Rating histories and ranking accuracy: model clarifications

Model readings

• Ratings keep accumulating: use a positive rating-arrival rate $g(\theta)>0$ for every seller type $\theta$, so the number of ratings $n_k(\theta)=\lfloor k g(\theta)\rfloor$ grows with time. This is the large-sample regime of the paper’s convergence and rate arguments. Source: cited publication:669–670, 745.
• Independent rating histories: conditional on a seller’s quality, ratings are independent draws from that quality’s rating distribution; different sellers’ histories are also independent. This makes explicit the independence used in the Appendix’s pairwise probability calculation. Source: cited publication:1638–1642.

Theorem 1 reading

• Adjacent pairs: the printed minimum through $i=M-1$ → a minimum through $i=M-2$, where $M$ is the number of ordered seller types. There is no next type after the last one.
• Strict quality comparisons: require upper-tail probabilities to increase strictly with quality only above the lowest rating. At the lowest rating, the upper-tail probability is one for every seller type.
• Impossible scores: allow the large-deviation cost to be $+\infty$ outside the rating support. Minimizing these costs gives the finite rate in Theorem 1. This avoids requiring every rating to have positive probability.

Average scores and the probability calculation

• Average of $n$ ratings: summing from $0$ through $n$ → summing from $0$ through $n-1$. The former has one extra term. This corrects the aggregate-score display.
• Appendix probability calculation: a continuum approximation → direct bounds for finite, independent rating histories. The upper and lower bounds differ by fixed multiplicative constants, which disappear when taking $-\log(\text{probability})/k$. The resulting exponential rate is unchanged.

From individual ratings to the seller population

• Population distribution: draw a seller’s type uniformly from the finite type set, draw that seller’s independent rating history, and average the ratings. The joint distribution of type and average score is the population state $\mu_k$. The proofs connect this description to the paper’s pairwise and overall ranking measures.
• Population-update notation: the printed recurrence mixes the time indices for adding a new rating and recomputing the seller’s average (cited publication:710–715). The formalization defines $\mu_k$ directly from these independent rating histories and proves convergence and ranking rates for that process.