Consider the social choice problem of deciding on an allocation of public tax dollars to public projects. This is a voting problem over budgets. Clearly, the number of voters in such situations can be large. More interestingly, unlike in traditional social choice theory, there is no reason to believe that the number of alternatives (budgets) is small. It is therefore unreasonable to assume that we can elicit full ordinal preferences over alternatives from every agent. For a voting mechanism to be practical in such a setting, one would ideally like it to require only an absolute constant number of simple queries, regardless of the number of voters and alternatives. We call this property constant sample complexity, and we explore mechanisms of this sort in this paper.
We define our model more formally in Section 4, but at a high level, we have a set of agents (or voters) and a set of alternatives , from which we must choose a single outcome. We assume that and are both large, and that eliciting the full ordinal rankings may be prohibitively difficult. Instead, we work with an ordinal query model, and a constant sample complexity mechanism uses only a constant number of these queries.
- Top- Query.
“What are your favorite alternatives, in order?” (We call a top- query a favorite query); and
- Comparison Query.
“Which of two given alternatives do you prefer?”
Query models are not just of theoretical interest. They can be used to reduce cognitive overload in voting. For example, in the context of Participatory Budgeting , the space of possible budget allocations is large, and one mechanism is to ask voters to compare two proposed budgets. Similarly, in a context like transportation policy for a city, a single alternative can be an entire transportation plan. In such examples, not only are there many alternatives, but it may be infeasible to expect voters to compare more than two alternatives at the same time. We stress that constant sample complexity is particularly important in settings where there may be a large number of possibly complex alternatives.
To evaluate the quality of our mechanisms, we adopt the implicit utilitarian perspective with metric constraints [7, 12, 3, 18, 2, 16]. That is, we assume that agents have cardinal costs over alternatives, and these costs are constrained to be metric, but asking agents to work with or report cardinal costs is impractical or impossible. We want to design social choice mechanisms to minimize the total social cost, but our mechanism is constrained only to use ordinal queries, that is, those that can be answered given a total order over alternatives. We therefore measure the efficiency of a mechanism as its Distortion (see Section 4), the worst case approximation to the total social cost.
The starting point for our inquiry is the constant sample complexity Random Dictatorship mechanism. The algorithm asks a single favorite query from an agent chosen uniformly at random, and has a tight Distortion bound of 3 . In this paper, we provide two new mechanisms (Random Referee and Random Oligarchy) that improve on this simple baseline in three different ways, outlined in each of our three technical sections. We hope that our work inspires future research on similarly lightweight mechanisms for social choice in large decision spaces.
Random Referee: Comparison Queries and Squared Distortion. In Section 5
, we show that one disadvantage of Random Dictatorship lies not in its Distortion, but in its variance. For randomized mechanisms, Distortion is measured as the expected approximation to the first moment of social cost. However, in many social choice problems, we might want a bound on the risk associated with a given mechanism. We capture this viaSquared Distortion, as suggested in . The Squared Distortion (see Definition 2 in Section 4) is the expected approximation to the second moment of social cost. A mechanism with constant Squared Distortion has both constant Distortion and constant coefficient of variation of the Distortion.
We show that mechanisms using only top- queries (including Random Dictatorship) have Squared Distortion . This motivates us to expand our query model to incorporate information about the relative preferences of agents between alternatives, i.e. use comparison queries. We define a novel mechanism called Random Referee (RR) that uses a random voter as a referee to compare the favorite alternatives of two other random voters (see Definition 3 in Section 5). Our main result in Section 5 is Theorem 2: The Squared Distortion of RR is at most 21. This also immediately implies that the Distortion of RR is at most 4.583.
Random Referee: Euclidean Plane. In Section 6, we show that top- only mechanisms (again, including Random Dictatorship) achieve their worst case Distortion even on benign metrics such as low dimensional Euclidean spaces. We analyze a special case on the Euclidean plane and prove that the Distortion of Random Referee beats that of any top- only mechanism. While the improvement we prove in Section 5 is quantitatively small, it is qualitatively interesting: we demonstrate that by using a single comparison query, Random Referee can exploit the structure of the metric space to improve Distortion, whereas Random Dictatorship or any other top- only mechanism cannot. We conjecture this result extends to Euclidean spaces in any dimension, and present some evidence to support this conjecture in Section 8.
Random Oligarchy: Favorite Only Mechanisms. In Section 7, we consider mechanisms that are restricted to favorite queries and show that constant complexity mechanisms are nearly optimal. We present a mechanism that uses only three favorite queries that has Distortion at most 3 for arbitrary ; however, it also has Distortion for small that improves upon the best known favorite only mechanism from  that uses at most favorite queries. Comparing with a lower bound for favorite only mechanisms, Random Oligarchy has nearly optimal distortion and constant sample complexity. Though this mechanism does not have constant Squared Distortion like Random Referee, we present it to demonstrate again the surprising power of constant sample complexity randomized social choice mechanisms in general, and of queries to just three voters in particular.
Techniques. We use different techniques to prove our different positive results. The proof of Squared Distortion (Theorem 2 in Section 5) relies heavily on Lemma 1, in which we prove (essentially) that Random Referee chooses a low social cost alternative as long as at least two of the three agents chosen at random are near the social optimum.
The proof of Distortion for Euclidean spaces (Theorem 5 in Section 6) is the most technical result. We show that we can upper bound the Distortion of a mechanism by the worst case “pessimistic distortion,” of just a constant size tuple of points, where the “pessimistic distortion” considers all permutations of the points as participating in Random Referee and allows OPT to choose the optimal point on just this tuple. This allows us to employ a computer assisted analysis by arguing that if a high Distortion instance exists, we can detect it as an instance with a small constant number of points on a sufficiently fine (but finite) grid in the Euclidean plane. This approach may be of independent interest for providing tighter Distortion bounds for mechanisms in specific structured metric spaces.
3 Related Work
Distortion of Randomized Social Choice Mechanisms in Metrics. The Distortion of randomized social choice mechanisms in metrics has been studied in [7, 3, 18, 20]. Of particular interest to us are the Random Dictatorship mechanism that uses a single favorite query and the 2-Agree mechanism  that uses at most most favorite queries. Random Dictatorship has an upper bound on Distortion of 3 , and 2-Agree provides a strong guarantee on Distortion when is small (better than Random Dictatorship for ). There is ongoing work on analyzing the Distortion of randomized ordinal mechanisms for other classic optimization problems like graph optimization  and facility location .
Squared Distortion and Variance. We are aware of two papers in mechanism design that consider the variance of mechanisms for facility location on the real line  and kidney exchange . Our work is more related to the former, but is not restricted to the real line, and does not focus on characterizing the tradeoff between welfare and variance. Using Squared Distortion as a proxy for risk was introduced in  along with the sequential deliberation protocol. Unlike sequential deliberation, Random Referee makes a constant number of ordinal queries. The most important baseline for Squared Distortion is the deterministic Copeland rule, which has Distortion  and therefore Squared Distortion 25. However, Copeland requires the communication of bits , essentially the entire preference profile. Our Random Referee mechanism has constant sample complexity, and has better bounds on Squared Distortion (21) and Distortion (4.583) than the Copeland mechanism.
Communication Complexity. For a survey on the complexity of eliciting ordinal preferences to implement social choice rules, we refer the interested reader to . Of particular interest to us is , in which the authors comprehensively characterize the communication complexity (in terms of the number of bits communicated) of common deterministic voting rules. A favorite query requires bits of communication, so our mechanisms have constant sample complexity, but logarithmic communication complexity.  and  design social choice mechanisms with low communication complexity when there are a small number of voters, but potentially a large number of alternatives. All of our mechanisms have guarantees that are independent of the number of voters.
Strategic Incentives. We do not consider truthfulness in this paper, and we do not use the term mechanism to imply any such property. While strategic incentives are not the focus of this work, we note that any truthful mechanism must have Distortion at least 3 . Random Dictatorship has a Distortion of 3, and is therefore in some sense optimal among exactly truthful mechanisms. Other works suggest that truthfulness is also incompatible with the weaker notion of Pareto efficiency in randomized social choice [9, 5]. Still other authors have considered the problem of truthful welfare maximization under range voting  and threshold voting .
We have a set of agents (or voters) and a set of alternatives, from which we must choose a single outcome. For each agent and alternative , there is some underlying dis-utility . Let , that is, is the most preferred alternative for agent . Ordinal preferences are specified by a total order consistent with these dis-utilities (i.e., an alternative is ranked above another only if it has lower dis-utility). A preference profile specifies the ordinal preferences of all agents. A deterministic social choice rule is a function that maps a preference profile to an alternative . A randomized social choice rule maps a preference profile to a distribution over .
We consider mechanisms that implement a randomized social choice rule using a constant number of queries of two types. A top- query asks an agent for the first preferred alternatives according to the order (ties can be broken arbitrarily). We refer to a top-1 query as a favorite query, that asks an agent for her most preferred alternative . A top- only mechanism uses only top- queries, for some constant (constant with respect to and ). Most of our lower bounds or impossibilities will be for any top- only mechanism (for constant ), whereas our positive results will only need favorite and comparison queries. A comparison query with alternatives and asks an agent for .
We use the term mechanism to clarify that our algorithms are in a query model. However, it is important to note that mechanisms so defined are still randomized social choice rules in the formal sense as long as they do not make queries based on exogenous information (e.g., names of participants). Our mechanisms will in fact be randomized social choice rules, and thus can be appropriately compared to other such rules in the literature that do not explicitly use a query model. By using the term mechanism, we do not mean to imply any strategic properties.
Distortion and Sample Complexity. We measure the quality of an alternative by its social cost, given by . Let be the minimizer of social cost. We define the commonly studied approximation factor called Distortion , which measures the worst case approximation to the optimal social cost of a given mechanism. We use the expected social cost if is the outcome of a randomized mechanism, and we seek to minimize Distortion.
The Distortion of an alternative is The Distortion of a social choice mechanism is
where is a preference profile consistent with .
We assume that is a set of points in a metric space such that dis-utility can be measured by the distance from an agent. Specifically, we assume there is a distance function satisfying the triangle inequality such that . The metric assumption is common in the implicit utilitarian literature [3, 18, 15, 20, 12, 2, 16]. It is also a natural assumption for capturing social choice problems for which there is a natural notion of distance between alternatives. For example, in our original motivating example of public budgets, there is are natural notions of distance between alternatives in terms of dollars.
We do not assume access to directly, which may be prohibitively difficult to elicit when there are many alternatives. Instead, we work with a query model. The queries are ordinal in the sense that they can be answered given only the information in . A mechanism has constant sample complexity if there is an absolute constant such that for all , , and , can be implemented using at most queries. In this paper, we consider top- (and the special case of favorite) and comparison queries, and explore mechanisms with constant sample complexity.
Squared Distortion. It is easy to see that randomization is necessary for constant sample complexity mechanisms to achieve constant Distortion. As grows large, any deterministic mechanism with constant sample complexity deterministically ignores (asks no queries of and receives no information from) an arbitrarily large fraction of . An adversary can therefore place an alternative with 0 dis-utility for arbitrarily many agents; this gives a lower bound for Distortion approaching as becomes large.
This naturally leads us to ask: If we look at the distribution of outcomes produced by the mechanism, is this distribution well behaved? Following , we capture this notion via Squared Distortion: essentially the approximation to the optimal second moment of social cost.
The Squared Distortion of an alternative is The Distortion of a social choice mechanism is
where is a preference profile consistent with .
Note that a mechanism with constant Squared Distortion has both constant Distortion (by Jensen’s inequality), and constant coefficient of variation. One way to interpret having constant Squared Distortion is that the deviation of the social cost around its mean falls off quadratically instead of linearly, which means that the social cost of such a mechanism is well concentrated around its mean value. We note that this interpretation gives randomized social choice mechanisms with constant Squared Distortion an interesting application in candidate selection. In particular, one can imagine running such a mechanism (like our Random Referee) to generate a candidate list on which one can use a deterministic (but potentially complex) voting mechanism like Copeland.
A related approach to understanding the distributional properties of a randomized mechanism is to characterize the tradeoff between Distortion (approximation to the first moment) and the variance of randomized mechanisms . Our specific goal in this paper is to develop a mechanism that achieves constant Distortion and constant variance, and this combination is captured by having constant Squared Distortion. We leave characterizing the exact tradeoff between the quantities as an interesting open direction.
5 Random Referee and Squared Distortion
Our first result is Theorem 1: Mechanisms that only elicit top- preferences, for constant , must necessarily have Squared Distortion that grows linearly in the size of the instance. This holds even for mechanisms that elicit the top- preferences of all of the voters, mechanisms which would not have constant sample complexity. The proof is in the appendix.
Any top- only social choice mechanism has Squared Distortion .
The problem with top- only mechanisms, exploited in the proof of Theorem 1, is that they treat agents as indifferent between their st favorite alternative and their least favorite alternative. This motivates the expansion of our query model to include comparison queries. Recall that a comparison query with alternatives and asks an agent for . We use a single comparison query in our Random Referee mechanism.
The Random Referee (RR) mechanism samples three agents independently and uniformly at random with replacement. and are asked for their favorite feasible alternatives and in , and then is asked to compare and . Output whichever of the two alternatives prefers.
Upper Bound for Random Referee. Our main result in this section is Theorem 2: The Squared Distortion of RR is at most 21. This also implies that the Distortion of RR is at most 4.583. As far as we are aware, no randomized mechanism has lower Squared Distortion in general.
We will need the following technical lemma. Let be the social cost minimizer. The basic intuition behind Random Referee is that it will choose a low social cost alternative as long as any two out of the three agents selected are near the optimal alternative . Lemma 1 makes this intuition formal. For convenience, let for , i.e., the dis-utility of for agent . Let , that is, the alternative Random Referee outputs when and are selected to propose alternatives and chooses between them.
For all ,
One should think of as the agents drawn to present their favorite alternative, as the referee, and as the agent from whom we measure the dis-utility of the resulting outcome. The triangle inequality implies that
We give two separate upper bounds on , thus yielding the min. We will frequently use the fact that for all , . This follows from the the definition of : the favorite alternative of , and thus no greater in distance from than . The first bound is straightforward: by definition of Random Referee, so
In a sense, this bound concerns the situation where both and are near , but is far away from . Now we argue for the second bound. Suppose without loss of generality that . chooses either or . If chooses , then , and so the bound holds. If chooses , then , which implies
and again the bound holds. In either case, is at most . Intuitively, this bound concerns the situation where and are close to , but is far away from . Taking the better of this bound with the bound through the min and factoring out the 2 yields the lemma. ∎
The Squared Distortion of Random Referee is at most 21.
Let be the optimal squared social cost, that is, the squared social cost of , where is the social cost minimizer. Recall that . Then
Let be the expected squared social cost of Random Referee. The expectation can be written out as
Let We apply Lemma 1 and simplify.
Noting that does not depend on , we can expand the square and simplify to find
Now we sum each term separately. Let
We will use the following basic facts: for any real numbers , , and .
Similarly, we analyze the second term using the fact that .
Adding together all of the terms, . ∎
This immediately yields the root mean square Distortion bound via Jensen’s inequality.
The Distortion of Random Referee is at most .
6 Distortion of Random Referee on the Euclidean Plane
Though the upper bound on the Distortion of Random Referee is slightly worse than that of Random Dictatorship, we now show another advantage of using a comparison query: Such mechanisms can exploit structure in specific metric spaces that favorite-only mechanisms cannot. In other words, we show that the Distortion of Random Referee improves significantly for more structured metric spaces, while top- only mechanisms do not share this property.
We examine the Distortion of Random Referee on a specific canonical metric of interest: the Euclidean plane when for every . The second assumption, functionally equivalent to assuming , simplifies our analysis considerably, and corresponds to the 0-decisive case in [3, 20]. We note that in examples where is very large, the assumption becomes more innocuous. If we consider our opening example of public budgets, the assumption is something like this: every agent is allowed to propose their absolute favorite over all budgets, and we assume that this budget has dis-utility of 0. We consider the Euclidean plane because the problem of minimizing distortion on the real line can be solved exactly , whereas we are unaware of any results for the Euclidean plane that are stronger than those for general metric spaces.
Lower Bounds for Distortion in the Restricted Model. We begin by giving lower bounds to demonstrate that our simplifying assumptions still result in a nontrivial problem in two senses: (1) the Distortion of randomized social choice mechanisms are still bounded away from 1 and (2) any top- only mechanism (one that elicits the top preferred alternatives of agents) for constant has Distortion at least 2 as and become large. The proofs of the lower bounds are in the appendix.
The Distortion of a randomized social choice mechanism is at least 1.2 generally, and at least 1.118 for the Euclidean plane, even when for every .
The Distortion of any top- only mechanism goes to as and become large, even on the Euclidean plane when for every .
Upper Bound for Distortion of Random Referee in the Restricted Model. Our positive result in this section demonstrates that a single comparison query is sufficient to construct a mechanism (Random Referee) with Distortion bounded below for arbitrary and . We note that the bound in the theorem seems very slack; our goal is simply to show that using a comparison query provably decreases Distortion. We conjecture the actual bound is below 1.75 based on computer assisted search, but leave proving this stronger bound as an interesting open question.
The worst case distortion of Random Referee is less than 1.97 when for every , , and for .
In the remainder of this section, we sketch the proof of Theorem 5. Proofs of some of the technical lemmas are in the appendix. Suppose for a contradiction that there is a set of agents in such that the Distortion of Random Referee is at least under the Euclidean metric. We will successively refine this hypothesis for a contradiction, finally arguing that it implies that some “bad” instance would appear on an exhaustive computer assisted search over a finite grid.
The crucial lemmas bound the pessimistic distortion of any set of five points in and relate this to the actual distortion. The quantity is pessimistic because it allows “OPT” to choose a separate point for every 5-tuple; the numerator of each 5-tuple is just a rewriting of the expected social cost of Random Referee. In this section, since we assume , it will not be necessary to refer to and separately, and it will be convenient to let .
The pessimistic distortion111Note that pessimistic distortion is technically a function of the mechanism used; we will use it exclusively in reference to Random Referee. of is defined as
where is the average social cost of Random Referee, which we can write as
and the average cost of the optimal solution is
Finally, if .
The first lemma relates this pessimistic distortion to the actual Distortion of Random Referee. The worst case (over 5-tuples) pessimistic distortion upper bounds the Distortion of Random Referee. Interestingly, this statement is not specific to the Euclidean plane, suggesting that our approach may be broadly applicable for proving stronger Distortion bounds on other specific metrics of interest.
If for all then the Distortion of Random Referee is at most on .
Observe that we can rewrite the Distortion of Random Referee as a summation over all possible 5-tuples of points. Let be a multiset of points (i.e., possibly with repeats). Let be an ordering of . Let
We can rewrite the Distortion of Random Referee over permutations of 5-tuples as
In words, we are considering all 5-tuples (with replacement) of agent points, and for each we consider the average distance over all orderings of the five agent points of the distance between the last two points and the outcome of Random Referee when the first three agents participate. This is in turn upper bounded by allowing OPT to choose a different for every 5-tuple, so that the Distortion is at most
To complete the proof, note that is equal to the numerator. We start with all 120 orderings of the points and avoid double counting the symmetric cases that arise from swapping the two points from which we take the argmin and swapping the two points from which we measure distance. ∎
Now we can refine our original hypothesis: without loss of generality, assume for a contradiction that there is a multiset with .
Lemma 3 provides the contradiction to our hypothesis and establishes Theorem 5. We outline the main ideas of the proof. First, we consider a grid on the Euclidean plane, and argue that we can assume a certain canonical structure of in relation to that grid by scaling, translating, and rotating. Next, we carefully argue for how much the pessimistic distortion would change if every point in were snapped to a grid point. We show that if there is some with pessimistic distortion at least 1.97, there must be a 5-tuple of points on a sufficiently fine (but finite) grid with a sufficiently large pessimistic distortion. However, we employ computer assisted analysis to brute force search over such a grid, and find no such bad example.
Our analysis is slack in two ways: (1) we have to interpolate between grid points, and (2) we consider pessimistic distortion over 5-tuples. Both are computational constraints: (1) because we cannot simulate an arbitrarily fine grid, and (2) because there is a combinatorial blow up in the search space when considering larger tuples (note that by considering 5-tuples, we implicitly allow OPT to choose a separate optimal solution for each 5-tuple). For Random Referee, the worst case example found by computer simulations for grid points is fairly simple: The 5 points lie on a straight line with pessimistic distortion 1.75. We conjecture the same example is the worst case even in the continuous plane, which suggests a distortion bound of at most 1.75. We leave finding the exact bound as an open question, as our result is sufficient to demonstrate that comparison queries can take advantage of structure that top-queries cannot.
7 Favorite Only Mechanisms: Random Oligarchy
Recall that a favorite query asks an agent for her favorite alternative . In this section, we return to the general model (arbitrary metrics and not assuming that ) and study mechanisms that are restricted to only use favorite queries. We show that essentially optimal Distortion as a function of is achieved by a simple mechanism that uses just queries. We call this mechanism Random Oligarchy.
The Random Oligarchy (RO) mechanism samples three agents independently and uniformly at random with replacement. All three are asked for their favorite alternatives , , and in . If the same alternative is reported at least twice, output that, else output one of the three alternatives uniformly at random.
We prove that Random Oligarchy has the best of both worlds with respect to the other favorite only mechanisms of Random Dictatorship and 2-Agree. Unlike 2-Agree, Random Oligarchy has constant sample complexity and the same Distortion bound of 3 for large as Random Dictatorship. However, like 2-Agree, it outperforms Random Dictatorship for small . The proof of the theorem is given in the appendix.
The Distortion of Random Oligarchy is upper bounded by for arbitrary , and by the following expression for particular .
Figure 1 shows the Distortion bounds of favorite only mechanisms. Comparing against the lower bound for any favorite only mechanism from  allows us to see that Random Oligarchy is essentially optimal among all favorite only mechanisms. In comparison to existing mechanisms, Random Oligarchy outperforms Random Dictatorship for small , and outperforms 2-Agree for large , while only using three favorite queries.
8 Open Directions
In this paper, we have considered constant sample complexity mechanisms. At a high level, we hope that our work inspires future research on lightweight mechanisms for social choice in large decision spaces. We also mention some natural technical questions raised by our work.
Compared to Distortion, there is much less understanding of the Squared Distortion of mechanisms. The only universal lower bound for Squared Distortion we are aware of is 4, a consequence of the lower bound of 2 for Distortion . Our Random Referee mechanism achieves Squared Distortion of at most 21 using just three ordinal queries. Closing this gap remains an interesting question even for mechanisms that elicit full ordinal information.
The analysis in Section 6 of Random Referee generalizes to higher dimensional Euclidean space, still only reasoning about 5-tuples of points. Computer search over a coarse grid in four dimensions again shows that the pessimistic distortion bound is better than . However, we have not been able to run the search on a fine enough grid to prove a Distortion bound formally. We leave proving this for higher dimensional Euclidean spaces as an interesting open question. Additionally, we hope that related methods may be of general interest for proving tighter Distortion bounds of mechanisms on restricted metric spaces.
It remains unclear whether using a constant number of queries greater than 3 would meaningfully improve our results. E.g., consider the natural extension of Random Referee: sample the favorite points of agents and ask a random referee to choose their favorite from among these. Using does not straightforwardly decrease the Squared Distortion bound of 21 for = 2. Also, as becomes large, this mechanism devolves to Random Dictatorship. Another natural question is whether comparison queries are necessary/sufficient to bound the ’th moment of Distortion. We leave these as additional open questions.
Brandon Fain is supported by NSF grants CCF-1637397 and IIS-1447554. Ashish Goel is supported by NSF grant CCF-1637418 and ONR grant N00014-15-1-2786. Kamesh Munagala is supported by NSF grants CCF-1408784, CCF-1637397, and IIS-1447554. Much of this work was done while Nina Prabhu was a student at The North Carolina School of Science and Mathematics, in cooperation with Duke University.
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Proof of Theorem 1
We want to construct a single ordinal profile over top- preferences such that for all randomized social choice mechanisms, there is some instantiation of these top- preferences in a metric space on which the mechanism has Squared Distortion . The top- ordinal profile is simple: Each agent has completely unique top- preferred feasible alternatives . There are no other alternatives, so (note that a top- only mechanism must have constant ).
Any randomized social choice mechanism must choose a distribution on , so in particular, there will be some such that the mechanism puts at least probability mass on the top- preferred alternatives of . Given this, consider a metric space, consistent with the top- preference profile, where forms a small clique well separated from all of the other alternatives which form a large clique. More formally, let . All pairwise distances are except those between the small and large clique. Instead, for all , and similarly for all .
Then the Squared social cost of choosing is simply , whereas the optimal solution chooses any for Squared social cost of . Since the mechanism chooses with probability at least , we can lower bound the Squared Distortion of any top- only mechanism as follows.
Proof of Theorem 3
We begin with the more general claim. Consider a star graph with three leaves and three agents, one agent at each leaf, as depicted in Figure 2. All four points are also alternatives, and the distance between an agent and the colocated alternative is 0. Call the central vertex and the set of outer vertices . We want to construct a single ordinal preference profile such that for all randomized social choice mechanisms, there is some metric space that induces this preference profile and for which the mechanism has Distortion at least 1.2. The preference profile is simple: each of the three agents prefer their colocated alternative to the central alternative to either of the other leaf alternatives. We induce this profile via two metric spaces. In both spaces, (that is, the solid lines in Figure 2); the spaces will differ in the distances between points inside of (that is, the dashed lines in Figure 2).
First, suppose that . Then the social cost of is 3, and the social cost of every vertex in is ; clearly the distortion of is 1 and the distortion of is . Let be the probability that a randomized ordinal social mechanism chooses some , and let be the probability that the same mechanism chooses . Then on this metric, the Distortion of the mechanism is (in expectation) just .
Second, suppose that for some . Then the social cost of is 3, and the social cost of every alternative in is , so the distortion of is and the distortion of is . Then on this metric, the Distortion of the mechanism is (in expectation) .
A social choice mechanism with only ordinal information cannot distinguish the cases, and must set some values for and that sum to 1, so such a mechanism cannot have Distortion less than
This expression is minimized by setting to (and ) yielding a Distortion of
The argument for the lower bound of on the Euclidean plane is similar. Now, the first case places the three points in on a unit circle around the central point , equidistant from one another. The law of cosines implies that . In the second case, the angular distance between consecutive points in is , so that two pairwise distances are , and one pairwise distance is . Completing the same argument as before yields the bound of 1.11.
Proof of Theorem 4
The construction for this lower bound follows that of Theorem 1. We briefly consider the argument for completeness. There are agents, and for each agent , there are unique alternatives that are distance away from them. are the top- preferred alternatives for . There are no other alternatives. Consider any top- only mechanism (recall that is constant). Such a mechanism chooses a distribution over alternatives, and thus with top- preferred alternatives such that the mechanism puts at least probability mass on .
Suppose that the other agents are arrayed in a circle such that the pairwise distance of diameter (clearly, if is small enough, such an arrangement need not violate our earlier construction). Further, suppose the distance from to the center of this circle is 1. Then the social cost of any alternative in is at least , and the social cost of any alternative not in is at most . Therefore, the Distortion of choosing any alternative in is at least
As and approach 0, the above approaches . The Distortion of choosing any alternative not in is clearly just . Since the mechanism chooses an alternative in with probability at least (say , where ) and an alternative not in with the remaining probability, the over Distortion is lower bounded by
Proof of Lemma 3
To begin, we argue that we can assume a canonical structure to in relation to a finite grid.
Let be a -fine grid: where and . There is a set of points such that and has the properties:
and these two maximally separated points and have the forms for some
We note that the pessimistic distortion of is invariant to scaling, rotation, and translation of . This follows because the pessimistic distortion is still defined in terms of euclidean distances. Given this, the argument for the Lemma is simply that we can construct by rotations, translations, and scaling.
First scale so that the maximum distance separating two points is 1 (note that at least two points are not equal). Next, rotate the points so that the line between these maximally separated points is vertical. Finally, translate the points until these maximally separated points have the appropriate forms - that is to say, until they lie exactly at grid points, and all other points are within . To see that this is possible, note that since they are the maximally separated points, no other points can lie outside in the vertical dimension. In the horizontal dimension, the total width spanned by the other points is at most 1, and we may need to expand by as much as in order to align the maximally separated points with grid points. ∎
The introduction of a -fine grid anticipates the computer analysis we employ. Lemma 4 allows us to further refine our assumption for a contradiction: Suppose without loss of generality222For simplicity of notation, we will assume lies in a rather than , and will simply run our computer analysis for the expanded grid. Similarly, relabeling and as the maximally separated points is just a notational convenience. that there is some such that , , and are of the form for some .
Define as . In words, maps a general point in to its nearest point on our -fine grid . We will argue (roughly) that the pessimistic distortion of closely approximates that of , where the pessimistic distortion of of is something we can compute directly in a brute force computer search. Recall Definition 4. We begin by bounding the numerator.
Let . Clearly we have from the definition of that for all (and and ). However, it is not necessarily true that is also small for all . It is possible that is slightly closer to than , but is slightly closer to than . We call such configurations indifferences.
Call indifferent with respect to and if .
Then we have the following fact: If is not indifferent with respect to and , then
This follows from observing that if is not indifferent, . Now, for our point set , we need to bound . We first consider the case where is not indifferent with respect to and . Then we have can upper bound by
where the bound tightens for and because they are already at grid points. Suppose instead that is indifferent with respect to and . Then we can upper bound by
which is at most
Furthermore, since , we know that
Call the set of indifferent cases . We can average over the indifferences, and over and , to upper bound by
Call the summations in this last line . Then
looks like a very involved quantity, but it is exactly what we will compute in our computer analysis. That is, when we are computing the pessimistic distortion on the grid for a set of five points, whenever we encounter an indifference, we take the worst case outcome over the two options. It is not hard to see that since there are least two points separated by a distance of , so we arrive at equation 1.
Now we need to bound