NIPS 2018
Sun Dec 2nd through Sat the 8th, 2018 at Palais des Congrès de Montréal
Paper ID: 2286 A Mathematical Model For Optimal Decisions In A Representative Democracy

### Reviewer 1

The paper models representative democracy for n voters by dividing voters into L groups (often homogenous with K = n/L + O(1) members), having each group pick a representative, and choosing the issue at hand via majority vote of the representatives. The representative choice mechanism is imagined to pick an "above average" voter in terms of voting skill, but only weak assumptions on this mechanisms are required for the main theorems. If the cost of voting is constant, the main theorem is that optimal group size is O(1): more accurate decisions are made by many small groups, and the ideal group size is bounded as the number of voters n -> infinity. If the cost of voting and the benefit of a correct decision are both polynomial (in L and n, respectively), the ideal number of representatives is O(log n), with more precision with extra assumptions. This is a nonstandard paper for NIPS, but it tackles interesting questions and the results are nonobvious. I would argue for acceptance, and would argue more strongly if concerns about field match are ignored. It is possible there are c onnections between these results and ensembling techniques, or in using majority vote or other schemes to improve data quality when gathering data from humans. My main high level comment is that the proofs are all effective or very close to effective, and in particular Theorem 2 computes the complicated exact constant c. The proofs could be simplified if this effectiveness was dropped, and it's unclear what the benefit of knowing the form of the constant is since I am not confident that it's tight (there is no discussion of this). I'll leave this decision up to the authors. Comments: 1. I'm not sure what the imagined final paper structure is, since the supplementary material is just a longer version of the whole paper rather than structured as appendices. 2. Line 98 has "Cost(n) = n^q2", but should be "Ben(n)". 3. In the proof of lemma 3, you say "at least a factor µ/(1 − µ) bigger than the previous term". This should be p/(1-p). 4. Lemma 5's proof is full of unnecessary parentheses around 2p-1. 5. In the proof of lemma 5, worth a note that one of the inequalities is because p/(1-p) > 1. This is pretty subtle since you drop a floor without comment, and p/(1-p) > 1 is needed to make that safe. 6. Typo: "resteiction" (there are likely others). 7. In the proof of theorem 5, "p(L)" is unfortunate notation given that p = mu is used elsewhere.

### Reviewer 2

The paper studies group-based majority voting. There is a binary issue with a "correct" answer. Each voter has some probability of voting the right way. In the proposed set up, voters are organized into groups of K, and the final decision is a majority vote among the highest-probability voter in each group. (This is the "max" aggregation function; the authors also study some other options.) The main finding are that with constant voting cost the optimal group size is constant, while with polynomial voting costs it is near-linear. The model is reasonably natural and topical. On a technical level, the arguments are elementary probability calculations --- not trivial, but not particularly illuminating, either.