SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The Distortion of Binomial Voting Defies Expectation*
Yannai A. Gonczarowski†
Gregory Kehne†
Ariel D. Procaccia $^{8}$
Ben Schiffer
Shirley Zhang
In computational social choice, the distortion of a voting rule quantifies the degree to which the rule overcomes limited preference information to select a socially desirable outcome. This concept has been investigated extensively, but only through a worst-case lens. Instead, we study the expected distortion of voting rules with respect to an underlying distribution over voter utilities. Our main contribution is the design and analysis of a novel and intuitive rule, binomial voting, which provides strong distribution-independent guarantees for both expected distortion and expected welfare.
In an election, voters report their preferences by casting ballots. Under the ubiquitous plurality rule, each voter names a single alternative, whereas other rules—such as the badly named ranked-choice voting,[7] whose adoption is rapidly expanding in the United States—require voters to rank the alternatives. However, even these ostensibly expressive ordinal ballots (whereby voters rank the alternatives) fail to capture voters' intensity of preference under truthful reporting.
If voters could report utility functions that are comparable to each other, then we would want to select socially desirable alternatives with respect to these utilities, but it is typically impractical to expect voters to compute and report such utilities. This creates a tension between the limited information available to the voting rule (through the report of ordinal ballots only) and its goal (good outcomes with respect to latent cardinal utilities).
A significant body of work in computational social choice aims to understand and alleviate this tension [Anshelevich et al., 2021]. It revolves around the notion of distortion, defined as the worst-case ratio between the utilitarian social welfare (sum of utilities) of the voting rule's outcome and that of the welfare-maximizing alternative. The worst case is taken over rankings, which serve as input to the voting rule, and over utilities that are consistent with these rankings.
*Gonczarowski gratefully acknowledges research support by the Harvard FAS Inequality in America Initiative. Procaccia gratefully acknowledges research support by the National Science Foundation under grants IIS-2147187, IIS-2229881 and CCF-2007080; and by the Office of Naval Research under grant N00014-20-1-2488. Schiffer was supported by an NSF Graduate Research Fellowship. Zhang was supported by an NSF Graduate Research Fellowship.
$^{\dagger}$ Department of Economics and Paulson School of Engineering and Applied Sciences, Harvard University | E-mail: yannai@gonch.name.
$^{\ddagger}$ Department of Computer Science, University of Texas at Austin | E-mail: gkehne@utexas.edu.
$^{\S}$ Paulson School of Engineering and Applied Sciences, Harvard University | E-mail: ariel-pro@seas.harvard.edu.
Department of Statistics, Harvard University | E-mail: bschiffer1@g.harvard.edu.
Paulson School of Engineering and Applied Sciences, Harvard University | E-mail: szhang2@g.harvard.edu.
7Also known as "instant-runoff voting" or "alternative vote."
As is often the case with worst-case analysis, however, the classic notion of distortion is arguably too conservative. In particular, nontrivial guarantees require restrictive assumptions (see Section 1.2), and so this type of analysis may not help identify appealing voting rules.
With this difficulty in mind, we focus on expected distortion. Its definition includes the same ratio as before, and we are still interested in the worst case over rankings. However, we now take the conditional expectation over utilities consistent with the rankings, given an i.i.d. distribution over the utilities. Of course, it might not be realistic for such a distribution to be known a priori, $^{8}$  and thus we search for voting rules that are distribution independent $^{9}$ : rules that choose the outcome in a way that does not depend on this distribution, even though their guarantees are stated in terms of this distribution.
Overall, our goal is to design distribution-independent voting rules that provide appealing expected distortion guarantees. We uncover novel, potentially practical, voting rules with these properties.
We start by considering the important case of two alternatives (e.g., US presidential elections or yes/no decisions) in Section 3. We show that, for any underlying distribution, the majority rule optimizes both expected distortion and expected social welfare.
This result suggests that maximizing expected social welfare may be a good approach for optimizing expected distortion. Indeed, our main result in Section 4 is that, under mild conditions on the underlying distribution and for a sufficiently large number of voters or alternatives, the expected-welfare-maximizing rule optimizes expected distortion almost perfectly.
The expected-welfare-maximizing rule, however, is tailored to the underlying distribution and relies on intimate knowledge thereof. Our aim is therefore to approximately optimize expected welfare via a distribution-independent rule.
In Section 5, we design and analyze such a rule, which belongs to the family of (positional) scoring rules. Under this rule, which we call binomial voting, each voter awards  $\sum_{\ell = k}^{m}\binom{m}{\ell}$  points to the alternative ranked in the  $k$ th position, where  $m$  is the number of alternatives, and the alternative awarded the largest number of points overall is selected; note that this rule is distribution independent. Our main result of Section 5 is that for any underlying distribution supported on  $[0,1]$  with (largest) median  $\nu$ , binomial voting provides a multiplicative  $\frac{\nu}{2}$ -approximation to the optimal expected welfare. Combining this result with that of Section 4, it follows that binomial voting gives almost the same  $\frac{\nu}{2}$  guarantee for expected distortion, when the number of voters or alternatives is sufficiently large.
It is worth noting that binomial voting is not an outlandish rule designed purely to achieve low expected distortion. On the contrary: as a positional voting rule, it inherits the desirable properties of this family. In fact, positional scoring rules are characterized by a number of natural axioms [Young, 1975]. Furthermore, we are aware of very few positional scoring rules that have received attention in their own right, as it is typically difficult to justify any specific choice of scores; in this sense, the expected distortion framework can be seen as a way of pinpointing particularly useful parameters. In summary, we identify binomial voting as an unusually attractive rule when viewed through the lens of expected distortion.
The literature on (worst-case) distortion [Anshelevich et al., 2021] can generally be partitioned into two threads. In the first thread [Procaccia and Rosenschein, 2006, Boutilier et al., 2015, Caragiannis et al., 2017, Mandal et al., 2019, Ebadian et al., 2022], it is assumed that voters have normalized utilities, that is, for each voter, the sum of utilities is one. Even with this restrictive assumption, deterministic voting rules cannot give nontrivial distortion bounds [Caragiannis et al., 2017], and the best possible distortion for randomized rules is  $\Theta(1/\sqrt{m})$  [Ebadian et al., 2022]. In the second thread [Anshelevich and Postl, 2017, Gross et al., 2017, Anshelevich et al., 2018, Gkatzelis et al., 2020, Kizilkaya and Kempe, 2022], known as metric distortion, it is assumed that utilities (rather, costs) are induced by an underlying metric space. While some well-known voting rules have constant
<sup>8</sup>For this reason, we avoid calling this distribution a Bayesian prior.
9This term is inspired by the literature on prior-independent mechanisms within mechanism design.
distortion in this setting [Anshelevich et al., 2018], the metric assumption is arguably difficult to justify in most domains of interest. By contrast, we make no assumptions on utilities.
Our work is most closely related to that of Boutilier et al. [2015]. While their most substantial results pertain to worst-case distortion, one of their results deals with a distributional setting that can be seen as the starting point for our work. They show that the voting rule that maximizes expected social welfare is a scoring rule whose scores depend on the underlying distribution; we will revisit and build on this result. However, they do not study expected distortion, nor do they explore (in this context) voting rules that are agnostic to the distribution.
Previous papers that analyze expected distortion include those of Cheng et al. [2017, 2018]. However, their papers are fundamentally different. For one, they study metric distortion. More importantly, they focus on one intuitive but very specific distribution, where the positions of alternatives in the underlying metric space are drawn uniformly at random from the voter positions. By contrast, we study general (i.i.d.) distributions over utilities and design distribution-independent voting rules. The work of Ghodsi et al. [2019] is more distantly related: they also analyze expected distortion in the metric setting, assuming that voters abstain with some probability.
By replacing worst-case analysis with expectation, our work introduces a Bayesian view canonical to economic theory into the literature on distortion. The Wilson doctrine [Wilson, 1987] advocates for the use of mechanisms that require as little prior information as possible. Distribution-independent (sometimes called prior-independent) mechanisms are, through this lens, the most desirable ones as they require no prior information at all, and have been studied within economics [see, e.g., McAfee, 1992, Segal, 2003] as well as within computer science [see, e.g., Hartline and Roughgarden, 2009, Devanur et al., 2011, Babaioff et al., 2018]. Our work can also be seen as belonging to a recent push within computer science on "beyond worst-case" analysis of algorithms [Roughgarden, 2021].
Let there be a set  $N = \{1,2,\dots ,n\}$  of voters and a set  $A = \{1,2,\ldots ,m\}$  of alternatives. Each voter  $i$  has utility  $u_{ij}$  for each alternative  $j$ . The voter utilities collectively form a utility profile  $\mathbf{u}$ , and the utilitarian social welfare for alternative  $j$  is denoted as  $\mathrm{sw}(j,\mathbf{u}) = \sum_{i\in N}u_{ij}$ . Each voter's utilities induce a ranking  $\sigma_{i} = \sigma_{i}(\mathbf{u})$  over alternatives, where  $j$  is ranked higher than  $k$  in  $\sigma_{i}$  only if  $u_{ij}\geq u_{ik}$ . Let  $\pmb {\sigma} = (\sigma_1,\dots ,\sigma_n)$  be the preference profile consisting of the rankings of all voters.
A (deterministic) voting rule  $f$  takes as input a preference profile and outputs a winning alternative  $f(\sigma) \in A$ . One can also define a randomized voting rule, which returns a distribution over alternatives. The definitions given below easily generalize to randomized rules, by considering their expected social welfare, where the expectation is taken over the randomness of the rule. We focus on deterministic rules, however, for two reasons: first, for ease of exposition, and second, because deterministic rules are in fact as powerful as randomized rules in our setting, as we show in Appendix A.
The worst-case distortion of  $f$  intuitively measures the loss in social welfare incurred by  $f$ , and is given by:
$$
\sup  _ {\mathbf {u}} \frac {\max  _ {j \in A} \operatorname {s w} (j , \mathbf {u})}{\operatorname {s w} (f (\boldsymbol {\sigma} (\mathbf {u})), \mathbf {u})}. \tag {1}
$$
The numerator is the optimal social welfare, whereas the denominator is the social welfare of the outcome under  $f$ ; the worst case of this ratio is taken over utility profiles  $u$ . Note that a voting rule  $f$  minimizes this expression if and only if it maximizes its inverse, and, more generally, that inverting this ratio does not change any of the results for worst-case distortion.
2.1 Distributional and Expected Distortion
Our point of departure is that we introduce an "average-case" setting; for the rest of this paper, we will assume that the voters' utilities for the alternatives are drawn i.i.d. from an underlying distribution  $\mathcal{D}$ . Unless otherwise noted, we will also assume that  $\mathcal{D}$  is supported on  $[0,1]$ . We will frequently use  $\mu$  and  $s^2$  to denote the mean and variance of  $\mathcal{D}$ , respectively.
${}^{10}$  Tie breaking can be performed arbitrarily, and is of no consequence for our results.
Let  $\mathbf{u} \triangleright \sigma$  denote that the utility vector  $\mathbf{u}$  is consistent with the preference profile  $\sigma$ . Given a preference profile  $\sigma$ , a utility vector  $\mathbf{u} \triangleright \sigma$  is drawn with respect to the distribution  $\mathcal{D}$  as follows. Each voter  $i \in N$  draws  $m$  i.i.d. utilities from  $\mathcal{D}$ , and these  $m$  utilities are assigned from highest to lowest to the  $m$  alternatives according to the order of  $\sigma_{i}$ , the ranking of voter  $i$  under  $\sigma$ . We define the distributional distortion under  $\mathcal{D}$  of a voting rule  $f$  for a preference profile  $\sigma$  as the following random variable, in the probability space defined by the above construction (for drawing  $\mathbf{u} \triangleright \sigma$  with respect to the distribution  $\mathcal{D}$ ):
$$
\operatorname {d d i s t} (f, \boldsymbol {\sigma}) = \frac {\operatorname {s w} (f (\boldsymbol {\sigma}) , \mathbf {u})}{\max  _ {j \in A} \operatorname {s w} (j , \mathbf {u})}. \tag {2}
$$
Note that this random variable representation of distributional distortion is in contrast to the single number representation in worst-case distortion. For the rest of this paper, when we refer to the distortion of a rule  $f$ , we are referring to the distributional distortion. We define the expected distortion of  $f$  for  $\sigma$  as  $\mathbb{E}\big[\mathrm{dist}(f,\sigma)\big]$ .
In Equation (2), we have defined distortion with the maximum in the denominator of the fraction. This means that the expected distortion of a voting rule lies in the range  $[0,1]$ , and a larger value of expected distortion (closer to 1) corresponds to a better voting rule. Note that this choice stands in contrast to Equation (1). As noted previously, inverting the ratio does not change the optimal rule  $f$  for worst-case distortion. However, this no longer holds true when considering expected distortion. For a random variable  $X$ , in general  $\mathbb{E}\left[\frac{1}{X}\right] \neq \frac{1}{\mathbb{E}[X]}$ , and therefore rules that minimize  $\mathbb{E}[X]$  can look very different compared to rules that maximize  $\mathbb{E}\left[\frac{1}{X}\right]$ . The reason for our choice is twofold. First, if the ratio were inverted then the random variable would be unbounded, which would imply that its expectation could be infinite even when the distribution  $\mathcal{D}$  is bounded on  $[0,1]$ . It turns out that this might be the case even for nice distributions such as beta distributions, as we prove in Appendix B. Second, if the ratio were inverted then there would exist distributions for which the best deterministic rule performs arbitrarily worse than the best randomized rule, even in situations where we would normatively expect the two rules to behave the same. That would prevent us from focusing on deterministic rules, as we are able to do here. For more details, see Appendix C.
We define an expected-distortion-maximizing rule (henceforth, an EDMR) as a rule  $f$  that, for every preference profile  $\sigma$ , maximizes the expected distortion under that profile. In other words, an expected-distortion-maximizing rule  $f$  satisfies
$$
\mathbb {E} \bigl [ \mathrm {d d i s t} (f, \boldsymbol {\sigma}) \bigr ] = \max  _ {g} \mathbb {E} \bigl [ \mathrm {d d i s t} (g, \boldsymbol {\sigma}) \bigr ]
$$
for all preference profiles  $\sigma$ , where the maximum is taken over all possible voting rules  $g$ . We write  $\operatorname{edmr}(\cdot)$  to refer to an EDMR.
2.2 Distributional and Expected Social Welfare
Similarly to Boutilier et al. [2015], we also wish to consider expected social welfare. To this end, we define the distributional social welfare of a rule  $f$  for preference profile  $\sigma$  as the random variable
$$
\mathrm {d s w} (f, \boldsymbol {\sigma}) = \mathrm {s w} (f (\boldsymbol {\sigma}), \mathbf {u}),
$$
in the same probability space as before (defined by the construction for drawing  $\mathbf{u} \triangleright \pmb{\sigma}$  with respect to a distribution  $\mathcal{D}$ ). We define the expected social welfare of  $f$  for  $\pmb{\sigma}$  as  $\mathbb{E}\left[\mathrm{dsw}(f, \pmb{\sigma})\right]$ .
An expected-welfare-maximizing rule (henceforth, an EwMR) is a rule  $f$  that, for all preference profiles  $\sigma$ , maximizes the expected social welfare, i.e.,  $f$  such that for all  $\sigma$ ,
$$
\mathbb {E} \left[ \mathrm {d s w} (f, \boldsymbol {\sigma}) \right] = \max  _ {g} \mathbb {E} \left[ \mathrm {d s w} (g, \boldsymbol {\sigma}) \right].
$$
We write  $\mathrm{ewmr}(\cdot)$  to refer to an EwMR.
The family of (positional) scoring rules consists of voting rules that assign a vector of scores  $(s_1, \ldots, s_m)$  to alternatives, from highest ranked through lowest ranked, respectively. The alternative that is chosen is the alternative that has the highest total score when adding up all of the rankings across the  $n$  voters, with ties broken arbitrarily. Boutilier et al. [2015] show that a scoring rule using scores
$$
\left. \left(s _ {1}, \dots , s _ {m}\right) = \left(\mathbb {E} \left[ X _ {(m)} \right], \mathbb {E} \left[ X _ {(m - 1)} \right], \dots , \mathbb {E} \left[ X _ {(1)} \right]\right)\right)
$$
is an EwMR, where  $\mathbb{E}\big[X_{(m - k)}\big]$  is the expected value of the  $m - k$  order statistic from  $m$  i.i.d. draws from  $\mathcal{D}$ . Note, however, that this rule is tailored to the specific distribution  $\mathcal{D}$ .
2.3 Approximately Optimal Rules
The goal of this paper is to explore rules that are approximately as good as an EDMR, but are distribution independent, i.e., are not tailored to  $\mathcal{D}$ . This will rely on approximations to an EwMR. We call a voting rule  $f$  an  $\alpha$ -Expected-Distortion-Maximizing Rule ( $\alpha$ -EDMR) for distribution  $\mathcal{D}$  if, for all preference profiles  $\sigma$ , the expected distortion of  $f$  is at least  $\alpha$  times the expected distortion of an EDMR. In other words,  $f$  is an  $\alpha$ -EDMR for  $\mathcal{D}$  if for all  $\sigma$ ,
$$
\mathbb {E} \bigl [ \mathrm {d d i s t} (f, \sigma) \bigr ] \geq \alpha \cdot \mathbb {E} \bigl [ \mathrm {d d i s t} (\mathrm {e d m r}, \sigma) \bigr ].
$$
If a voting rule is a  $(1 - \varepsilon)$ -EDMR, then we refer to it as an  $\varepsilon$ -approximation of an EDMR.
We use the analogous terminology of  $\alpha$ -EWMR for rules  $f$  such that, for all  $\sigma$ ,
$$
\mathbb {E} \left[ \mathrm {d s w} (f, \sigma) \right] \geq \alpha \cdot \mathbb {E} \left[ \mathrm {d s w} (\operatorname {e w m r}, \sigma) \right].
$$
To make our goal more tangible, consider a policy maker in charge of choosing a voting rule for a certain municipality. There are various guarantees such a policy maker might wish for the voting rule to satisfy with respect to (expected) distortion. If said policy maker is choosing a voting rule to use for years to come, they are likely to be interested in distribution-independent guarantees, since demographic distributions might shift over time.
A first guarantee in which such a policy maker might be interested is that ex ante, before an election is held, the distortion is expected to be low. Note that this expectation is over the entire space of voter utility profiles. While this is an appealing guarantee, it is rather weak (and easy to satisfy). To see the sense in which it is weak, consider the policy maker not at the day on which they choose the voting rule, but rather a while later, in a specific election that uses this rule, after the votes have been cast and officially tallied. At this point in time, the ordinal preferences  $\sigma$  are public knowledge, and, depending on what they are, it might be the case that despite the expected distortion having been low ex ante, it turns out that the expected distortion is high ex post, that is, it turns out that conditioned on all of the public information so far—i.e., conditioned on the ordinal preference profile  $\sigma$ —the (conditioned) expected distortion is high. While the policy maker had good reasons to choose the voting rule ex ante, a news outlet might still run a news story noting that given the tallied (realized, possibly low probability) ordinal preferences  $\sigma$ , there is reason to expect very high distortion. A forward-looking (and negative-press-averse) policy maker might want a (worst-case) guarantee that this scenario cannot happen, i.e., a worst-case guarantee on the ex post expected distortion. This is of course a much stronger guarantee (in particular, it also implies the same guarantee on ex ante expected distortion), and is considerably harder to satisfy. Seeking rules that give such a guarantee is precisely the goal of this paper.
We note that such an ex post guarantee is in a sense the strongest guarantee that such a negative-press-averse policy maker might wish for. Indeed, an even stronger guarantee would be for the distortion to be low given the actual realized cardinal preferences  $\mathbf{u}$ , which coincides with "traditional" worst-case distortion, but since the policy maker cares about press coverage, and since the realized cardinal preferences never become public knowledge (and hence, never become the basis for a news story), such a strong guarantee would be an overkill for the policy maker. This is of course fortunate, because as noted in Section 1.2, there are impossibilities that preclude getting such an overly strong guarantee in a distribution-independent setting like ours.
3 The Case of Two Alternatives
To develop some intuition, let us start with a naive attempt to compute an EDMR: calculate the expected distortion for each alternative  $j \in A$ , and choose the alternative that optimizes the expected distortion. This algorithm has two major shortcomings, however. First, calculating the expected distortion of a fixed alternative requires nm concentric integrals, and therefore this algorithm would take time that is exponential in both  $n$  and  $m$ . Second, the naive algorithm requires knowledge of the underlying distribution  $\mathcal{D}$ .
Even for the case of two alternatives, the above algorithm is exponential in  $n$  and therefore not computationally feasible—not to mention that it still requires knowledge of the distribution. However, it turns out that for this special case, majority—which selects the alternative preferred by the majority of voters (with ties broken arbitrarily)—is both an EDMR and an EWMR. Majority is furthermore computationally efficient and distribution-independent (i.e., agnostic to the distribution), so it satisfies all of the desiderata discussed so far. As a bonus, it is also indisputably practical.
Theorem 3.1 Let  $m = 2$ . For every distribution  $\mathcal{D}$ , majority is both an expected-distortion-maximizing rule and an expected-welfare-maximizing rule.
Proof. We first show that majority is an EwMR. Let  $\mu_{1}$  and  $\mu_{2}$  be the expected values for the minimum and maximum, respectively, of two i.i.d. draws from  $\mathcal{D}$ . Let  $0 \leq k \leq n$  be the number of voters that prefer alternative 1 to alternative 2. Then, by linearity of expectation, the expected welfare for alternative 1 and alternative 2 are  $k\mu_{2} + (n - k)\mu_{1}$  and  $(n - k)\mu_{2} + k\mu_{1}$ , respectively. Since  $\mu_{2} \geq \mu_{1}$ , alternative 1 is an expected-welfare-maximizing alternative if and only if  $k \geq n - k$ .