Nine narrated, beat-synced visual lessons, three per paper. Each opens in its own player: space to play, arrow keys for slides, comma and period to step through the beats of a figure, square brackets for speed, d for links into the notes and the PDF at the right page.

Every spoken sentence was fact-checked against the PDF pages by a skeptic agent before narration, and anything that is our inference and not the paper's is flagged out loud. The core lessons end with two recall questions; the player pauses itself before each answer.

Suggested path: the three core lessons first (about 27 minutes at 1×, 18 at 1.5×), then the deep dive that pulls at you, then the report pages under each paper, then the PDF.

Three PDFs from Omer Tamuz's Caltech page, surveyed on 2026-09-17. All three have Omer Tamuz (Caltech, economics and mathematics) as a co-author. They share an author and nothing else: one is pure math (amenability of a monoid), one is a lab experiment (social learning), one is economic theory with random-matrix tools (taxes and subsidies under noisy demand estimates).

These pages are a reader's survey: the goal was to get the core of each paper into one reader's head, fast, with the central points first and the logical next place to read after each. The notes were drafted in markdown and rendered into this page with compiled LaTeX. Survey agents wrote the notes and the report pages; nothing here was typed into the HTML by hand.

Status: pass 2. Seven survey agents have read every page of all three papers, including appendices and figures, and written the report pages that sit under each paper in the left rail. Their corrections to the pass 1 notes are merged. Every claim on these pages carries a PDF page link so it can be checked in one click. Page numbers are PDF page numbers (what the viewer's page box shows).

From public pages (agent C1, full map in tamuz-map): Omer Tamuz is a professor at Caltech, primary division HSS (economics) with PMA (mathematics) listed second. His homepage and the admissions office page say he chairs the undergraduate First-Year Admissions Committee.

All three PDFs are byte-identical to the public copies on his Caltech page, so none is a private draft. Status: the taxes paper is forthcoming in the Journal of Political Economy (arXiv 2411.03026); the imitation experiment is published in the Journal of Economic Theory 235 (2026); the face-maps paper is published in Math. Proc. Cambridge Philos. Soc. 181(1).

1. Taxes first. For a reader whose native tools are linear systems, eigenmodes and identification under noise, it is the closest to home, and about 80% of the value of the set sits here.
2. Learning experiment second. Fast to absorb, one clean result, one three-parameter model. It also bears on how to run multi-agent systems (should workers see each other's conclusions or only the raw sources).
3. Deletion last. Shortest paper, furthest from that background. The payoff is a clean idea (amenability that "hides at infinity") and a brush with Thompson's group , a famous open problem.

Each paper has a result that only appears in the large limit and fails at every finite stage.

• Deletion: each finitely generated submonoid is non-amenable; the union is amenable.
• Taxes: with finitely many products and bounded eigenvalues the regulator can do nothing robust; with and a top eigenvalue that outgrows the noise, robust intervention exists.
• Learning: the weakest instance of the pattern. Group size helps only when raw signals are shared; with actions alone 8 people do about as well as 4, which the authors loosely connect to a bound that is uniform in group size.

This is a reading aid. It is not a claim the papers make.

• Left rail or keys 1 2 3 jump to a paper. h is this hub, g the glossary, s how this was made. [ and ] step through sections. ? shows keys.
• Dotted-underlined terms are glossary terms. Hover for the definition, click to open the entry.
• Small p.N chips open the PDF at that page.
• "Check yourself" boxes are closed by default. Answer in your head, then open.
• The depth pills at the top of a paper page filter it: L0 one paragraph, L1 mechanism, L2 results ledger, L3 proofs and appendices.

Narrated, beat-synced visual lessons. Start with the core lesson, then go deeper where you want.

• Core lesson: steering a market you can barely see (13 slides, about 8 minutes at 1×)
• Deep dive: the proof of Theorem 1 (the no-gap subspace argument)
• Deep dive: limits, the phase transition, and the data diagnostic

Reports behind the lessons: taxes-core (the argument, fully derived) and taxes-limits (Monte Carlo, Proposition 3, the diagnostic, hedonic models).

Claim. A regulator (or a marketplace operator like Amazon) sees the matrix of cross-price demand effects only through heavy noise. Entry by entry, the estimate is useless. The regulator can still pick per-unit subsidies and taxes that raise total surplus with probability near 1, provided the true matrix has a few very large eigenvalues and the noise has a small operator norm by comparison. p.2

Mechanism. Subsidize along a top eigenvector of the normalized Slutsky matrix . In that direction prices barely move, quantities move a lot, and each dollar spent returns about two dollars of producer surplus and about zero change in consumer surplus. Top eigenvectors survive noise (Davis–Kahan); the rest of the matrix does not need to be known. p.14 p.19

Consequence. The policy-relevant object in a huge market is a low-dimensional subspace, and the condition for it to exist ("significant structure") requires large-scale complementarities across families of products. Models built mainly on substitutability, the usual focus in hedonic demand work, do not have it. p.5

A control-theory dictionary for this paper. The right column is the survey's translation, not the authors' language.

Everything is first order. " " means the derivative of outcome in direction at . The paper never computes a finite intervention. p.7

Normalization: rescale units so every own-price effect is , that is . Then each firm's first-order condition reads , which is what makes the algebra short. p.10

For any intervention, with consumer surplus, producer surplus, the regulator's spend: p.11

Read it as a frontier. If consumers are not to be hurt ( ), the most the regulator can buy is , reached when all the incidence goes to producers. Total surplus is , and the identity gives . So with consumers held harmless ( ) the best rate is one dollar of net surplus per dollar spent. This is a full-information identity; nothing about robustness enters yet. Without the constraint on consumers the rate is unbounded: tax a low- mode and subsidize a high one at zero net spend. With full information, for generic , every surplus triple on the plane is reachable as long as has one nonzero off-diagonal entry (Proposition 1, part 2). p.11

Monopoly sanity check: one firm, linear demand. A subsidy drops the price by . Consumers get half the spend, the producer gets all of it (half on old volume, half on new volume). . p.11

Why information is the binding constraint: . The inverse is sensitive to every entry of . With a noisy the regulator cannot even sign . p.12

Write with eigenvalues . Think of eigenvector as a bundle of products. A subsidy proportional to moves only that bundle's price and quantity; the other bundles do not respond. Lemma 1: p.14

So per dollar spent in direction : consumers get , producers get , net total surplus is . p.19 Two forces oppose each other as grows. Strategic interaction among the firms makes the bundle's equilibrium price less sensitive to the subsidy. Demand for the bundle gets more sensitive to its price (the slope of bundle demand in bundle price is exactly , a fact about demand alone). The second force wins, so quantity pass-through rises with while price pass-through falls. p.14

Surplus effects add across modes, each weighted by how much status-quo quantity sits on that bundle (Lemma 2 on p.15, combined with Lemma 1 as equation (10) on p.20):

Put all of on modes with huge and : the frontier from Step 1. Two things must hold: is not orthogonal to those modes (otherwise the intervention spends nothing and does nothing), and the regulator can find them.

Definition 2. There is a threshold such that: p.16

1. quantity noise is no bigger than quantities, ;
2. demand noise is small next to the threshold in operator norm, ;
3. has eigenvalues above in absolute value, and keeps a fixed fraction of its norm inside their eigenspace.

Why this is plausible: for independent bounded-variance entry noise, grows like , while a matrix with complementarities of fixed size spread across categories has a top eigenvalue that grows like . The running example is a block model: products within a category are substitutes, across categories complements (tennis rackets and tennis shoes). In the homogeneous version its top eigenvector is exactly the all-ones vector and ; in the general block model Lemma 3 gives a positive, category-constant eigenvector. The noise bound needs independent, mean-zero, uniformly bounded errors. p.12 p.17 p.39

Recovery uses the subspace ( ) form of Davis–Kahan, and it does not assume the large eigenvalues are separated from the rest of the spectrum or from each other. The proof manufactures its own gap with three thresholds: true eigenvalues above (the space ), estimated eigenvalues above (the computable space ), and true eigenvalues above (a buffer space ). Each comparison has a gap of , which dwarfs , so up to a vanishing tilt. No individual eigenvector is ever estimated. p.41 p.42 p.43 Full derivation: taxes-core.

The rule is a projection, so no sign is ever chosen: in normalized units. Estimated spend equals exactly. Quantity noise may be as large in norm as itself; it washes out because has at most dimensions (from and negative semidefiniteness), and independent noise spread over coordinates has almost no energy in so few. p.42 p.44 The sign rule belongs only to the one-eigenvector sketch in the main text. p.20

Theorem 1. Under significant structure and Assumptions 1 to 4, for any and target spend , for large enough a single rule achieves all of the following -robustly: (i) ; (ii) , and the same for every individual household; (iii) . p.18

The rule: normalize , project onto the span of eigenvectors of with eigenvalue above (the main text says ; the appendix proof needs the lower value), divide by the squared norm of the projection and multiply by . p.21 p.42

The figure to remember is the Monte Carlo in Section 7 ( , two-block model with unknown block membership, Wigner noise). The horizontal axis is with ; the noise has operator norm about . Alignment and welfare-per-dollar switch on at . (Survey inference, confirmed by two independent re-simulations: that is half the noise norm and exactly the spiked-matrix (BBP) threshold, with squared overlap above it, and the Davis–Kahan bound is vacuous until about 4.) Reading Figure 3B: the 5th percentile of welfare is a loss from the left edge until about 3.7 and the 10th until about 2.4, including a stretch where the median is already above 0.5; the skeptic's re-run puts the share of losing runs at 42%, 18%, 7% at . The same band has a tail where consumers lose. Note the simulated example flips the sign pattern of Section 4: complements within blocks, substitutes across. p.22 p.23 p.24

A skeptic pass (taxes-verify) checked this note, the two reports and the core lesson against the PDF; its must-fix items are applied. Its one new observation: with observed exactly, weakly raises first-order welfare in every state with no knowledge of (from equation (9)), so what Theorem 1 buys is the rate, consumer neutrality and the spending target.

All eight are answered in taxes-core (questions 2 and 5) and taxes-limits (1, 3, 4, 6, 7). Question 8 (what the referees pushed on) was not pursued. One new criticism from T1: Appendix A.3 never invokes Assumption 4, and the normalization error is not automatically . p.41

1. Proposition 3, both parts: what exactly is impossible without significant structure, and why consumers cannot be robustly favored.
2. The no-gap subspace argument in Appendix A: what replaces per-eigenvector Davis–Kahan.
3. Section 8.2 diagnostic: statement, how it is computed, what it would take to run on real data.
4. Assumption 2 (own-price local linearity) and Section 8.4: how much breaks without it.
5. Which assumptions carry the "every household" clause of Theorem 1(ii).
6. Relation to Galeotti, Golub, Goyal (2020) "Targeting interventions in networks", which did the known-network version.
7. BBP question above.
8. What did the four referees likely push on? (Compare with the 2021 working paper if it can be found.)

Drafts after pass 1, to be sharpened or killed.

• The guarantee is first order in . How large can the intervention get before the eigenbasis it was designed in has moved?
• Is there an adversarial-noise version? The operator-norm condition seems to want independence across entries; a platform's estimates of would have structured errors.
• Does significant structure show up in any real demand system you have looked at (the acknowledgments mention Pellegrino's data)?

Narrated, beat-synced visual lessons. Start with the core lesson, then go deeper where you want.

• Core lesson: why redundant guesses make a group smarter (13 slides, about 9 minutes at 1×)
• Deep dive: the theory benchmark and the three-parameter model
• Deep dive: does the headline survive an audit?

Reports behind the lessons: learning-main (main text, every figure read) and learning-appendix (the 50-page appendix, audited).

Claim. Give a group the full public record of evidence. Then also show them each other's past guesses. The guesses are a deterministic-or-noisy function of evidence everyone already has, so they carry zero extra information, and the standard behavioral worries (herding, correlation neglect, overload) predict they should hurt. They help: about 5 percentage points more correct guesses on average across rounds, and about 10 points more consensus. p.13 Read the first number with its uncertainty: Table A.1 gives 0.051 with a session-clustered standard error of 0.028 (significant at the 10% level, 11 sessions). The consensus effect, 0.095 with the same standard error, is the statistically firm one. p.34

Mechanism. People count evidence badly when the tally is close. In exactly those rounds they lean on what others did last round; when the tally is lopsided they ignore others. Other people's guesses are a noisy copy of the correct count, so mixing them in denoises your own. p.16 p.17

Consequence. Imitation is a cheap error-correcting code on top of bad individual inference. Too much of it is still harmful (everyone copying everyone ignores the data), so the benefit lives at a moderate weight p.32, and the paper estimates that weight p.29.

An urn is majority red (6 red, 4 green) or majority green, fair coin. Groups of 8. Twenty rounds. Each round every player guesses the urn's color, then gets one private draw with replacement, correct with probability 0.6. One random round of one random game pays $20 if correct and $5 if not, so there is nothing to hedge and myopic best-guessing is incentive compatible. Ten games per session, groups reshuffled each game. An on-screen table keeps the whole history, so memory is not a factor. 606 subjects, 31 sessions, UCSD lab and OSU online. p.6 p.8

Four information treatments, all on top of your own signals: p.7

The headline comparison is signals against all. A Bayesian does identically in both: guess the color that is ahead in the pooled tally. The Bayesian benchmark hits 0.96 by round 10 and above 0.99 by round 18. p.10

Both groups fall well short of the Bayesian curve, ending near 0.83 to 0.88 where theory says 0.99. The all group sits above the signals group from the early rounds on. p.12

This is the mechanism figure. Flat lines at top and bottom: when the pooled tally is lopsided, what others did has almost no effect on your guess. The steep yellow line: when the tally is close (difference within 14, the middle 50% of end-of-game histories; cutoffs on p.11 p.12), your probability of guessing red runs from about 0.13 to about 0.86 as the share of others guessing red goes from 0 to 1. p.16 Caution from agent L1: inside the Weak bin the tally and the peers' share still move together, so this 0.73 swing overstates the causal effect. The structural model puts the effect of flipping one sampled peer at 0.31. The paper's own Figure 9 shows the same artifact. p.30

The paper's four numbered observations, compressed:

1. Seeing signals and actions beats seeing signals alone, in accuracy and in consensus. p.14
2. In both treatments people use the signals worse than a Bayesian would, mostly when the tally is close. In all they patch those errors with others' actions. p.17
3. High-IQ subjects (six ICAR items) process the raw signals well with or without social information. Low-IQ subjects do not, and do badly without it. Both groups condition on others' actions, almost only when signals are weak, and both gain. p.20 The low-IQ gain is no larger than the high-IQ gain (interaction +0.010, SE 0.031, Table J.2). p.64
4. With signals and actions visible, groups of 8 learn faster than groups of 4 (all minus all4 = 0.065, SE 0.023). With actions only the difference is 0.008 (SE 0.022). p.35 The authors link this to Harel, Mossel, Strack, Tamuz (2021, "Rational Groupthink"), and call the link "quite a loose interpretation" themselves. p.22 The interval for the actions effect also contains the 0.04 to 0.05 gap that myopic Bayesians would show (Table 2), so the data cannot separate "no effect" from "the Bayesian effect".

Private-information side result: actions beats no info by a wide margin (the footnote gives 5.3 points early, 9.6 points late), although extracting private signals from repeated actions is a hard inference problem in theory. p.22

Each round, player looks at two numbers. is the pooled signal tally (red , green ) divided by the number of signals raised to a power . is the last-round action of one randomly chosen other player. The player guesses red with logistic probability: p.22

That is the likelihood used for estimation in Section 6.2, where the sampled peer is latent and integrated out against the share of others who guessed red. p.26

is the Bayesian sufficient statistic (raw difference). is the sample proportion, the usual choice in the behavioral literature. Estimates land between 0.52 and 0.63, which the paper calls closer to a square-root scaling than to the sample proportion (0.5 itself lies outside the credible intervals for signals and all): two reds alone feel stronger than 102 reds against 100 greens, but less drastically than proportion-thinking implies. p.23 p.28 L1 adds that may be absorbing inattention: the model has no lapse rate, and about a quarter of subjects in two treatments act as if random (Appendix N).

Posterior medians from Table 3. In all, flipping the sampled peer from green to red moves by 0.31, about the same as a one standard deviation move in (0.29). p.29

Why helps here: the mistake probability depends on the sign and size of . is correlated with the sign of because peers mostly follow the tally. Adding it usually pushes the sum the right way, and it matters most when , where your own response is close to a coin flip. p.24

Survey replication, 3,000 simulated games at the Table 3 medians: all ends at 0.86 and signals at 0.83 in round 20, close to the data in Figure 1. Holding and at the all estimates and sweeping , round-20 accuracy climbs from 0.82 at to about 0.90 on a flat top from 1.75 to 2.6, then collapses (0.68 at ). Under the paid objective (mean accuracy over rounds) the optimum is about 1.5 and the gain over the subjects' 0.65 is about 2 points (skeptic re-run, 80,000 games per point). That is a simulation by the survey on the paper's model, and the paper does not claim it.

A skeptic pass (learning-verify) checked this note, both reports and the core lesson against the PDF, re-ran the units ruling (agrees: doubles, does not) and the sweep. Two facts it added: the paper's subject-level measure finds high-IQ subjects rely on others more ( ) p.20, and Appendix L finds the treatment effect largest in the lower quantiles of realized performance p.72.

Answered in learning-main (1, 5, 7) and learning-appendix (2, 3, 4, 6). The short version of the audit: the lab-versus-online confound is never directly tested (Appendix F compares only all at UCSD with all at OSU and prints no numbers; L2's digitized within-online gap is about 5.4 points, with no usable standard error; by subjects all is 64 UCSD and 88 OSU); Appendix N covers only no info and signals and never re-runs the headline without the random-acting quarter; the screen shows no running tally, so part of the counting noise comes from the interface (agent inference). p.8 p.51 p.56

1. Exact model normalization and estimation (Section 6.2): hierarchical? per-subject? How was "one random peer" handled in the likelihood?
2. Is the +5 point effect robust to the lab-versus-online split? Signals ran only online at OSU; all ran half in the UCSD lab, half online. That is a confound worth reading Appendix F for. p.8
3. Appendix K: how much of the gap to the Bayesian benchmark does imitation close, in RMSE terms?
4. Appendix N: how many subjects are effectively random, and do results survive dropping them?
5. Is there an optimal ? the survey's sweep says yes, near 2 for . The authors mention a formal analysis and conjectures as future work; find where and what, and whether they note that subjects under-imitate.
6. What do subjects say they did (Appendix B, open-ended strategies)?
7. Observation 3 details: size of the low-IQ gain.

Drafts after pass 1.

• With a peer who is confidently wrong on purpose, where does the benefit of imitation break? (They flag this as future work.)
• Does the normalization suggest subjects are doing significance testing in their heads?
• For a fixed decoder noise , what maximizes group accuracy, and do subjects sit near it?

Narrated, beat-synced visual lessons. Start with the core lesson, then go deeper where you want.

• Core lesson: random sets you can delete from (12 slides, about 9 minutes at 1×)
• Deep dive: the proof of Theorem A (means, the failure of Fubini, the walk theorem, Goldstine and Mazur)
• Deep dive: the facial monoid, amenability, and Thompson's group F

Reports behind the lessons: deletion-proofs (line-by-line proof read) and deletion-context (Appendix B, amenability, Thompson's , Moore, Ryll-Nardzewski).

Claim. Take a finite set of integers and delete its th smallest element; call that map . For any and any there is a finitely supported random finite set whose law moves by less than in total variation under every one of (Theorem A). The companion result: any infinite family of continuous affine maps of a compact convex set that satisfies the face relations for has a common fixed point (Theorem B). The paper calls the two "two visages of the same phenomenon" and derives both from one idempotent mean; it does not claim they are equivalent statements. p.3 p.1 p.2

Mechanism. Build a "random walk" whose step distribution is an idempotent mean on , a finitely additive probability measure sitting at infinity with . Deleting any entry of its -point trajectory gives exactly its -point trajectory, because merging two adjacent steps is a convolution and the convolution changes nothing. Then approximate that impossible object by honest finitely supported distributions using Day's Hahn–Banach argument. p.5 p.6

Consequence. The monoid presented by the face relations is amenable. Each submonoid generated by with is non-amenable. For groups that cannot happen, since every subgroup of an amenable group is amenable. The authors' phrase: the amenability of "hides at infinity." is a quotient of the positive monoid of Thompson's group , whose amenability is a famous open problem. p.2 p.3

Day's theorem: for a monoid the following are equivalent. (1) Every action of by continuous affine maps on a nonempty compact convex set has a fixed point. (2) has a left-invariant mean. (3) For every finite and there is a finitely supported probability on with for all . p.12

Theorem B is (1). Theorem A is the same kind of statement as (3), for the deletion maps acting on finite sets. Precision from agent D1: the paper proves Theorem A directly on finite sets in Section 2 and never passes through ; Sections 3 and 4 exist to move the same mean onto and get Theorem B. The identification of with finite sets:

• Every element of has a unique flat normal form with (rewrite until stuck; termination and local confluence, then Newman's lemma). So as a set. p.10
• Under that identification, left multiplication by is : fill the th hole of the set (adjoin the th smallest integer not in it). p.11
• Taking complements inside swaps "fill the th hole" with "delete the th element" on a suitable class of sets ("pin-headed": nonempty, and ). p.7 p.8

So the level-lowering means become level-lowering means for hole-filling, and a Cesàro limit over all levels is a left-invariant mean on , which is amenability. p.9

1. Pick . Take any invariant mean on (an accumulation point of uniform distributions on , equivalently a Banach limit flavored Cesàro limit). It satisfies : convolving it with itself gives it back. It gives mass zero to every finite set, so think of it as "a random very large number." p.4 p.5
2. Walk. Define on increasing -tuples by iterating: with . Loosely, is the law of the first positions of a walk with i.i.d. steps. Only loosely: Fubini fails for means, and the definition is the (non-commutative) Arens product. p.4
3. Theorem 2.3: for every . Deleting the last position obviously gives the -step walk. Deleting position in the middle merges steps and into one step of law , so again a -step walk. The written proof is this idea done carefully with the order of the nested means respected, plus an induction for . p.5 p.6
4. Average over . is within of each . An accumulation point is exactly invariant under . p.6
5. Come back to earth. Finitely supported probabilities are weak-* dense in means (Goldstine). Weak convergence of upgrades to norm convergence after passing to convex combinations (Mazur's trick, which is Hahn–Banach). Norm in is twice total variation. p.6 p.7