Is Sortition Both Representative and Fair?
Anonymous Author(s)
AffiliationAddress email
Abstract
Sortition is a form of democracy built on random selection of representatives.
Two of the key arguments in favor of sortition are representation (a random panelreflects the composition of the population) and fairness (everyone has a chance toparticipate).
Uniformly random selection is perfectly fair, but is it representative?
To answer this question, we introduce the notion of a representation metric onthe space of individuals, and assume that the cost of an individual for a panel isdetermined by the q -th closest representative; the representation of a (random) panelis measured by the ratio between the (expected) sum of costs of the optimal panelfor the individuals and that of the given panel.
For k/ 2 < q ≤ k − Ω( k ) , where kis the panel size, we show that uniform random selection is indeed representativeby establishing a constant lower bound on this ratio.
By contrast, for q ≤ k/ 2 ,no random selection algorithm that is almost fair can give such a guarantee.
Wetherefore consider relaxed fairness guarantees and develop a new random selectionalgorithm that sheds light on the tradeoff between representation and fairness.
Introduction
Most people think of democracy as synonymous with elections.
But that has not always been thecase: From the inception of democracy in ancient Athens until the American and French revolutions,democracy had typically been associated with random selection of representatives [1], a paradigmknown as sortition .
Nowadays, sortition is mainly seen in the form of citizens’ assemblies — randomly selected groupsof people who deliberate on central questions, with the goal of informing policy.
The impact andprevalence of citizens’ assemblies around the world have motivated computational work on how tofairly and transparently select assembly members [2–4].
But there are signs that sortition is becomingeven more widely accepted, including its recent institutionalization in Belgium, where permanentsortition-based bodies are now working alongside the parliaments of the German-speaking region andthe Brussels region.
In light of this progress, it may only be a matter of time until one of the manyblueprints for sortition-based democracy [5] is implemented at the level of an entire country.
The excitement about sortition is driven by several appealing qualities, which are seen as providingsolutions to some of the problems plaguing electoral democracy.
We briefly present two of them in
the context of uniform selection , which selects a uniformly random panel and is considered to be theideal sortition method [6].
• Descriptive representation : A panel selected uniformly at random is likely to reflect thecomposition of the population from which it was drawn.
Representation lends legitimacyto the process [7, 8], as citizens are able to identify some panelists who are similar tothemselves.
• Fairness : Under uniform selection, each citizen has an equal chance to participate.
Po-
litical theorists have argued that this quality realizes philosophical ideals like equality of
opportunity and allocative justice [
By any reasonable measure of the fairness of selection probabilities — e.g., the minimum selectionprobability of any individual [2] — uniform selection achieves perfect fairness, as selection probabilities are equalized.
We ask: Is uniform selection also representative in a rigorous sense?
If wehad a similar measure of representation, we would be able to evaluate whether this is the case.
Butquantifying representation poses a conceptual challenge.
Our approach.
We address this challenge by assuming that there exists a representation metric onindividuals, which measures to what degree one individual represents another (smaller distance meansbetter representation).
Readers familiar with the algorithmic fairness literature will no doubt makethe connection to the similarity metric of Dwork et al.
[11], which has been criticized on the groundsthat it is difficult to explicitly construct [12]; a major obstacle is that the question of whether certainfeatures should be used to determine similarity is domain-specific and tied to legal interpretation.
By contrast, a representation metric is a more viable object, as it can be defined as a function of acommon set of features that are routinely used by practitioners for this purpose, such as gender, age,ethnicity and education.
Moreover, some of our main results, which pertain to uniform selection, arefully independent of the metric — for these it sufficies that such a metric exists .
Note, however, that a distance metric (on individuals) does not directly tell us to what degree anindividual is represented by a panel.
Following very recent work by Caragiannis et al.
[13], weassume that the cost of a panel for an individual is determined by the q -th closest member of thepanel, and our results are parameterized by q .
We can now define representation by taking a page from the literature on distortion in socialchoice [14].
Specifically, for a given selection algorithm, we measure its representation via theratio between the social cost (sum of costs) of the optimal panel and that of the panel chosen by thealgorithm, in the worst case over underlying representation metrics .
Our results.
Returning to the question of whether uniform selection is also representative, and, moregenerally, the eponymous question of whether sortition is both representative and fair, our answer isthat “it depends” — on the value of q .
When k/ 2 < q ≤ k − Ω( k ) , where k is the size of the panel, we show that uniform selection (whichis perfectly fair) achieves constant representation.
Qualitatively, we view this as providing positiveanswers to our questions in the regime of q > k/ 2 .
Note that this regime has a natural interpretation:each individual wants a majority of the panel to be representative of themselves.
This is especiallyjustifiable when the panel makes decisions or recommendations through voting, which is often thecase in citizens’ assemblies.
By contrast, for the regime of q ≤ k/ 2 , we prove that any selection algorithm that chooses eachindividual with probability somewhat higher than q/n , where n is the number of individuals, musthave representation of precisely 0.
This result clearly applies to uniform selection, where the minimumselection probability is k/n , and it motivates us to consider weaker fairness guarantees.
We design analgorithm, R ANDOM R EPLACE , which selects each individual with probability at least q/n and has anontrivial representation guarantee of 1 / ( q + 1) for any value of q .
Finally, we evaluate the average-case representation of uniform selection and R ANDOM R EPLACEbased on two real demographic datasets.
For q > k/ 2 , the experiments show that uniform selectionachieves representation consistently greater than 55%.
Even for q ≤ k/ 2 , in contrast to our worstcase result, uniform selection achieves good average-case representation.
These results suggestthat, in practice, uniform selection may be a good choice for all q -cost distance functions. While R ANDOM R EPLACE outperforms uniform selection across the board in terms of representation (at theexpense of fairness), its advantage is small for values of q around k/ 2 and close to k , which means itdoes not offer a good representation-fairness tradeoff in those regimes. Lastly, for both algorithms,we observe a spike in
Related work.
The design of practical, fair and transparent algorithms for selecting citizens’assemblies was explored in several previous papers [2–4] — two of which appeared in previous
NeurIPS conferences.
Assemblies are required to be representative of the population with respectto features like gender, age, ethnicity, education and geography.
This is done by setting quotas onindividual features; for example, a panel of 100 people might be required to include at least 48men, at least 48 women, and at least 2 people who identify as non-binary.
The challenge is that anassembly is selected from a pool of volunteers, which is typically unrepresentative of the populationdue to self-selection bias.
Uniformly random selection, therefore, would likely itself result in anunrepresentative panel.
Instead, the primary selection algorithm advocated by Flanigan et al.
[2]computes a distribution over quota-compliant panels that (roughly speaking) maximizes the minimumselection probability of any volunteer, thereby maximizing fairness subject to the demographicconstraints.
By contrast, like Benadè et al.
[15] and the political theory literature, we take a longerterm view: We are interested in random selection directly from the population, which is a hallmarkof some plans for sortition-based democracy [5].
In addition, we take a fundamentally different,and arguably more nuanced, view of representation.
Indeed, our framework can accommodatequestions of intersectionality (to what degree is a rural, college-educated man represented by a rural,college-educated woman?) and also provides a more holistic analysis of the composition of thepanel (to what degree is a rural, college-educated man represented by a panel that includes 10 rural,college-educated men and 90 urban women with no college education?). Lang and Skowron [16], Celis et al. [17], and Do et al.
[18] also study the problem of creating committees or assemblies inwhich the features of the participants satisfy some desired quotas, but in the absence of an underlyingrepresentation metric and any fairness constraints.
Our approach to evaluating representation through a metric is rooted in spatial theories of votingfrom the political theory literature [19, 20].
The idea of measuring how poorly a panel representsan individual by the distance of the q -th closest panel member to the individual was introduced by
Caragiannis et al.
[13] in the context of committee elections.
Finally, aiming to minimize the totalmisrepresentation to all people and comparing that to the optimal panel makes our representationmeasure the inverse of distortion in voting theory [21, 22], where distortion is the inverse of ourrepresentation measure.
In voting, it is assumed that we only have partial access to the metric inthe form of voters’ ranked preferences over the candidates induced by distance comparisons.
Incontrast, our results for uniform selection use no knowledge of the underlying metric, while our otheralgorithms assume complete access.
When selecting a single candidate as the winner, it is known thatthe best distortion achievable by deterministic selection is 3 (which maps to 1 / 3 representation in ourformulation) [23].
Our setting is closer to committee selection, where a committee of k candidatesis selected.
Here, Caragiannis et al.
[13] show that when each voter measures her distance to theq -th closest committee member, there is a trichotomy: the best possible distortion is infinite whenq ≤ k/ 3 , linear in the number of voters when q ∈ ( k/ 3 , k/ 2] , and 3 for deterministic selection whenq > k/ 2 .
Sortition is a special case of committee elections in which the set of candidates is thesame as the set of voters.
Hence, all the positive results from Caragiannis et al.
[13] carry over inthe absence of any fairness constraints.
However, our results show that when (perfect) fairness inselection probabilities is sought in conjunction with representation, the distortion becomes infinite(zero representation) for all q ≤ k/ 2 but constant distortion can still be achieved for q > k/ 2 .
Theidea of the set of voters acting as the set of candidates was explored by Cheng et al.
However,they model infinitely many voters using a continuous distribution over the metric space.
Preliminaries
For all t ∈ N , define [ t ] = { 1 , . .
. , t } .
Let N =
[ n ] be the set that constitutes the underlyingpopulation.
A panel P is a subset of the population.
Let S k ( N ) denote the set of all subsets of N ofsize k .
We omit N when it is clear form the context.
The population lies in an underlying metricspace endowed with distance d , which we think of as the representation metric discussed earlier.
Foreach i, j ∈ N , d ( i, j ) denotes the distance between i and the following properties are satisfied: (a)d ( i, j ) ≥ 0 , and d ( i, j ) =
if and only if i = j , (b) d ( i, j ) =
d ( j, i ) , and (c) for each i, j, ℓ ∈ N ,d ( i, ℓ ) + d ( ℓ, j ) ≥ d ( i, j ) .
The last property is known as the triangle inequality.
An instance of ourproblem is given by the underlying population along with distances as defined by d ; hereinafter, wesimply denote such an instance by d .
Given a panel P of size k and a positive integer q ∈ [ k ] , the q -cost of individual i for P , denotedby c q
( i, P ; d ) , is equal to the distance of i from her q -th closest representative in P .
Note that forq = 1 , we have c 1 ( i, P ; d ) =
min j ∈ P d ( i, j ) and for q =
k we get c k
( i, P ; d ) = max j ∈ P d ( i, j ) .
Let top q
( i, P ; d ) be the set of q closest members of P to i (ties broken arbitrarily).
The q -social costof panel P is given by SC q ( P ; d )
=
(cid:80) i ∈ N c q
( i, P ; d ) , i.e., the sum of the q -costs over all individuals.
We omit d from the notation when it is clear from the context.
In this setting, a selection algorithm
A k,q that is defined over k and q takes as input d and outputs adistribution over all panels of size k .
We are especially interested in the uniform selection algorithm,denoted by U k , that always outputs a uniform distribution over S k , independently of the value of q .
In other words, it does not take into account the underlying metric space or q , but instead outputs acommittee of size k chosen uniformly at random.
Fairness: As discussed in the introduction, the main property of uniform selection is that eachindividual is selected to be part of the panel with probability exactly equal to k/n , i.e., Pr[ i ∈
U k ]
=kn .
In other words, all the individuals are treated equally since they have the same chances to bechosen.
We call this property perfect fairness , and in general an algorithm A k,q provides perfectfairness when for each instance d , it ensures that min i ∈ N Pr[ i ∈ A k,q ] = k/n .
When perfect presentation is too restrictive, we relax this constraint by allowing some individuals tobe selected with probability less than k/n .
Then, we measure the fairness of an algorithm A k,q as theworst-case ratio of the minimum probability of an individual to be selected by the algorithm and theideal selection probability k/n .
More formally,
[Equation]
Representation: As we have discussed above, another key property we are interested in measuringis representation.
To do so, we consider a panel to be a good representative of the whole populationwhen the q -social cost is not much larger than the optimum.
In other words, a selection algorithm
A k,qthat outputs a distribution over the different committees of size k provides good representation whenthe expected q -social cost of the panel is similarly small.
More formally, we define the representationof a selection algorithm
A k,q as the worst-case ratio of the minimum possible social q -cost of anypanel and the expected q -social cost of the panel chosen by A k,q over all possible instances, i.e.,repr q ( A k,q ) = inf d min P ′ ∈S k ( N ) SC q ( P ′ ; d ) E [SC q ( A k,q ( d )] .
Representation with Perfect Fairness for q > k/
We start with the case that q > k/ 2 .
We show that in this case uniform selection is asymptoticallyoptimal with respect to representation among all selection algorithms that are perfectly fair.
Moreover,the representation of uniform selection is constant for any q = c · k for 1 / 2 < c < 1 .
Theorem 1.
For q > k/ 2 , uniform selection satisfies repr q ( U k ) ≥ 12 · k − q +1 k .
A crucial property that we exploit in this section is that the q -costs of the individuals satisfy thetriangle inequality when q > k/ 2 .
This observation was first made by Caragiannis et al.
[13]; wepresent the lemma below for completeness.
Lemma 1.
For q > k/ 2 , individuals i, j ∈ N , and a panel P ,
c q ( i, P ; d ) + c q ( j, P ; d ) ≥ d ( i, j ) .
Proof.
Let T i =
top q ( i, P ) and T j = top q
( i, P ) be the q closest neighbors of i and j , respectively,in the panel P .
As | T i | =
| T j | > k/ 2 , there exists an individual k ∈ T i ∩ T j .
Therefore,d ( i, j ) ≤ d ( i, k ) + d ( k, j ) ≤ c q ( i, P ) + c q ( j, P ) .
This observation enables us to show the following lower bound on the social cost of the optimalcommittee.
Lemma 2.
For q > k/ 2 , the q -social cost of the optimal panel P ∗ is at least
[Equation]
Proof.
By applying Lemma 1 for all pairs of individuals ( i, j ) , we get(cid:88)
[Equation]
The q -cost of each person appears exactly 2( n − 1) times on the left hand side.
Thus,
[Equation]
We are now ready to prove the theorem.
Proof of Theorem 1.
For any committee P of size k ,
[Equation]
where in the first inequality the minimum is upper bounded by the average.
Therefore, the expectedsocial cost of uniform selection is at most
[Equation]
By Lemma 2 and the upper bound shown above, we haverepr [Equation] U
[Equation]
In the proof above, the only property of uniform selection we use is that the marginal probabilitiesare equal to k/n .
Hence, this lower bound also holds for any perfectly fair selection algorithm.
We next establish an upper bound on the representation of any perfectly fair selection algorithm.
It shows that the lower bound of Theorem 1 is tight up to a factor of 4.
[-Proof. First, note that if q = k +12 , the statement trivially holds, since repr q ( A k,q ) ≤ 2 · k − q +1 k +
 =-]The proof is relegated to
Appendix A.
Theorem 2.
For any q > k/ 2 , every selection algorithm
A k,q with fairness ( A k,q ) = 1 satisfies
Representation with Relaxed Fairness for q ≤ k/
In stark contrast to the case of q > k/ 2 , uniform selection and, more generally, any perfectly fairselection algorithm, cannot obtain bounded representation when q ≤ k/ 2 .
In fact, the followingtheorem shows that selection algorithms with fairness strictly more than q +( k mod q ) k
(which itself isupper bounded by (2 q − 1) /k ) suffer from unbounded representation.
The proof is in Appendix A.
Theorem 3.
For q ≤ k/ 2 , ϵ > 0 , and any selection algorithm
A k,q with fairness ( A k,q ) ≥q +( k mod q ) k
+ ϵ , repr q ( A k,q ) is 0.
Although bounded representation is not feasible with fairness slightly larger than qk , we designa selection algorithm that, given an α -representative panel, can achieve representation of at leastαq +1 with fairness of qk .
In particular, starting from the optimal panel ( α = 1 ), it achieves 1 q +1representation.
Although finding the optimal panel given the metric space is an NP-hard problemfor most metric spaces, computing a constant-factor approximation (i.e., constant α ) is feasible inpolynomial time [26], as we explain in Section 5.
Our Algorithm, R ANDOM R EPLACE q , is given in Algorithm 1. It starts from a panel P , randomlyselects a panel S of q individuals, and replaces individuals in P with individuals in S as follows.
First, individuals in S ∩ P remain in the final panel.
Then, for i ∈ S \ P , swap i with one of itsq -closest neighbors in the optimal panel top q ( i, P ) that has not been replaced by the algorithm yet.
The next theorem establishes the fairness and representation guarantees of R ANDOM R EPLACE .
Theorem 4.
For any q ∈ [ k ] and any panel P with repr q ( P ) = α , we have thatrepr q ( R ANDOM R EPLACE q ( P )) ≥ αq +1 and fairness ( R ANDOM R EPLACE q ) ≥ qk .
Proof.
Let S ⊆
N be a set of size q chosen uniformly at random.
We denote with P S the panel thatis returned from the algorithm.
First, we show that Line 4 of the algorithm is valid.
The algorithmreaches this line when it considers i ∈ S \ P S , meaning that i is not included in the panel P S andtherefore is not included in P .
Hence, top q ( i, P ) \ S cannot be empty since | top q ( i, P ) | = q , | S | =
qand there exists i in S but not in top q ( i, P ) .
We see that every individual in S is included in P S as in Line 5 the algorithms ensures that each suchindividual is included and is never excluded afterwards.
Hence, as each individual is chosen in Swith probability at least q/n , we can see that fairness ( R ANDOM R EPLACE q ) ≥ q/k .
Now, we prove that for any S ∈ S q and any individual i ′ ∈ N ,
[Equation]
The second inequality holds as the maximum is at most the sum.
Therefore, we focus on the firstinequality.
If c q ( i ′ , P S ) ≤ c q ( i ′ , P ) , then it trivially holds.
Otherwise, c q ( i ′ , P S ) >
c q ( i ′ , P ) .
In this case, we can show that there exists some i ∈ S \ P such that d ( i ′ , i ) ≥ c q ( i ′ , P S ) andj i ∈ top q ( i ′ , P ) .
First, note if for each i ∈ S \ P ∗ , j i does not belong in top q ( i ′ , P ) , then it isnot possible that c q ( i ′ , P S ) >
c q ( i ′ , P ) , since top q ( i ′ , P ) ⊆
P S .
Next, suppose for contradictionthat for every i that was included in P S when j i ∈ top q ( i ′ , P ) was excluded from it, it holds thatd ( i ′ , i ) <
c q ( i ′ , P ) .
Then, in P S there are | top q ( i ′ , P ) \∪ i ∈ S \ P ( { j i }∩ top q ( i ′ , P ))
| individuals thathave distance at most c q ( i ′ , P ) <
c q ( i ′ , P S ) from i ′ and |{ i ∈ S \ P : j i ∈ top q ( i ′ , P ) }| individualsthat have distance less than c q ( i ′ , P S ) from i ′ .
Note that| ∪ i ∈ S \ P ( { j i } ∩ top q ( i ′ , P ))
| = |{ i ∈ S \ P : j i ∈ top q ( i ′ , P ) }| , and hence we get that there are at least | top q ( i ′ , P ) | =
q individuals in P S with distance strictly lessthan c q ( i ′ , P S ) from i ′ , which is a contradiction.
From the above observation , we have that
[Equation]
where the penultimate inequality holds because j i ∈ top q ( i ′ , P ) and j i ∈ top q ( i, P ) from thedefinition of j i .
This proves (1).
Summing (1) over all i ′ ∈ N , we have SC q ( P S ) ≤ SC q ( P ) + n · (cid:80) i ∈
S c q ( i, P ) .
Taking theexpectation of this equation with respect to S , and using the fact that Pr[ i ∈ S ] = q/n , we have
E [SC q ( P S )] ≤ SC q ( P ) + n · (cid:88) i ∈ N q n · c q ( i, P ) =
( q + 1) · SC q ( P ) , that is, 1 q +1 · E [SC q ( P S )] ≤ SC q ( P ) . Combined with the assumption that P is α -representative, i.e.,α · SC q ( P ) ≤ min P ′ ∈ S k ( N ) SC q ( P ′ ) , we get repr q ( R ANDOM R EPLACE q ( P )) ≤ αk +1 .
The next theorem shows that when q = Ω( k ) , R ANDOM R EPLACE q attains an asymptotically optimalrepresentation-fairness tradeoff.
The proof is in Appendix A.
Theorem 5.
For any q ≤ k/ 2 such that k mod q = 0 , every selection algorithm
A k,q withfairness ( A k,q ) ≥ q/k satisfies repr q
( A k,q ) ≤ k/q 2 .
From the above theorem, it follows that Algorithm 1 achieves the highest possible fairness of q/ksubject to positive representation.
Experiments
Next, to complement the worst-case analysis of the selection algorithms we have considered so far, weconduct an empirical comparison between the average-case q -social cost and representation ( repr q )of the different selection methods. Datasets.
Data on the metric-structure preferences of groups in their full richness are difficult tocome by, but it is reasonable to expect that the extent to which individuals feel well-represented byone another is at least partly a function of their relative characteristics along some observable features.
We therefore begin with two datasets that express the characteristics of populations along a range offeatures, and randomly construct synthetic metric preferences from these feature signatures. Adult dataset.
Our first source of demographic data is the UCI Adult dataset, which was derivedfrom the 1994 Current Population Survey of the US Census Bureau, and is made available bythe UCI Machine Learning Repository under a CC BY 4.0 license [27].
It contains a range ofdemographic variables principally related to employment. Our experiments do not require Adult tobe representative of any actual population, nor should this an assumption be made lightly [28]. For Adult we choose the features workclass , education , marital status and sex .
Adult containsn =
30162 individuals with values for each feature, who may be viewed as a distribution over the721 unique feature vectors which they collectively hold.
ESS dataset.
Our second source of demographic data is the European Social Survey ( ESS ) [29], whichis made available by the Norwegian Centre for Research Data under a CC BY 4.0 license.
We usethe ESS Round 9 (2018) data, which consists of 46,276 people in 27 countries, and contains ∼ 1450features regarding socioeconomic demographic, political beliefs, geographical region, house-holdcomposition, personal values, media use and trust, etc.
Most of the features are country-specific,which leaves roughly 250 features available per country, while each country has between 781 and2745 entries (with a mean of 1713).
Each entry is assigned an analysis weight which is aimed tocorrect the differential selection probabilities.
In contrast to our experiments with Adult , we use allof the available features available in ESS .
We report the experiment results based on the data of the
United Kingdom (2204 entries).
Similar results were obtained using the data from other countries.
Metric construction.
We choose some set of features along which to evaluate the individuals inthe population.
These are the features which will inform our metric.
For each feature s , F s is theset of possible values that this feature can take.
Each individual i is then represented as a vector offeature values, where f s i is the value that i has for feature s .
For categorical features (e.g., maritalstatus, sex), we define d ( i, j ; s )
{+= 
+}
− 1 { f si = f sj } .
For range (e.g., income) features we defined ( i, j ; s ) =
| f si − f sj | / (max i ′ ,j ′ | f si ′ − f sj ′ | ) , where the normalization makes d ( i, j ; s ) ∈ [0 , 1] .
Then,for each feature we sample a weight w t ∼ U [0 , 1] uniformly at random from the interval [0 , 1] .
Finallythe distance between individuals i and j and our metric d is defined to be d ( i, j ) :
= (cid:80) s w s · d ( i, j ; s ) .
20 k 0.
S C _ q q=1 q=2 q= q=10 q= (a) U NIFORM S ELECTION   20 k 0.4 0.
S C _ q q=1 q=2 q= q=10 q=
[Figure]
10 q 0. S C _ q opt opt
[Figure]
apx
[Figure]
For the empirical evaluation of our selection algorithms on these randomly generated metrics, wesuppose that this is in fact the distribution for a population large enough that there are at least kindividuals with any given feature vector.
As these metrics represent populations, the q -social costsin Figures 1 and 2 are normalized by population size.
Figures 1 to 3 depict data averaged over 100random metrics constructed in this manner, and the error bars show the standard error of the mean.
Proxy for the optimum.
In evaluating U NIFORM S ELECTION and R ANDOM R EPLACE , we use aproxy for the optimal q -social cost, since n and k are too large to support finding SC q ( P ∗ , d ) exactly.
This selection algorithm O PT P ROXY (indicated as ‘opt_apx’ in legends) is an implementation of thefault-tolerant metric k -medians algorithm of Kumar et al.
[26], which guarantees a constant-factorapproximation to SC q ( P ∗ , d ) .
This algorithm uses a constant-factor metric k -medians algorithmas a primitive; we implement the local search algorithm of Arya et al.
[30] with single swaps.
More specifically, O PT P ROXY works as follows.
It finds the (approximately) optimal
⌊ k/q ⌋ -mediansolution and picks q individuals from each of these ⌊ k/q ⌋ positions, and then selects the remainingk mod q people uniformly at random.
In order to evaluate the fidelity of O PT P ROXY we compare it with SC q ( P ∗ , d ) for 100 metrics dconstructed by drawing 30 randomly chosen feature vectors from the supports of Adult instancesdescribed above.
For panels of size k = 10 and all values of q we find that O PT P ROXY recovers SC q ( P ∗ , d ) exactly (Figure 2).
The fact that the q -social cost attained by O PT P ROXY for fixed qand q ∈ ( k/ 2 , k ] appears constant, on the other hand, is not universally true of SC q ( P ∗ , d ) .
Theseplateaus are due to the way O PT P ROXY selects panels by returning q copies of the optimal 1 -medianlocation.
Since P ∗ q need not be of this form, it is interesting to note that it exhibits this same step-likebehavior in Figure 2.
Experimental results.
In Figure 1a, we see U NIFORM S ELECTION behaving as expected.
For afixed k , the q -social cost of a uniformly random panel is reliably higher for larger q , and for a fixed q ,it decays smoothly as we increase the panel size k .
Under the same conditions, O PT P ROXY behavessimilarly (Figure 1b) but the decay is not as smooth due to the reasons explained above.
Here, all ofour approximately optimal panels P q start at the same value of SC q ( P q , d ) when q = k and decay asq remains fixed and the size of the panel k increases.
Figure 3 shows the ratio between the q -social cost obtained by different selection algorithms and theq -social cost of O PT P ROXY , which is an approximation of their representation, averaged over 100 randomly generated metrics. Specifically, we evaluate U NIFORM S ELECTION and R ANDOM R EPLACE r(indicated as ‘unif_select’ and ‘rand_replace_ r ’ in legends, respectively), the latter being a generalization of R ANDOM R EPLACE q that replaces r individuals instead of q in the input panel and achievesrepresentation at least α/ ( r + 1) (when the input panel has representation α ) and fairness at least r/k .
We compare these algorithms for a range of r and q , on the Adult dataset (Figure 3a) with panels ofsize k = 20 and on the ESS (Figure 3b) dataset with panels of size k = 40 .
Representation of U NIFORM S ELECTION .
As expected, the representation of U NIFORM S ELECTIONis the lowest for all k and q . However, it still achieves consistently good representation across theboard, with representation dipping below 50% only on the Adult dataset with q close to k .
[Figure] We find it especially interesting that, on both datasets, U NIFORM S ELECTION exhibits good representation in the regime of q ≤ k/ 2 , as this contrasts with our worst-case result.
We conclude that,in real-world datasets, U NIFORM S ELECTION may be quite representative even with respect to theq -social cost measure when q ≤ k/ 2 . Overall, we view these results a providing empirical supportfor the use of U NIFORM S ELECTION in practice.
Representation of R ANDOM R EPLACE . Figure 3 shows that R ANDOM R EPLACE
[-U NIFORM S ELECTION increases very smoothly from q = 1 to q = k , which is suggestive of a metricthat is not comprised of a few well-separated groups. Additionally, we see R ANDOM R EPLACE r,qinterpolating nicely from O PT P ROXY at q = 
 up-]r achieves significantimprovements in representation compared to U NIFORM S ELECTION for small values of r — at thecost of fairness, of course. This tradeoff may be justified for some values of q , but the advantage of
R ANDOM R EPLACE is much smaller around q = k/ 2 and when q approaches k , so for these regimes
R ANDOM R EPLACE should not be in the running.
Shift at q = k/ 2 . In both Figures 3a and 3b, there is a sudden improvement in the representation of
U NIFORM S ELECTION at q = k/ 2 . This provides empirical support for the peculiar transition in theq -social cost occurring q = k/ 2 , which is predicted by the theory and is the reason we separated Section 3 and Section 4.
Discussion
Our results in Section 4 show that in some cases, relaxing fairness requirements allows improvingrepresentation dramatically.
More generally, it is interesting to understand the tradeoff betweenrepresentation and fairness, and to chart the Pareto frontier.
In Appendix B, we take some first stepsin this direction.
One observation is the aforementioned generalization of R ANDOM R EPLACE thatreplaces r individuals instead of q .
We also show that a random-dictatorship-like algorithm givesnontrivial fairness and representation guarantees in the regime of q > k/ 2 .
However, there remainseveral open questions: for example, when q ≤ k/ 2 , what level of fairness can be achieved if we seekconstant representation?
We focused our attention on the q -cost formulation of Caragiannis et al.
[13], in which each individualmeasures their distance to the q -th closest panel member.
One can analyze the representation-fairnesstradeoff with other cost functions.
For example, what if different individuals use different values of q ?
Another appealing choice is when each individual measures the average distance to all panel members;in Appendix C, we show that for this cost function, uniform selection achieves representation of atleast 1 / 2 for all q > k/ 2 .
More broadly, one can use measures of representation other than the total cost to all individuals.
For example, one may wish to use a selection algorithm that minimizes the maximum cost to anyindividual, or strikes a balance between maximum cost and total cost.
We hope that answers to someof these questions will lead to a better understanding of the strengths of sortition, and to new ways ofrealizing this democratic paradigm.
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