Source
Miller, Resnick & Zeckhauser (2005), LIT-1518, read in NOTE-1139 from the authors’ final pre-publication draft: Proposition 1 and its proof (p. 8), the worked example (pp. 6–8) and footnote 12 (p. 9).
What was actually shown
- Truthful equilibrium. If the reference rater reports truthfully, rater i’s expected score from announcing a is Σ_n R(s_n | a) g(s_n | s), where s is what she saw. A strictly proper rule maximizes this only at a = s, so truth is a strict best reply. Stochastic relevance makes the posterior differ across announcements, so every lie is strictly worse.
- The example. g(h | l) = .536 and g(h | h) = .712. Under the log rule a rater who saw l expects −0.690 from reporting l and −0.760 from reporting h, although she thinks h the likelier report from her peer. Rewarding agreement would have paid her to say h.
- Uninformative equilibria. Against a reference rater who always says h, reporting h scores ln .712 and l scores ln .536, whatever was seen, so always reporting h is an equilibrium; likewise always l. The authors call the truthful equilibrium focal and propose that the center punish all raters when it detects an uninformative one.
Checked step by step when filed, and again on 2026-10-30 (notes below).
What this does not say
That raters reach the truthful equilibrium, or that the mechanism resists collusion, which the authors leave open. That it works when the center’s prior or signal model is wrong, or when raters hold private information the center does not elicit (their Yes-Man case). Or that the scores measure the quality of a report in any sense beyond its fit to the center’s model of how honest reports correlate.
Note of 2026-10-30: the second check and a corrected figure
The second check at the status review (ADR-065 §5) followed each step.
Proposition 1. If the reference rater is truthful, announcing a earns Σ_n g(n | s) S(g(· | a), n), where s is what the rater saw. A strictly proper rule makes this largest only at g(· | a) = g(· | s). Stochastic relevance makes g(· | a) differ from g(· | s) for every a ≠ s, so every lie is strictly worse.
The example. P(h) = .5 × .85 + .5 × .45 = .65, and P(H | h) = .425/.65 = .6538, so g(h | h) = .6538 × .85 + .3462 × .45 = .7115. P(H | l) = .075/.35 = .2143, so g(h | l) = .2143 × .85 + .7857 × .45 = .5357. A rater who saw l expects .5357 ln .5357 + .4643 ln .4643 = −0.6906 from reporting l, and .5357 ln .7115 + .4643 ln .2885 = −0.7595 from reporting h. The file had −0.754 for the second, which is what the posteriors rounded to .54 and .71 give (−0.7544), and how the paper reaches its −0.75. The conclusion holds with a wider margin.
The uninformative equilibria. Against a reference rater who always says h, reporting h scores ln .7115 = −0.340 and l scores ln .5357 = −0.624, whatever was seen. Against one who always says l, l scores ln .4643 = −0.767 and h scores ln .2885 = −1.243.
A script (numpy, in the session’s scratchpad) reproduced these figures and ran the search reported in the next note.
Note of 2026-10-30: the clause the narrowing drops
The title said, under its general hypotheses, that equilibria in which every rater reports the same signal regardless of what she observed also exist. The paper says only that “there may be non-truthful equilibria” (p. 9), and its example has them. Stated generally, the clause is false.
Against a reference rater who always announces m, a rater’s expected score from announcing a is S(g(· | a), m), whatever she saw. So all raters always announcing m is an equilibrium only if a = m maximises it.
A counterexample, under the quadratic rule S(p, m) = 2p_m − Σ_n p_n², which is strictly proper. There are two types with prior (.375, .625) and four signals, drawn independently given the type, with f(· | t₁) = (.220, .094, .023, .663) and f(· | t₂) = (.675, .022, .130, .173). The posteriors over a reference rater’s signal are g(· | 1) = (.6006, .0338, .1125, .2531), g(· | 2) = (.3477, .0738, .0530, .5255), g(· | 3) = (.6313, .0289, .1197, .2200) and g(· | 4) = (.3579, .0722, .0554, .5145). They are distinct, so stochastic relevance holds. Against a reference rater who always announces 1, the scores by announcement are (.7626, .2901, .8005, .3147), so the best reply is 3. Against one who always announces 2 they are (−.3710, −.2577, −.4043, −.2567), best reply 4. Against 3 they are (−.2136, −.2992, −.2227, −.2902), best reply 1. Against 4 they are (.0677, .6457, −.0221, .6279), best reply 2. No signal is its own best reply, so no equilibrium has every rater report one signal.
A random search over models with 2 to 4 signals found 8 such models in about 20,000 under the quadratic rule, none of them with two signals, and none at all under the log rule. For two signals the clause does hold in general, but that is the record’s own derivation and is not filed. The general clause is shown false, so it is recorded here only (ADR-047’s kind 3), and the title now states the authors’ example.