Appendix

Proof of Proposition 1:

Assuming that the nearest neighbor set consists of n_f = (1 - δ) n fair ratings and n_u = δ n unfair ratings (0 ≤δ < 0.5), the most disruptive unfair ratings strategy—in terms of influencing the sample median—is one where all unfair ratings are higher than the sample median of the contaminated set. In that case and for δ < 0.5, all the ratings that are lower than or equal to the sample median will have to be fair ratings. Then, the sample median of the contaminated set will be identical to the k^th order statistic of the set of n_f fair ratings, where k = (n+1)/2.

It has been shown (Cadwell, 1952) that, as the size n of the sample increases, the k^th order statistic of a sample drawn from a normal distribution N( ,σ) converges rapidly to a normal distribution with mean equal to the q^th quantile of the parent distribution where q = k/n. Therefore, for large rating samples n, under the worst possible unfair ratings strategy, the sample median of the contaminated set will converge to x_q where x_q is defined by:

Where

and Φ^-1(q) is the inverse standard normal CDF.

Given that , the asymptotic formula for the average reputation bias, achievable by δ 100% unfair ratings when fair ratings are drawn from a normal distribution N( ,σ) and unfair ratings follow the most disruptive possible unfair ratings distribution, is given by:

Proof of Proposition 2:

Assume that the entire buyer population is n, unfair raters are δ n, and the width of the reputation estimation time window is a relatively small W (so that, each rating within W typically comes from a different rater). Then, after applying frequency filtering to the nearest-neighbor set of raters, in a typical time window we expect to find:

• fair ratings, where ϕ(u) is the probability density function of fair ratings frequencies, and at most

• W δ n α f_cutoff unfair ratings, where 0 ≤ α ≤ 1 is the fraction of unfair raters with submission frequencies below fcutoff.

Therefore, the unfair/fair ratings ratio in the final set would be equal to:

(9)

where denotes the inflation of the unfair/fair ratings ratio in the final set relative to its value in the original set. The goal of unfair raters is to strategically distribute their rating frequencies above and below the cutoff frequency in order to maximize I. In contrast, the goal of the market designer is to pick the cutoff frequency f_cutoff so as to minimize I.

The cutoff frequency has been defined as the (1 - D) n^th order statistic of the sample of buyer frequencies, where D ≥ δ. For relatively large samples, this converges to the q-th quantile of the fair rating frequencies distribution, where q satisfies the equation:

(10)

From this point on, the exact analysis requires some assumptions about the probability density function of fair ratings frequencies. I assume a uniform distribution between F_min = f₀/1+s) and F_max = f₀ )1+s). Let S = F_max - F_min. Then, by applying the properties of uniform probability distributions to equation (9), I get the following expression of the inflation I of unfair ratings:

(11)

After some algebraic manipulation I find that and . This means that unfair raters will want to maximize a, the fraction of ratings that are less than or equal to fcutoff , while market makers will want to minimize D, i.e., set D as close as possible to an accurate estimate of the ratio of unfair raters in the total population. Therefore, at equilibrium, α = 1, D = δ and:

(12)

The above expression for the unfair/fair ratings inflation depends on the spread S of fair ratings frequencies. At the limiting cases we get and .

By substituting the above limiting values of I in equation (9), we get the final formula for the equilibrium relationship between δ, the ratio of unfair raters in the total population of buyers, and δ', the final ratio of unfair ratings remaining in the nearest-neighbor set using time windowing and frequency filtering: