Don’t Roll the Dice, Ask Twice: The Two-Query Distortion of Matching Problems and Beyond
Abstract
In most social choice settings, the participating agents express their preferences over the different alternatives in the form of linear orderings.
While this clearly simpliﬁes preference elicitation, it inevitably leads to poor performance with respect to optimizing a cardinal objective, such as the social welfare, since the values of the agents remain virtually unknown.
This loss in performance because of lack of information is measured by the notion of distortion .
A recent array of works put forward the agenda of designing mechanisms that learn the values of the agents for a small number of alternatives via queries , and use this limited extra information to make better-informed decisions, thus improving distortion.
Following this agenda, in this work we focus on a class of combinatorial problems that includes most well-known matching problems and several of their generalizations, such as One-Sided Matching, Two-Sided Matching, General Graph Matching, and kConstrained Resource Allocation.
We design two-query mechanisms that achieve the best-possible worst-case distortion in terms of social welfare, and outperform the best-possible expected distortion achieved by randomized ordinal mechanisms.
Introduction
The notion of distortion in social choice settings was deﬁned to capture the loss in aggregate objectives due to the lack of precise information about the preferences of the participants [Procaccia and Rosenschein, 2006].
More concretely, the distortion was originally deﬁned as a measure of the deterioration of the total happiness of the agents when access is given only to the (ordinal) preference rankings of the agents, rather than to the complete numerical (cardinal) information about their preferences.
This research agenda has successfully been applied to a plethora of different settings, giving rise to a rich and vibrant line of work in major venues at the intersection of computer science and economics.
For a comprehensive overview, see the survey of Anshelevich et al.
Out of all of these scenarios, some of the most fundamental are matching problems, in which agents are matched to items or other agents, aiming to maximize the social welfare of the matching (the total value of the agents).
An example is the classic One-Sided Matching setting [Hylland and Zeckhauser, 1979], where the goal is to match n items to n agents based on the preferences of the agents over the items. For this setting, Filos-Ratsikas et al.
[2014] showed that the best achievable distortion is Θ( √ n ) .
Importantly, this guarantee is only possible if one is allowed to use randomization and the values of the agents are normalized .
Moving on from merely preference rankings, Amanatidis et al.
[2021b] recently put forward the agenda of studying the tradeoffs between information and efﬁciency, when the employed mechanisms are equipped with the capability of learning the values of the agents via queries .
The rationale is that asking the agents for more detailed information about only a few options is still cognitively not too burdensome, and could result in notable improvements on the distortion.
This was indeed conﬁrmed in that work for general social choice, and in a follow-up work for several matching problems [Amanatidis et al., 2021a].
Speciﬁcally, the latter work shows that it is possible to obtain distortion O ( n 1 /k ) with O (log n ) queries per agent for any constant integer k , and distortion O (1) with O (log 2 n ) queries per agent.
Crucially, the mechanisms achieving these bounds do not use randomization nor demand the values to be normalized.
While these works make a signiﬁcant ﬁrst step, they leave some important questions unanswered.
The mechanisms they propose require a logarithmic number of queries to achieve any signiﬁcant improvement.
Answering that many queries might still be cognitively too demanding for the agents, especially when there is a large number of possible options.
The main high-level motivation of this research agenda, from our perspective, is that a small amount of information can be more valuable than randomization.
But what does really constitute a “small amount”?
Ideally, we would like to design mechanisms that make only a few queries per agent, independently of the size of the input parameters.
Since with a single query, sub-linear distortion bounds are not possible [Amanatidis et al., 2021a,b], the ﬁrst fundamental question that we would like to answer is the following:
What is the best achievable distortion when we can only ask two queries per agent?
1.1 Results and Technical Overview
We settle the aforementioned question for several matching problems, including One-Sided Matching ,
Two-Sided Matching , General Graph Matching , and other more general graph-theoretic problems.
For all matching problems considered, we show that there is a deterministic mechanism that makes two queries per agent, runs in polynomial time, and achieves a distortion of O ( √ n ) .
This upper bound is based on a novel mechanism, which we call M ATCH -T WO Q UERIES in the case of OneSided Matching (see Mechanism 1).
The mechanism asks two queries per agent and computes a maximum-weight matching based of the revealed values due to these queries.
It starts by querying the agents at the ﬁrst position of their preference rankings.
For the second query, it computes a certain type of assignment A of agents to items (or agents to agents in more general matching problems), to which we refer as a sufﬁciently representative assignment , and queries the agents about the items they are assigned to in A .
The existence of such an assignment for all instances is far from trivial, and one of our main technical contributions is to show its existence and efﬁcient computation for the wide range of problems we consider.
We also show that no deterministic mechanism for these settings that makes two queries per agent can achieve a distortion better than Ω( √ n ) .
This lower bound follows by a more general construction yielding a lower bound of Ω( n 1 /λ ) on the distortion of any mechanism that makes a constant number λ of queries for any of these mechanisms.
This mirrors the corresponding lower bounds of Amanatidis et al.
[2021a] for One-Sided Matching.
While our results apply to general matching settings, their most impressive implications are for OneSided Matching: We show that by using only two cardinal queries per agent, we can match the bound of Θ( √ n ) for purely ordinal mechanisms, without requiring randomization or any normalization .
M ATCH -T WO Q UERIES clearly also outperforms another mechanism of Amanatidis et al. [2021a], which uses two queries and achieves a distortion of O ( n 2 / 3 √ log n ) assuming that the values of each agent sum up to 1 .
In contrast, our mechanism works for unrestricted values, and achieves the best possible distortion of O ( √ n ) based on conceptually much simpler ideas.
Results for general social choice.
Given that our approach works for a wide variety of matching problems, one might be curious as to whether similar arguments could be used to show bounds for the general social choice setting, where n agents have preferences over m alternatives, and the goal is to select an alternative with high social welfare; this was after all the original setting that Amanatidis et al.
[2021b] studied in the introduction of the information-distortion tradeoff research agenda.
In this setting, the situation is quite similar: the upper bounds follow by mechanisms that ask O (log m ) queries, and nothing positive is known for smaller numbers of queries.
We show that a mechanism with structure similar to that of M ATCH -T WO Q UERIES can indeed achieve a distortion of O ( √ m ) using only two queries, subject to being able to compute a sufﬁciently representative set of alternatives, which is analogous of the sufﬁciently representative assignment in matching problems.
It turns out that this property is very closely connected to the notion of an (approximately) stable committee [Jiang et al., 2020, Cheng et al., 2020], and it follows that it exists when m = Ω( n ) , thus allowing us to obtain the desired bound of O ( √ m ) when this is true.
This case is quite natural, as it captures instances where a group of people need to decide over a large set of possible options (e.g., shortlisting candidates for a job, deciding the best paper for a conference, etc.).
Interestingly, in contrast to the matching setting for which we show that sufﬁciently representative assignments can be found via a simple greedy algorithm, computing sufﬁciently representative sets of alternatives in general social choice requires rather involved techniques [Jiang et al., 2020, Cheng et al., 2020].
An obvious open question here is whether the O ( √ m ) bound can also be achieved by asking only two queries when m = o ( n ) .
This seems to be a more challenging task to prove; we discuss it further in Section 6.
We also show that the bound of O ( √ m ) is the best possible, as part of a more general distortion lower bound of Ω( m 1 /λ ) for mechanisms that make a constant number λ of queries per agent; the latter result signiﬁcantly improves the previously known lower bound of Ω( m 1 /
( λ +1) )
[Amanatidis et al., 2021b].
Roadmap.
For the sake of presentation, we fully demonstrate how our methodology works for the One-Sided Matching problem in Section 3.
Before doing so, we start with some necessary notation and terminology in Section 2.
In Section 4, we brieﬂy discuss other graph-theoretic problems for which our methodology can be applied; all missing details are presented in the full version [Amanatidis et al., 2022].
Our results for general social choice are presented in Section 5; again, the detailed proofs are deferred to the full version.
We conclude with some interesting open problems in Section 6.
1.2 Additional Related Work
The literature on the distortion of ordinal mechanisms in social choice is long and extensive, focusing primarily on settings with normalized utilities (e.g., [Boutilier et al., 2015, Caragiannis et al., 2017, Filos-Ratsikas et al., 2020]), or with metric preferences (e.g., [Anshelevich et al., 2018, Anshelevich and Postl, 2017, Gkatzelis et al., 2020]) ; see the survey of Anshelevich et al.
[2021] for a detailed exposition.
The distortion of mechanisms for One-Sided Matching and more general graph-theoretic problems has been studied in a series of works for a variety of preference models, but solely with ordinal information [Anshelevich and Zhu, 2018, Abramowitz and Anshelevich, 2018, Anshelevich and Zhu, 2017, Anshelevich and Sekar, 2016, Filos-Ratsikas et al., 2014, Caragiannis et al., 2016].
Besides the papers of Amanatidis et al.
[2021b,a], the effect of limited cardinal information on the distortion has also been studied in other works [Abramowitz et al., 2019, Mandal et al., 2019, 2020, Benadè et al., 2021].
Mostly related to us is the paper of Ma et al.
[2021] which considered the One-Sided Matching problem with a different type of cardinal queries, and showed qualitatively similar results to Amanatidis et al. [2021a] for Pareto optimality (rather than social welfare).
Our upper bound for the general social choice setting makes use of the results of Cheng et al.
[2020] and Jiang et al.
[2020] for (approximately) stable committees (see also Aziz et al.
[2017]); a stable committee is very similar to a representative set of alternatives in our terminology.
Cheng et al.
[2020] showed that, while exactly stable committees do not always exist [Jiang et al., 2020], ﬁnding a random version of such committees, coined stable lotteries , is always possible and can be done in polynomial time.
Later on, Jiang et al.
[2020] showed that, via an intricate derandomization process, stable lotteries can yield approximately stable committees, where the approximation is a small multiplicative constant; for our purposes, this is sufﬁcient.
Interestingly, very recently, Ebadian et al.
[2022] used stable lotteries to construct a purely ordinal randomized social choice mechanism that achieves the best possible distortion under unit-sum normalized values.
Preliminaries on One-Sided Matching, Mechanisms, and Distortion
In One-Sided Matching, there is a set N of n agents and a set A of n items .
Each agent i ∈ N has a value v i,j for each item j ∈ A ; we refer to the matrix v
= ( v i,j ) i ∈N ,j ∈A as the valuation proﬁle .
A (one-sided) matching X : N → A is a bijection from the set of agents to the set of items, i.e., each agent is matched to a different single item.
Our goal is to choose a matching X to maximize the social welfare , deﬁned as the total value of the agents for the items they have been matched to according to X : SW ( X | v ) =
(cid:80) i ∈N v i,X ( i ) .
Usually v is clear from the context, so we then simplify our notation to SW ( X ) for the social welfare of matching X .
As in most of the related literature, we assume that we do not have access to the valuation proﬁle of the agents.
Instead, we have access to the ordinal preference (cid:31) i of each agent i , which is derived from the values of the agent for the items, such that a (cid:31) i b
if v i,a ≥ v i,b ; we refer to the vector (cid:31) v =
( (cid:31) i ) i ∈N as the ordinal proﬁle of the agents.
A mechanism M in our setting operates as follows: It takes as input the ordinal proﬁle (cid:31) v of the agents.
It then makes a number λ ≥ 1 of queries per agent to learn part of the valuation proﬁle.
In particular, each agent is asked her value for at most λ items.
Given the answers to the queries, and also using the ordinal proﬁle, M computes a feasible solution (here a matching)
M ( (cid:31) v ) .
In this paper we focus on mechanisms that make two queries per agent, i.e., λ = 2 , and compute a solution of high social welfare.
However, pinpointing an (approximately) optimal solution without having full access to the valuation proﬁle of the agents can be quite challenging; the ordinal proﬁle may be consistent with a huge number of different valuation proﬁles, even after the queries.
Nevertheless, we aim to achieve the best asymptotic performance possible, as quantiﬁed by the notion of distortion .
Deﬁnition 1.
The distortion of a mechanism M is the worst-case ratio (over the set V of all valuation proﬁles in instances with n agents and n items) between the optimal social welfare and the social welfare of the solution chosen by M :
[Equation]
where X is the set of all matchings between N and A .
An Optimal Two-Query Mechanism
In this section, we present a mechanism for One-Sided Matching that makes two queries per agent and achieves a distortion of O ( √ n ) .
Due to the lower bound of Ω( n 1 /λ ) on the distortion of any mechanism that can make up to λ queries per agent shown by Amanatidis et al.
[2021a], our mechanism is asymptotically best possible when λ
= 2 .
Without any normalization assumptions about the valuation functions, it is easy to see that a mechanism cannot have any guarantee unless it queries every agent about her favorite item.
However, there are no obvious criteria suggesting how to use the second query.
Before we present the details of our mechanism, we deﬁne a particular type of assignment of agents to items that will be critical for deciding where to make the second queries.
Deﬁnition 2.
A many-to-one assignment A of agents to items (i.e., each agent is assigned to one item, but multiple agents may be assigned to the same item) is a sufﬁciently representative assignment if (a) For every item j ∈ A , there are at most √ n agents assigned to j ; (b) For any matching X , there are at most √ n agents that prefer the item they are matched to in X to the item they are assigned to in A .
A natural question at this point is whether a sufﬁciently representative assignment exists for any instance, and if so, whether it can be efﬁciently computed.
In Section 3.2, we present a simple polynomial-time algorithm for this task.
3.1 The Mechanism
Our mechanism M ATCH -T WO Q UERIES (Mechanism 1)
ﬁrst queries every agent about her favorite item.
Next, it computes a sufﬁciently representative assignment A (see Section 3.2) and queries each agent about the item she is assigned to in A .
Finally, it outputs a matching that maximizes the social welfare based only on the revealed values (all other values are set to 0 ).
Although computational efﬁciency is not our primary focus here, if we use a polynomial-time algorithm for computing a maximum weight matching (e.g., the Hungarian method [Kuhn, 1956]), M ATCH -T WO Q UERIES runs in polynomial time as well.
Mechanism 1 M ATCH -T WO Q UERIES ( N , A , (cid:31) v ) 1: Query each i ∈ N about her favorite item w.r.t.
(cid:31) i 2: Compute a sufﬁciently representative assignment A 3: Query each agent about the item she is assigned to in A 4: Set all non-revealed values to5: return a maximum-weight perfect matching Y
Of course, if we compare the mechanism’s behaviour to an actual optimal matching X , we expect to see that we asked agents about the “wrong” items most of the time: for many agents the second query is about better items than what they are matched to in X , and for many agents it is about worse items.
The desired bound of O ( √ n ) on the distortion of M ATCH -T WO Q UERIES is established by balancing the loss due to each of these two cases.
Theorem 1.
M ATCH -T WO Q UERIES has distortion O ( √ n ) .
Proof.
Consider any instance with valuation proﬁle v .
Let Y be the matching computed by the
M ATCH -T WO Q UERIES mechanism when given as input the ordinal proﬁle (cid:31) v , and let X be an optimal matching.
Let SW R ( Y ) be the revealed social welfare of Y , i.e., the total value of the agents for the items they are matched to in Y and for which they were queried about.
We will show that
SW ( X ) ≤ (1 + 2 √ n ) · SW R ( Y ) , and then use the fact that SW ( Y ) ≥ SW R ( Y ) to directly get the desired bound on the distortion.
We can write the optimal social welfare as
[Equation]
where SW R ( X ) is the revealed social welfare of X that takes into consideration only the values revealed by the queries, whereas SW C ( X ) is the concealed social welfare of X that takes into consideration only the values not revealed by any queries.
Since Y is the matching that maximizes the social welfare based only on the revealed values, we have that
[Equation]
To bound the quantity SW C ( X ) , let S be the set of agents who are not queried about the items they are matched to in X .
We partition S into the following two subsets consisting of agents for whom the second query of the mechanism is used to ask about items that the agents consider better or worse than the items they are matched to in X .
Recall that an agent i is queried about the item A ( i ) she is assigned to according to the sufﬁciently representative assignment A .
So, S is partitioned into
[Equation]
and
[Equation]
Given these sets, we can now write
[Equation]
where
[Equation]
and
[Equation]
For every item j , let S ≥ j = { i ∈ S ≥ :
A ( i ) =
j } be the set of all agents in S ≥ that are queried about j by the mechanism using the second query.
Thus, S ≥ = (cid:83) j ∈A S ≥ j .
Since A is a sufﬁciently representative assignment, | S ≥ j | ≤ √ n for every item j .
Therefore,
[Equation]
For the last inequality, recall that A assigns every agent to a single item, and thus the sets S ≥ j are disjoint.
In addition, the values of all the agents for the items they are matched to according to A are revealed by the second query of the mechanism.
Since we can match the agent in S ≥ j that has the maximum value for j to j , and Y maximizes the social welfare based on the revealed values, we obtain that SW R ( Y ) ≥ (cid:80) j ∈A max i ∈ S ≥ j v i,j .
Next consider the quantity SW <C
( X ) .
By the fact that A is a sufﬁciently representative assignment, it follows that | S < | ≤ √ n ; otherwise X would constitute a matching for which there are strictly more than √ n agents that prefer the item they are matched to in X to the item they are assigned to by A .
Combined with the fact that all agents are queried at the ﬁrst position of their ordinal preferences, we obtain
[Equation]
The bound follows directly by (1), (2) and (3).
3.2 Computing Sufﬁciently Representative Assignments
To establish the correctness of M ATCH -T WO Q UERIES , we need to ensure that a sufﬁciently representative assignment exists for any ordinal proﬁle and that it can be computed efﬁciently.
For this we present a simple polynomial time algorithm, which we call √ n -S ERIAL D ICTATORSHIP (Mechanism 2).
This algorithm creates √ n copies of each item and then runs a serial dictatorship algorithm, which ﬁrst ﬁxes an ordering of the agents and then assigns each agent to her most preferred available item according to her ordinal preference.
It is easy to see that the running time of √ n -S ERIAL D ICTATORSHIP is polynomial (in particular, it is O ( n 1 . 5 ) ).
Mechanism 2 √ n -S ERIAL D ICTATORSHIP ( N , A , (cid:31) v ) 1: Let B be a multiset containing √ n copies of each j ∈
A 2: for every agent i ∈ N do 3:
Let α i be a most preferred item of agent i in B 4:
Remove α i from B 5: end for 6: return A
= ( α i ) i ∈N
Theorem 2.
For any instance, the output of √ n -S ERIAL D ICTATORSHIP is a sufﬁciently representative assignment.
Proof.
Let A be the output of the algorithm.
During the execution of the algorithm, whenever every copy of an item has been assigned, we say that such an item is exhausted .
Assume, towards a contradiction, that A is not a sufﬁciently representative assignment.
By construction, every item is assigned to at most √ n agents, so there must be a matching violating the second condition of
Deﬁnition 2.
That is, there is a subset of items A (cid:48) and a subset of agents N (cid:48) , such that |A (cid:48)
| =
|N (cid:48) | > √ n , and each agent i ∈ N (cid:48) prefers to be assigned to a distinct item β
i ∈ A (cid:48)
(i.e., β i (cid:54) =
β j for i (cid:54) =
j ) instead of the item she is assigned to in A .
Consider any agent i ∈ N (cid:48) .
The fact that this agent was not assigned to β i by the algorithm implies that when the agent was picked, item β i was exhausted.
Since this is true for all agents in N (cid:48) , at the end of the algorithm all items of A (cid:48) must be exhausted.
However, an item is exhausted when all its √ n copies have been assigned and there are n agents in total, so we can only have as many as n/ √ n = √ n exhausted items.
This means that |A (cid:48) | ≤ √ n , a contradiction.
Further Combinatorial Optimization Problems
The approach of Section 3 can be extended to a much broader class of graph-theoretic problems.
Informally, it works when the objective is to maximize an additive function over subgraphs of a given graph which contain all “small” matchings and have constant maximum degree.
To make things more concrete, given a constant k ∈ N and a weighted graph G on n nodes, we say that a family F of subgraphs of G is a matching extending k
-family if:
• Graphs in F have maximum degree at most k ; • For any matching M of G of size at most (cid:98) n/ 3 k (cid:99) , there is a graph in F containing M .
Clearly, the set of matchings of a graph is a matching extending 1 -family, but it is not hard to see that different constraints are also captured, e.g., the set of subgraphs that are unions of disjoint paths and even cycles is a matching extending 2 -family.
We are ready to introduce the general problem that we tackle here.
As this is a special case of the class of problems captured by Ordinal-Max-on-Graphs (introduced by Amanatidis et al.
[2021a]), we use a similar formulation and name.
Ordinal- k -Max-on-Graphs : Fix a constant k ∈ N and let N be a set of n agents.
Every agent i ∈ N has a (private) valuation function v i : N → R ≥ 0 .
We are given a graph G =
( N , E ) , an ordinal proﬁle (cid:31) v =
( (cid:31) i ) i ∈N consistent with v =
( v i ) i ∈N , and a concise description of a matching extending k -family F .
The goal is to ﬁnd some H ∈ F that maximizes (cid:80) { i,j }∈ E ( H ) v i ( j ) .
Besides One-Sided Matching, a large number of problems that are relevant to computational social choice are captured by Ordinal-
k -Max-on-Graphs.
We give a few examples:
General Graph k -Matching :
Given a graph G , ﬁnd a k -matching of maximum value, i.e., F contains the subgraphs of G of maximum degree at most k and is a matching extending k -family.
The problem becomes the General Graph Matching problem when k = 1 ; by also assuming that G is bipartite, we have the celebrated Two-Sided Matching problem [Gale and Shapley, 1962, Roth and Sotomayor, 1992]. k -Constrained Resource Allocation :
Given a bipartite graph G =
( N 1 ∪ N 2 , E ) , assign at most k nodes of N 2 to each node in N 1 so that the total value of the corresponding edges is maximized.
That is, F contains all 1 -to- k matchings of G and is again a matching extending k -family.
The problem for k = 1 becomes One-Sided Matching.
Here N is partitioned into the set N 1 of “actual agents” and the set N 2 of “items”, and v i ( j ) can be strictly positive only for i ∈ N 1 , j ∈ N 2 .
Short Cycle Packing :
Given an integer (cid:96) and a weighted complete graph G , ﬁnd a collection of node-disjoint cycles of length at most (cid:96) so that their total weight is maximized.
Here, F is a matching extending 3 -family (note that k
= 2 for any (cid:96) ).
It is worth mentioning that Short Cycle Packing is not a generalization of any of the matching problems above.
It is also closely related to Clearing Kidney (cid:96) -Exchanges
[Abraham et al., 2007].
It is straightforward to extend the notion of distortion (Deﬁnition 1) for Ordinal- k -Max-on-Graphs by taking the supremum over all instances of a certain size n and letting X be the set of feasible solutions of each instance.
As already discussed, for One-Sided Matching there is a lower bound of Ω( n 1 /λ ) on the distortion of deterministic mechanisms that make up to λ ≥ 1 queries per agent [Amanatidis et al., 2021a].
We can get the analogous result for any problem captured by Ordinal-
k -Max-on-Graphs using a reduction that preserves distortion up to a constant factor.
Further, by appropriately using the ideas of Section 3, we show that this lower bound is asymptotically tight for the case of two queries.
For the statements of the theorems below, we assume that k ∈ N is a constant and that for every (general / bipartite) graph G , a matching extending k -family F ( G ) is speciﬁed.
Theorem 3.
No deterministic mechanism using at most λ ≥ 1 queries per agent can achieve a distortion better than Ω( n 1 /λ ) for Ordinal-
k -Max-on-Graphs.
For the positive result below, we rely on the existence of a generalization of sufﬁciently representative assignments.
While the construction is very similar, the counting argument here is less intuitive compared to the case of One-Sided Matching.
Theorem 4.
There is a deterministic mechanism for Ordinal- k -Max-on-Graphs which uses at most two queries per agent and has distortion O ( √ n ) .
Our mechanisms run in polynomial time whenever there is a polynomial-time algorithm for the full information version of the corresponding optimization problem.
Luckily, all variants of matching problems we presented above can be solved efﬁciently by Edmond’s algorithm [Edmonds, 1965] or its extensions [Marsh III, 1979].
Corollary 5.
There are deterministic polynomial-time mechanisms for General Graph Matching,
Two-Sided Matching, General Graph k-Matching, and k-Constrained Resource Allocation which all use at most two queries per agent and have distortion O ( √ n ) .
Towards Tight Bounds for General Social Choice
Here we consider the general social choice setting where a set N of n agents have preferences over a set A of m alternatives .
As in the One-Sided Matching problem, there is a valuation proﬁle v
= ( v i,j ) i ∈N ,j ∈A specifying the non-negative value that each agent i has for every alternative j .
The goal is to choose a single alternative x ∈ A to maximize the social welfare, that is, the total value of the agents for x : SW ( x | v ) =
(cid:80) i ∈N v i,x .
Again, when v is clear from context, we will drop it from notation.
Similarly to One-Sided Matching and the problems discussed in the previous section, v is unknown, and we are only given access to the ordinal proﬁle (cid:31) v that is induced by v .
Social choice mechanisms must decide a single alternative based only on (cid:31) v and the values they can learn by making a small number of queries.
The notion of distortion (Deﬁnition 1) can be extended for this setting as well, by taking the supremum over all instances with n agents and m alternatives, and letting X be the set A of alternatives.
For this general social choice setting, Amanatidis et al.
[2021b] showed a lower bound of Ω (cid:0) m 1 / (2( λ +1)) (cid:1) on the distortion of mechanisms that make at most λ ≥ 1 queries per agent.
We improve this result by showing a lower bound of Ω( m 1 /λ ) for any constant λ .
The full proof of the following theorem can be found in the full version.
Theorem 6.
In the social choice setting, the distortion of any deterministic mechanism that makes at most a constant number λ ≥ 1 of queries per agent is Ω( m 1 /λ ) .
Proof sketch.
Let M be an arbitrary mechanism that makes at most λ ≥ 1 queries per agent.
Consider the following instance with n agents and m = n alternatives.
We assume that m satisﬁes the condition m ≥ 1 2 (cid:80) λ
(cid:96) =
1 m ( λ − (cid:96) +1) /λ + 2 , and also that it is superconstant; otherwise the theorem holds trivially.
We partition the set of alternatives A into λ + 2 sets A 1 , A 2 , ...
A λ +1 , A λ +2 , such that
• | A (cid:96) | = 12 m ( λ − (cid:96) +1) /λ for (cid:96) ∈
[ λ ] ; • | A λ +
|
= 2 ; • | A λ +2 | = m − 12 (cid:80) λ(cid:96) =
1 m ( λ − (cid:96) +1) /λ − 2 .
The ordinal proﬁle has the following properties:
• For every (cid:96) ∈ [ λ + 1] , each alternative j ∈ A (cid:96) is ranked at position (cid:96) by a set T j,(cid:96) of m | A (cid:96) | =
Θ (cid:0) m ( (cid:96) − 1) /λ (cid:1) agents.
• For every (cid:96) ∈
• For every agent, the alternatives that she does not rank in the ﬁrst λ + 1 positions are ranked arbitrarily from position λ + 2 to m .
For every agent i , a query of M for alternative j reveals a value of m − (cid:96)/λ if i ranks j at position (cid:96) ∈
[ λ + 1] , and a value of 0 if i ranks j at any other position.
Let y be the alternative that M chooses as the winner for this instance.
No matter the choice of y , we will deﬁne the cardinal proﬁle so that it is consistent to the information revealed by the queries of M , and the values of the agents for alternative y are also consistent to the information that would have been revealed, irrespective of whether those values have actually been revealed.
That is, any agent has a value of m − (cid:96)/λ for y if she ranks y at position (cid:96) ∈ [ λ + 1] , and a value of 0 if she ranks y at any other position.
Hence, the social welfare of y is Θ (cid:0) m
( (cid:96) − 1) /λ (cid:1) · m − (cid:96)/λ =
Θ( m − 1 /λ ) if y ∈ A (cid:96) for (cid:96) ∈ [ λ + 1] , or 0 if
y ∈
A λ +2 .
Consequently, to show the desired bound of Ω( m 1 /λ ) on the distortion of M , it sufﬁces to assume that
y ∈ A (cid:96) for some (cid:96) ∈ [ λ + 1] , and prove that the values of the agents that have not been revealed and do not correspond to alternative y can always be deﬁned such that there exists an alternative x (cid:54) =
y with social welfare Ω(1) .
The remaining details showing that such an x always exists can be found in the full version.
Our approach for all the problems discussed in the previous sections can also be applied to the much more general social choice setting, subject to being able to compute a particular set of alternatives.
Deﬁnition 3.
Let c ≥ 1 be any constant.
A subset of alternatives B ⊆
A with | B | ≤ c · √ m is a sufﬁciently representative set if, for every alternative j ∈ A , at most √ m agents prefer j over their favorite alternative in B .
We now present a mechanism that works under the assumption that sufﬁciently representative sets of alternatives can be (efﬁciently) computed; we discuss this assumption right after the statement of
Theorem 7.
Mechanism 3 SC-T WO Q UERIES ( N , A , (cid:31) v )
1: Query each agent about her favorite alternative 2: Compute a sufﬁciently representative set B 3: Query each agent for her favorite alternative in B 4: For every j ∈ A , compute the revealed welfare SW R ( j ) 5: return y ∈ arg max j ∈A SW R ( j )
In particular, SC-T WO Q UERIES (Mechanism 3)
ﬁrst queries each agent about her overall favorite alternative (the one ranked ﬁrst).
Then, given a sufﬁciently representative set of alternatives B , it queries each agent for her favorite alternative in B .
Given the answers to these two queries per agent, the mechanism outputs an alternative that maximizes the revealed social welfare which is based only on the values learned from the queries.
Theorem 7.
The mechanism SC-T WO Q UERIES has distortion O ( √ m ) , when restricted to the social choice instances for which a sufﬁciently representative set of alternatives exists.
A sufﬁciently representative set of alternatives trivially exists when m is much larger than n (namely, when m = Ω( n 2 ) ).
In contrast, when m is much smaller than n , sufﬁciently representative sets of alternatives do not always exist.
2 Jiang et al.
[2020] showed the following useful result:
Theorem 8 ([Jiang et al., 2020]) .
For any ξ ∈ [ n ] , there exists a set S of alternatives with | S | ≤ 16 · n/ξ such that for every j ∈ A , there are at most ξ agents that prefer j over their favorite alternative in S .
A set S as in the theorem above is called an approximately stable committee Cheng et al.
Jiang et al.
Clearly, when m = Ω( n ) and ξ = √ n , an approximately stable committee is also , for any k > √ m , consider an instance with n = k · m !
agents, such that for each possible ordering of the m alternatives there are exactly k agents that have it as their preference.
Then, for any subset B of at most √ m alternatives and any alternative j ∈
A \ B , there are at least k > √ m agents that prefer j over any alternative in B .
a sufﬁciently representative set with c = 16 .
Therefore, combining Theorems 7 and 8, we obtain the following.
Corollary 9.
When m = Ω( n ) , SC-T WO Q UERIES has distortion O ( √ m ) .
Conclusion and Open Problems
In this paper, we showed that for a large class of problems, which includes One-Sided Matching and many other well-studied graph-theoretic problems, it is possible to achieve a distortion of O ( √ n ) using a deterministic mechanism that makes at most two queries per agent, and that this is best possible asymptotically.
Our whole methodology is based on computing assignments of agents to items or other agents that exhibit a very particular structure.
In addition, in the social choice setting, when m = Ω( n ) , sets of alternatives with analogous properties can be computed, and our methodology yields a two-query mechanism with best possible distortion for this setting as well.
It is an interesting open problem to design a mechanism that makes two queries and achieves the best possible distortion of O ( √ m ) when m = o ( n ) , or show that this is impossible.
We suspect that to obtain a positive result one would need to come up with an adaptive mechanism, which decides where to ask each query based on the answers to all previous ones.
Another question, about any of the problems we considered, is whether one can design mechanisms that make at most a constant λ ≥ 3 queries per agent and their distortion matches the lower bound of Ω( n 1 /λ )
(or, in the case of social choice, Ω( m 1 /λ ) ).
Acknowledgments and Disclosure of Funding
This work was supported by the ERC Advanced Grant 788893 AMDROMA “Algorithmic and Mechanism Design Research in Online Markets”, the MIUR PRIN project ALGADIMAR “Algorithms,
Games, and Digital Markets”, and the NWO Veni project No. VI.Veni.192.153.