Wisdom of the Crowd Voting: Truthful Aggregation of Voter Information and Preferences

1 Introduction . Social choice theory studies how to aggregate participants ’ heterogeneous opinions/preferences and output a collective decision from a set of alternatives . Typically , though not always , it is assumed that each participant has a clear preference over the alternatives , e.g. , a preference order over all the alternatives , a valuation for each alternative , etc . However , even with only two alternatives , this is not the typical case for all participants . In addition to the participants who have clear , predetermined preferences for one alternative over the other , typically there are also contingent participants who only have partial information on which alternative is “ preferable for them ” and yet would like to select the alternative that is “ preferable for them. ” A standard example would be an election with two candidates a and b coming from political parties A and B respectively . Voters are normally partitioned into three types : some partisans for Party A prefer candidate a based on his support for the platform of party A ; other partisans for Party B prefer candidate b based on her support for the platform of party B ; finally , there are swing voters who are largely indifferent between the parties ’ platforms and would like to elect whichever candidate can make more progress on non-partisan issues . However , swing voters do not have perfect information about which candidate is better suited for addressing the non-partisan needs of the community . Instead , each voter has a hunch of which candidate will perform better on the non-partisan problems facing the community based on both public information and their private experiences and beliefs . Additional examples where participants have preferences , but may or may not know what is “ preferable for them ” , abound . In votes for corporate strategies , for hiring decisions , and for policy decisions typically some participants would like to select according to some truth they are collectively trying to A full version of this paper is available at https : //arxiv.org/abs/2108.03749 . This material is based upon work supported by the National Science Foundation under Grant No . 2007256 . ∗corresponding author 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . discern ( e.g. , impact on future profits , suitability for the position , efficacy of policy , etc . ) while others may have predetermined preferences , for example , because of the way they are uniquely affected ( e.g. , prominence of their position in future corporate strategy , vision/skills of the job candidate , who in particular the policy benefits/harms ) . 1.1 Informal Setting . In this paper , we consider a two-alternative social choice setting where voters ’ preferences may depend on a state variable . In the above example , the state is which candidate will make more progress on non-partisan issues . We consider binary state variables in the main body of this paper . In general , the state need not be binary . For example , we could generalize the above example so which candidate would preform better on non-partisan issues is on a scale of 1 to 10 ( where 1 indicates candidate a is much better and 10 indicates candidate b is much better ) . Predetermined voters ’ preferences would still not depend on the state . However , contingent voters ’ may have different thresholds on the state where they would transfer their support from candidate a to candidate b . In Appendix E , we discuss the non-binary setting and extend our results to this setting . The state variable is not directly observable in the election phase , as the performance of a new government official , a new hired employee , a new policy , etc. , may not be revealed until many years after the vote . Instead , each voter receives a signal that is correlated to the hidden state which models the information voters received from difference sources . Our goal is to select the majority wish , the alternative that would be preferred by the majority if they knew the state of the world . If some type of predetermined voters forms a majority , this is rather easy . However , in the case where the predetermined voters of neither alternative forms a majority , the mechanism needs to aggregate the information and preferences of the contingent voters and selects the alternative which , for a majority of voters , is “ preferable for them ” . 1.2 Imperfectly Informed Voters . Social Choice Social choice theory with imperfectly informed voters dates back to Condorcet ’ s jury theorem in 1785 [ De Condorcet , 1785 ] , and has also been widely studied [ Miller , 1986 , Young , 1988 , Ladha , 1992 , Nitzan and Paroush , 2017 ] . In Condorcet ’ s setting , there are two alternatives , one of which is “ correct ” , and each voter votes for the correct alternative with probability p. Condorcet ’ s jury theorem states that the probability that the majority voting scheme outputs the correct alternative goes to 1 as the number of voters increases when p > 0.5 , and , conversely , this probability goes to 0 when p < 0.5 . Two unfortunate limitations for Condorcet ’ s jury theorem are 1 ) It fails to output the correct alternative in the case voters ’ beliefs are aligned to the incorrect alternative ( i.e. , p < 0.5 ) , and 2 ) It assumes voters vote truthfully and disregards voters ’ potential strategic behaviors . However , even in the case all voters have the same preference for the correct alternative , voting truthfully still may not be a Nash equilibrium [ Austen-Smith and Banks , 1996 ] . To circumvent the first limitation , Feddersen and Pesendorfer [ 1997 ] consider the scenario where voters play a Nash equilibrium strategy profile , while the voting rule is still the majority scheme . Feddersen and Pesendorfer [ 1997 ] show that when the number of voters tends to infinity , the probability that a majority voting scheme outputs the correct alternative approaches to 1 if voters play the equilibrium strategy profile , while this probability is bounded away from 1 if voters play the truthful strategy profile instead . Feddersen and Pesendorfer ’ s model assigns each voter a preference parameter x ∈ [ −1 , 1 ] describing his/her alignment to the two alternatives . In their model , the unique ( Bayes ) Nash equilibrium is characterized by two thresholds x0 , x1 ∈ [ −1 , 1 ] with x0 < x1 such that voters with preferences below x0 always vote for one alternative , voters with preferences above x1 always vote for the other , and voters with preferences between x0 and x1 vote truthfully . Although Feddersen and Pesendorfer ’ s solution guarantees that the correct alternative is output with high probability , it requires sophisticated voters . The voters need to calculate the values of x0 and x1 to decide their actions . The values x0 and x1 are each the zero point of a continuous monotone function involving a complicated Riemann integral . This is usually too demanding for voters in practice , especially those who do not have a mathematical background . Moreover , it needs to be common knowledge that all agents can and will perform this computation . In this paper , we take a different approach . Instead of asking voters to play the Nash equilibrium for majority voting , we seek to design a more sophisticated voting scheme , or a mechanism , than the majority voting scheme , such that voters are incentivized to vote truthfully under the mechanism , while guaranteeing the correct alternative is output with high probability . Our social choice mechanism thus elicits truthful information from the voters and then aggregates it . Information Aggregation The information aggregation literature considers how to obtain a “ correct answer ” by aggregating individuals ’ partial information—the crowd ’ s wisdom . The straightforward procedure of outputting the answer that is believed to be correct by the majority does not always work [ Chen et al. , 2004 , Simmons et al. , 2011 ] . An example where this fails is when the crowd has a strong prior belief for the incorrect answer while novel specialized knowledge is only shared among a minority of the agents . It is also known that further calibration based on collecting participants ’ confidences ( the posterior of their beliefs ) does not always solve this problem [ Hertwig , 2012 , Prelec et al. , 2017 ] . In a seminal work by Prelec et al . [ 2017 ] , a new “ surprisingly popular ” approach was proposed : the participants ’ predictions over the other remaining participants ’ reported answers are collected , and the answer that is reported by more participants than predicted is output ( we will review this in Section 3.1 ) . They justified this approach both theoretically and through experimentation . In particular , they demonstrate the viability of approaches that require agents to predict reports of the other agents . Hosseini et al . [ 2021 ] empirically extend the surprisingly popular approach to the non-binary setting , where the goal is to learn the correct ranking over many options instead of the correct answer in two options . The work by Prelec et al . [ 2017 ] and Hosseini et al . [ 2021 ] does not fit into the social choice context in two aspects . Firstly , the objective for an information aggregation mechanism is to output the correct answer . Participants who collaboratively contribute their knowledge/information do not have preferences on which answer is finally selected . This is fundamentally different from the social choice setting where the whole point of a social choice mechanism is to select an alternative favored by the majority . Secondly , as agents care about the outcome , agents may be strategic and manipulate their reports in order to make their preferred alternatives win , while Prelec et al . [ 2017 ] and Hosseini et al . [ 2021 ] do not put the problem in a game theory setting . 1.3 Our Results . In this paper , we study the social choice problem in a game theory setting with the existence of imperfectly informed voters who only have partial information regarding which alternative is more favorable . For various settings with two alternatives , we propose a mechanism that aggregates participants ’ private information and outputs the alternative favored by more than half of the participants—the majority wish . Our mechanisms are truthful , in the sense that the truthful strategy profile forms a strong Bayes Nash Equilibrium . Our main wisdom-of-the-crowd-voting mechanism , presented in the paper , applies to the case of two worlds/states , where each agent receives a binary signal . We show it has strong truthfulness and aggregation properties , even for relatively small numbers of agents ( Section 3 ) . This result requires that the distribution of agents is a common knowledge . In the full version , we show several additional results . First , the common knowledge assumption is necessary to attain a strongly truthful mechanism that outputs the majority wish with high probability ( Appendix D ) . Additionally , we extend our results to the setting of more than two worlds ( Appendix E.2 ) . Finally , we show how to extend our results to the case of more than two signals ( Appendix E.3 ) . Our mechanism can easily be implemented using a simple questionnaire that elicits voters ’ information and preferences . The questions in the questionnaire are friendly to those voters who do not have relevant backgrounds in mathematics , game theory , etc. , and they require participants to predict the responses of other agents , which is empirically validated by the surprisingly popular method from Prelec et al . [ 2017 ] . We ensure our mechanisms have a group truthfulness property by employing a “ median trick. ” Intuitively , by the Median Voter Theorem [ Black , 1948 , Hotelling , 1929 ] , the median voter ’ s vote ( in a binary choice ) is favored by the majority . By a careful design , our mechanism ensures that the voters who are “ below ” the median have a conflict of interest to the voters who are “ above ” the median , which makes sure less than half of the voters have an incentive to deviate and those voters can only change the outcome in the unfavorable direction by the property of median . From a high level , our work can be understood as a revelation principal applied to plurality voting.4 However this view is not entirely accurate . First , our equilibrium concept is strong Bayes Nash equilibrium while we only know that plurality voting implements the majority wish outcome in ( Bayesian ) Nash equilibrium [ Feddersen and Pesendorfer , 1997 ] . Second , the revelation principal requires that agents report all their knowledge . In our case , this would include the entire prior , which is not realistic . In contrast , our mechanisms only require that agents report a preference and a prediction of other agents ’ preferences . Such reporting requirements have previously been shown to be pragmatic [ John et al. , 2012 , Prelec et al. , 2017 , Hosseini et al. , 2021 ] . Third , our setting is different than prior work [ Feddersen and Pesendorfer , 1997 ] , and this makes our results incomparable . In particular , we deal with a discrete state space . This difference also allows us to achieve some of our results not just in the limit , but for finite sets of agents . Appendix A contains an additional comparison of our work with Feddersen and Pesendorfer [ 1997 ] ’ s .

Summary: The authors consider elections that have two alternatives and three distinct classes of voters: candidate-friendly, candidate-unfriendly, and contingent agents. Candidate-friendly and candidate-unfriendly agents have made up their minds before the election, but each contingent voter's opinion depends on an unobservable state variable; they each receive a private signal about this state variable. The authors present a mechanism that elicits private signals from voters and which outputs the correct answer based on the state of the world. This mechanism is also strategyproof against even coalitions in the sense that truthful reporting is a strong Bayes Nash equilibrium. The mechanism draws on the "surprisingly popular" method of Prelec et al.