SlowGuess/NeurIPS_2023_OCR huggingface dataset:
To prove that majority is an EDMR, we use a coupling argument to show that the expected distortion of an alternative that is ranked first by  $k$  out of  $n$  voters is increasing in  $k$ . A direct consequence of this is that a majority winner is an expected-distortion-maximizing alternative.
Recall that  $u_{ij}$  is defined as the utility that voter  $i$  has for alternative  $j$  for  $i \in N$  and  $j \in \{1,2\}$ . Let  $\sigma^k$  be the preference profile for  $n$  voters that has alternative 1 ranked first by the first  $k$  voters (and only by them). Similarly, let  $\sigma^{k+1}$  be the preference profile for  $n$  voters that has alternative 1 ranked first by the first  $k$  voters and by the last voter (and only by them). To show that the majority winner is an EDMR, it is sufficient (by symmetry) to show that
$$
\underset {\mathbf {u} \triangleright \boldsymbol {\sigma} ^ {k}} {\mathbb {E}} \left[ \frac {\sum_ {i = 1} ^ {n} u _ {i 1}}{\max \left\{\sum_ {i = 1} ^ {n} u _ {i 1} , \sum_ {i = 1} ^ {n} u _ {i 2} \right\}} \right] \leq \underset {\mathbf {u} \triangleright \boldsymbol {\sigma} ^ {k + 1}} {\mathbb {E}} \left[ \frac {\sum_ {i = 1} ^ {n} u _ {i 1}}{\max \left\{\sum_ {i = 1} ^ {n} u _ {i 1} , \sum_ {i = 1} ^ {n} u _ {i 2} \right\}} \right].
$$
Note that we used  $\mathbb{E}_{\mathbf{u} \triangleright \sigma^k}$  to denote the expectation over a draw of a utility profile  $\mathbf{u}$  consistent with  $\sigma^k$ , and similarly for  $k + 1$ .
Next, define  $Z_{j} = \sum_{i=1}^{n-1} u_{ij}$  for  $j \in \{1,2\}$ . Furthermore, define  $\sigma_{-n}^{k}$  as a preference profile on  $n-1$  voters that has the first  $k$  voters ranking alternative 1 higher than alternative 2. Finally, let  $\mathbf{u}_{-n}$  be the truncated version of  $u$  that excludes the  $n$ th voter. By the law of total expectation, we can take an outer expectation over the utilities of the first  $n-1$  voters and an inner expectation over the utilities of the  $n$ th voter:
$$
\underset {\mathbf {u} \triangleright \boldsymbol {\sigma} ^ {k}} {\mathbb {E}} \left[ \frac {\sum_ {i = 1} ^ {n} u _ {i 1}}{\max  \left\{\sum_ {i = 1} ^ {n} u _ {i 1} , \sum_ {i = 1} ^ {n} u _ {i 2} \right\}} \right] = \underset {\mathbf {u} _ {- n} \triangleright \boldsymbol {\sigma} _ {- n} ^ {k}} {\mathbb {E}} \left[ \underset {u _ {n 1} \leq u _ {n 2}} {\mathbb {E}} \left[ \frac {Z _ {1} + u _ {n 1}}{\max  \left\{Z _ {1} + u _ {n 1} , Z _ {2} + u _ {n 2} \right\}} \middle | \mathbf {u} _ {- n} \right] \right],
$$
$$
\underset {\mathbf {u} \triangleright \boldsymbol {\sigma} ^ {k + 1}} {\mathbb {E}} \left[ \frac {\sum_ {i = 1} ^ {n} u _ {i 1}}{\max  \left\{\sum_ {i = 1} ^ {n} u _ {i 1} , \sum_ {i = 1} ^ {n} u _ {i 2} \right\}} \right] = \underset {\mathbf {u} _ {- n} \triangleright \boldsymbol {\sigma} _ {- n} ^ {k}} {\mathbb {E}} \left[ \underset {u _ {n 1} \geq u _ {n 2}} {\mathbb {E}} \left[ \frac {Z _ {1} + u _ {n 1}}{\max  \left\{Z _ {1} + u _ {n 1} , Z _ {2} + u _ {n 2} \right\}} \middle | \mathbf {u} _ {- n} \right] \right].
$$
By the definition of  $\sigma^k$  and  $\sigma^{k + 1}$ , the two final expressions only differ in the inequality between the random variables  $u_{n1}$  and  $u_{n2}$ . Furthermore, since  $Z_{1}$  and  $Z_{2}$  are both functions of  $\mathbf{u}_{-n}$ , they are both constants in the inner expectation. Therefore, to prove the desired result it is sufficient to show that for any constants  $Z_{1}, Z_{2} \geq 0$ ,
$$
\underset {u _ {n 1} \leq u _ {n 2}} {\mathbb {E}} \left[ \frac {Z _ {1} + u _ {n 1}}{\max \left\{Z _ {1} + u _ {n 1} , Z _ {2} + u _ {n 2} \right\}} \right] \leq \underset {u _ {n 1} \geq u _ {n 2}} {\mathbb {E}} \left[ \frac {Z _ {1} + u _ {n 1}}{\max \left\{Z _ {1} + u _ {n 1} , Z _ {2} + u _ {n 2} \right\}} \right].
$$
Assume first that  $f$  is a continuous distribution. Let  $f$  be the PDF of  $\mathcal{D}$ . The joint PDF of the minimum and maximum of two draws from  $\mathcal{D}$  is  $2f(x)f(y)$  for  $x \geq y$ . This means that the expectation can be rewritten as a double integral giving the desired result:
$$
\begin{array}{l} \underset {u _ {n 1} \geq u _ {n 2}} {\mathbb {E}} \left[ \frac {Z _ {1} + u _ {n 1}}{\operatorname* {m a x} \left\{Z _ {1} + u _ {n 1} , Z _ {2} + u _ {n 2} \right\}} \right] = \int_ {0} ^ {\infty} \int_ {y} ^ {\infty} \frac {Z _ {1} + x}{\operatorname* {m a x} \left\{Z _ {1} + x , Z _ {2} + y \right\}} 2 f (x) f (y) d x d y \\ \geq \int_ {0} ^ {\infty} \int_ {y} ^ {\infty} \frac {Z _ {1} + y}{\max  \left\{Z _ {1} + y , Z _ {2} + x \right\}} 2 f (x) f (y) d x d y \\ = \underset {u _ {n 1} \leq u _ {n 2}} {\mathbb {E}} \left[ \frac {Z _ {1} + u _ {n 1}}{\max  \left\{Z _ {1} + u _ {n 1} , Z _ {2} + u _ {n 2} \right\}} \right]. \\ \end{array}
$$
The inequality in the second line follows from a simple lemma in Appendix D, which takes advantage of the fact that  $x \geq y$ .
A similar derivation using the Lebesgue decomposition or the Riemann-Stieltjes integral gives the same result for general (not necessarily continuous)  $f$ .
When  $m \geq 2$ , majority is no longer defined, and simple extensions (like plurality) are neither an EDMR nor an EwMR. We therefore require a different strategy for such  $m$ .
4 From Expected Distortion to Expected Welfare
Despite failing to extend beyond the case of two alternatives, Theorem 3.1 does give some hope that perhaps an expected-welfare-maximizing rule, which is equivalent to an expected-distortion-maximizing rule when  $m = 2$ , can provide good expected distortion for larger  $m$ . This is not the case in general, however: If we consider the space of all distributions  $\mathcal{D}$  supported on  $[0,1]$ , an EwMR may not provide a good approximation to an EDMR. In fact, there exist distributions and preference profiles for which the expected distortion of an EwMR is an  $\Omega(m)$  factor worse than that of an EDMR, as shown in Appendix E.
Theorem 4.1 There exists a constant  $C > 0$  such that for every  $m \geq 5$  there exist a number of voters  $n$ , a distribution  $\mathcal{D}$  supported on  $[0,1]$ , and a preference profile  $\sigma$ , such that  $\mathbb{E}[\mathrm{ddist}(\mathrm{ewmr},\sigma)] \leq \frac{C}{m} \cdot \mathbb{E}[\mathrm{ddist}(\mathrm{edmr},\sigma)]$ .
It is worth noting that the rule that chooses one of the  $m$  alternatives uniformly at random has expected distortion no more than a factor  $O(m)$  worse than that of an EDMR. Therefore, this theorem shows that for some distributions and preference profiles, an EwMR performs almost as poorly as the rule that chooses an alternative uniformly at random.
One distinctive trait of the distribution used to prove Theorem 4.1 is that both the values that the distribution can take and their respective probabilities depend on  $n$  and  $m$ . This is perhaps unnatural, as we might expect that as the numbers of voters and alternatives increase, the i.i.d. distribution from which each voter draws utilities is fixed. In fact, with such an assumption that the distribution  $\mathcal{D}$  is independent of  $n$  and  $m$ , an EwMR will be an  $\varepsilon$ -approximation for an EDMR for sufficiently large  $n$  (regardless of the value of  $m$ ). Furthermore, with the additional assumptions that the distribution is continuous and that the derivative of the CDF is bounded from below, an EwMR will also be an  $\varepsilon$ -approximation for an EDMR for sufficiently large  $m$  (regardless of the value of  $n$ ). Therefore, asymptotically in both  $n$  and  $m$ , an EwMR does perform well by the metric of expected distortion; the proof is relegated to Appendix F.
Theorem 4.2 Assume  $\mathcal{D}$  is supported on  $[0,1]$  and has constant (relative to  $n,m$ ) mean  $\mu$  and variance  $s^2$ . Then for every  $\varepsilon > 0$ , there exists  $n_0$  such that if  $n \geq n_0$ , then  $\mathbb{E}[\mathrm{ddist}(\mathrm{ewmr},\pmb{\sigma})] \geq (1 - \varepsilon)\mathbb{E}[\mathrm{ddist}(\mathrm{edmr},\pmb{\sigma})]$ .
Furthermore, if  $\mathcal{D}$  has a continuously differentiable CDF  $F$  satisfying  $\frac{dF}{dx} > 0$ , then there exists  $m_0$  such that the same result holds if  $m \geq m_0$ .
Note that it is not a priori obvious that either large  $n$  or large  $m$  are sufficient conditions in Theorem 4.2, and in fact the proofs rely on different forms of concentration. For large  $n$ , the key to the proof is that sums of i.i.d. bounded random variables concentrate by Hoeffding's inequality. For large  $m$ , the key idea is that, for continuous distributions, the variance of the order statistics goes to 0 as the number of samples  $m$  grows.
5 Distribution-Independent Expected-Welfare Maximization
Theorem 4.2 gives a computationally simple way to calculate an "almost EDMR" for fixed  $\mu$  and  $s^2$  by simply calculating an EwMR, which—as noted in Section 2—can be done by using a scoring rule based on the expected order statistics of  $\mathcal{D}$ . This is satisfying because it has a practical form and performs well for both metrics, namely expected distortion and expected welfare. A fundamental flaw with such a rule, however, is that it relies on knowing the entire underlying distribution  $\mathcal{D}$ , which might not be realistic in practice. Our strategy for approximating an EDMR, therefore, is to seek a distribution-independent approximation to an EwMR, that is, a rule that simultaneously provides
guarantees for every distribution (at least under some assumptions). To this end, we leverage the following observation.
Observation 5.1 If a voting rule  $f$  is a  $\beta$ -EWMR, and an EWMR is a  $\gamma$ -EDMR, then  $f$  is a  $\beta \gamma$ -EDMR.
In particular, for any  $\varepsilon > 0$ , Theorem 4.2 yields  $\gamma = (1 - \varepsilon)$  for sufficiently large  $n$  or  $m$ . Our goal in this section, therefore, is to seek a rule  $f$  for which we can establish a value of  $\beta$  in a distribution-independent way.
Our first result is negative: It is impossible for a single voting rule to achieve high expected social welfare for all distributions supported on  $[0,1]$ .
Theorem 5.2 For every  $0 < \alpha \leq 1$  there exist  $n, m, \sigma$  and two distributions  $\mathcal{D}_1, \mathcal{D}_2$  that are both supported on  $[0,1]$  such that no alternative is an  $\alpha$ -EWMR for both  $D_1$  and  $D_2$  under  $\sigma$ .
We prove Theorem 5.2 in Appendix G. We will circumvent Theorem 5.2 in two natural ways. Our first way will do so by restricting the class of distributions. Specifically, we first focus on the class of symmetric distributions supported on  $[0,1]$ , that is, distributions  $\mathcal{D}$  such that  $\operatorname{Pr}_{x\sim \mathcal{D}}(x\leq \frac{1}{2} -\varepsilon) = \operatorname{Pr}_{x\sim \mathcal{D}}(x\geq \frac{1}{2} +\varepsilon)$  for all  $\varepsilon \in [0,\frac{1}{2} ]$
The fact that an EwMR is given by a scoring rule [Boutilier et al., 2015], albeit one that depends on the distribution, suggests that defining a scoring rule in a distribution-independent way may be a good approach. Consider the following scoring rule, whereby voters give their approval to the top half of their ranking.
Definition 5.3 Top-half approval is the scoring rule that assigns a score of 1 to alternatives ranked in the highest  $\left\lceil \frac{m}{2} \right\rceil$  positions and 0 to all other alternatives.
It turns out that top-half approval gives good guarantees for the subclass of symmetric distributions, as we prove in Appendix H.
Theorem 5.4 For all  $n, m$ , top-half approval is a  $\frac{1}{3}$ -EWMR for all symmetric distributions  $\mathcal{D}$ .
Given the attractiveness of top-half approval, a natural follow-up question is whether another distribution-independent rule can give an even better EwMR approximation for every symmetric distribution. While we cannot preclude this possibility altogether, we do give an upper bound (negative result) on the feasible approximation in Appendix I, which holds not just for scoring rules but for any voting rule.
Theorem 5.5 No distribution-independent voting rule is an  $\alpha$ -EWMR for all symmetric distributions  $\mathcal{D}$ , for any  $\alpha > \sqrt{\frac{1}{3}}$ .
5.1 Binomial Voting
Top-half approval clearly leaves some information on the table, as it ignores the ranking of alternatives within each half. Let us refine it, then, by considering a variant that accounts for this additional information. As we will see, this refinement will allow us to dispense with the assumption of a symmetric distribution.
Definition 5.6 Binomial voting is the scoring rule that assigns a score of  $\sum_{\ell = k}^{m}\binom{m}{\ell}$  to an alternative that is ranked  $k$
Binomial voting can be seen as a "smoother" version of top-half approval. Indeed, because the binomial coefficients are largest in the vicinity of  $\frac{m}{2}$ , the top, say,  $\frac{m}{2} - \sqrt{m}$  positions have scores close to the maximum, and the bottom  $\frac{m}{2} - \sqrt{m}$  positions have scores close to the minimum, with the transition occurring rapidly in between.
Our main result for binomial voting is the following, which we prove in Appendix J. This result circumvents Theorem 5.2 not by restricting the class of distributions, but rather by giving a guarantee that depends on the median of the distribution.
Theorem 5.7 Let  $\mathcal{D}$  be a distribution supported on  $[0,1]$  whose largest median is  $\nu$ , i.e.,  $\nu = \sup \left\{y \mid \operatorname*{Pr}_{x \sim \mathcal{D}}(x \leq y) \leq \frac{1}{2}\right\}$ . Then binomial voting is a  $\frac{\nu}{2}$ -EWMR for  $\mathcal{D}$ .
Note that any symmetric distribution has a largest median of  $\nu \geq \frac{1}{2}$ , so for such distributions Theorem 5.7 almost recovers the guarantee given by top-half approval for symmetric distributions. Needless to say, the guarantee for binomial voting (Theorem 5.7) is much more powerful than that for top-half approval (Theorem 5.4), as the former applies to any distribution and not only to symmetric ones. Note that while Theorem 5.7 gives a guarantee that depends on the median of the distribution  $\mathcal{D}$ , because binomial voting is distribution-independent, it achieves this guarantee even when nothing is known about the distribution, including its median.
By putting together Theorem 4.2, Observation 5.1, and Theorem 5.4, we get the following corollary, which ties these results back to expected distortion.
Corollary 5.8 Let  $\mathcal{D}$  be supported on  $[0,1]$  with constant (relative to  $n,m$ ) mean, variance, and largest median  $\nu$ . Then for every  $\varepsilon$ , there exists  $n_0$  such that if  $n \geq n_0$ , binomial voting is a  $(\frac{1}{2} - \varepsilon)\nu$ -EDMR for  $\mathcal{D}$ .
Furthermore, if  $\mathcal{D}$  has a continuously differentiable CDF  $F$  satisfying  $\inf_x\frac{dF(x)}{dx} >0$ , then there exists  $m_0$  such that the same result holds if  $m\geq m_0$ .
5.2 Beyond the Median of the Distribution
A natural generalization of binomial voting is to consider quantiles other than the median. For example, one might wish to guarantee a  $\beta$ -EWMR where  $\beta$  depends on the third quartile. The following corollary generalizes binomial voting to other distribution-free scoring rules with expected social welfare guarantees based on the quantile of interest. All proofs for this section can be found in Appendix K.
Corollary 5.9 Let  $p \in (0,1]$ . Consider the scoring rule that assigns a score of  $\sum_{\ell = k}^{m} \binom{m}{\ell} (1 - p)^{\ell} p^{m - \ell}$  to an alternative ranked  $k$ . This scoring rule is a  $(1 - p)Q$ -EWMR, where  $Q$  is the  $p$ -quantile of  $\mathcal{D}$ , i.e.  $Q = \sup \{y \mid \operatorname{Pr}_{x \sim \mathcal{D}}(x \leq y) \leq p\}$ .
So far we have focused on distribution-independent rules where the rules stay the same but the bounds change depending on the underlying distribution. By contrast, the known EwMR of Boutilier et al. [2015] that requires complete knowledge about the underlying distributions looks very different for each distribution. It turns out that we can interpolate between these two extremes by incorporating partial information that might be known about the distribution  $\mathcal{D}$ . If we are given access to multiple quantiles of the distribution, then we can extend binomial voting to a distribution-dependent rule as follows.
Definition 5.10 Suppose we have access to quantiles  $Q_{1}, \ldots, Q_{T}$  corresponding to fractions  $0 < p_{1} < p_{2} < \dots < p_{T} \leq 1$ , where  $Q_{k} = \sup \{y \mid \operatorname{Pr}_{x \sim \mathcal{D}}(x \leq y) \leq p_{k}\}$ . Define generalized binomial voting as the scoring rule that gives to the alternative ranked  $k$  a score of
$$
s _ {k} := \sum_ {t = 1} ^ {T} Q _ {t} \left(\sum_ {\ell = k} ^ {m} \binom {m} {\ell} (1 - p _ {t}) ^ {\ell} p _ {t} ^ {m - \ell} - \sum_ {\ell = k} ^ {m} \binom {m} {\ell} (1 - p _ {t + 1}) ^ {\ell} p _ {t + 1} ^ {m - \ell}\right).
$$
Theorem 5.11 If the distribution  $\mathcal{D}$  has quantiles  $Q_{1},\ldots ,Q_{T}$  corresponding to fractions  $0 < p_1 < p_2 < \dots < p_T\leq 1$  (define  $p_{T + 1} = 1$ ), then generalized binomial voting is a  $\beta$ -EWMR for  $\beta = \sum_{s = 1}^{T}Q_{s}(p_{s + 1} - p_{s})$ .
As an example, suppose the only information known about the underlying distribution is that the values  $Q_{1}, Q_{2}, Q_{3}$  are the three quartiles (25%, 50%, 75%). Then generalized binomial voting is a  $\frac{Q_{1} + Q_{2} + Q_{3}}{4}$ -EWMR. Whenever  $Q_{3} > Q_{2}$  or  $Q_{1} > 0$ , this is strictly stronger than vanilla binomial voting, which is guaranteed to be a  $\frac{Q_{2}}{2}$ -EWMR. Given more quantiles, generalized binomial voting gives weakly stronger approximations of an EWMR. Intuitively, generalized binomial voting is approximating the distribution  $\mathcal{D}$  with left-oriented rectangles based on the quantile information. In fact, with sufficient quantile information, the rule will exactly approach an EWMR in the limit as  $T \to \infty$ . This result follows from generalized binomial voting being an approximation based on quantile rectangles, which in the limit becomes exactly the integral for expectation.
Corollary 5.12 For the quantiles  $(p_1, p_2, \ldots, p_T) = (\frac{1}{T}, \frac{2}{T}, \ldots, \frac{T}{T})$ , generalized binomial voting approaches an expected-welfare-maximizing rule as  $T \longrightarrow \infty$ .
In this section we discuss limitations of our work and present some extensions thereof.
In our view, the main limitation of our work is the simplifying assumption that voters' utilities for all alternatives are independent and identically distributed. In reality, some alternatives are inherently stronger than others. In addition, utilities are typically correlated; for example, in US politics, if a voter's favorite candidate is a Republican, they are likely to prefer another Republican candidate to a Democratic candidate.
That said, similar i.i.d. assumptions are commonly made in computational social choice and mechanism design [see, e.g., Dickerson et al., 2014]. Moreover, there is a large body of literature that relies on the impartial culture assumption [Tsetlin et al., 2003, Pritchard and Wilson, 2009], under which rankings are drawn independently and uniformly at random; when rankings are induced by utilities, the i.i.d. assumption on utilities gives rise to impartial culture. While these assumptions are admittedly strong, they lead to clean insights and solutions that often generalize well.
6.2 Benchmarks
We chose to define our benchmarks of expected distortion and expected welfare with respect to an arbitrary fixed preference profile  $\sigma$ . Our main results therefore hold for any choice of preference profile  $\sigma$ . In other words, our results should be interpreted as being in the worst-case regime for preference profiles while being in the average-case regime for utilities. We chose to use the EDMR and EwMR as benchmarks in our main theorems because, by definition, these are the rules which maximize our objectives. However, the proofs of our results do not fundamentally rely on the use of these benchmarks. In fact, many of our main results (including Theorem 4.2 and Theorem 5.7) can be adapted to directly lower bound the actual values of expected distortion and expected welfare. More details can be found in Appendix M.
6.3 Extensions
We have shown that an EwMR can be a good approximation to an EDMR. One follow-up question is whether it is possible to do better by directly computing an EDMR. As noted earlier, a potential EDMR is to choose the deterministic alternative with the highest expected distortion. Unfortunately, this rule requires exponential computation time. One way to circumvent this is to use i.i.d. samples from the underlying distribution to empirically approximate the expected distortion of each alternative, as shown in Appendix L. With polynomially many samples from  $\mathcal{D}$  and polynomial time, such a rule is a  $(1 - \varepsilon)$ -EDMR with high probability. However, it may be unrealistic to expect i.i.d. sample access to the underlying distribution, which limits the practicality of this approach.
Theorem 5.2 states that for a given constant  $\alpha$ , no rule can be an  $\alpha$ -EWMR for all distributions. Despite this, one might hope that a rule that is an EWMR for one distribution could also be a good approximation for an EWMR for another similar distribution. This is in fact not the case. Specifically, for any  $\alpha, \varepsilon > 0$ , an EWMR for a distribution  $\mathcal{D}$  may not even be an  $\alpha$ -EWMR for another distribution that has total variation distance only  $\varepsilon$  away from  $\mathcal{D}$ . Surprisingly, though, a voting rule called Borda count, which is an EWMR for the uniform distribution, is in fact a  $(1 - \varepsilon)$ -EWMR for all distributions that are sufficiently close to uniform. For more details, see Appendix N.
Finally, Theorem 5.7 gives a strong positive result for the expected welfare of binomial voting that depends on the median of the distribution. In Appendix O, we show that for any distribution (regardless of its median) and any  $n, m$ , plurality is an  $\alpha$ -EWMR for  $\alpha = \max \left\{\frac{1}{n}, \frac{1}{m}\right\}$ . We conjecture that plurality may be "tight" in the sense that no rule can provide a better guarantee, but we leave this question open for future work.
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A Randomized Rules
As mentioned in Section 2, the definition of distributional distortion (2) can be extended to randomized voting rules by defining it to be the random variable
$$
\operatorname {d d i s t} (f, \boldsymbol {\sigma}) := \frac {\mathbb {E} [ \operatorname {s w} (f (\boldsymbol {\sigma}) , \mathbf {u}) ]}{\max  _ {j \in A} \operatorname {s w} (j , \mathbf {u})}, \tag {3}
$$