016

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.