Learning to Elect

1 Introduction . Voting systems are highly prevalent in our daily lives . Examples range from large scale democratic elections to company or family-wide decision making , recommender systems and product design [ Boutilier et al. , 2015 ] . As with any social decision-making process , the goal of designing voting rules is to reconcile differences and maximize some collective objective . The area of research that studies different voting rules and the approaches to designing them is called voting theory . A vast number of voting rules have been proposed over the years . Among them is the widely applied plurality rule . Despite being simple and intuitive , the plurality rule is very limited in that it does not consider the strength of voters ’ preferences . Other examples of voting rules , such as Borda and Copeland , take into consideration the ranked preferences of the voters . Voting theorists have developed different approaches to designing voting rules . For example , the axiomatic approach constrains the voting rules to satisfy certain desired properties ( axioms ) such as anonymity ( treating all voters equally ) and neutrality ( treating all candidates equally ) . The utilitarian approach , on the other hand , aims to maximize a pre-defined notion of social welfare — a scalar quantity that measures the quality of the elected candidate in the eyes of the electorate . There are major hurdles to overcome in the traditional way of designing and implementing voting rules . First , the celebrated Arrow ’ s Theorem states the nonexistence of non-dictatorship voting rules that simultaneously satisfy a set of seemingly sensible axioms [ Arrow et al. , 1951 ] . Second , for some voting rules such as the ones based on pairwise comparisons , finding the winner can be computationally expensive , making them infeasible for large-scale applications . Last but not least , for the utilitarian approach , it is not obvious how to design voting rules that maximize a given notion ∗Equal contribution . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . of social welfare.2 There might be a diverse set of social welfare functions of interest , but theory is lacking in finding their corresponding optimal voting rules . In this paper , we aim to tackle the latter two limitations using neural networks . While doing so , we also seek to preserve certain desired properties such as voter anonymity . In particular , we identify three permutation-invariant neural network ( PIN ) architectures : DeepSet [ Zaheer et al. , 2017 ] , Set Transformer [ Lee et al. , 2019 ] , and Graph Isomorphism Network ( GIN ) [ Xu et al. , 2018 ] , and apply them to learn the voting rules . As compact and universal function approximators , such trained neural networks not only address the computational burden of some voting rules , but also provide a flexible approach to maximize a diverse set of social welfare functions . The main contributions of this paper include : • We show that PIN architectures are theoretically and empirically well-suited for learning to represent voting rules . Theoretically , we show that the three PIN architectures are universal approximators in the space of permutation-invariant functions . This includes a novel proof on the universality of Graph Isomorphism Networks with fully-connected graphs ( as in learning voting rules ) . • We apply the aforementioned PIN models to mimic five classical voting rules : plurality , Borda , Copeland , Maximin and Kemeny . We show that they can mimic these voting rules to compelling accuracy and can generalize seamlessly to unseen real datasets and elections with an unseen number of voters . • We train the networks to maximize two different social welfare functions — utilitarian and egalitarian — on elections sampled using three different underlying voter utility distributions . We show that the PIN models discover novel voting rules that maximize social welfare better than classical voting rules . In cases where theoretical optimal voting rules are known ( i.e . for the utilitarian social welfare function ) , the PIN models match the optimal performance . The organization of this paper is as follows . Section 2 provides background on voting theory . Section 3 describes our method of using PINs to learn voting rules . Specifically , we introduce the permutation-invariant network architectures ( Section 3.1 ) , show the universality results ( Section 3.2 ) , and describe the proposed the training procedure for learning voting rules in detail . In Section 4 , we show comprehensive experiment results on the effectiveness of PIN models in learning voting rules . We discuss related works in Section 5 , limitations in Section 6 and conclude with Section 7 . 2 Background . 2.1 Voting theory preliminaries . We adopt the formalism used by Boutilier et al . [ 2015 ] . Let N = { 1 , . . . , n } be a set of voters , A = { a1 , . . . , am } be a set of candidates and R = { 1 , . . . , m } be the integer rankings each voter can assign to a candidate . Each vote i is represented as a bijection from candidates to rankings : σi : A 7→ R. The vector of candidate preferences # σ = ( σi , . . . , σn ) ∈ ( Sm ) n is called a preference profile . A voting rule f : ( Sm ) n 7→ A maps preference profiles to candidates . The utilitarian approach to voting makes the assumption that voters have utility functions ( u : A 7→ R+ ) over the alternatives which quantify how much a voter prefers a candidate . Utility functions are consistent with the rankings — that is , candidates with higher utilities are ranked higher by voters . The vector of all voters ’ utility functions # u = ( u1 , . . . , un ) is called a utility profile . The social welfare of an alternative sw ( a , ~u ) : A× Rn×m 7→ R quantifies the “ desirability '' of a candidate a under a given preference profile ~u . The utilitarian approach posits that the ultimate goal of a voting rule is to pick the candidate that maximizes social welfare . Note that since voting rules only have access to rankings and not the utilities , this is often impossible to achieve without further assumptions . A popular notion of social welfare is the utilitarian social welfare function , which computes the sum of all the utilities the voters assign a candidate swutil ( a , ~u ) = ∑ i∈N ui ( a ) . There exists , however , many different social welfare functions . For example , the Rawlsian social welfare function aims to make even the least happy voter as happy as possible : swrawl ( a , ~u ) = mini ui ( a ) , and the egalitarian social 2We consider average-case social welfare maximization , unlike the worst-case analysis in the distortion literature [ Boutilier et al. , 2015 , Caragiannis et al. , 2017 ] . welfare function aims to maximize the utilitarian welfare , regularized by a penalty term promoting equality : swegal ( a , ~u ) = ∑ i∈N ui ( a ) − λ ∑ i∈N ( ui ( a ) −minj ( uj ( a ) ) ) [ Allcott , 2011 ] . It is up to the designer of the voting system to pick a notion of social welfare best suited for the task at hand . 2.2 Classical voting rules . Most classical voting rules can be classified under two groups : score based rules and comparison based rules . Score based rules ( also called scoring functions ) are defined by a score vector ~s = ( s1 . . . sm ) . Each candidate a ∈ A is assigned the score ∑ i∈N sσi ( a ) and the candidate with the largest score is picked as the winner [ Boutilier et al. , 2015 ] . Famous examples include Plurality ( with a score vector of ~splr = ( 1 , 0 , . . . , 0 ) ) and Borda ( with a score vector of ~sborda = ( m − 1 , m − 2 , . . . , 0 ) ) . Comparison based rules operate using pairwise comparison matrices R ∈ Rm×m whose entries are filled based on how candidates fare against each other in pairwise comparisons . Well-known examples include the Copeland , Maximin and Kemeny rules . The Copeland rule picks the candidate who fares better in pairwise comparisons the largest number of times : argmaxi ( ∑ j rij ) where rij stands for the number of times candidate i fares better against candidate j in the voter preferences . Maximin , also known as Simpson ’ s rule , picks the candidate for whom the candidate who fares the best against him/her in pairwise comparisons has the least pairwise score : argmini ( maxj rji ) [ Bubboloni et al. , 2020 ] . The Kemeny rule [ Kemeny , 1959 ] first computes a ranking that maximizes the sum of all pairwise wins : σ∗ = argmaxσ ∑ i σj rij , where i σ j means that candidate i is preferred against j according to ranking σ . The Kemeny winner is the candidate that ranks the first in σ∗ . Note that computing the Kemeny ranking or Kemeny winner is NP-hard [ Bartholdi et al. , 1989 ] . 2.3 Average-case optimal voting rules . In cases where the social welfare function takes a very simple algebraic form , it might be possible to derive the voting rule that achieves the largest expected social welfare under a given utility distribution Pu . For example , it is known that the average-case optimal voting rule for the utilitarian social welfare funtion is a score-based voting rule with the score vectors s∗k = Eu∼Pu [ u ( a ) | ( σ ( u ) ) ( a ) = k ] ( or , the average utility of the candidates that are ranked at kth position ) [ Boutilier et al. , 2015 ] . 3 Permutation-Invariant Networks ( PINs ) to learn set-input functions . We review the fundamentals of learning set-input functions through the use of permutation-invariant networks ( PINs ) . Learning set-input functions is desirable in the context of learning voting rules : 1 ) it enables processing elections of varying sizes ( i.e . different number of voters ) 2 ) it makes it possible to satisfy desirable properties such as anonymity ( invariance to the shuffling of voter identities ) through architectural constraints . We review three permutation-invariant architectures ( DeepSets , fully-connected graph networks and Set Transformers ) and show that they ’ re universal approximators of set-input functions . We also detail the training procedure for learning voting rules . 3.1 Constructing permutation-invariant architectures . Functions defined on sets are by definition permutation-invariant [ Zaheer et al. , 2017 ] . Property 1 . Let 2X represent the powerset of X . Any set-input function f : 2X 7→ Y must be invariant to the reordering of its inputs by any permutation σ : f ( { x1 , . . . , xM } ) = f ( { xσ ( 1 ) , . . . , xσ ( M ) } ) Zaheer et al . [ 2017 ] showed that one can practically train expressive set-input neural networks by chaining together a permutation-equivariant feature extractor ( encoder ) and a permutation-invariant decoder.3 permutation-equivariance is the property that when the inputs of a function are permuted , the outputs get permuted the same way : Property 2 . A function f : RM×din 7→ RM×dout is permutation-equivariant if f ( [ xσ ( 1 ) , . . . , xσ ( M ) ] ) = [ fσ ( 1 ) ( x ) , . . . , fσ ( M ) ( x ) ] 3Composition of permutation-equivariant and permutation-invariant functions results in permutation-invariant functions . All of the architectures described below follow the central design principle of composing permutationequivariant building blocks with permutation-invariant ones to build expressive set-input networks . DeepSets [ Zaheer et al. , 2017 ] This architecture first encodes each element of the set independently using an encoder network ( e.g . a multilayer perceptron ) , pools the outputs of the encoder ( e.g . using sum , max or mean pooling ) and finally passes the result through a decoder network ( e.g . another multilayer perceptron ) . Since the encoder treats the elements of the set independently to achieve permutation-equivariance , the pooling operation is the only step that can model interactions between the elements in the set . ( Fully-connected ) graph neural networks ( GNNs ) Instead of achieving permutation-equivariance through the independent processing of the set elements , we can instead view the input set as a fully-connected graph and build a GNN-based [ Scarselli et al. , 2008 ] encoder that takes into account all of the interactions between different elements of the set . Each GNN layer transforms the nodes in the graph by concatenating a permutation-invariant aggregation of features of the neighbour nodes and a transformation that produces the next-layer features given the current-layer features and the result of aggregation . A GNN consists of multiple GNN layers , and the graph-level embedding is obtained by pooling all the node features using a permutation-invariant function . In our experiments , we use a powerful variant of GNN — the Graph Isomorphism Network ( GIN ) [ Xu et al. , 2018 ] . Detailed background on GNNs and GIN can be found in Supplementary Material . For a fully-connected graph , the graph structure remains unchanged under node permutation . Since both the neighbourhood aggregation and the graph-level pooling operations are permutation-invariant , the whole GNN represents a permutation-invariant function . Set Transformers [ Lee et al. , 2019 ] The computational building block of set-transformers is the Query-Key-Value ( QKV ) attention [ Vaswani et al. , 2017 ] , which can be interpreted as a differentiable dictionary retrieval operation . The neural network controls both the input and output space of the QKV operation using multihead attention , which multiplies all the input and output tensors by learnable weight matrices . The multihead attention possess desirable permutation-equivariance/invariance properties for learning set-input functions . Set Transformer builds highly expressive encoder-decoder architecture with multiple multihead attention blocks , resulting in the overall permutation-invariance [ Lee et al. , 2019 ] . Detailed background and the full encoder-decoder architecture are found in Supplementary Material .

Summary: The paper studies two problems related to using neural networks as single winner voting rules. (1) Training networks to mimic well-known existing rules, and (2) training networks to act as novel rules that select utility-maximizing alternatives.  The major novel contribution of the paper is in their choice of network architecture. "Permutation-invariant networks" are used, which are agnostic to the order of input data, thus ensuring that the resultant voting rules are anonymous by default.  Experiments show that the 3 PIN architectures considered are able to learn the studied voting rules with high accuracy after a large amount of training. These results are shown for testing done on each of real and synthetic data. Similarly, when the networks are trained with an oracle to learn the welfare maximizing alternative they perform quite well.