SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Next, borrowing an insight from the prior distortion literature [37, 42, 43], we round the agent utilities to the nearest power of two and ignore utilities below  $1 / m^2$ , reducing the number of distinct utility values to  $O(\log m)$  while losing at most a constant factor in the analysis. This allows us to subdivide voters in each group  $N_r$  into  $O(\log m)$  subgroups such that every agent in a subgroup with the same utility, say  $\tau$ , for  $a^*$ . We then employ three strategies to bound the distortion within each subgroup. Finally, we show that the overall distortion can be upper bounded, up to logarithmic factors, by the worst of these  $O(\log^2 m)$  distortion bounds across all subgroups of all the  $N_r$  groups.
Strategy 1 (Welfare above  $\mathbf{a}^*$ ). Voters in a subgroup of  $N_{r}$  who have utility  $\tau$  for  $a^*$  also have utility at least  $\tau$  for their top  $\ell_{r-1}$  alternatives. This helps us derive social welfare guarantees for the randomized positional scoring rule.
Strategy 2 (Probability of  $\mathbf{a}^*$ ). We use the well-known observation (see, e.g., [22]) that the distortion is always upper bounded by the inverse of the probability of selecting  $a^*$ .
Strategy 3 (Absolute Welfare Lower Bounds). Another novel insight from our work is that proving an absolute lower bound on the welfare achieved by a rule across all instances can be useful in bounding the distortion, even though the latter needs to compare the welfare achieved in each instance to the optimum welfare in that instance. We carefully analyze and approximate, up to logarithmic factors, the minimum welfare achieved by the randomized positional scoring rules we study, and use it to bound its distortion. A similar idea has been used in other domains (see, e.g. [47]), but to the best of our knowledge, we are the first to successfully apply it to distortion analysis.
Definition 1 (Minimum Welfare). Define the minimum welfare of a distribution over alternatives  $p \in \Delta(A)$  on a preference profile  $\vec{\sigma}$  as  $\min\text{-sw}(p, \vec{\sigma}) = \inf_{\vec{u} \in \mathcal{C}(\vec{\sigma})} \text{sw}(p, \vec{u})$ , which is the minimum social welfare of  $p$  across all consistent utility profiles. The minimum welfare of a voting rule  $f$  is the minimum welfare of its output, minimized over all preference profiles:  $\min\text{-sw}_{n,m}(f) = \min_{\vec{\sigma}} \min\text{-sw}(f(\vec{\sigma}), \vec{\sigma})$ , where the minimum is taken over all preference profiles with  $n$  agents and  $m$  alternatives. We drop  $n$  and  $m$  when clear from the context.
Due to  $\mathcal{C}(\vec{\sigma})$  being compact, the infimum in the min-sw  $(p,\vec{\sigma})$  definition is indeed attained. In the supplementary material, we make an structural observation that for any preference profile and distribution  $p\in \Delta (A)$ , the minimum welfare is at most  $n / m$ , attained at a dichotomous utility profile. Furthermore, every randomized positional scoring rule  $f_{\vec{s}}^{\mathrm{rand}}$  satisfies min-sw  $(f_{\vec{s}}^{\mathrm{rand}})\in [n / (4m^{2}),n / m]$ . We also show how to approximate minimum welfare better (up to constants or logarithmic terms); see Table 1 for tight bounds for the rules induced by common scoring vectors.
To this end, we present our most intricate technical lemma to derive a generic welfare lower bound, which we use to apply Strategies 1 and 3. Instead of focusing only on  $\operatorname{Pr}[f_{\vec{s}}^{\mathrm{rand}}(\vec{\sigma}) = a^{*}]$ , it bounds the overall welfare expression  $\sum_{a \in A} \operatorname{Pr}[f_{\vec{s}}^{\mathrm{rand}}(\vec{\sigma}) = a] \cdot \mathsf{sw}_T(a)$ ; the product of probability of selection and welfare of an alternative leads to a quadratic program, where the variables encode the worst case, and this is analytically solved using the Karush-Kuhn-Tucker (KKT) conditions.
Lemma 2. Fix any scoring vector  $\vec{s}$ , preference profile  $\vec{\sigma}$ , subset of agents  $T \subseteq N$ , threshold  $\tau \geqslant 0$  and rank  $\ell \in [m]$ . For a partial utility profile  $\vec{u}$  in which every agent in  $T$  has utility at least  $\tau$  for each of her top  $\ell$  alternatives and all other utilities are 0, we have:
$$
\mathsf {s w} _ {T} (f _ {\vec {s}} ^ {\text {r a n d}} (\vec {\sigma}), \vec {u}) \geqslant \tau \cdot \frac {| T | \ell}{2 n \| \vec {s} \| _ {1}} \min  _ {h \in [ m ]} \frac {1}{h} \Big (2 s _ {\ell} \cdot | T | \ell + (n - | T |) \cdot \sum_ {j = 1} ^ {h} s _ {m - j + 1} \Big).
$$
Instead of tediously explaining the lemma, we will later show how its straightforward application wondrously gives us the desired welfare lower bound for the example of randomized Borda rule.
3.2 Analyzing Common Rules
We are ready to present our main result, which uses the aforementioned insights to pinpoint the asymptotic distortion of common randomized positional scoring rules.
Theorem 2. For  $f \in \{ \text{plurality}, \text{Borda}, \text{harmonic}, \text{veto}, k \text{-approvals} \}$ , the minimum welfare (min-sw) and the distortion (dist) of the 'randomized  $f$ ' rule are as shown in Table 1.
Due to space limitations, we only provide a proof for the distortion upper bound of the randomized Borda rule, and defer the rest to the supplementary material. For conciseness, we use the notation  $\mathsf{Borda}(a) \triangleq \mathsf{score}(a, \vec{s}_{\mathsf{Borda}})$ . First, we need the following lower bound on its minimum welfare.
Lemma 3. The minimum welfare of the randomized Borda rule is min-sw  $(f_{\vec{s}_{\mathrm{Borda}}}^{\mathrm{rand}}) = \Omega \left(\frac{n}{m\sqrt{m}}\right)$ .
Proof. Fix any preference profile  $\vec{\sigma}$  and consistent utility profile  $\vec{u} \in \mathcal{C}(\vec{\sigma})$ . Our goal is to show that  $\mathrm{sw}(f_{\vec{s}_{\mathrm{Borda}}}^{\mathrm{rand}}(\vec{\sigma}), \vec{u}) = \Omega(\frac{n}{m\sqrt{m}})$ . First, we make a few modifications to the scoring vector and the preference profile that are guaranteed to not increase the welfare, and then invoke Lemma 2.
Simplify the scores. Let us consider the scoring vector  $\vec{s}'$  which is equal to  $\vec{s}_{\text{Borda}}$  except the top  $m/2$  scores are all equal to  $m/2$ . Note that  $s_j' \leqslant (s_{\text{Borda}})_j \leqslant 2s_j'$  for all  $j \in [m]$ . Hence, invoking Lemma 1 with  $\alpha = 1$  yields  $\mathsf{sw}(f_{\vec{s}_{\text{Borda}}}^{\text{rand}}(\vec{\sigma}), \vec{u}) \geqslant 1/2 \cdot \mathsf{sw}(f_{\vec{s}'}^{\text{rand}}(\vec{\sigma}), \vec{u})$ .
Simplify the preference and utility profiles. Next, let us lower bound  $\mathsf{sw}(f_{\vec{s}'}^{\mathrm{rand}}(\vec{\sigma}), \vec{u})$ . For each agent  $i$ , let  $A_i$  be the set of top  $m/2$  alternatives in  $\sigma_i$  and  $a_i \in \arg \min_{a \in A_i} \mathsf{score}(a, \vec{s}')$  be the alternative in  $A_i$  with the lowest score (equivalently, probability of selection under  $f_{\vec{s}'}^{\mathrm{rand}}(\vec{\sigma})$ ). Now,
$$
u _ {i} \left(f _ {\vec {s} ^ {\prime}} ^ {\text {r a n d}} (\vec {\sigma})\right) \geqslant \sum_ {a \in A _ {i}} \frac {\operatorname {s c o r e} \left(a , \vec {s} ^ {\prime}\right)}{n \| \vec {s} ^ {\prime} \| _ {1}} \cdot u _ {i} (a) \overset {(1)} {\geqslant} \frac {\operatorname {s c o r e} \left(a _ {i} , \vec {s} ^ {\prime}\right)}{n \| \vec {s} ^ {\prime} \| _ {1}} \cdot \left(\sum_ {a \in A _ {i}} u _ {i} (a)\right) \overset {(2)} {\geqslant} \frac {\operatorname {s c o r e} \left(a _ {i} , \vec {s} ^ {\prime}\right)}{n \| \vec {s} ^ {\prime} \| _ {1}} \cdot \frac {1}{2},
$$
where (1) follows from the definition of  $a_i$  and (2) uses the fact that each agent has a total utility of at least  $1/2$  for her top  $m/2$  alternatives (due to the pigeonhole principle).
Invoking Lemma 2. The final expression above can be written as  $u_{i}^{\prime}(f_{\vec{s}^{\prime}}^{\mathrm{rand}}(\vec{\sigma}^{\prime}))$ , where  $\vec{\sigma}^{\prime}$  is a preference profile in which each agent  $i$  ranks  $a_{i}$  first, and  $\vec{u}^{\prime}$  is a partial utility profile in which each agent  $i$  has utility  $1/2$  for her top alternative and 0 for the rest. Summing the above for all agents, we have  $\mathsf{sw}(f_{\vec{s}^{\prime}}^{\mathrm{rand}}(\vec{\sigma}), \vec{u}) \geqslant \mathsf{sw}(f_{\vec{s}^{\prime}}^{\mathrm{rand}}(\vec{\sigma}^{\prime}, \vec{u}^{\prime}))$ . Thus, to lower bound it, we invoke Lemma 2 with  $\vec{s} \gets \vec{s}^{\prime}$ ,  $\vec{\sigma} \gets \vec{\sigma}^{\prime}$ ,  $T$  being an arbitrary subset of  $n/2$  agents,  $\tau \gets 1/2$ , and  $\ell \gets 1$ . Using  $\| \vec{s}^{\prime} \|_1 = \frac{m}{2} \cdot \frac{m}{2} + \binom{m/2}{2} = \frac{m(3m-2)}{8}$ , this gives us
$$
\begin{array}{l} \mathsf {s w} \left(f _ {\vec {s} ^ {\prime}} ^ {\text {r a n d}} (\vec {\sigma} ^ {\prime}), \vec {u} ^ {\prime}\right) \stackrel {(1)} {\geqslant} \frac {1}{2} \cdot \frac {\frac {n}{2} \cdot 1}{2 n \cdot \frac {m (3 m - 2)}{8}} \cdot \min  _ {h \in [ m / 2 ]} \frac {1}{h} \left(2 \cdot \frac {m}{2} \cdot \frac {n}{2} + \frac {n}{2} \cdot \frac {h (h - 1)}{2}\right) \\ = \frac {n}{4 m (3 m - 2)} \cdot \min _ {h \in [ m / 2 ]} \left(\frac {2 m}{h} + h - 1\right) \stackrel {(2)} {\geqslant} \frac {n \cdot (2 \sqrt {2 m} - 1)}{4 m (3 m - 2)} \stackrel {(3)} {\geqslant} \frac {n}{6 m \sqrt {m}}, \\ \end{array}
$$
where the restriction to  $h \in [m/2]$  in (1) is based on the fact that the bound would be  $\Omega(n/m)$  when  $h > m/2$ , (2) is due to the AM-GM inequality, and (3) uses  $m \geqslant 2$ . Connecting the dots, we have
$$
\mathsf {s w} \left(f _ {\vec {s} _ {\text {B o r d a}}} ^ {\text {r a n d}} (\vec {\sigma}), \vec {u}\right) \geqslant \frac {1}{2} \cdot \mathsf {s w} \left(f _ {\vec {s} ^ {\prime}} ^ {\text {r a n d}} (\vec {\sigma}), \vec {u}\right) \geqslant \frac {1}{2} \cdot \mathsf {s w} \left(f _ {\vec {s} ^ {\prime}} ^ {\text {r a n d}} (\vec {\sigma} ^ {\prime}), \vec {u} ^ {\prime}\right) \geqslant \frac {n}{1 2 m \sqrt {m}}.
$$
To translate the welfare lower bound from Lemma 3 into a distortion upper bound, we need the following relation between the Borda score of an alternative and its social welfare.
Lemma 4. For any preference profile  $\vec{\sigma}$ , consistent utility profile  $\vec{u} \in \mathcal{C}(\vec{\sigma})$ , and alternative  $a \in A$ , we have  $\mathsf{sw}(a) \leqslant (\mathsf{Borda}(a) + n)/m$ .
The desired distortion upper bound can now be derived using a standard analysis. The crux of our proof for the randomized Borda rule lies in our intricate derivation of its minimum welfare in Lemma 3, for which Lemma 2 does the heavy lifting.
Lemma 5. The distortion of the randomized Borda rule is  $O(m^{5/4})$ .
Proof. Fix any preference profile  $\vec{\sigma}$  and consistent utility profile  $\vec{u} \in \mathcal{C}(\vec{\sigma})$ . Let  $a^* \in \arg \max_{a \in A} \mathsf{sw}(a, \vec{u})$  be an optimal alternative. By Strategy 2 from Section 3.1, we know that the distortion is at most  $\frac{n \| \vec{s}_{\text{Borda}} \|_1}{\text{Borda}(a^*)}$ . Following Strategy 3 from Section 3.1 and the minimum welfare analysis in Lemma 3, we have that distortion is at most  $\frac{\mathsf{sw}(a^*)}{n / (12m\sqrt{m})}$ , which, using Lemma 4, is at
most  $12\sqrt{m}\cdot \left(\frac{\text{Borda}(a^{*})}{n} +1\right)$ . Putting everything together, and using the fact that  $\min \{a,b\} \leqslant \sqrt{ab}$  for all  $a,b\in \mathbb{R}_{\geqslant 0}$ , the distortion is upper bounded by
$$
\begin{array}{l} \min  \left\{\frac {n \| \vec {s} _ {\text {B o r d a}} \| _ {1}}{\text {B o r d a} \left(a ^ {*}\right)}, \frac {\text {B o r d a} \left(a ^ {*}\right) \cdot 1 2 \sqrt {m}}{n} + 1 2 \sqrt {m} \right\} \\ \leqslant 1 2 \sqrt {m} + \sqrt {\frac {n \cdot m (m - 1) / 2}{\mathsf {B o r d a} (a ^ {*})} \cdot \frac {\mathsf {B o r d a} (a ^ {*}) \cdot 1 2 \sqrt {m}}{n}} \leqslant 8 \sqrt {m} + \sqrt {6} m ^ {5 / 4} = O (m ^ {5 / 4}). \\ \end{array}
$$
4 Random Committee Member Rules
Next, we focus on our second class of explainable randomized voting rules that select an alternative uniformly at random from a shortlisted committee of size  $k \in [m]$ . We call them random  $k$ -committee member rules. For  $k = 1$ , we are left with deterministic rules, among which plurality achieves the optimal distortion of  $\Theta(m^2)$ . For  $k = m$ , we are left with uniform selection among all alternatives, which has distortion  $\Theta(m)$ . Is it possible that, for some intermediate value of  $k$ , we in fact achieve sublinear distortion? Could we achieve distortion at most logarithmic factors worse than the optimal  $\Theta(\sqrt{m})$ , like with randomized positional scoring rules? We answer the former positively but the latter negatively. First, we present a lower bound proving that any random  $k$ -committee member rule, for any value of  $k$ , incurs a distortion of at least  $\Omega(m^{2/3})$ .
Theorem 3. For  $k \in [m]$ , any random  $k$ -committee member rule incurs  $\Omega(\max(k, m^2 / k^2))$  distortion. This lower bound is at least  $\Omega(m^{2/3})$  for all  $k$ .
As a result, this class of rules is less powerful than the class of randomized positional scoring rules, and thus, the class of all randomized rules. However, especially for small values of  $k$ , we gain the benefit of randomizing over a small support, which could translate to greater explainability.
To derive upper bounds, one might be tempted to turn again to positional scoring rules, and consider selecting uniformly at random from the  $k$  alternatives with the highest score according to some scoring vector. In the supplementary material, we show that using plurality scoring vector yields  $\Theta(m(m - k + 1))$  distortion. While it nicely interpolates between the extremes of  $\Theta(m^2)$  at  $k = 1$  and  $\Theta(m)$  at  $k = m$ , it fails to achieve sublinear distortion, which we prove to be achievable. What about other scoring vectors? Unfortunately, it is relatively easy to see that using scoring vectors such as Borda, harmonic, or veto results in unbounded distortion. Despite the disappointing worst-case performance, we show in Section 5 that these rules perform relatively well empirically.
Next, we design a novel random  $k$ -committee member rule, which, with the right value of  $k$ , allows us to achieve sublinear distortion.
Theorem 4. There is a polynomial-time computable random  $k$ -committee member rule with distortion  $O(\max \{k, m^2 / (k\sqrt{k})\})$ . This is minimized at  $k = m^{4/5}$ , where the bound becomes  $O(m^{4/5})$ .
To achieve the above, we concoct a three-way mixture of an approximately stable committee [48], a powerful notion which has been used to derive optimal randomized rules [19], alternatives with high plurality scores, and alternatives picked carefully to guarantee high minimum welfare from sufficiently many agents. We refer to the committee thus formed as a top-biased stable  $k$ -committee. The rule that returns this committee is presented as Algorithm 1 in the supplementary material.
Theorems 3 and 4 leave open the question of the optimal distortion that can be achieved by a random committee member rule, sandwiching this value between  $O(m^{4/5})$  and  $\Omega(m^{2/3})$ . It is also interesting to wonder which value of  $k$  is optimal. Our upper bound is optimized at  $k = m^{4/5}$ , and our lower bound implies that the optimal  $k$  must be in  $[m^{3/5}, m^{4/5}]$  as the distortion outside of that range is  $\Omega(m^{4/5})$ . See Section 5 for an empirical evaluation of the optimal  $k$ .
Next, we empirically evaluate the efficiency of explainable rules studied in the previous sections.
Rules. We consider three classes of rules: deterministic positional scoring rules, randomized positional scoring rules from Section 3, and, from Section 4, rules that select uniformly at random from the  $k$  alternatives with the highest scores (henceforth, uniform random  $k$ -positional scoring rules).
(a) Average distortion with  $\phi = 0.1$
(b) Average distortion with  $\phi = 0.5$
(c) Average distortion with  $\phi = 1$
(d) Best value of  $k$  based on  $\phi$ , for  $m = 25$ .
Figure 1: All figures show results averaged over 150 runs along with the standard error. Figures 1a to 1c share the legend on the left.
We consider four representative scoring vectors  $f \in \{\text{Plurality}, \text{Borda}, \text{Harmonic}, 3\text{-Approval}\}$ , and denote the corresponding rules in the three classes by 'D  $f$ ', 'R  $f$ ', and 'UR $_k$ f'', respectively. Thus, overall, we test 12 voting rules. As benchmarks, we also add the Uniform rule, which selects an alternative uniformly at random from the set of all alternatives, and the Instance Optimal rule, which selects the lottery over alternatives minimizing distortion on the preference profile. Boutilier et al. [22] show how to use linear programming to compute the latter in polynomial time.
Data Generation. We generate preference profiles by sampling  $n$  rankings over  $m$  alternatives iid from the Mallows model [49], which is widely used in machine learning and statistics. The model takes as input an underlying reference ranking  $\sigma^*$  (which can be set arbitrarily) and a dispersion parameter  $\phi \in [0,1]$ . When  $\phi = 1$ , the model converges to a uniform distribution over all  $m!$  rankings (also known as impartial culture), whereas  $\phi \rightarrow 0$  converges to the point distribution where  $\sigma^*$  is sampled with probability 1, so all the agents have the same preference ranking in the sampled profile. For a precise definition of the model and an efficient algorithm to sample from it (which we use in our experiments), see the work of Lu and Boutilier [50]. For each combination of  $n = 100$  agents,  $m \in \{5,10,\dots,50\}$  alternatives, and dispersion parameter  $\phi \in \{0,0.1,\dots,1\}$ , we sample 150 instances, and report averages along with the standard error. In the supplementary material, we report the results for two other statistical models namely the Polya-Eggenberger urn model [51] and the Plackett-Luce model [52, 53].
Evaluation. For each rule  $f$  under consideration and each instance  $\vec{\sigma}$ , we evaluate the efficiency of the output  $f(\vec{\sigma})$  by measuring its instance-specific distortion  $\mathrm{dist}(f(\vec{\sigma}),\vec{\sigma})$ . Note that this still takes a worst case over the utility profiles consistent with  $\vec{\sigma}$ , but unlike in Sections 3 and 4 where we also take a worst case over  $\vec{\sigma}$ , here we compute the expected distortion over  $\vec{\sigma}$  drawn from the Mallows model. Again, we compute  $\mathrm{dist}(f(\vec{\sigma}),\vec{\sigma})$  using the LP-based approach of Boutilier et al. [22].
Results. Figures 1a to 1c show the average distortion of different rules for  $\phi \in \{0.1, 0.5, 1\}$ , respectively, fixing  $m = 25$ . For large  $\phi$  (impartial culture), the alternatives are spread out in agents' preference rankings. Thus, it becomes likely that some alternative is ranked slightly lower than another alternative by many agents, but has much higher social welfare in total. We see that in this case, randomized positional scoring rules outperform deterministic and random committee member rules as well as the uniform benchmark since it is more efficient to give a chance of winning to each alternative. As  $\phi$  decreases to 0.5 and agent rankings become somewhat correlated, random committee member rules start to outperform some of the randomized positional scoring rules (though randomized plurality and randomized 3-approval still perform quite well). But crucially, both families
of rules still outperform deterministic rules and the improved performance of random committee member rules now allows them to outperform the uniform benchmark as well. At  $\phi = 0.1$ , when agent preferences are highly correlated, deterministic rules gain some traction. Nonetheless, at least one rule from one of the two randomized classes still outperforms all deterministic rules (randomized plurality for low  $m$  and any random committee member rule for high  $m$ ). Overall, randomized plurality and randomized 3-approval perform reasonably well, and randomized plurality outperforms all four deterministic rules across almost all values of  $m$  and  $\phi$ . The evidence suggests that one can almost always choose an explainable randomized rule that achieves better efficiency than deterministic rules, though the choice of the rule may have to depend on the setting at hand. A detailed comparison of rules in each class can be found in the supplementary material.
In the above experiments, for random committee member rules (specifically, the uniform random  $k$ -positional scoring rules), we use a committee size of  $k = 3$ . Figure 1d shows the best value of  $k$  (one that yields the minimum distortion) for different scoring vectors as a function of  $\phi$ . It turns out that the best  $k$  is indeed very small ( $\leqslant 5$ ) unless  $\phi$  is really close to 1. Thus,  $k = 3$  is a reasonable choice that helps our random committee member rules achieve high efficiency. Still, it is possible to further optimize the efficiency of these rules by pairing them with their corresponding optimal value of  $k$ ; the resulting average distortion is presented in the supplementary material.
Explainability. We focus on the families of randomized positional scoring rules and random committee member rules as two examples of explainable randomized voting rules. While we argue in the introduction that these families admit intuitive procedural explanations and provide example explanations, checking whether stakeholders find these explanations reasonable and satisfactory in the context of a real-life application requires an in-depth investigation, possibly via user studies. Our work also treats explainability as a qualitative attribute, but different rules — even within the same family — may differ in the degree to which they are explainable. Quantifying the degree of explainability, both theoretically and empirically, remains to be tackled.
Efficiency. Our work uses distortion as a yardstick for efficiency, leaving open exciting technical questions. While our analysis of randomized multi-level approval rules in the supplementary material takes a step towards characterizing the distortion of all randomized positional scoring rules, it still remains an unresolved challenge. For random committee member rules, even the more basic question of identifying the optimal distortion they can achieve remains open, though we are able to pinpoint it to be between  $\Omega(m^{2/3})$  and  $O(m^{4/5})$ . Taking a step back, while distortion is a reasonable theoretical measure for efficiency, it remains to be seen whether it is also correlated for other measures of efficiency one may care about in practice. For example, in the context of food donations, Lee et al. [15], who use the deterministic Borda rule to make decisions, suggest a number of important decision factors other than the social welfare of the stakeholders, such as whether the donations are distributed equitably, how long the drivers have to travel to deliver donations, and whether organizations with higher poverty rates, lower median incomes, and worse food access are receiving sufficient donations. An important next step would be to measure the efficiency of explainable randomized voting rules in real-life applications such as food donation.