Revisiting the Distortion of Distributed Voting

We consider a setting with agents that have preferences over alternatives and are partitioned into disjoint districts. The goal is to choose one alternative as the winner using a mechanism which first decides a representative alternative for each district based on a local election with the agents therein as participants, and then chooses one of the district representatives as the winner. Previous work showed bounds on the distortion of a specific class of deterministic plurality-based mechanisms depending on the available information about the preferences of the agents in the districts. In this paper, we first consider the whole class of deterministic mechanisms and show asymptotically tight bounds on their distortion. We then initiate the study of the distortion of randomized mechanisms in distributed voting and show bounds based on several informational assumptions, which in many cases turn out to be tight. Finally, we also experimentally compare the distortion of many different mechanisms of interest using synthetic and real-world data.


Introduction
Voting is a ubiquitous method for making decisions with a large number of applications, such as electing political representatives, deciding how to split a public budget between projects, or choosing which services (restaurants, hotels, etc) to recommend to new users based on past user experiences.As such, it has been at the epicenter of research within multiple disciplines including political sciences, economics and computer science [Brandt et al., 2016].e most prominent question in this research agenda is to identify the best voting rule to use to collectively aggregate the preferences of agents over alternative options into a single winning alternative, with most of the earlier literature focusing on axiomatic properties that good voting rules should have.An alternative way to tackle this question that has been proposed in computer science is through the distortion framework [Anshelevich et al., 2021] which allows to compare different voting rules based on how well they approximate the optimal choice as measured in terms of a social objective function like the utilitarian social welfare.Procaccia and Rosenschein [2006], the distortion framework has been applied to several utilitarian social choice se ings (e.g., [Boutilier et al., 2015, Anshelevich et al., 2018, Gkatzelis et al., 2020]).e lion's share of previous work has focused on centralized models with a single pool of agents whose preferences are directly given as input to a voting rule, which thus can utilize all the given information at once to make a decision.However, there are many applications with multiple pools of agents which make independent local decisions that can be thought of as recommendations for the final decision.To give a concrete example, in most political election systems, the citizens are partitioned into districts based on geographic or other criteria, and vote within their districts to propose the candidate (party) they would like to be selected as the winner.

Since its inception in 2006 by
Inspired by situations like the one described above, Filos-Ratsikas et al. [2020] initiated the study of the distortion of mechanisms in a distributed single-winner se ing where a set of n agents with Deterministic Randomized-of-Deterministic Randomized-of-Randomized Ordinal Θ(km 2 ) Θ(km Table 1: An overview of our results.Specific details can be found in the appropriate sections. cardinal preferences over a set of m alternatives are partitioned into k disjoint districts.e authors focused on deterministic mechanisms of the form P f , which first choose a representative alternative for each district according to some rule f , by holding a local election with the agents of the district as the voters, and then picking the winner to be the alternative that is representative of the most districts (i.e., using the P rule).Filos-Ratsikas et al. considered mechanisms for which the rule f can be cardinal or ordinal, i.e., it may use the actual numerical information about the preferences of the agents within the districts or just consistent rankings.
e authors showed that, when the districts are symmetric (that is, each of them contains the same number of agents), the distortion of a cardinal mechanism, namely P R V is O(km), and provided an asymptotically matching lower bound of Ω(km) on the distortion of any P f mechanism.For ordinal mechanisms, they showed that P P achieves a distortion of O(km 2 ), and that this is asymptotically best among all ordinal P f mechanisms.

Revisiting the distortion of distributed voting
A first observation about the results of Filos-Ratsikas et al. [2020] is that there is a-priori no reason to restrict our a ention to only mechanisms in the class P f , as using other over-districts rules could potentially lead to be er distortion.Indeed, follow-up work considered distributed social choice se ings with metric preferences [Anshelevich et al., 2022, Filos-Ratsikas andVoudouris, 2021] without such restrictions on the over-districts rule.In addition, all of the previous work on these se ings only considered deterministic mechanisms that use deterministic in-district and over-districts rules.Randomization has proven out to be a very useful tool in achieving be er (expected) distortion bounds in the centralized se ing (see Boutilier et al. [2015], Ebadian et al. [2022]), so it is only natural to consider randomized mechanisms in the distributed se ing as well.Finally, an important question is how the distortion bounds are affected in case the participants act selfishly, and whether there are strategyproof mechanisms with good distortion bounds.
is question has been considered in the centralized se ing [Filos-Ratsikas and Miltersen, 2014, Bhaskar and Ghosh, 2018, Bhaskar et al., 2018, Ebadian et al., 2022] and also in the distributed metric se ing [Filos-Ratsikas and Voudouris, 2021]; we consider it in the context of the normalized se ing of Filos-Ratsikas et al. [2020] as well.

Our Contributions
We consider the class of all mechanisms for distributed voting in the se ing of [Filos-Ratsikas et al., 2020].In particular, we consider the f over -of-f in class of mechanisms, where f in is an in-district rule that takes as input the preferences of the agents within each district and outputs a representative alternative for the district, while f over is a rule that takes as input the representative alternatives of all districts and chooses one of them as the overall winner.We consider several different cases depending on the nature of f over and f in (deterministic or randomized), and the type of information they can utilize (cardinal or ordinal).We show the following results; see Table 1 for an overview.
Deterministic Mechanisms.When f over and f in are both deterministic and the districts are symmetric, we show that the best possible distortion is Θ(km) when the valuation functions of the agents are accessible (cardinal mechanisms), and is Θ(km 2 ) when only ordinal information about the preferences of the agents is available (ordinal mechanisms).e upper bounds were shown by Filos-Ratsikas et al. [2020] and here we provide asymptotically tight lower bounds.
ese results show that for general, unstructured (normalized) valuations, employing different over-district rules in fact does not result in improvements on the distortion.We present these results in Section 3. Randomized Mechanisms.In Section 4, we consider for the first time the distortion of randomized mechanisms in distributed voting.We first prove a simple composition theorem, which shows that using an in-district rule with known distortion δ in the centralized se ing and then selecting the winner uniformly at random from the set of representatives, defines a distributed mechanism with distortion O(kδ).Using this, complemented with new lower bounds, we show that the best possible distortion for cardinal unanimous mechanisms is Θ(k); in fact, this is true even when the districts are asymmetric and when f over is randomized but f in is deterministic.
For ordinal mechanisms, we consider two cases: (a) mechanisms that use deterministic in-district rules f in , and (b) fully-randomized mechanisms, where both f over and f in are randomized rules.For (a), we show that the best possible distortion is Θ(km 2 ).e upper bound follows from the bound on P P proven in [Filos-Ratsikas et al., 2020]; here, we provide an asymptotically matching lower bound assuming a natural universal tie-breaking rule.For (b), we prove a simple but very interesting result: For a well-studied class of randomized centralized voting rules called pointvoting schemes (e.g., see Gibbard [1977], Barbera [1978]), there exists a distributed implementation so that there is no effect on the induced probability distribution, even for asymmetric districts.Simply put, using such rules it is possible to escape the ill effects of districts in terms of the distortion, even when the districts are asymmetric.From this result, it follows that there exists a distributed implementation of a well-known mechanism of Boutilier et al. [2015]  Experiments.Finally, in Section 5, we perform experiments using real-world data and synthetic data to evaluate the effect of distributed decision making to the distortion in se ings closer to practice.e main conclusions of our experimental results mirror that of our theoretical results in Sections 3 and 4.
Besides the aforementioned works on distributed voting, Borodin et al. [2019] studied a related two-stage se ing in which the voters participate in a central election, but the candidates themselves come from local elections within the political parties' electorates.Beyond distortion, in the context of district-based elections, there have also been other works that have considered the degree of deviation from proportional representation (e.g., see [Bachrach et al., 2016] and references therein), and some works that have studied the complexity of manipulation (e.g., see [Elkind et al., 2021, Lewenberg et al., 2017, Lev and Lewenberg, 2019, Borodin et al., 2018]).

Preliminaries
An instance I of our problem is given by a tuple I = (N, A, v, D). ere is a set N of n agents (or voters) that have preferences over a set A of m alternatives (or candidates).
e preferences of each agent i ∈ N are captured by a valuation function v i : A → R ≥0 that maps every alternative a ∈ A to a real non-negative value v i (a) = v ia .Following previous work, we assume that the valuation functions are normalized such that a∈A v ia = 1 for every i ∈ N (unit-sum assumption).Let v = (v i ) i∈N be the valuation profile consisting of the valuation functions of all agents.e agents are also partitioned into a set D of k disjoint districts.
For every district d ∈ D, let N d be the set of agents it contains, such that d∈D N d = N .In the symmetric case, each district d contains exactly λ = n/k agents.In the asymmetric case, each district d contains a number n d of agents.All our lower bounds follow by instances consisting of symmetric districts, whereas our upper bounds in Section 4 hold for asymmetric districts.

Mechanisms
Our goal is to choose an alternative to satisfy several criteria of interest.
is choice must be done using a distributed mechanism that uses an in-district voting rule f in and an over-districts voting rule f over to implement the following two independent steps: • Step 1: For each district d, choose a representative alternative a d ∈ A by holding a local election based on f in . • Step 2: Choose a district representative as the winner based on f over by considering the districts as voters and their representatives as the candidates they approve.
For simplicity we refer to such mechanisms as f over -of-f in .Different choices of f in and f over lead to different distributed mechanisms.Note that the in-district rule can in general use various types of information about the preferences of the agents.For instance, it may be able to use exact cardinal information about the valuation functions, or only ordinal information that is induced by the values (i.e., rankings of alternatives that are consistent to the values of the agents for them).In the la er case, we will use σ i to denote the preference ranking of agent i ∈ N so that σ i (a) is the rank of alternative a ∈ A in the ranking of i, and ) i∈N be the ordinal profile consisting of the preference rankings of all agents.To be concise in the definitions below, let δ(I) be the information about the preferences of the agents in instance I = (N, A, v, D) that is used by a mechanism; that is, δ(I) = v in case of cardinal information, or δ(I) = σ in case of ordinal information.
We will focus on different classes of distributed mechanisms depending on the available information about the preferences of the agents at the district level (cardinal or ordinal), and also on whether their decision is deterministic or randomized (that is, they choose the district representatives or final winner based on probability distributions).

Social Welfare and Distortion
Given an instance I, the social welfare of an alternative a ∈ A is the total value that the agents have for a, that is, SW(a|I) = i∈N v ia .So, the expected social welfare achieved by a randomized distributed mechanism M that chooses alternative a ∈ A as the winner w with probability Pr e efficiency of a distributed mechanism is measured by the notion of distortion.e distortion of a distributed mechanism M is the worst-case ratio (over all possible instances with n agents, m alternatives, and k districts) of the maximum social welfare achieved by any alternative over the (expected) social welfare of the alternative chosen by the mechanism as the winner w, that is, .
Clearly, dist(M ) ≥ 1.When the denominator in the definition of the distortion tends to 0, we will say that the distortion is infinite or unbounded.Our goal is to identify the best possible distributed mechanisms in terms of distortion.

Strategyproofness
Another important property that we would like our mechanisms to satisfy is that of strategyproofness.A strategyproof mechanism makes decisions such that providing false information never leads to the selection of an alternative that an agent prefers over the alternative chosen when the agent provides truthful information.In particular, for any instance I, it must be the case that v i (M (δ(I))) ≥ v i (M (δ(I ′ ))) for any agent i ∈ N , where I ′ is the instance obtained when only agent i reports information different than that in I.

Some useful observations and properties
Before we present our technical results, let us briefly discuss some useful properties.
Locality of distributed mechanisms: First, observe that any distributed mechanism f over -of-f in satisfies a locality property in the following sense.A district d (that is, the preferences of a number of agents) appears in different instances if in each of these instances there is a district with the same number of agents and the same information about theirs preferences as in d (depending on what is required by the mechanism).Since the information is the same, the in-district rule f in must decide the same alternative as the representative of the district in all these instances.Similarly, in all instances where the mechanism has decided the same set of district representatives, the over-districts rule f over must decide the same final winner.
Distortion of distributed vs centralized: Another useful observation is that the distortion of a distributed mechanism f over -of-f in is at least as much as the distortion of the in-district centralized voting rule f in .Indeed, when k = 1, there is only one representative alternative chosen by f in , and thus this alternative must be chosen as the winner by f over ; this is also true for instances with k ≥ 2 districts which are all copies of one district.Consequently, the distortion of f in is a lower bound on the distortion of f over -of-f in .
Strategyproofness: Observe that for a distributed mechanism f over -of-f in to be strategyproof it is necessary that the in-district rule f in is strategyproof. is again follows by how the mechanism would work in instances with a single district, in which case the over-districts rule f over does not play any role in the selection of the final winner.
Unanimity: A few of our results will require the in-district rules f in to be unanimous.Unanimity stipulates that if all of the agents have the same alternative as the top preference, that alternative must be selected (with probability 1).Unanimity is a very natural property of "reasonable" voting rules, especially deterministic ones.For randomized rules, there might be reasons to consider nonunanimous choices, e.g., see Gibbard [1977], Filos-Ratsikas and Miltersen [2014].

Deterministic mechanisms
We start with deterministic distributed mechanisms and focus explicitly on the case of symmetric districts in this section (that is, the size of each district is λ).When full information about the valuations of the agents is known at the district level, Filos-Ratsikas et al. [2020] showed that the mechanism P R V , which chooses the representative of each district to be the alternative with maximum social welfare for the agents in the district, has distortion O(km).We show that this mechanism is asymptotically best possible over all possible deterministic distributed mechanisms that use unanimous in-district rules (but may not use P as the over-districts rule).
eorem 3.1.e distortion of any deterministic distributed mechanism with a unanimous in-district rule is Ω(km).
Proof.Let M be some deterministic distributed mechanism with a unanimous in-district rule.Without loss of generality, whenever there are k distinct district representatives {a 1 , . . ., a k }, we assume that M chooses a 1 as the overall winner.Let ε > 0 be some positive infinitesimal and consider the following instance with k districts {d 1 , . . ., d k } and m > k alternatives: • In district d 1 , all agents have value 1/m + ε for alternative a 1 , and value 1/m − ε/(m − 1) for any other alternative.
Since the in-district rule is unanimous, the district representatives are alternatives {a 1 , . . ., a k }, and the overall winner is thus a 1 .e social welfare of alternative a 1 is approximately λ/m, whereas the social welfare of alternative x is approximately k • λ/2, leading to distortion Ω(km).
When only ordinal information about the preferences of the agents is available, Filos-Ratsikas et al. [2020] showed that P P , which chooses the favorite alternative of most of the agents in a district as its representative and then the alternative that represents the most districts as the winner, has distortion O(km 2 ).We show that this mechanism is asymptotically best possible among all ordinal distributed mechanisms (without any restrictions), thus improving upon the result of Filos-Ratsikas et al. [2020] who showed that P P is best possible only within the class of mechanisms they studied.
We first prove an easy but important lemma showing that when only ordinal information is available, to achieve finite distortion, it is necessary the representative of each district to be some alternative that is the favorite of at least one agent in the district.
Lemma 3.2.e representative of any district must be some top-ranked alternative, otherwise the distortion is infinite.
Proof.Let d be a district and let T be the set of top-ranked alternatives.Suppose that the representative of d is chosen to be some alternative x ∈ T .en, in any instance consisting of copies of d, the winner must be x.However, the valuation profile might be such that all agents have value 1 for their favorite alternative and 0 for any other alternative.Consequently, the social welfare of x might be 0, whereas the social welfare of any top-ranked alternative is positive, leading to infinite distortion.
We say that a district is divided if its λ agents are partitioned into m/2 equal-sized sets such that all the 2λ/m agents in each set rank the same alternative first and different sets of agents have different top-ranked alternatives.By Lemma 3.2, the representative of such a district must be one of the topranked alternatives.e following lemma shows that choosing the representative of a divided district as the winner is, under some circumstances, a bad choice.
Lemma 3.3.Suppose that some alternative y 1 is chosen as the winner by a deterministic ordinal distributed mechanism when the set of representatives is {y 1 , . . ., y k }.If there exists a divided district that is represented by y 1 , then there are k − 1 districts with representatives y 2 , . . ., y k , and altogether these k districts define an instance such that the distortion of the mechanism is Ω(km 2 ).
Proof.Let M be a deterministic ordinal distributed mechanism that selects y 1 as the winner when the set of representatives is {y 1 , . . ., y k }, and let d be the divided district that is represented by y 1 .Consider the following k districts: • e first district is a copy of d.
• For every ℓ ∈ {2, . . ., k}, the ℓ-th district is such that all agents therein rank y ℓ first, x ∈ {y 1 , . . ., y k } second, and then all other alternatives.By Lemma 3.2, M must choose y ℓ as the representative of the ℓ-th district, as this is the only top-ranked alternative.
So, indeed the set of representatives is {y 1 , . . ., y k } and M chooses y 1 as the winner by assumption.
One possible valuation profile is the following: • In the first, divided district, the 2λ/m agents that rank y 1 first have value 1/m for all alternatives, and the remaining agents all have value 1 for their favorite alternative.
Consequently, the social welfare of y 1 is λ/m 2 whereas the social welfare of x is approximately k •λ/2, and thus the distortion is Ω(km 2 ).
Lemma 3.3 shows that deterministic ordinal distributed mechanisms with distortion o(km 2 ) must not output the representative of a divided district as the winner when it is given a set of districts with different representatives.However, as we show in the proof of the next theorem, there are instances where such a choice is inevitable, and thus the distortion is Ω(km 2 ).eorem 3.4.e distortion of any deterministic ordinal distributed mechanism is Ω(km 2 ).
Let d 1 be a divided district with set of top-ranked alternatives {a 1 , b 1 , . . ., b m/2−1 }.By Lemma 3.3, if a 1 is the representative of d 1 , then there exists an instance such that the distortion of M is Ω(km 2 ).So, suppose that the representative of d 1 is some other top-ranked alternative, say b 1 .Again by Lemma 3.3, if b 1 is chosen as the winner whenever she is part of a representative set consisting of k distinct alternatives, then the distortion of M would be Ω(km 2 ).So, let us assume that when the district representatives are {b 1 , a 2 , . . ., a k }, the winner is an alternative different than b 1 , say a 2 .
We can now repeat this argument step by step for each alternative a ℓ , ℓ ∈ {2, . . ., k}.In particular, let d ℓ be a divided district with top-ranked alternatives {a ℓ , b ℓ , . . ., b m/2+ℓ−2 } (note that alternatives b 1 , . . ., b ℓ−1 do not appear as top-ranked alternatives in d ℓ ).By Lemma 3.3, if a ℓ is the representative of d ℓ then the distortion of M is Ω(km 2 ), so the representative is some other alternative from the set {b ℓ , . . ., b m/2+ℓ−2 }, say b ℓ .Again by Lemma 3.3, if b ℓ is chosen as the winner whenever she is part of a representative set consisting of k distinct alternatives, then the distortion of M would be Ω(km 2 ).So, when the district representatives are {b 1 , . . ., b ℓ , a ℓ+1 , . . ., a k }, the winner is an alternative not in {b 1 , . . ., b ℓ }, say a ℓ .e last step of this repeated argument leads to the lower bound of Ω(km 2 ): We have reached an instance with set of representatives {b 1 , . . ., b k } all of whom are representative of some divided district, and thus no ma er who of them is chosen as the winner, by Lemma 3.3 there exists an instance that includes the corresponding divided district and k − 1 unanimous districts (like in the proof of the lemma) such that the distortion is Ω(km 2 ).
Finally, let us discuss the case of deterministic strategyproof distributed mechanisms.Bhaskar and Ghosh [2018] showed that the distortion of any deterministic centralized strategyproof voting rule (including those that have access to the valuation functions) is Θ(nm).From the discussion Section 2.4, we directly obtain a lower bound of Ω(nm) for the distributed se ing as well.A tight upper bound is also not hard to derive by considering the straightforward F F mechanism which works as follows: • For each district d, choose the favorite alternative of the first agent therein as the representative.
• Choose the representative of the first district as the winner.eorem 3.5.F F is strategyproof and achieves an asymptotically best possible distortion of Θ(nm) within the class of deterministic strategyproof distributed mechanisms. Proof.
e mechanism is clearly strategyproof since the winner is the favorite alternative of the first agent of the first district who acts as a dictator.Since the winner is ranked first by an agent, the social welfare of the mechanism is at least 1/m.e maximum possible social welfare is n, and thus the distortion is O(nm).

Randomized mechanisms
We start our discussion on randomized distributed mechanisms by analyzing a general class of mechanisms that we call U δ A .A mechanism M in this class works as follows: • For each district d, M chooses the representative a d according to some centralized voting rule f in that has distortion at most δ.
• M chooses the winner uniformly at random from the set of representatives.
Picking the winner uniformly at random from the representatives that have been selected seems to be the most natural choice as there is not much information about the preferences of the agents in the districts, and essentially all we can do is assign higher proportional probability to an alternative that is representative of more districts.We have the following result.
Since a d is chosen based on a voting rule with distortion at most δ, we have that Combining this together with the fact that SW(o) = d∈D SW d and using the linearity of expectation, we obtain Hence, the distortion of the mechanism is at most kδ.
eorem 4.1 is a simple composition theorem, analogous to the one presented by Anshelevich et al. [2022] for the metric se ing.Based on it, we can define randomized distributed mechanisms with proven distortion guarantees by appropriately choosing the in-district rule.Before we continue, observe that the sizes of the districts do not appear in the proof of eorem 4.1, and thus the distortion of any U δ A mechanism is O(kδ) even if the districts are asymmetric.So, the distortion of the mechanism depends on the number of agents only if the distortion δ of the in-district rule depends on the number of agents.
If cardinal information is available at the district level, by using R V with δ = 1 as the in-district rule, we obtain the following.
If only ordinal information about the preferences of the agents is given at the district level, then we can use P with δ = O(m 2 ) and the randomized rule S L mechanism of Ebadian et al. [2022] with δ = O( √ m) as the in-district rule to obtain the following results.
Consider an instance with k symmetric districts such that in district d ℓ there is a set of 2λ/m agents with preference ordering a ℓ ≻ x ≻ [A \ {a ℓ , x}], a set of 2λ/m agents with preference ordering b 1 ≻ x ≻ [A \ {b 1 , x}], . .., and a set of 2λ/m agents with preference ordering b m/2−1 ≻ x ≻ [A \ {b m/2−1 , x}].By Lemma 3.2, the representative of d ℓ must be one of the top-ranked alternatives (otherwise the distortion of the mechanism would be infinite).Since a ℓ is ranked above the other alternatives in the tie-breaking ordering, she chosen as the representative of d ℓ .Hence, the set of representatives is {a 1 , . . ., a k }, and the winner is chosen according to some probability distribution over this set.e valuation profile may be such that the 2λ/m agents in district d ℓ that rank a ℓ first have value 1/m for all alternatives, while all other agents in d ℓ have value 1/2 for their two favorite alternatives.Consequently, the social welfare of alternative a ℓ is 2λ/m 2 , and thus the social welfare of the mechanism is also this much, no ma er the probability distribution over the district representatives.In contrast, the social welfare of x is approximately kλ/2, leading to a distortion of Ω(km 2 ).
When randomization at the district level can be leveraged by ordinal distributed mechanisms, then we achieve distortion much be er than what is implied by Corollary 4.4, while also achieving strategyproofness.In particular, there are several centralized voting rules that can be implemented as distributed mechanisms, in the sense that they define the same probability distribution over the alternatives.One such important class of voting rules is that of point-voting schemes, which is part of a larger class of strategyproof mechanisms [Barbera, 1978, Hylland, 1980, Gibbard, 1977] and includes rules with almost best possible distortion guarantees [Boutilier et al., 2015, Ebadian et al., 2022].

Point-voting schemes
A point-voting scheme chooses an agent uniformly at random and then outputs her t-th favorite alternative with probability p t , where p 1 ≥ . . .≥ p m ≥ 0 and m t=1 p t = 1.Hence, the probability according to which the point-voting scheme using the probability vector p = (p 1 , . . ., p m ) chooses alternative a ∈ A as the winner w is Pr[w = a] = 1 n i∈N p σ i (a) , where σ i (a) is the position that i ranks a in her preference ranking σ.
ere are many point-voting schemes of interest.For every positional scoring rule using the scoring vector s = (s 1 , . . ., s m ), we can define a point-voting scheme f (s) by normalizing the scoring vector, that is, define p t = s t / j∈[m] s j for every t ∈ [m] so that the winning probability of alternative a is Another important point-voting scheme is the rule that chooses each alternative uniformly at random; in this case, we have For any point-voting scheme f that uses a probability vector p, we consider the distributed mechanism P f P V , which works as follows: • For every district d, choose the representative a d to be alternative a ∈ A with probability • Choose the winner to be the representative of district d with probability n d /n.eorem 4.8.P f P V defines the same probability distribution as the pointvoting scheme f . Proof.
e probability that alternative a is chosen as the winner by P f P V is that is, P f P V chooses a with the same probability as f .eorem 4.8 shows that P f P V achieves the same distortion bound as the point-voting scheme f it uses as the in-district rule, and also that it inherits its strategyproofness property.is is extremely useful, as there are centralized voting rules that are based on point-voting schemes and achieve almost the best possible distortion.Boutilier et al. [2015] considered a voting rule that is a convex combination of two point-voting schemes: With probability 1/2 choose an alternative uniformly at random, and with probability 1/2 run the point-voting scheme defined by normalizing the harmonic scoring rule H = (1, 1/2, . . ., 1/m).We will refer to this mechanism as BCHLPS.Boutilier et al. [2015] showed that this voting rule has distortion O( √ m log m).An important property of point-voting schemes is that any rule that is a convex combination of point-voting schemes is also a point-voting scheme.e following lemma is similar to lemmas proved before in the literature (e.g., see Filos-Ratsikas and Miltersen [2014], Barbera [1978]); we provide a proof for completeness.
Lemma 4.9.Let f 1 , . . ., f κ be point-voting schemes defined by the probability vectors p 1 , . . ., p κ .For any non-negative numbers q 1 , . . ., q κ such that j∈[κ] q j = 1, the voting rule f that chooses the outcome of f j with probability q j is a point-voting scheme.
Proof.Let σ be an arbitrary preference profile.For any j ∈ [κ], denote the t-th coordinate of p j as p j,t , and let P j (a) = Pr[a = f j (σ)] be the probability of choosing a as the winner according to point-voting scheme f j .en, the voting rule f chooses alternative a as the winner w with probability Hence, f is a point-voting scheme defined by the probability vector p with p t = j∈[κ] q j • p j,t .
Consequently, by eorem 4.8 and Lemma 4.9, we can construct a randomized ordinal distributed mechanism based on the point-voting scheme of Boutilier et al. [2015] that achieves the same distortion bound and is strategyproof.We also use the rule of Boutilier et al. [2015] (we refer to it as BCHLPS in the following); recall that this is a point-voting scheme that with probability 1/2 selects an alternative at random and with probability 1/2 runs the PropHarmonic rule defined above.As established in Corollary 4.10 (and the discussion before the statement of the corollary), this is best possible in terms of the worst-case distortion.
e results of our experiments can be seen in Table 2.In the table we only present the results where as f over , we used P for deterministic rules and U for randomized rules.is is in accordance to our approach in the theoretical results in previous sections.
e bounds for the cases not shown are quite similar, and slightly larger in general.For each of the randomized rules, we perform 300 runs and calculate their expected social welfare, which we then use to calculate the distortion.
From the results of Table 2 we observe that, as expected, the existence of multiple districts has an adverse effect on the distortion of deterministic mechanisms, which becomes worse compared to the centralized case k = 1.For these rules, we can also observe that the distortion generally increases as k increases.In contrast, the distortion of randomized rules remains virtually unchanged for any value of k. is is in complete accordance with our theoretical findings, where we established that these rules induce the same probability distribution.
e experiments showcase that this does not only hold in expectation, but also in practice (given sufficiently many runs).
Another crucial observation is that, in terms of the absolute distortion numbers, randomization does not seem to help; if anything, it makes the distortion bounds worse! is can be justified by the fact that real-world instances like those from the Jester dataset display a large degree of homogeneity, which results in the simple deterministic rules performing quite well.On the other hand, randomization o en leads to suboptimal choices even on such "well-behaved" instances, demeaning the distortion bounds on average.Surprisingly, among ordinal voting rules, B seems to perform best across the board even though the theoretical distortion of B is in fact unbounded.

Experiments with Synthetic Datasets
We also perform experiments with datasets that are generated from probability distributions.In particular, and to be consistent with the Jester experiment presented above, we create instances with 100 agents and 8 alternatives, by first drawing the values of the agents from a certain distribution, and then constructing the induced ordinal preference profile from those values.We use the following distributions: • Uniform distribution in [1,100]. is is the simplest case, where all possible values are equally likely.
• Beta distribution with α = 1/10 and β = 1/10.is distribution has a symmetric convex pdf function centered around a mean of 1/2, assigning higher probabilities to values very close to 1 or 0.
• Exponential distribution with exponent 4, i.e., the pdf is f (x) = 4e 4 for x ≥ 0 and f (x) = 0 otherwise.is distribution generates values close to 0 with high probability, and as the values increase, the probability of them being generated decreases exponentially.
For the rest of the experiment, we perform similar steps as in the case of the Jester dataset: We normalize the values to sum up to 1, and run the set of mechanisms described above.For each randomized mechanism we now perform 150 individual runs and calculate its expected welfare.We calculate the average distortions over 500 runs of the experiment for k symmetric districts, where k ∈ {1, 2, 5, 20, 25}.Note that the number of runs and the number of district sizes is slightly smaller in this experiment, because it is more computationally intensive (as we need to calculate bounds for 3 different distributions).Again, we use P as f over for deterministic and U for randomized mechanisms; the results for the other cases were similar and are not reported.e results can be found in Table 3. Similarly to the Jester experiment, it is evident that the distortion of the deterministic mechanisms becomes worse for k ≥ 2, whereas it remains pre y much the same for randomized mechanisms.Again, we observe that randomization results in worse distortion bounds overall, and that B performs best among deterministic mechanisms.Interestingly, contrary to the Jester dataset, here we do not see a clear pa ern of the distortion increasing as k increases for deterministic mechanisms (other than the jump from k = 1 to k = 2).is is probably due to the fact that the synthetic instances are highly homogeneous, and with uniform random district partitions, the districts end up being quite uniform, regardless of their number and size.
e role of unit-sum.We remark here that normalizing the values to sum up to 1 effectively makes the Uniform and Exponential distributions pre y similar, and this is reflected in the results.To get a sense of the effect of normalization, we also ran the experiments without it.We observe that the distortions for the exponential distribution are now larger than those of the uniform distribution.In general, the distortion bounds still lie in the range [1.03, 1.15] for all distributions, but their average values (over all documented distortion bounds) are larger for all distributions except Uniform.It is also the case that for the Beta distribution, the bounds of deterministic mechanisms are much closer to those of randomized ones.e distortion of randomized mechanisms is still almost the same for any number of districts.

Open Problems
From our results, an interesting technical challenge is to remove the requirement for a consistent tiebreaking ordering from the statement of eorem 4.7.Similarly, we could a empt to remove unanimity from the lower bound of eorem 3.1; although unanimity is usually pre y natural, removing it would make the theorem stronger.More interestingly, our result about point-voting schemes in eorem 4.8 crucially does not depend on the normalization of the valuations, and hence also could be applied verbatim to the metric distributed social choice se ing studied by Anshelevich et al. [2022], where randomized mechanisms have never been considered; this seems like a natural starting point for such an investigation.
eorem 4.1.e distortion of any U δ A mechanism is O(kδ).Proof.Consider an arbitrary instance.Let o be the optimal alternative, a d the representative of district d, and w the final winner.Denote by SW d (x) the social welfare of alternative x only from the agents in d; clearly, SW(x) = d∈D SW d (x).e expected social welfare of the mechanism is

Table 2 :
Distortion bounds of various voting rules based on valuations defined by the provided scores of the Jester dataset and random district partitions.

Table 3 :
Distortion bounds of various voting rules based on valuations defined according to several probability distributions and random district partitions.Results for deterministic mechanisms are presented at the le of the bold vertical line, and results for randomized mechanisms are at the right of the bold vertical line.