A short recap. An urn holds six balls of one color and four of the other. Eight players guess the majority color for twenty rounds. After each guess, each player gets one private draw that matches the truth sixty percent of the time.

Four treatments vary what you see of the others: nothing, their guesses, their signals, or both. The last is called all.

By round twenty, the all treatment is right about eighty eight percent of the time and signals about eighty three. A Bayesian would pass ninety nine in both.

This lesson takes that benchmark apart, then builds the model the authors use to explain the gap.

Every signal carries the same evidence, up or down. So the posterior depends only on the tally, reds minus greens, and the best guess is the color ahead.

That makes the benchmark exact: a binomial tail plus half the chance of a tie. Here it is for one player, four, and eight.

One worked entry: round two, eight signals. Five or more correct has probability zero point five nine four. Half of the ties adds zero point one one six. That gives zero point seven one, matching Table two.

The lone player's curve moves in pairs: an even numbered signal can never flip the lead. At most it creates a tie.

Here is the causal chain. The urn generates signals. Signals generate actions, along with noise unrelated to the urn.

In the all treatment, a Bayesian sees every signal. Given the signals, actions carry no further news about the urn. So all and signals share one benchmark.

In actions, signals are private. In round two everyone plays her own first signal, so each action reveals one signal. Our survey's hand check: nine signals in round three give zero point seven three, matching Table two.

After that, each guess blends private signals with what everybody already inferred. The authors call the case analytically intractable, and simulate myopic players with common knowledge of rationality.

With public signals, a bigger group means more draws per round. Table two puts that gain at three to nine points.

With actions only, the same table gives four to six points. The authors cite Harel, Mossel, Strack and Tamuz: for Bayesian myopic agents watching actions, the speed of learning stays bounded as the group grows.

They add a caveat. The theorem concerns the long run, and mapping it onto twenty rounds is, in their words, quite a loose interpretation.

Measured: six and a half points in the all treatment, under one point in actions. Our survey adds that the second interval still overlaps the benchmark gap.

Each player gets two stimuli. S is the tally divided by a power of the number of signals. A is the last guess of one random other player, coded plus or minus one.

She guesses red with a logistic probability in beta S plus gamma A. At gamma zero, this is one soft step in the tally.

A red peer slides the curve one way, a green peer the other. These use the estimates for the all treatment.

The gap between the curves is the peer's influence: zero point three one at a tied tally, under a tenth at S equal to two. Imitation switches on when evidence is close.

The third parameter is an exponent, psi. The motivation is Griffin and Tversky: a given lead persuades less as the sample grows.

Take a lead of four signals. At psi zero, the Bayesian case, it convinces equally at any sample size. At psi one, the sample proportion, it fades fast. One half is the square root case.

The estimates run from zero point five two to zero point six three. One half lies outside the intervals for signals and for the all treatment.

Our survey's reading: psi produces the accuracy plateau. By round twenty there, the typical index is only one point six five, which maps to zero point eight four.

Fix a weak tally, beta S at zero point four one. Alone, she errs forty percent of the time.

Her peer reads the same tally with the same noise. He is right sixty percent of the time and shifts her index by gamma. The paper argues likewise.

These curves are ours. Her error falls to zero point three seven six near gamma of one, then returns to forty percent: pure copying inherits his error.

If everyone imitates and the group settles, the error is zero point three seven at the estimated gamma. It nears zero point three one, as if beta doubled, only for huge gamma. No urn news entered.

The data show who guessed red, not which peer a subject sampled. That peer is latent.

The paper integrates it out. The peer is red with probability p, the share of others on red, which gives a mixture of two logits, equation six.

It is linear in p, and at a tied tally the slope gives gamma. Inverting the mixture gives the index. Two sample sizes at one tally give psi.

The fit is Markov chain Monte Carlo. Beta is near one without actions on screen and one point two seven with them. Gamma is zero point nine seven in actions and zero point six five in all.

This is a finding of our survey. Section six says red has probability proportional to an exponential, and simulates at beta one half and gamma one. Section six point two writes a plain logistic. The first wording fits two rules.

We simulated each reading against the paper's Figure six. Without imitation there is no freedom: the literal reading misses, and doubling beta hits.

With imitation, doubling gamma too overshoots every curve. A gamma of zero point eight to one fits.

So in Table three units the illustration has beta near one and gamma near zero point nine. The all treatment estimate, zero point six five, sits a little below.

In the paper's mechanism figure, the Weak line swings by zero point seven three as the others go from all green to all red.

The gold line is the same counterfactual in the estimated model, at a tied tally: zero point three one.

Our survey's explanation is a confound. Inside the Weak bin, tallies range over plus or minus fourteen, and in our simulation the mean tally climbs from minus nine to plus nine with peers' share.

Grouped by share, that simulated model swings by two thirds from a true effect of zero point three one. The paper's Figure nine agrees. Imitation is real. The picture doubles it.

Figure eight feeds the estimates back through the model. At round twenty it predicts zero point eight seven five for the all treatment and zero point eight three for signals, within a point of the data.

Round two is weaker. Reading the figures, our survey finds the model low by three to six points.

Figure ten refits that treatment with separate weights by I Q group. Beta is zero point nine nine for low I Q and one point six four for high, and the difference excludes zero.

Gamma is zero point six one and zero point six eight, and that difference includes zero. Both groups lean on peers about equally.

The paper conjectures that some positive gamma beats zero and that the benefit fades. The rest is our simulation and arithmetic, in Table three units at the all treatment estimates.

Final round accuracy is zero point eight two without imitation and zero point eight six at the estimate. It peaks near zero point nine around gamma of two, then collapses.

Pay is for a random round, so the mean matters more. It peaks near one and a half, two points above the estimate.

Heavy imitation fails because the group's share of red remembers its past for about half of one plus e to the gamma rounds. At gamma four that exceeds the game.

The authors name their open problems. They promise proofs that some imitation beats none, and conjecture that group size matters little under the heuristic when only actions are visible.

They ask how imitation fares against a well informed central source, ideological agents who ignore the data, and unequal information quality.

Start with Appendix F: signals ran only online and the headline effect is marginal. Then Appendix N, on random subjects, which bears on whether psi absorbs lapses.

Then Figure H one, which narrows the Weak band. By our reading the swing drops to zero point four six, as the confound predicts. Last, Harel and coauthors, for the group size bound.