Fair allocation of resources is a widely studied problem in social choice literature. In several societal decision-making scenarios, the resources are public goods, meaning that they can be enjoyed by multiple agents simultaneously. For instance, consider the problem of locating a fixed number of parks or libraries to serve a population [ChenFLM19]. Each such resource provides shared utility to several members of society. Viewed as a facility location problem, a standard objective involves locating the facilities to minimize the total distance traveled by the population to its nearest open facility. However, such a solution need not be fair: In a city with a dense urban core and sprawling suburbs, it can lead to the algorithm placing many more facilities in the suburbs, causing the locations at the urban core to become overcrowded. In other words, the globally optimal solution many produce disparate outcomes for different demographic slices.
Similarly, consider the Participatory Budgeting problem [PBP, FairKnapsack, implicitPB, knapsack1, knapsack2, Fain2016]. Recently, many cities and wards across the world have a process to put part of their budget to vote. The city chooses several projects such as repaving streets, installing lights, etc, and each voter indicates preferences over these projects. The goal of the city is to choose a subset of projects that is feasible within the budget to fund. Again, simple schemes to aggregate voter preferences may overly bias the outcome towards majority preferences, and may ignore entirely the preferences of a sizable, coherent, minority.
1.1 Committee Selection and Fairness Model
In this paper, we consider an abstract resource allocation model – committee selection – that captures not only the above two settings, but also several others considered over many decades in social choice literature. In this setting, we study a general, classical notion of group fairness, and show that solutions that approximately satisfy this notion always exist.
Using social choice parlance, the committee selection problem models the common scenario of determining a winning subset, i.e. a committee, from a set of candidates. A set of voters (or agents) and a set of candidates are given, and each candidate is associated with a weight . Let denote the weight of the committee . The goal is to find a committee of weight at most a given value . We note that our results extend to the setting where the weight is a subadditive function of ; we present the additive model for simplicity of exposition.
Each voter explicitly or implicitly specifies an ordinal ranking over all possible committees (that is, all possible subsets of ). Given voter and two committees and , we use the notation to indicate the voter strictly prefers to . We indicate weak preference by . We assume preferences are complete, so that for every voter and every two committees and , the voter either (weakly or strongly) prefers to , or vice versa.
The only condition we impose on the preferences is the following:
Monotonicity: If , then for all .
In Section 1.2, we will show that both the facility location and participatory budgeting settings described above are special cases of committee selection.
Fairness via Stability.
The notion of group fairness or proportionality is a central objective in committee selection. The general idea appeared in literature more than a century ago [Droop] and various incarnations of this notion have gained significant attention recently [Chamberlain, Monroe, Brams2007, Brill, Sanchez, Fain2016, FainMS18, PJR2018]. Here, each group of voters should feel that their preferences are sufficiently respected, so that they are not incentivized to deviate and choose an alternative committee of proportionally smaller weight. In the common scenario that we do not know beforehand the exact nature of the demographic coalitions, we adopt the robust solution concept which requires the committee to be agnostic to any potential subset of voters deviating.
Formally, we study fairness via the notion of core stability from economics literature [lindahlCore, scarfCore, coreConjectureCounter, LindahlPaper, SamuelsonPaper]. This uses the notion of capture count defined below.
Definition 1 (Capture Count).
Given two committees , the capture count of over is the number of voters who strictly prefer to : .
Given the above definition, we are now ready to define fairness via core stability as follows.
Definition 2 (Stable Committees).
Given a committee of weight at most , the weight limit, a committee of weight blocks iff . A committee is stable if no committee blocks it.
In other words, for any and any voters, there should not be another committee of weight at most , so that all these voters are strictly better off with the new committee. It is easy to check that a stable outcome is (weakly) Pareto-optimal among committees of weight at most , by considering the deviating coalition of all the voters. Furthermore, for every coalition of voters, a stable committee is also Pareto-optimal relative to committees with proportionally scaled-down weight.
In economics, this notion of core stability can be justified with fair taxation [lindahlCore, LindahlPaper, SamuelsonPaper]: Each voter has an endowment of , so the society together has a budget of . Candidate costs , and we select a committee whose weight is at most . If no subset of voters (blocking committee) with size can deviate and use their endowment () to purchase an alternative committee, then the committee is said to be stable.
1.2 Some Special Cases
Though our results hold for any monotone purely ordinal preference structure over committees, to place them in context, we now define some special cases of committee selection. We note that each of these special cases have a rich history in social choice and related domains.
- Participatory Budgeting.
This models the civic budgeting application described above. Each candidate is a public project, and its weight equals its cost. Voter has utility for project . The value is the total budget available to the city. The utility of the voter for committee is , and the voter prefers committees that provide her higher utility. This can be generalized to utility functions that capture complements and substitutes.
- Approval Set.
This is a special case of the setting described above that has been widely studied in multi-winner election literature. In this model [Brams2007, Brill, Sanchez, PJR2018], we assume each . Each voter specifies an approval set of candidates. Given two committees and , iff , i.e., the voter prefers committees in which she has more approved candidates.
In this model [ElkindFSS17], each candidate has unit weight. Each voter has a preference ordering over candidates in . In this case, iff ’s favorite candidate in is ranked higher (in her preference ordering) than her favorite candidate in .
- Facility Location.
This is a special case of ranking that is motivated by the problem of locating public facilities described above, and was recently considered by [ChenFLM19]. Here, the preferences over candidates are dictated by distances in an underlying metric space. Formally, there is a metric space over . Each location in is a potential facility. Given a subset of locations, the cost of voter is the distance to the closest facility in . A voter prefers to if it incurs smaller cost in the former than in the latter.
1.3 Approximate Stability and Main Result
As shown in [ChengJMW19, FainMS18], there are simple instances with cyclic preferences where stable committees may not exist; we present such an instance in Appendix A. This motivates us to consider an approximate notion of stability.
Definition 3 (-Approximately Stable Committees).
Given a parameter , and a committee of weight at most , the weight limit, we say that a committee of weight -blocks iff . A committee is -approximately stable if there are no committees that -block it.111Any -approximately stable committee can trivially be made Pareto-optimal while preserving the value of . Therefore, Pareto-optimality comes for free in our setting.
Note that when , the solution is exactly stable, and our goal is to find the minimum for which a -approximately stable solution exists. Theorem 4 (Appendix A) shows that for any constant even in the Ranking setting. Our main result is the following general and somewhat surprising theorem, that we prove in the main body of the paper.
For any monotone preference structure over any number of voters, and of candidates with arbitrary weights, and any weight , a -approximately stable committee of weight at most always exists.
It is worth noting that prior to our work, no non-trivial result was known for the existence of approximately stable committees even in the very special cases of Approval Set and Facility Location preferences described above.
1.4 Techniques and Other Results
Our proof of Theorem 1 proceeds by first constructing a lottery (or randomization) over committees of weight that is -approximately stable, and iteratively rounding this solution. The first challenge is to define the appropriate notion of randomized stability. As discussed in Section 1.5, though stability when committee members are chosen fractionally is a classical concept, these notions require convex and continuous preferences over fractional allocations, and it is not clear how to relate them to deterministic (or integer) solutions that we desire.
We proceed via a different randomized notion of stability that was first defined in [ChengJMW19]. We define this notion next. Given a weight , we let denote a distribution (or lottery) over committees of weight at most .
Definition 4 (Stable Lotteries [ChengJMW19]).
A distribution (or lottery) over committees of weight at most is said to be -approximately stable iff for all committees of weight , we have: .
In [ChengJMW19], it was shown that an exactly stable lottery under this definition exists for Approval Set and Ranking settings, via solving the dual formulation. However, it was not clear either how to extend this technique even to Participatory Budgeting preferences, or what a stable lottery implied about deterministic stable committees that is our main focus here. In this paper, we resolve both these questions. As our first contribution, in Section 2, we show the following.
For any weight and all monotone preferences, a -approximately stable lottery over committees of weight at most always exists.
The key insight is to reduce the general case to a Ranking instance and with candidates of different weights, where the committees in the original instance become the candidates in the new instance. This allows us to develop a new dual construction that provides a simple proof for all monotone preference structures.
Once we construct this lottery, our main contribution (Section 3) is rounding it to show Theorem 1. The randomized stability condition implies the existence of a committee that satisfies a certain fraction of voters simultaneously, in the sense that it lies not too far down the preference ordering of these voters. We iteratively eliminate such voters and re-compute the lottery, with the non-trivial aspect being to ensure that this process preserves approximate stability.
In Section 4, we show that our results extend to the more general setting where is a subadditive set function, and also to the setting where there are multiple weight constraints. We also discuss some settings in which an approximately stable committee can be efficiently computed.
Exactly Stable Lotteries.
When considering lotteries, we haven’t been able to find an instance of a monotone preference structure where an exactly stable solution does not exist. The loss of factor of in Theorem 2 seems to be an artifact of our analysis. Indeed, in Appendix B, we show a different way of constructing the dual solution that leads to the following result. We conjecture that this results extends to all , and we discuss this and other open questions in Section 5.
Theorem 3 (Proved in Appendix B).
For unit-weight candidates and any number of voters with arbitrary monotone preferences, when , an exactly stable lottery always exists.
1.5 Related Work
Committee selection is omnipresent in political and economic activities of a society: We see it in parliamentary elections, in group-hiring processes, and in participatory budgeting. Recent work in social choice [Procaccia2008, Meir2008, Lu2011, Brill, Sanchez, PJR2018] has extensively studied the properties of committee selection rules and established axiomatization in this field. Furthermore, group fairness in committee selection arises in areas outside social choice: In a shared-cache system with multiple users, consider the problem of deciding which parts of the data to keep in the cache that has only limited storage [ROBUS, Psomas]. Users gain utility from their data being cached. We can model this as committee selection where each atomic piece of data corresponds to a candidate. In this context, a fair caching policy provides proportional speedup to each user.
We now compare our stability notions with some closely related notions in literature. This will place our technical work in context.
The Lindahl Equilibrium.
Stability is the same the notion of core
in cooperative game theory. Scarf[scarfCore] first phrased it in game-theoretic terms, and it has been extensively studied in public-good settings [LindahlPaper, SamuelsonPaper, lindahlCore, coreConjectureCounter, Fain2016]. Much of this literature considers convex and continuous preferences, which in our setting implies convex preferences over fractional allocations (that is, when candidates can be chosen fractionally). The seminal work of Foley [lindahlCore] considers the Lindahl market equilibrium. In this equilibrium, each candidate is assigned a per-voter price. If the voters choose their utility maximizing allocation subject to spending a dollar, then (1) they all choose the same fractional outcome; and (2) for each chosen candidate, the total money collected pays for that candidate. It is shown via a fixed point argument that such an equilibrium pricing always exists when fractional allocations are allowed, and this outcome lies in the core. Though this existence result is very general, it needs the preferences over fractional outcomes to be convex and continuous.
For instance, in the case of facility location, a fractional allocation satisfies and . One possible convex disutility of voter for is
Though Foley’s result shows there exists an allocation that is a core outcome, it is not clear (a) how to compute this fractional solution efficiently; and (b) more importantly, how to round this allocation to an approximately stable integer solution. The difficulty in rounding is because we cannot relax the distances when considering when a voter can deviate; indeed, if we could relax distances, the problem becomes very different, and there is an approximately stable solution (that only relaxes distances and not the size of the deviating coalition) via a simple greedy algorithm [ChenFLM19].
This motivates using the new notion of randomized stability, where a deterministic outcome is first drawn from the lottery, and subsequently the voters who see higher utility deviate. This notion does not correspond to underlying convex preferences over the space of lotteries; however, as we show, a stable lottery can now be converted to an approximately stable committee. Furthermore, for facility location and more generally, Ranking, this stable lottery can be efficiently computed [ChengJMW19], while we do not know how to compute the Lindahl equilibrium efficiently.
Nash Welfare and its Variants.
There is extensive work (see [ElkindFSS17]) on voting rules where we construct a score for each voter and committee, and choose the committee that maximizes . For instance, for Approval Set preferences, the classic Proportional Approval Voting (PAV) method that dates back more than a century [Thiele], assigns . More generally, the Nash Welfare objective [Portioning, Fain2016, Thiele, PJR2018, FairKnapsack, FainMS18] assigns score , where is the utility of the voter for committee . These methods compute a stable solution when the utility of voters in a deviating coalition is scaled down. This requires knowing cardinal utility functions of voters (and does not work with disutilities), and is otherwise incomparable to the more widely studied and classical notion of core stability that we consider, where the committee size on deviation is scaled down and the utilities of voters are unchanged. Further, for Approval set preferences, the PAV method is no better than an -approximation to a stable outcome [ChengJMW19].
A recent line of work [Sanchez, PJR2018, Brill] has considered a special case of stability with Approval set preferences, when the coalition that deviates is not arbitrary, but is cohesive in terms of preferences. They term this Justified Representation (with several generalizations known). It is shown that the PAV method and its variants achieve or closely approximate these notions of stability. However, as mentioned above, the PAV method do not approximate the core outcome, so that stability is very different in structure from Justified Representation. Furthermore, it is not clear how to generalize the definition of Justified Representation beyond the case of Approval set preferences, to either Participatory Budgeting or Ranking.
2 Existence of -Approximately Stable Lotteries
In this section, we consider choosing a stable lottery over committees of weight at most . Recall the definition of stability in this setting from Definition 4.
We take the approach in [ChengJMW19] and consider the dual formulation of selecting stable lotteries. The existence of a -approximately stable lottery is equivalent to deciding:
where is a distribution (lottery) over committees of weight at most . Viewing as a mixed strategy over the “defending” committees and as the “attacking” strategy, we treat (1) as a zero-sum game. Duality now allows us the swap the order of actions by allowing the attacker to use a mixed strategy. (1) is thus equivalent to
where is a lottery over committees of weight at most chosen by the attacker. This dual view provides a convenient tool for showing the existence of approximately stable lotteries. The rest of the section is devoted to proving Theorem 2, that we restate here.
For any value and all monotone preferences, a -approximately stable lottery over committees of weight at most always exists.
2.1 Per-Voter Guarantee
Assume we are given a lottery . If there is a committee in with , then for any . This implies . Therefore, the attacker can remove these strategies from its lottery, and we can assume only has committees with weight at most .
Given any distribution over committees of weight at most , we need to show there is a defending committee with weight at most , such that
Suppose that the strategy chooses
with probability, committee with probability , …, with , where . Let
be the ratio between the expected total weight of the attacking strategy and , the allowable weight for the defending strategy. We need to show an of weight at most so that:
We will construct a distribution over committees that satisfies a stronger property:
Summing over all voters implies the existence of a lottery satisfying (3), and hence a deterministic committee satisfying the same. This will imply the theorem statement.
2.2 Dependent Rounding
Let for . We have:
for all ; and
We will construct the defending committee by including each attacker committee with probability
; the details of which are below. We use the random variableto denote whether we include in our defending committee , so that . We therefore obtain a distribution over committees .
We use the dependent rounding procedure in [byrka] to construct given the . This has the following properties.
(Almost-Integrality) For any realization of ’s, all but at most one of them takes value in (the remaining one takes value in ).
(Correct Marginals) for all .
(Preserved Weight) .
(Negative Correlation) , .
To have full integrality instead of the almost-integrality, in any realization, we include in our as long as (instead of only fully including it when ). Since we assumed for all , using the almost-integrality and preserved-weight conditions, for any realization of , the weight of the resulting satisfies
Fix a voter . W.l.o.g. assume her preference over the sets in are .
Here, the first equality follows because when the adversary chooses set , it only beats if included none of . Here, we are using the monotonicity of the preference structure: Since for , this implies when . The second equality follows since the realization of the adversary’s lottery is independent of that of the defender, and since . The first inequality follows by negative correlation property of . To see the third equality, note that if , then , so that . Otherwise, .
3 Existence of Approximately Stable Committees
For any monotone preference structure over any number of voters, and of candidates with arbitrary weights, and any weight , a -approximately stable committee of weight at most always exists.
For the proof, fix a deviating committee of weight . Suppose our final committee is of weight at most . Our goal is to show that:
3.1 Good and Bad Committees
Throughout the proof, we fix two constants , whose choice will be determined at the very end. To begin, we define a subroutine that returns a -approximately stable lottery (via Theorem 2) for any subset of voters, and any committee size.
Given candidate set , voter set , and committee size , let return a lottery over committees of weight at most that is -approximately stable for the set of voters . Similarly, let
Let be the probability that committee is included in constructed above.
Given a voter , we define the set of good and bad sets of committees relative to , and respectively, as follows:
The idea is that the good committees appear sufficiently high up in ’s ranking, while the bad committees are lower down in the ranking. The notion of high and low is relative to the probability mass . The following lemma is immediate.
If , then only if .
If , then . This implies . Since , we have , so that . ∎
The next lemma implies that (a) Any committee cannot lie in too many good sets relative to its weight; and (b) There is some committee (with non-zero support in ) that does not lie in more than a constant fraction of the bad sets. The previous claim rules out the possibility where too many voters prefer to such a committee, relative to the weight of , which will be crucial for the algorithm we subsequently design.
Given Lottery, we have the following upper and lower bounds:
For all committees , we have
There exists with non-zero support in such that
To see the first part, for any committee , we have: Summing over ,
where the inequality comes from the fact that is -approximately stable. Thus there are fewer than voters with , which is necessary for .
To see the second part, suppose . Then for each , since , we have: . Therefore, the expected number (over the choice ) of such that is at least . Therefore, there exists an for which the claim holds. ∎
Algorithm 1 shows our full procedure. The main idea is the following: If we pick a committee that does not lie in for most voters , then by Claim 1, is forced to lie in for these voters if beats . But since is -approximately stable, by Lemma 1, there are only a small number of where can lie in . We can therefore remove these set of voters for whom , since makes sure no can capture too many of these voters. This eliminates a constant factor of the voters. For the remaining voters, we recursively find another committee of smaller (but not too smaller) weight, this eliminates another constant factor of the voters; and so on. The key point is that the total weight of all these committees is a geometric sequence, and the number of voters who can be captured by in each round is also a geometric sequence, showing a constant-approximately stable solution.
We finally show that the resulting set is approximately stable, completing the proof of Theorem 1.
When and , then is a -approximately stable committee of weight at most .
We now present some extensions of the above results to the setting where weights are subadditive, and there are multiple weight constraints. We also show settings in which the Algorithm 1 has an efficient implementation.
4.1 General Weight Functions
There is a -approximately stable committee for any subadditive weight function over committees, and any monotone preferences of the voters.
Multiple Constraints. We note that Theorems 1 and 2, naturally extend to the following setting with multiple weight constraints. In the multi-constraint setting, there are types of resources, and the weight limit of the -th resource is . Given a subadditive weight for committee and resource , we select a committee so that all constraints are respected. A coalition should only have access to amount of resource .
Definition 7 (Stable Committees with Multiple Constraints).
Given a committee of weight at most , a committee of weight blocks iff for all . A committee is stable if no committee blocks it.
Notions of -approximately stable committees and lotteries can be similarly generalized. By normalizing the weights, we can assume the cost limits are . Redefine the weight of a committee to be the maximum weight across the resources, i.e., . This weight function is also subadditive. Further, any (-approximately) stable solution in the new single-resource instance would also be -approximately stable in the original multi-resource one: It is straightforward to verify all constraints are satisfied in the original setting and the no-deviation requirements are exactly the same in both settings. We therefore have:
There is a -approximately stable committee in the setting with resources.
4.2 Running Time
Our main result above is that of existence of approximately stable committees. If preferences are arbitrary, then we can find this solution by brute-force calculation of for all pairs of feasible committees , which takes time exponential in . Our algorithm has comparable running time, and the bottleneck is constructing a stable lottery efficiently. Indeed, Algorithm 1 runs in poly time if we can find an approximately stable lottery with polynomial size support in polynomial time. Achieving this for Approval Set or Participatory Budget setting is still an open question. We now present some settings where a more efficient implementation is possible.
We first define the following notion of -approximately stable committee, generalizing the notion defined in [ChengJMW19] to arbitrary weights.
Definition 8 (-Approximately Stable Committee).
A committee of weight at most an integer value is -approximately stable for if there is no committee with at most candidates such that .
In the Approval set setting, a -stable committee is exactly a committee that satisfies Justified Representation [Brill], which in itself is a non-trivial property. If we restrict the attacking committees to have at most candidates, the number of such committees is . It is now easy to show that the -approximately stable lottery in Section 2 can be computed in time poly via the multiplicative weight update (MWM) method for solving zero-sum games. The idea is that given a distribution over , the defending strategy involves dependent rounding over this distribution and hence runs in poly time. The number of rounds of MWM will be polynomial in the number of attacking strategies and , and the resulting distribution over defending strategies will have a support of size poly. This implies Algorithm 1 is efficient for constant .
For any , a -approximately stable committee can be computed in poly time.
In the Ranking setting with additive weights, it is easy to observe that a committee is -approximately stable iff it is -approximately stable. This directly implies the following.
For sufficiently small constant , a -approximately stable committee for Ranking and Facility Location preferences, even when candidates have arbitrary (additive) weights, can be computed in poly time.
This can be improved to a -approximation when the candidates are unweighted, since an exactly stable lottery exists in this setting [ChengJMW19]. We emphasize that prior to this, no such result was known, even for existence of an approximately stable solution for these preferences.
5 Open Questions
There are several challenging open questions, both for existence and computation.
Does an exactly stable lottery always exist for all monotone preference structures? We show its existence for committees of weight and unit-weight candidates in Appendix B. Though there is some intuition that the problem in Section 2 resembles fractional knapsack and hence must overflow the knapsack while rounding, our proof for shows that this intuition is misleading. Indeed, our proof uses a different rounding procedure than standard dependent rounding, and it is an open question whether such a procedure always exists.
Does an exactly stable committee exist for the Approval set setting? For this specific setting, no counterexample to exact stability is known.
For Approval Set or Participatory budgeting, can an approximately stable lottery (and hence a deterministic approximation) be efficiently computed? Unlike ranking and Facility Location settings, it may be the case where a solution is not reasonably approximately stable, but no deviating coalition is small. (See, e.g., the lower bound example for PAV rules in [ChengJMW19].) On the other hand, though there are exponentially many committees, the preference structure in these settings is simple and we cannot rule out polynomial time algorithms.
We thank Yu Cheng for many helpful discussions.
Appendix A Lower Bound on Approximation
It is relatively easy to construct a Participatory Budgeting preference profile where a -approximately stable committee does not exist. This instance has cyclic preferences. There are candidates of unit weight, and voters . Let . The preference of is:
Any feasible committee is some single candidate , but all voters except can deviate and choose (or if ). Therefore, the approximation ratio is at least when is large enough.
We now strengthen this example to show that even for the Ranking setting with unit-weight candidates and integral committee weight , there exist instances where a -approximately stable committee does not exist.
In the unweighted Ranking setting, -approximately stable deterministic committees of integral size may not exist for any constant .
For any positive integer and , we construct the following instance with voters and candidates. We view the candidates as a matrix with rows and columns. The voters are , where has the following preference:
For candidates not in the same row, she thinks
for any .
For candidates in the same row , she thinks
Let . For any deterministic committee of size , there must be some , so that no candidate from the -th row is in , and at most one candidate from the -th (or first if ) row is in . Otherwise, every row where no candidate is selected must be followed by a row where at least candidate is selected. Thus, on average, at least candidate is selected from each row, contradicting with .
Let the candidate in the -th row in be if there is one. Notice that is preferred to by at least voters: those in the -th row and the -th row except . Therefore, the approximation ratio is at least , which is close to when and are large. ∎
Appendix B Existence of Exactly Stable Lottery for
In this section, we strengthen the result in Section 2 in the following special case: Each candidate has unit weight, and . There are candidates and voters with arbitrary monotone preferences over committees. In this setting, we show that there is a different way of constructing a dual solution that yields an exactly stable lottery. This opens up the possibility that the analysis in Section 2 is not tight even for larger values of . Indeed, we conjecture that there is an exactly stable lottery for any and any monotone preference structure.
For unit-weight candidates and any number of voters with arbitrary monotone preferences, when , an exactly stable lottery always exists.
The case is trivial. In the case, w.l.o.g. we can assume the attacking strategy only comprises committees of size . This is because having a size- committee in does not help the attacker even if all voters prefer to . Then the case is covered by Lemma 4 below. Therefore we focus on the case. We adopt the duality view introduced in Section 2. Given any attacking strategy , w.l.o.g. assume it only comprises committees of size and size for the same reason that having a size- committee in does not help.
Let , so that . Note that the expected weight of is .
Suppose . In that case, all committees in the support of have the same weight. The following lemma now shows the existence of an exactly stable lottery.
For any , if every committee in the support of has exactly the same weight , then an exactly stable lottery over committees of size at most always exists.
Given , we draw independently from where . Let so . Now we need to show
For any voter , is the probability that is strictly most preferred among , each of which is independently drawn from . Thus
Summing over gives the desired result. ∎
From now on, we will assume . For convenience, we use and to denote the conditional distributions of when is of weight and , respectively. That is, for any ,
We construct a defending committee using the following procedure:
With probability , independently draw two committees from and let be their union.
Otherwise (with probability ), independently draw one committee from and one from and let be their union.
Denote the distribution of as , which is the defending strategy. To prove Theorem 3, we need to show
Fix voter . Consider the committees in decreasing order of voter preference. We say that a committee appears at position if the total probability mass in of committees is . For convenience, we assume is continuous; this will only help the attacking strategy in the proof below. Similarly, let denote the total probability mass in <