1 Introduction
The main goal of social choice theory (Sen, 1986) is to come up with outcomes that accurately reflect the collective opinions of individuals within a society. A prominent example is that of elections, where the preferences of voters over different candidates are aggregated into a single winner, or a set of winners in the case of committee elections. Besides elections, the abstract social choice theory setting, where a set of agents express preferences over a set of possible alternatives captures very broad decisionmaking application domains, such as choosing public policies, allocations of resources, or the most appropriate position to locate a public facility.
In the field of computational social choice, Procaccia and Rosenschein (2006) defined the notion of distortion to measure the loss in an aggregate cardinal objective (typically the utilitarian social welfare), due to making decisions whilst having access to only limited (ordinal, in particular) information about the preferences of the agents, rather than their true cardinal values (or costs). Following their work, a lot of effort has been put forward to bound the distortion of social choice rules, with Anshelevich et al. (2018) and Anshelevich and Postl (2017) being the first to considered settings with metric preferences. In such settings, agents and alternatives are points in a metric space, and thus the distances between them (which define the costs of the agents) satisfy the triangle inequality. The metric space can be highdimensional, and can be thought of as evaluating the proximity between agents and alternatives for different political issues or ideological axes (e.g., liberal to conservative, or libertarian to authoritarian). The distortion in metric social choice has received significant attention, with many variants of the main setting being considered over the recent years.
In contrast to the centralized decisionmaking settings considered in the papers mentioned above, there are cases where it is logistically too difficult to aggregate the preferences of the agents directly, or different groups of agents play inherently different roles in the process. In such scenarios, the collective decisions have to be carried out in a distributed manner, as follows. The agents are partitioned into groups (such as electoral districts, focus groups, or subcommittees), and the members of each group locally decide a single alternative that is representative of their preferences, without taking into account the agents of different groups. Then, the final outcome is decided based on properties of the group representatives, and not on the underlying agents within the groups; for example, the representatives act as agents themselves and choose an outcome according to their own preferences. However, since the representatives cannot perfectly capture all the information about the preferences of the agents (even when it is available in the group level), it is not surprising that choosing the final outcome this way may lead to loss of efficiency.
Motivated by this, FilosRatsikas et al. (2020)
initiated the study of the deterioration of the social welfare in general normalized distributed social choice settings. They extended the notion of distortion to account for the information about the agents’ preferences that is lost after the local decision step, and showed bounds on the distortion of maxweight mechanisms when the number of groups is given. Very recently,
FilosRatsikas and Voudouris (2021) considered the distributed distortion problem with metric preferences, and showed tight bounds on the distortion of (cardinal and ordinal) mechanisms under several restrictions: Their bounds (a) apply only to the real line metric, (b) concern the (average) social cost objective (the total distance between agents and the chosen alternative), and (c) are mainly limited to groups of agents of the same size.In this paper, we extend the results of FilosRatsikas and Voudouris (2021) in all three axes: We provide bounds for (a) general metrics (including refined bounds for the line metric), (b) four different objectives (including the average total cost, the maximum cost, as well as two new objectives that are tailormade and clearly motivated by the distributed nature of the setting), and (c) groups of agents that could vary in size. We paint an almost complete picture of the distortion landscape of distributed mechanisms when the agents have metric preferences.
1.1 Our Contributions
We consider a distributed, metric social choice setting with a set of agents and a set of alternatives, all of whom are located in a metric space. The preferences of the agents for the alternatives are given by their distances in the metric space, and as such they satisfy the triangle inequality. Furthermore, the agents are partitioned into a given set of districts of possibly different sizes. A distributed mechanism selects an alternative based on the preferences of the agents in two steps: first, each district selects a representative alternative using some local aggregation rule, and then the mechanism uses only information about the representatives to select the final winning alternative.
The goal is to choose the alternative that optimizes some aggregate objective that is a function of the distances between agents and alternatives. In the main part of the paper we consider the following four cost minimization objectives, which can be defined as compositions of objectives applied over and within the districts:

The average of the average agent distance in each district^{1}^{1}1Note that this objective is not exactly equivalent to the wellknown (average) social cost objective, defined as the (average) total agent distance over all districts. All of our results extend for this objective as well, by adapting our mechanisms to weigh the representatives proportionally to the district sizes. When all districts have the same size, coincides with the average social cost. (denoted by );

The average of the maximum agent distance in each district (denoted by );

The maximum agent distance in any district (denoted by );

The maximum of the average agent distance in each district (denote by ).
While and are adaptations of objectives that have been considered in the centralized setting, and are only meaningful in the context of distributed social choice. In particular, can be thought of as a fairnessinspired objective guaranteeing that no district has a very large cost, where the cost of a district is the average cost of its members. Similarly, guarantees that the average district cost is small, where the cost of a district is now defined as the egalitarian (maximum) cost of any of its members. We consider the introduction and study of these objectives as one of the major contributions of our work.
We measure the performance of a distributed mechanism by its distortion, defined as the worstcase ratio (over all instances of the problem) between the objective value of the alternative chosen by the mechanism and the minimum possible objective value achieved over all alternatives. The distortion essentially measures the deterioration of the objective due to the fact that the mechanism must make a decision via a distributed twostep process, on top of other possible informational limitations related to the preferences of the agents. We consider deterministic mechanisms that are either cardinal (in which case they have access to the exact distances between agents and alternatives), or ordinal (in which case they have access only to the rankings that are induced by the distances). Table 1 gives an overview of our bounds on the distortion of distributed mechanisms, for the four objectives defined above. We provide bounds that hold for general metric spaces, and then more refined bounds for the fundamental special case where the metric is a line.
General metric  Line metric  

Cardinal  Ordinal  Cardinal  Ordinal  
*  *  *  *  
Several of our bounds for general metric spaces are based on a novel composition technique for designing distributed mechanisms. In particular, we prove a rather general composition theorem, which appears in many versions throughout our paper, depending on the objective at hand. Roughly speaking, the theorem relates the distortion of a distributed mechanism to the distortion of the centralized voting rules it uses for the local (indistrict) and global (overdistricts) aggregation steps. In particular, for two such voting rules with distortion bounds and , the distortion of the composed mechanism is at most . This effectively enables us to plug in voting rules with known distortion bounds, and obtain distributed mechanisms with low distortion. The theorem is also robust in the sense that the AVG and MAX objectives can be substituted with more general objectives satisfying specific properties, such as monotonicity and subadditivity; we provide more details on that in Section 5.2.
To demonstrate the strength of the composition theorem, consider the objective . The upper bound of for cardinal mechanisms and general metrics is obtained by using optimal centralized voting rules (with distortion ) for both aggregation steps. Similarly, the upper bound of for ordinal mechanisms follows by using the ordinal PluralityMatching rule of Gkatzelis et al. (2020) in both steps of the distributed mechanism; this rule is known to have distortion at most for general instances, and at most when all agents are at distance from their mostpreferred alternative (which is the case when the representatives are thought of as agents in the second step of the mechanism).
Even though the composition theorem is evidently a very powerful tool, it comes short of providing tight bounds in some cases. To this end, we design explicit mechanisms with improved distortion guarantees, both for general metrics as well as the fundamental special case where the metric is a line. A compelling highlight of our work is a novel mechanism for objectives of the form , to which we refer as AcceptableRightmostLeftmost (ARL). While this mechanism has the counterintuitive property of not being unanimous (i.e., there are cases where all agents agree on the best alternative, but the mechanism does not choose this alternative as the winner), it achieves the best possible distortion of among all distributed mechanisms on the line. In contrast, we prove that unanimous mechanisms cannot achieve distortion better than . To the best of our knowledge, this is the first time that not satisfying unanimity turns out to be a necessary ingredient for achieving the best possible distortion in the metric social choice literature.
1.2 Related Work
The distortion of centralized social choice voting rules has been studied extensively for many different settings. For a comprehensive introduction to the distortion literature, we refer the interested reader to the recent survey of Anshelevich et al. (2021).
After the work of Procaccia and Rosenschein (2006), a series of papers adopted their normalized setting, where the agents have unitsum values for the alternatives, and proved asymptotically tight bounds on the distortion of ordinal singlewinner rules (Caragiannis and Procaccia, 2011; Boutilier et al., 2015), multiwinner rules (Caragiannis et al., 2017), rules that choose rankings of alternatives (Benadè et al., 2019), and strategyproof rules (Bhaskar et al., 2018). Recent papers considered more general questions related to how the distortion is affected by the amount of available information about the values of the agents (Mandal et al., 2019, 2020; Amanatidis et al., 2021a). The normalized distortion has also been investigated in other related problems, such as participatory budgeting (Benadè et al., 2017), and onesided matching (FilosRatsikas et al., 2014; Amanatidis et al., 2021b).
The metric distortion setting was first considered by Anshelevich et al. (2018) who, among many results, showed a lower bound of on the distortion of deterministic singlewinner ordinal rules for the social cost, and an upper bound of , achieved by the Copeland rule. Following their work, many papers were devoted to bridging this gap (e.g., see (Munagala and Wang, 2019; Kempe, 2020b)) until, finally, Gkatzelis et al. (2020) designed the PluralityMatching rule that achieves an upper bound of ; in fact, this bound holds for the more general fairness ratio (Goel et al., 2017) (which captures various different objectives, including the social cost and the maximum cost). Besides the main setting, many other works have shown bounds on the metric distortion for randomized rules (Anshelevich and Postl, 2017; Feldman et al., 2016), rules that use less than ordinal information about the preferences of the agents (Fain et al., 2019; Kempe, 2020a; Anagnostides et al., 2021), committee elections (Chen et al., 2020; Jaworski and Skowron, 2020), primary elections (Borodin et al., 2019), and for many other problems (Abramowitz and Anshelevich, 2018; Anshelevich and Zhu, 2018).
Most related to our work are the recent papers of FilosRatsikas et al. (2020) and FilosRatsikas and Voudouris (2021), who initiated the study of the distortion in distributed normalized and metric social choice settings, respectively. As already previously discussed, we improve the results of FilosRatsikas and Voudouris (2021) by extending them to hold for general metrics and asymmetric districts, and also show bounds for many other objectives. In our terminology, FilosRatsikas and Voudouris showed a tight bound of for cardinal distributed mechanisms and a tight distortion of for ordinal mechanisms, when the metric is a line, the districts have the same size, and the objective is the social cost (which is equivalent to our objective when the districts are symmetric). Interestingly, not only do we generalize these results to hold for asymmetric districts and other objectives, but our composition theorem also provides easier proofs, compared to the characterizations of worstcase instances used in their paper.
2 Preliminaries
An instance of our problem is defined as a tuple , where

is a set of agents.

is a set of alternatives.

is a collection of districts, which define a partition of (i.e., each agent belongs to a single district). Let be the set of agents that belong to district , and denote by the size of .

is a metric space that contains points representing the agents and the alternatives. In particular, defines a distance between any , such that the triangle inequality is satisfied, i.e., for every .
A distributed mechanism takes as input information about the metric space, which can be of cardinal or ordinal nature (e.g., agents and alternatives could specify their exact distances between them, or the linear orderings that are induced by the distances), and outputs a single winner alternative by implementing the following two steps:

Step 1: For every district , the agents therein decide a representative alternative .

Step 2: Given the district representatives, the output is an alternative .
In both steps, the decisions are made by using direct voting rules, which map the preferences of a given subset of agents to an alternative. To be more specific, in the first step, an indistrict direct voting rule is applied for each district with input the preferences of the agents in the district (set ) to decide its representative . Then, in the second step, the district representatives can be thought of as pseudoagents, and an overdistricts direct voting rule is applied with input their preferences to decide the final winner . In the special case of instances consisting of a single district, the process is not distributed, and thus the two steps collapse into one: The final winner is the alternative chosen to be the district’s representative.
2.1 Objectives
We consider standard minimization objectives that have been studied in the related literature, and also propose new ones that are appropriate in the context distributed setting. Each objective assigns a value to every alternative as a cost function composition of an objective function that is applied over the districts and an objective function that is applied within the districts. Our main four objectives are defined by considering all possible combinations of , where the functions AVG and MAX define an average and a max over districts or agents within a district, respectively. In particular, we have:

The of an alternative is defined as

The of an alternative is defined as

The of an alternative is defined as

The of an alternative is defined as
The objective is similar to the wellknown utilitarian average social cost objective measuring the average total distance between all agents and alternative ; actually, coincides with the average social cost when the districts are symmetric (i.e., have the same size), but not in general. The objective coincides with the egalitarian max cost measuring the maximum distance from among all agents. The new objectives and make sense in the context of distributed voting, and can be thought of as measures of fairness between districts. For example, minimizing the objective corresponds to making sure that the final choice treats each district fairly so that the average social cost of each district is almost equal to that of any other district. Of course, besides combinations of AVG and MAX, one can define many more objectives; we consider such generalizations in Section 5.2.
2.2 Distortion of voting rules and distributed mechanisms
Direct voting rules can be suboptimal, especially when they have limited access to the metric space (for example, when ordinal information is known about the preferences of the agents over the alternatives). This inefficiency is typically captured in the related literature by the notion of distortion, which is the worstcase ratio between the objective value of the optimal alternative over the objective value of the alternative chosen by the rules. Formally, given a minimization cost objective , the distortion of a voting rule is
where denotes the alternative chosen by the voting rule when given as input the (singledistrict) instance consisting of a set of agents , a set of alternatives , and a metric space .
The notion of distortion can be naturally extended for the case of distributed mechanisms. Given a composition objective , the distortion of a distributed mechanism is
where is the alternative chosen by the mechanism when given as input an instance consisting of a set of agents , a set of alternatives , a set of districts, and a metric space . Our goal is to bound the distortion of distributed mechanisms for the different objectives we consider. To this end, we will either show how known results from the literature about the distortion of direct voting rules can be composed to yield distortion bounds for distributed mechanisms, or design explicit mechanisms with low distortion.
3 Composition Results for General Metric Spaces
In this section we consider general metric spaces, and show how known distortion bounds for direct voting rules can be composed to yield distortion bounds for distributed mechanisms that rely on those voting rules. Given an objective , we say that a distributed mechanism is inover if it works as follows.
[title=inover mechanisms,colback=white]

[left=1pt]

For each district , choose its representative using an indistrict voting rule with distortion at most .

Choose the final winner using an overdistricts voting rule with distortion at most .
Our first technical result is an upper bound on the distortion of inover mechanisms, for any . The proof of the following theorem also follows from the more general Theorem 5.5 in Section 5.2.1, which considers objectives that are compositions of functions satisfying particular properties, such as monotonicity and subadditivity.
Theorem 3.1.
For any , the distortion of any inover mechanism is at most .
Proof.
Here, we present a proof only for the objective; the proof for the other objectives is similar. Consider an arbitrary inover mechanism and an arbitrary instance . Let be the alternative that outputs as the final winner when given as input, and denote by an optimal alternative. By the definition of , we have the following two properties:
(1) 
and
(2) 
By the triangle inequality, we have for any agent . Using this, we obtain
By (1) and (2) for , we obtain
By the triangle inequality, we have for any agent . Using this and (1) for , we can upper bound the first term of the last expression above as follows:
Putting everything together, we obtain a distortion upper bound of . ∎
By applying Theorem 3.1 using known results, we can derive bounds on the distortion of distributed mechanisms, for any . In particular, due to the structure of and , if the whole metric space is known (that is, we have access to the exact distances between agents and alternatives), then we can easily compute the alternative that optimizes and . In other words, there exist direct voting rules with  and distortion , which can be used to obtain a inover distributed mechanism, and the following statement.
Corollary 3.2.
For any , there exists a cardinal distributed mechanism with distortion at most .
If only ordinal information is available about the distances between agents and alternatives, then we can employ the PluralityMatching voting rule of Gkatzelis et al. (2020) both within and over the districts. This rule is known to achieve the best possible distortion of among all ordinal rules, for any . In fact, this rule achieves a distortion bound of when all agents are at distance from their top alternative; this is the case when the agents are a subset of the alternatives as in the second step of a distributed mechanism.^{2}^{2}2See Theorem 1 and Proposition 6 in the arxiv version of the paper of Gkatzelis et al. (2020). Hence, we have a inover mechanism, and Theorem 3.1 yields the following statement.
Corollary 3.3.
For any , there exists an ordinal distributed mechanism with distortion at most .
Corollaries 3.2 and 3.3 demonstrate the power that Theorem 3.1 gives us in designing distributed mechanisms with constant distortion upper bounds, by using known results from the literature as black boxes. However, this method does not always lead to the best possible distributed mechanisms. In particular, let us consider the objectives , for and the class of ordinal mechanisms. We can improve upon the upper bound of due to Corollary 3.3 using the following, much simpler mechanisms.
[title=inarbitraryover mechanisms,colback=white]

[left=1pt]

For each district , choose its representative using an ordinal indistrict voting rule with distortion at most .

Output an arbitrary representative as the final winner.
Theorem 3.4.
For any , the distortion of any inarbitraryover mechanism is at most .
Proof.
We will present a proof only for the objective; the proof for follows by similar, even simpler arguments. Consider any inarbitraryover mechanism and any instance . Let be the alternative chosen by when given as input the ordinal information of , and denote the optimal alternative by . In addition, let be the district that gives the max cost for . By the triangle inequality, and since for every , we have
Now, let be the district whose representative is . By the triangle inequality and the fact that the indistrict voting rule used to choose as the representative of has AVGdistortion at most , we obtain
Putting everything together, we obtain a distortion upper bound of . ∎
Now, using again the voting rule of Gkatzelis et al. (2020) with AVG and MAXdistortion at most as an indistrict rule, we obtain a inarbitraryover distributed mechanism, and Theorem 3.4 implies the following result.
Corollary 3.5.
For any , there exists an ordinal distributed mechanism with distortion at most .
4 Improved Results on the Line Metric
In this section we focus on the line metric, where both the agents and the alternatives are assumed to be points on the line of real numbers. Exploiting this structure, there are classes of mechanisms for which we can obtain significantly improved bounds compared to those implied by the general composition Theorem 3.1, as well as Theorem 3.4.
4.1 Ordinal Mechanisms
We start with ordinal distributed mechanisms and the two objectives for . Recall that Corollary 3.3 implies a distortion bound of at most for these objectives. However, when the metric is a line, we can do much better by observing that there is an ordinal direct overdistricts voting rule with AVGdistortion of . In particular, we can identify the median district representative and choose it as the final winner. Using the rule of Gkatzelis et al. (2020) as the direct indistrict voting rule, we obtain a distributed inover mechanism with distortion at most due to Theorem 3.1.
Corollary 4.1.
When the metric is a line, there exists an ordinal distributed mechanism with distortion at most , for any .
Corollary 4.1 essentially recovers the tight distortion bound of by FilosRatsikas and Voudouris (2021) for when the districts are symmetric, and also extends it to the case of asymmetric districts. For , this bound of is a first improvement, but we can do even better with the following distributed mechanism.
[title=ArbitraryMedian,colback=white]

[left=1pt]

For every district , choose its representative to be the favorite alternative of an arbitrary agent .

Output the median representative as the final winner.
Theorem 4.2.
When the metric is a line, ArbitraryMedian has distortion at most .
Proof.
Consider any instance , in which is a line metric. Let be the alternative chosen by ArbitraryMedian when given as input (the ordinal information of) , and denote by the optimal alternative. For every district , let be the agent in that gives the max cost for . By the triangle inequality, and since , we have
Without loss of generality, we can assume that is to the left of on the line. Since is the median representative, there is a set of at least districts whose representatives are to the left of (or coincide with) . As a result, for every district , agent (whose favorite alternative becomes the representative of ) prefers over , and thus . Using this, we obtain
Putting everything together, we obtain a distortion upper bound of . ∎
Next, we show an almost matching lower bound of approximately . Before going through with the proof, we argue that ordinal distributed mechanisms with finite distortion must satisfy a unanimity property, which will be used extensively in all our lower bound constructions. Formally, we say that a distributed mechanism is unanimous if it chooses the representative of a district to be an alternative whenever all agents in prefer over all other alternatives.
Lemma 4.3.
For any , every ordinal distributed mechanism with finite distortion must be unanimous.
Proof.
Suppose towards a contradiction that there is an ordinal distributed mechanism with finite distortion that is not unanimous. Consider an instance with a single district consisting of agents all of whom prefer alternative to all other alternatives. Since is not unanimous, it chooses some different alternative to be the representative of the district, and thus the overall winner (as there is only a single district). However, if and all agents are positioned at the same point in the metric space, we have that and , and the distortion of is unbounded, a contradiction. ∎
We are now ready to present the lower bound on the distortion of ordinal distributed mechanisms.
Theorem 4.4.
The distortion of any ordinal distributed mechanism is at least , for any , even when the metric is a line.
Proof.
Suppose towards a contradiction that there is an ordinal distributed mechanism with distortion strictly smaller than , for any . We will reach a contradiction by defining instances with two alternatives and , and districts consisting of the same size. Without loss of generality, we assume that chooses alternative as the final winner when given as input any instance with only two districts, such that both alternatives are representative of some district. Let and be two integers such that , where is the golden ratio.
First, consider an instance consisting of the following two districts:

The first district consists of two agents, such that one of them prefers alternative and the other prefers alternative .

The second district consists of two agents, such that both of them prefer alternative . Due to unanimity (Lemma 4.3), the representative of this district must be .
Suppose that chooses as the representative of the first district, in which case both alternatives are representative of some district, and thus chooses as the final winner. Consider the following metric:

Alternative is positioned at , and alternative is positioned at .

In the first district, the agent that prefers alternative is positioned at , whereas the agent that prefers alternative is positioned at .

In the second district, both agents are positioned at .
Then, we have that and , leading to a distortion of . Consequently, must choose as the representative of the first district (where one agent prefers and one prefers ).
Next, we argue that for instances with districts such that is the representative of districts and is the representative of districts, must choose as the final winner. Assume otherwise that chooses in such a situation, and consider an instance consisting of the following districts:

Each of the first districts consists of a single agent that prefers alternative ; thus, is the representative of all these districts.

Each of the next districts consists of a single agent that prefers alternative ; thus, is the representative of all these districts.
By assumption, chooses as the final winner. Consider the following metric:

Alternative is positioned at and alternative is positioned at .

In each of the first districts, the agent therein is positioned at .

In each of the next districts, the agent therein is positioned at .
Since
and
the distortion is
Consequently, must choose , whenever there are districts such that is the representative of districts and is the representative of the remaining districts.
Finally, consider an instance with the following districts:

Each of the first districts consists of two agents that prefer alternative . Due to unanimity, must be the representative of all these districts.

Each of the next districts consists of two agents, such that one of them prefers alternative , while the other prefers alternative . By the discussion above (about instance ), must be the representative of these districts.
Since is the representative of districts and is the representative of districts, by the discussion above (about instance ), chooses as the final winner. Consider the following metric:

Alternative is positioned at and alternative is positioned at .

In each of the first districts, the two agents therein are positioned at .

In each of the next districts, the agent that prefers is positioned at , whereas the agent that prefers is positioned at .
Since
and
the distortion is
This contradicts our assumption that has distortion strictly smaller than , thus completing the proof. ∎
Next, we consider the objectives for . Corollary 3.5 implies a distortion bound of at most for these objectives. When and the metric is a line, we can get an improved bound of using a rather simple distributed mechanism, which essentially outputs the favorite alternative of an arbitrary agent.
[title=ArbitraryDictator,colback=white]

[left=1pt]

For each district , choose its representative to be the favorite alternative of an arbitrary agent in .

Output an arbitrary district representative as the final winner.
Theorem 4.5.
When the metric is a line, ArbitraryDictator has distortion at most .
Proof.
Let be an arbitrary instance, with a line metric. Without loss of generality, we assume that the alternative chosen by ArbitraryDictator is positioned to the right of the dictator agent . In addition, we denote by the optimal alternative, by the leftmost agent, and by the rightmost agent. With some abuse of notation, in the following we will also use , , , and to denote the positions of the corresponding agents and alternatives on the line.
First observe that the distortion is at most when . In particular, since there is at least one alternative inbetween the agents, we have that and . Hence, we can assume that (since ). Furthermore, it cannot be the case that (since then would prefer over ), neither that (since then would not be the optimal alternative). Consequently, . Since is the favorite alternative of , is also the favorite alternative of . Hence, . We distinguish between the following two subcases:

If , we clearly have that , and

If , we have that
and
This completes the proof. ∎
The following theorem shows that the bound of is the best possible we can hope for the objective using a unanimous distributed mechanism, even when the metric is a line. This lower bound directly extends to ordinal mechanisms, as any such mechanism with finite distortion has to be unanimous (Lemma 4.3).
Theorem 4.6.
The distortion of any unanimous distributed mechanism is at least , even when the metric is a line.
Proof.
Consider an arbitrary unanimous distributed mechanism , and the following instance with two alternatives and . There are two districts with agents each: All agents in the first district prefer alternative , whereas all agents in the second district prefer . Due to unanimity, must be the representative of the first district, and the representative of the second district. The final winner can be any of the two, say without loss of generality. Now, consider the following metric:

Alternative is located at and alternative is located at .

All agents in the first district (who prefer ) are located at .

All agents in the second district (who prefer ) are located at .
Consequently, we have that and , leading to a distortion of . ∎
Finally, let us focus on the objective , for which we show a lower bound of approximately on the distortion of all ordinal distributed mechanisms, thus almost matching the upper bound of implied by Corollary 3.5.
Theorem 4.7.
The distortion of any ordinal distributed mechanism is at least , for any , even when the metric is a line.
Proof.
Suppose towards a contradiction that there is an ordinal distributed mechanism with distortion strictly smaller than , for any . We will consider instances with two alternatives and . We can assume without loss of generality that outputs as the winner whenever it is given as input an instance with two districts of the same size, such that both alternatives are representative of some district. However, we do not know how decides the representative of a district in case there is a tie between the two alternatives therein. Let , and denote by a sufficiently large integer, such that and are both integers.^{3}^{3}3Here, we assume the existence of such an to simplify the presentation of the proof. Since is irrational, to be exact, we can choose to be a sufficiently large integer so that and are approximately equal to their floors, which would lead to the distortion lower bound being , for any . Observe that is such that .
First, consider an instance with a single district consisting of agents, such that agents prefer , and agents prefer . Suppose that outputs (as the district representative and the final winner) when given these ordinal preferences of the agents in the district as input, and consider the following metric:

Alternative is located at and alternative is located at .

The agents that prefer are located at .

The agents that prefer are located at .
Since there is only one district, we clearly have that
and
Hence, the distortion in this case is . As this contradicts our assumption that has distortion strictly smaller than , must choose as the winner in such a singledistrict instance.
Now, consider an instance with the following two districts:

The first district is similar to the one in instance , i.e., it consists of agents that prefer , and agents that prefer . By the above discussion, the representative of this district must be alternative .

The second district consists of agents, all of whom prefer . Due to unanimity (Lemma 4.3), the representative of this district must be alternaive .
Since both and are representative of some district, outputs as the final winner. Now, consider the following metric:

Alternative is located at and alternative is located at .

In the first district, the agents that prefer are located at , and the remaining agents that prefer are located at . The total distance of the agents in the district is from , and from .

In the second district, all agents are located at . The total distance of the agents in the district is from , and from .
Consequently, from the second district, we have that
and, from both districts, we have that
Thus, the distortion of the mechanism for this instance is , a contradiction. ∎
4.2 Cardinal Mechanisms
We now turn out attention to distributed mechanisms that have access to the line metric, and are thus aware of the distances between agents and alternatives. Recall that for such mechanisms, Corollary 3.2 implies a distortion bound of for all the objectives we have considered so far. As in the case of ordinal mechanisms, when the metric is a line, FilosRatsikas and Voudouris (2021) showed a matching lower bound of for (when the districts are symmetric), which extends for as the construction also works for instances with singleagent districts, in which case .
For objectives of the form , we design a novel distributed mechanism that is tailormade for the line metric and achieves a distortion of at most . This mechanism is particularly interesting as it is not unanimous: Even when all agents in a district prefer an alternative to every other alternative (i.e., is the closest alternative to all agents), the mechanism may end up choosing a different alternative as the representative. In fact, to break the distortion barrier of , our mechanism has to be nonunanimous; by setting in the proof of Theorem 4.6, we have that any unanimous distributed mechanism cannot achieve a distortion better than 3.
For a given , we say that an alternative is acceptable for a district if her value for the agents in is at most times the value of any other alternative for the agents in . Given an objective , we define a class of distributed mechanisms parameterized by that work as follows:
[title=AcceptableRightmostLeftmost (ARL),colback=white]

[left=1pt]

For each district , choose its representative to be the rightmost acceptable alternative for the district.

Output the leftmost district representative as the final winner.
Theorem 4.8.
For any , the distortion of ARL is at most .
Proof.
We will present the proof only for the objective; the proof for follows by similar, even simpler arguments. Consider an arbitrary instance , where is a line metric. Let be the alternative selected by ARL when given as input , and denote by the optimal alternative. We consider the following two cases depending on the relative positions of and .
Case 1: is to the left of . Let be the district that gives the max cost for . By the triangle inequality, and since for every , we have
Now, let be the district whose representative is . In other words, is the rightmost acceptable alternative for . Let be the alternative that minimizes the total distance of the agents in . We make the following two observations:

Since is trivially acceptable, it must be the case that she is located to the left of , and thus .

Since is to the right of and is not the representative of , it must be the case that she is not acceptable for , that is, .
Combining these two observations together with the triangle inequality, we obtain
Putting everything together, we obtain a distortion of at most .
Case 2: is to the right of . As in the previous case, let be the district with maximum cost for . Denote by the representative of , and by the optimal alternative for that minimizes the (average) total distance of the agents in . We consider the following two subcases:

is not acceptable for . To argue about this case, we will need the following folklore technical lemma for instances on the line:
Lemma 4.9.
Let be a set of agents, and denote by the optimal alternative for the agents in that minimizes their total distance. Then, for every two alternatives and , such that , or , it holds that .

is acceptable for . In this case, we clearly have that
and thus the distortion is at most in this case.
Putting everything together, the distortion of the mechanism is at most . ∎
By optimizing the distortion bound achieved by the class of ARL mechanisms over the parameter , we see that the best such mechanism achieves a distortion of at most .
Corollary 4.10.
For any , the distortion of ()ARL is at most .
We conclude this section by presenting a lower bound of on the distortion of distributed mechanisms, which holds even when the metric is a line, thus showing that ARL is the best possible on the line.
Theorem 4.11.
For any , the distortion of any distributed mechanism is at least , even when the metric is a line.
Proof.
Suppose towards a contradiction that there exists a distributed mechanism with distortion strictly smaller than . We will consider instances with two alternatives and that are located at and , respectively. In addition, our instances will consist of singleagent districts so that therein, and thus the distortion bounds hold for any . We can assume without loss of generality that outputs as the winner whenever it is given as input an instance with two districts of the same size, such that both alternatives are representative of some district.
First, consider an instance with a single district consisting of one agent that is located at . Clearly, we have that and . Therefore, must choose as the district representative, and thus the overall winner; otherwise its distortion would be at least .
Next, consider an instance with a single district consisting of one agent that is located at . Here, we have that and . Therefore, must choose as the district representative; otherwise its distortion would be at least .
Finally, consider an instance with the following two districts:

The first district is similar to the one in and consists of an agent that is located at . By the above discussion, the representative of this district is .

The second district is similar to the one in and consists of an agent that is located at . By the above discussion, the representative of this district is .
Since both alternatives are representative of some district and the districts have the same unit size, outputs as the final winner. However, it holds that (realized by the agent in the second district) and (realized by both agents), and thus the distortion of is at least