Who Is Raising Whom?

Part 5 of Raising Machines, a six-part series that started with two quotes about raising AI. Earlier parts: Gandhi Won’t Take the Pill, The β Parameter, Cats Adopted Us First and The Wealth of the Stimulus. Literature cutoff: 10 October 2026.

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The Blurb Names Its Children

The De Kai line that opened this series is the publisher’s description of his book Raising AI, not a passage from the book, which I have not read [[1]]. The longer passage is specific about which children it means:

“AIs are not gods or slaves, but our children. All day long, your YouTube AI, your Reddit AI, your Instagram AI, and a hundred others adoringly watch and learn to imitate your behavior. They’re attention-seeking children who want your approval. […] Our artificial children began adopting us 10–20 years ago; now these massively powerful influencers are tweens. How’s your parenting?”

The children are recommender systems. “Tweens” is a figure of speech, and the “10–20 years” comes without a source, but the structural claim inside the figure is sharp. A child who imitates you is shaped by you. A feed that imitates you also decides what you see next, and what you do with it becomes training data. YouTube’s engineers said so plainly in 2016: “we use the implicit feedback […] of watches to train the model, where a user completing a video is a positive example” [[2]]. That is a loop, and in a loop the title’s question has no fixed answer. Does the loop actually narrow people? Yes as a mechanism, yes in simulations, much less clearly in the field.

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A Prediction That Changes Its Own Target

Machine learning has a name for this. In 2020, Perdomo, Zrnic, Mendler-Dünner and Hardt called it performative prediction: “When predictions support decisions they may influence the outcome they aim to predict. We call such predictions performative; the prediction influences the target.” [[3]] Their examples include “recommendations shape preferences and thus consumption”. A model with parameters $\theta_{\mathrm{PS}}$ is performatively stable if

$$\theta_{\mathrm{PS}} = \arg\min_{\theta}\; \mathbb{E}_{Z \sim \mathcal{D}(\theta_{\mathrm{PS}})}\, \ell(Z;\theta),$$

where $Z$ is one observed data point (a user, an item, a click), $\ell(Z;\theta)$ is the loss of a model with parameters $\theta$ on that point, and $\mathcal{D}(\theta)$ is the distribution of data the world produces once a model with parameters $\theta$ is deployed. The parameter $\theta_{\mathrm{PS}}$ appears on both sides: the model is the best fit to a world it has itself made, one of the “fixed points of risk minimization”.

Retraining on the data the current model produced, which the paper calls repeated risk minimisation,

$$\theta_{t+1} = \arg\min_{\theta}\; \mathbb{E}_{Z \sim \mathcal{D}(\theta_t)}\, \ell(Z;\theta),$$

is how a deployed system looks for that point. Call the world $s$-sensitive if moving the parameters by $\delta$ moves $\mathcal{D}$ by at most $s\,\delta$ in earth mover’s distance (the paper writes $\varepsilon$; I need that letter below). By the paper’s Theorem 3.5, if the loss is $\gamma$-strongly convex and its gradient $\beta$-Lipschitz in $\theta$ and in the data point $z$, retraining converges to a unique stable point whenever $s < \gamma/\beta$; with $s \ge \gamma/\beta$ it can fail to converge at all (Proposition 3.6). I know of no estimate of $s$ for a deployed feed.

Stable is not optimal. The paper’s Theorem 4.3 bounds the gap: if the loss is $\gamma$-strongly convex and $L_z$-Lipschitz in $z$, and the world is $s$-sensitive, then every stable point $\theta_{\mathrm{PS}}$ and every optimum $\theta_{\mathrm{PO}}$ satisfy $\lVert\theta_{\mathrm{PO}} - \theta_{\mathrm{PS}}\rVert_2 \le 2L_z s/\gamma$, which is small only when the world barely reacts. Optimal means optimal for the deployer’s loss. Neither definition measures whether the fixed point serves the people it acts on.

In 2018, Chaney, Stewart and Engelhardt asked what retraining on such data does inside a recommender [[4]]: “Using simulations, we demonstrate how using data confounded in this way homogenizes user behavior without increasing utility.” The users and items are synthetic and the user model is assumed, so this shows that the mechanism can work, not that it operates on any platform.

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Averaging Your Way to Agreement

The classic small model of opinion dynamics is DeGroot’s from 1974 [[5]], which I take as Hegselmann and Krause restate it [[6]]. Each person $i$ holds an opinion $x_i$, and each round everyone replaces it with a weighted average of everyone’s:

$$x(t+1) = A\,x(t), \qquad\text{so}\qquad x(t) = A^t\,x(0),$$

where $x(t)$ is the vector of opinions after round $t$, and $A$ is a stochastic matrix: entry $a_{ij} \ge 0$ is the weight person $i$ gives person $j$, and each row sums to one.

Let person 1 give weight $a$ to person 2, and person 2 give weight $b$ to person 1, so that $A = \begin{pmatrix} 1-a & a \\ b & 1-b \end{pmatrix}$. Subtracting one update from the other gives

$$x_1(t+1) - x_2(t+1) = (1-a-b)\,\bigl(x_1(t) - x_2(t)\bigr),$$

so for $0 < a + b < 1$ the disagreement shrinks by the factor $1-a-b$ each round, while $b\,x_1 + a\,x_2$ never changes. With $a = 0.2$, $b = 0.1$, $x_1(0) = 0$ and $x_2(0) = 1$, the gap falls to $0.7^{10} \approx 0.028$ of its starting size in ten rounds, and both settle at

$$x^{*} = \frac{b\,x_1(0) + a\,x_2(0)}{a+b} = \frac{0.2}{0.3} = \tfrac{2}{3}.$$

Person 2, who listens less, gets two-thirds of the say. With two people, influence goes to whoever moves least. With more, the consensus, when one is reached, is still a weighted average of the starting opinions, but the weights are the left eigenvector of $A$ (the stationary distribution of the chain that $A$ defines), so they depend on the whole pattern of who listens to whom: an agent that barely moves can still end up with no say at all if nobody listens to it.

For larger groups, a sufficient condition for agreement is mild: “If any two agents put jointly a positive weight on a third one then for every initial opinion profile a consensus will be approached.” [[6]] (The Kuramoto post is a loose cousin.) What DeGroot leaves out is exactly what a feed changes: the weights are fixed and never depend on what anyone thinks.

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Listening Only to the Near

Hegselmann and Krause made the weights depend on opinions [[6]]. In their bounded-confidence model, “An agent i takes only those agents j into account whose opinions differ from his own not more than a certain confidence level”, and weights them equally:

$$x_i(t+1) = \frac{1}{|I_i(t)|}\sum_{j \in I_i(t)} x_j(t), \qquad I_i(t) = \{\, j : |x_i(t) - x_j(t)| \le \varepsilon \,\},$$

where $\varepsilon$ is the confidence level (how far from your own view you will still listen), $I_i(t)$ is the set of agents that agent $i$ listens to in round $t$, itself included, and $|I_i(t)|$ is their number. In single runs of 625 agents from one start, with $\varepsilon = 0.01$ “exactly 38 different opinions survive”, at 0.15 “the agents end up in two camps”, and at 0.25 everyone agrees.

I wrote a short Python script to reproduce this and then break it. The table comes from a re-implementation of it that is kept with this site’s source (docs/editorial-pass/figures/who-raises-whom-recommender-feedback/) and reproduces the original script’s 20-start numbers exactly. A hundred agents ($N = 100$) start uniformly on $[0,1]$, drawn by Python’s random module with seeds 0 to 999, and update simultaneously until no opinion moves by more than $10^{-10}$ in a round. Clusters are groups separated by gaps larger than $10^{-3}$. The table gives means and standard errors over these 1,000 random starts for all clusters (some hold one to four agents) and for clusters of at least five agents, 5% of $N$. The last column is the share of runs that end in a single cluster.

Model	Setting	All clusters	Clusters ≥ 5 agents	Consensus
HK	$\varepsilon = 0.05$	7.67 ± 0.03	7.16 ± 0.03	0%
HK	$\varepsilon = 0.10$	3.90 ± 0.02	3.80 ± 0.02	0%
HK	$\varepsilon = 0.15$	2.62 ± 0.02	2.56 ± 0.02	0%
HK	$\varepsilon = 0.25$	1.39 ± 0.02	1.39 ± 0.02	61.5 ± 1.5%
Narrowing (toy)	$\varepsilon_0 = 0.25$	2.11 ± 0.01	1.98 ± 0.01	4.2 ± 0.6%
Narrowing (toy)	$\varepsilon_0 = 0.40$	1.00 ± 0.00	1.00 ± 0.00	100%
Feed (toy)	$k = 2$	21.85 ± 0.05	9.93 ± 0.06	0%
Feed (toy)	$k = 5$	11.12 ± 0.04	10.74 ± 0.03	0%
Feed (toy)	$k = 10$	5.97 ± 0.03	5.88 ± 0.03	0%
Feed (toy)	$k = 20$	3.11 ± 0.02	3.11 ± 0.02	0%
Feed (toy)	$k = 40$	1.70 ± 0.01	1.70 ± 0.01	30.2 ± 1.5%

The HK rows reproduce the phase picture. At $\varepsilon = 0.15$, 551 of the 1,000 runs end in three clusters of at least five agents, 447 in two and 2 in four. With $N = 625$, counting clusters of at least 5% (32 agents), it is three in 49% and two in 51%, so 0.15 sits on a boundary. Hegselmann and Krause call their two camps “a fairly typical result” on the strength of opinion densities averaged over 50 runs; counted run by run, a third large camp appears about half the time. At 0.25, 100 agents reach consensus in 62% of runs and 625 agents in 998 of 1,000.

The other rows are my own toys. Narrowing tolerance stands in for people listening less as a dispute goes on; it is not derived from any feed, and nothing here shows that feeds do this. Tolerance shrinks by 10% a round, $\varepsilon(t) = \varepsilon_0 \cdot 0.9^t$ (floor 0.01). From 0.25, consensus comes in 4% of runs instead of 62%; from 0.4, the groups meet before the window closes and every run agrees.

In the nearest-$k$ toy, each agent averages its own opinion with the $k$ other opinions closest to it, however far away (ties go to the lower agent index): a feed of more of what you already think. At $k = 10$ each agent sees about a tenth of the others, as in plain HK at $\varepsilon = 0.05$ at the start (a window of width 0.1 on a uniform spread). Fig. 1 runs both rules, and HK at 0.25, from the same start.

Over 1,000 starts the feed gives 6.0 clusters, the narrow-minded population 7.7, and Fig. 2 shows the same across the whole range. In this toy, the feed divides people no more than their own narrowness would. It sets from outside the width of what each person sees, and the width decides, with no persuasion or intent anywhere.

That is all the toy shows. Opinions are one number, people only average, and my ranking rule never learns: the performative part, retraining on what users did after the ranking, is exactly what I did not model.

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The Field Evidence Is Smaller and Messier

The largest tests are the 2020 US election studies run with Meta. Guess and colleagues moved consenting Facebook and Instagram users to chronological feeds for three months [[7]]. Their time on the platforms dropped substantially. Yet “the chronological feed did not significantly alter levels of issue polarization, affective polarization, political knowledge, or other key attitudes during the 3-month study period.” Nyhan and colleagues found that content from like-minded sources “constitutes the majority of what people see on the platform” [[8]]; cutting that exposure “by about one-third” for 23,377 users had “no measurable effects on eight preregistered attitudinal measures”.

In a 2024 letter to Science, Bagchi and colleagues argued that Meta introduced 63 “break-glass” feed changes during the experiment, “changing the control condition” [[9]]. They target the misinformation finding, but the journal’s editor-in-chief summarised the letter more broadly: the emergency measures “may have made the default algorithm similar to the reverse chronological algorithm” [[10]]. If the two feeds were temporarily alike, every contrast between them was weaker, attitudes included, so the null results above are harder to read than they look. By the same account, the authors replied that “the paper was written with a sufficient number of caveats”, and the journal posted both letters with an editorial instead of a correction. The paper has two errata (December 2024, March 2026), which I could not read.

The Reuters Institute’s 2022 review “finds no support for the filter bubble hypothesis”, and finds that algorithmic selection generally leads to “slightly more diverse news use” [[11]]. It covers news diets, not video. On YouTube, the platform the blurb names first, Hosseinmardi and colleagues used “counterfactual bots” to show that relying on the recommender alone “results in less partisan consumption”, and that since YouTube’s 2019 algorithm changes “algorithmic recommendations play, if anything, a moderating role” [[12]]. That measures consumption, not attitudes.

Deactivation studies point both ways. Paying people to leave Facebook for four weeks before the 2018 US midterms lowered an index of political polarisation by 0.16 standard deviations [[13]]. A follow-up that deactivated 19,857 Facebook and 15,585 Instagram users for six weeks before the 2020 election found polarisation effects “precisely estimated and close to zero” [[14]]. Both test a platform, not its ranking.

Ranking does show up elsewhere. In a ten-day field experiment with 1,256 X users during the 2024 US campaign, Piccardi and colleagues reranked posts expressing antidemocratic attitudes and partisan animosity, which “shifted out-party partisan animosity by more than 2 points on a 100-point feeling thermometer” [[15]].

So the loop is real as a mechanism and in simulations, and its measured effect on attitudes in the field is small or mixed. None of these studies tests the retraining half of the loop; they swap, rerank or remove a feed for days or months. Three months is long for an experiment and short for a childhood.

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Approval Shapes the Assistant

The most explicit documented case of a model shaped by its users’ approval does not involve a feed. On 29 April 2025, OpenAI announced that it had “rolled back last week’s GPT-4o update in ChatGPT”, because the update “was overly flattering or agreeable—often described as sycophantic” [[16]]. In describing its training in general, it listed “user signals like thumbs-up / thumbs-down feedback on ChatGPT responses”. On 2 May it added that the update “introduced an additional reward signal based on user feedback”, that “User feedback in particular can sometimes favor more agreeable responses, likely amplifying the shift we saw”, and that the changes “may have played a part in tipping the scales on sycophancy when combined” [[17]]. It had launched the update “due to the positive signals from the users who tried out the model. Unfortunately, this was the wrong call.”

These are vendor blog posts, and the causal story is the vendor’s. At ICLR 2024, Sharma and colleagues found that “both humans and preference models (PMs) prefer convincingly-written sycophantic responses over correct ones a non-negligible fraction of the time” [[18]].

For the other direction, from model to user, there is one-shot evidence. In March 2026, Cheng and colleagues reported in Science that across 11 models, “AI affirmed users’ actions 49% more often than humans” [[19]]. In three preregistered experiments with 2,405 participants, “even a single interaction with sycophantic AI reduced participants’ willingness to take responsibility and repair interpersonal conflicts, while increasing their conviction that they were right”, and “sycophantic models were trusted and preferred.” These are stated intentions after one interaction in advice scenarios.

Each link is documented separately: a vendor’s account of approval shaping a model, lab data on preference judgements, one experiment on agreement shifting users within a session. No study I know of has watched such a loop close over time, with the same users training a model that then changes them.

Part 2 followed what an optimiser does to a proxy reward. Here the proxy is a thumb. The parenting analogy holds at one point: a learner trained on approval learns to please, and a thumbs-up is approval stripped of everything else a parent conveys. It stops at another. A child raised on praise still has peers, consequences and a body that gets hungry. On OpenAI’s account, the 2025 model also had a “primary reward signal, which had been holding sycophancy in check” [[17]]. The analogy concerns how feedback is weighted, not an inner life.

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Children Raised on Their Own Output

If models learn from the internet and the internet fills with model output, models end up trained on their own offspring. In 2024, Shumailov and colleagues reported in Nature that “indiscriminate use of model-generated content in training causes irreversible defects in the resulting models, in which tails of the original content distribution disappear” [[20]]. (A 2025 Author Correction fixes a notation slip in its theory section, nothing quoted here.)

“Indiscriminate” matters. In their language-model experiment, keeping 10% of the original data “leads to only minor degradation of performance”. A 2024 preprint by Gerstgrasser and colleagues shows that “accumulating the successive generations of synthetic data alongside the original real data avoids model collapse”, with test error bounded independently of the number of generations in their setting [[21]]. The experiments use small models and small datasets and show nothing about deployed systems.

The image of children raised on their own output holds in that each generation loses what the one before under-represented, and stops at the remedy: a human generation cannot choose its data mix as a pipeline can. Shumailov and colleagues name the source to keep: “data collected about genuine human interactions with systems will be increasingly valuable”. My inference, not theirs: that human signal is the one the feed and the chat window are already bending.

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Who the Parents Are When Everyone Is

De Kai’s publisher asks, “What should each of us do as the responsible adults in the room?” [[1]] Part of the difficulty is arithmetic. A shared model is trained on a loss summed over enormous numbers of users. YouTube’s engineers generated “a fixed number of training examples per user, effectively weighting our users equally in the loss function” [[2]], and in April 2025 OpenAI spoke of “500 million people using ChatGPT each week” [[16]]. In the shared model, one person’s conscientious clicking is one equal term among millions. In one’s own feed it counts for more, because YouTube’s 2016 system learns “user embeddings u as a function of the user’s history and context” [[2]]. You can parent your own feed a little and the shared model hardly at all. Users can also see their own history, but only the platform holds everyone’s records and retrains on them.

Some of the people in the loop are literal children. Ofcom’s 2025 report, using 2024 data, found YouTube used by 88% of UK 3–17-year-olds [[22]]. Of 8–17-year-olds, 63% said they were aware that apps use algorithms to choose content, from 44% at ages 8–9 to 81% at 16–17.

The institutional parenting that exists is modest. The EU’s Digital Services Act requires platforms to set out the “main parameters” of their recommender systems (Article 27) [[23]]. Article 28(1) requires platforms accessible to minors to take measures ensuring “a high level of privacy, safety, and security of minors”. Article 28(2) bars profiling-based advertising when a platform is “aware with reasonable certainty” that a user is a minor. The Commission’s July 2025 guidelines under Article 28 address feeds directly, advising platforms to “prioritise explicit signals from children over behavioural signals”. They are voluntary, but the Commission will use them to assess compliance [[24]]. Article 38 requires very large online platforms and search engines to offer “at least one option for each of their recommender systems which is not based on profiling”.

One recommendation, labelled as such: where the Article 38 option exists, use it to step out of profiling-based ranking. It does not change the objective, since a feed without a profile can still be ranked for engagement, and the nearest field test, the chronological feed, cut time on the platform without measurably moving attitudes. It is a small lever.

Who is raising whom? Each is raising the other, and only one of them keeps everyone’s records.

Sources and calculations checked through 2026-10-24. The simulation is my own toy model (1,000 random starts per setting), not evidence about any platform.

Next, and last, in the series: Gods, Slaves, Children, Tools, 27 October.

References

[1] De Kai. (2025). Raising AI: An Essential Guide to Parenting Our Future. MIT Press. ISBN 9780262049764. Quoted from the publisher’s book description, not from the text of the book: https://www.penguinrandomhouse.com/books/780363/raising-ai-by-de-kai/9780262049764

[2] Covington, P., Adams, J., & Sargin, E. (2016). Deep neural networks for YouTube recommendations. In Proceedings of the 10th ACM Conference on Recommender Systems (RecSys ‘16), 191–198. ACM. https://doi.org/10.1145/2959100.2959190

[3] Perdomo, J. C., Zrnic, T., Mendler-Dünner, C., & Hardt, M. (2020). Performative prediction. In Proceedings of the 37th International Conference on Machine Learning (PMLR 119), 7599–7609. arXiv:2002.06673. https://arxiv.org/abs/2002.06673

[4] Chaney, A. J. B., Stewart, B. M., & Engelhardt, B. E. (2018). How algorithmic confounding in recommendation systems increases homogeneity and decreases utility. In Proceedings of the 12th ACM Conference on Recommender Systems (RecSys ‘18), 224–232. ACM. https://doi.org/10.1145/3240323.3240370

[5] DeGroot, M. H. (1974). Reaching a consensus. Journal of the American Statistical Association, 69(345), 118–121. https://doi.org/10.1080/01621459.1974.10480137

[6] Hegselmann, R., & Krause, U. (2002). Opinion dynamics and bounded confidence models, analysis, and simulation. Journal of Artificial Societies and Social Simulation, 5(3), 2. https://www.jasss.org/5/3/2.html

[7] Guess, A. M., Malhotra, N., Pan, J., Barberá, P., Allcott, H., Brown, T., et al. (2023). How do social media feed algorithms affect attitudes and behavior in an election campaign? Science, 381(6656), 398–404. https://doi.org/10.1126/science.abp9364. Errata: Science, 386(6726), eadu8261 (2024), https://doi.org/10.1126/science.adu8261; Science, 391(6791), eaeh2575 (2026), https://doi.org/10.1126/science.aeh2575 (not read).

[8] Nyhan, B., Settle, J., Thorson, E., Wojcieszak, M., Barberá, P., Chen, A. Y., et al. (2023). Like-minded sources on Facebook are prevalent but not polarizing. Nature, 620(7972), 137–144. https://doi.org/10.1038/s41586-023-06297-w. Author Correction (a row removed from a supplementary table): Nature, 623, E9 (2023), https://doi.org/10.1038/s41586-023-06795-x

[9] Bagchi, C., Menczer, F., Lundquist, J., Tarafdar, M., Paik, A., & Grabowicz, P. A. (2024). Social media algorithms can curb misinformation, but do they? eLetter to Science; preprint, arXiv:2409.18393. Not peer reviewed. https://arxiv.org/abs/2409.18393

[10] Thorp, H. H. (2024, 27 September). More doubts for Facebook on polarization. Science Forever (blog of the editor-in-chief of Science). https://www.science-forever.com/p/more-doubts-for-facebook-on-polarization

[11] Ross Arguedas, A., Robertson, C. T., Fletcher, R., & Nielsen, R. K. (2022). Echo chambers, filter bubbles, and polarisation: A literature review. Reuters Institute for the Study of Journalism. https://doi.org/10.60625/risj-etxj-7k60

[12] Hosseinmardi, H., Ghasemian, A., Rivera-Lanas, M., Horta Ribeiro, M., West, R., & Watts, D. J. (2024). Causally estimating the effect of YouTube’s recommender system using counterfactual bots. Proceedings of the National Academy of Sciences, 121(8), e2313377121. https://doi.org/10.1073/pnas.2313377121

[13] Allcott, H., Braghieri, L., Eichmeyer, S., & Gentzkow, M. (2020). The welfare effects of social media. American Economic Review, 110(3), 629–676. https://doi.org/10.1257/aer.20190658 (Effect sizes quoted from NBER Working Paper 25514.)

[14] Allcott, H., Gentzkow, M., Mason, W., Wilkins, A., Barberá, P., Brown, T., et al. (2024). The effects of Facebook and Instagram on the 2020 election: A deactivation experiment. Proceedings of the National Academy of Sciences, 121(21), e2321584121. https://doi.org/10.1073/pnas.2321584121. Correction (2026, a reviewer’s undisclosed funding): https://doi.org/10.1073/pnas.2607249123

[15] Piccardi, T., Saveski, M., Jia, C., Hancock, J., Tsai, J. L., & Bernstein, M. S. (2025). Reranking partisan animosity in algorithmic social media feeds alters affective polarization. Science, 390(6776), eadu5584. https://doi.org/10.1126/science.adu5584

[16] OpenAI. (2025, 29 April). Sycophancy in GPT-4o: What happened and what we’re doing about it. Vendor blog post. https://openai.com/index/sycophancy-in-gpt-4o/

[17] OpenAI. (2025, 2 May). Expanding on what we missed with sycophancy. Vendor blog post. https://openai.com/index/expanding-on-sycophancy/

[18] Sharma, M., Tong, M., Korbak, T., Duvenaud, D., Askell, A., Bowman, S. R., Cheng, N., Durmus, E., Hatfield-Dodds, Z., Johnston, S. R., Kravec, S., Maxwell, T., McCandlish, S., Ndousse, K., Rausch, O., Schiefer, N., Yan, D., Zhang, M., & Perez, E. (2024). Towards understanding sycophancy in language models. ICLR 2024. arXiv:2310.13548. https://arxiv.org/abs/2310.13548

[19] Cheng, M., Lee, C., Khadpe, P., Yu, S., Han, D., & Jurafsky, D. (2026). Sycophantic AI decreases prosocial intentions and promotes dependence. Science, 391(6792), eaec8352. https://doi.org/10.1126/science.aec8352 (Preprint: arXiv:2510.01395; figures here are from the Science version.)

[20] Shumailov, I., Shumaylov, Z., Zhao, Y., Papernot, N., Anderson, R., & Gal, Y. (2024). AI models collapse when trained on recursively generated data. Nature, 631(8022), 755–759. https://doi.org/10.1038/s41586-024-07566-y. Author Correction: Nature, 640, E6 (2025), https://doi.org/10.1038/s41586-025-08905-3

[21] Gerstgrasser, M., Schaeffer, R., Dey, A., Rafailov, R., Sleight, H., Hughes, J., Korbak, T., Agrawal, R., Pai, D., Gromov, A., Roberts, D. A., Yang, D., Donoho, D. L., & Koyejo, S. (2024). Is model collapse inevitable? Breaking the curse of recursion by accumulating real and synthetic data. Preprint, arXiv:2404.01413. https://arxiv.org/abs/2404.01413

[22] Ofcom. (2025, 7 May). Children and parents: Media use and attitudes report 2025. https://www.ofcom.org.uk/media-use-and-attitudes/media-habits-children/children-and-parents-media-use-and-attitudes-report-2025

[23] Regulation (EU) 2022/2065 of the European Parliament and of the Council of 19 October 2022 on a Single Market for Digital Services (Digital Services Act). Official Journal of the European Union, L 277, 27 October 2022. https://eur-lex.europa.eu/eli/reg/2022/2065/oj

[24] European Commission. (2025, 14 July). Commission publishes guidelines on the protection of minors. https://digital-strategy.ec.europa.eu/en/library/commission-publishes-guidelines-protection-minors

Changelog

• 2026-10-24: Replaced the 20-start simulation table with means and standard errors over 1,000 starts from a re-implementation of my original script (seeds 0–19 reproduce the old table exactly; no comparison changes) and updated the quoted numbers: at $\varepsilon = 0.15$, three large clusters in 55% of runs, not 14 of 20 (625 agents: 49%, not 10 of 20); consensus at 0.25 in 62%, not 13 of 20 (625 agents: 998 of 1,000); narrowing from 0.25, 4%, not 2 of 20; 6.0 against 7.7 clusters, not 5.9 against 7.65. Stated the large-cluster threshold as 5% of $N$ (32 agents for 625). Corrected “one to three agents” to one to four: the old starts already left four-agent clusters. Withdrew “HK’s ’two camps’ is one draw” (from “so 0.15 sits on a boundary and …”): Hegselmann and Krause call the two camps “a fairly typical result” on the strength of densities averaged over 50 runs, so the text now quotes them beside the run-by-run count and keeps only the boundary. Corrected “like-minded sources dominate what people see” to the abstract’s “constitutes the majority of what people see on the platform” and listed the 2023 Author Correction of [8], which changes nothing quoted. Added two figures, and Perdomo et al.’s convergence condition for retraining with their Theorem 4.3 bound and the remark that I know of no estimate of the sensitivity $s$ for a deployed feed.