Extended Abstract: The Swing Voter s Curse in Social Networks

Extended Abstract: The Swing Voter s Curse in Social Networks Berno Buechel & Lydia Mechtenberg January 20, 2015 Summary Consider a number of voters with common interests who, without knowing the true state of the world, have to decide in a majority vote which of two alternative policies is the right one. There are informed and uninformed voters. Informed voters have imperfect private information indicating the right policy; uninformed voters only know that both policies are equally likely to be correct. We show that, contrary to both intuition and the largest part of the literature, communication prior to the vote can reduce the informational efficiency of the voting outcome, depending on the structure of the communication network. We study communication networks in which informed voters give private voting recommendations to uninformed voters and each uninformed voter listens to only one informed voter. In such networks, information aggregation through voting becomes inefficient if (i) information transmission occurs prior to the vote and (ii) the degree distribution in the communication network is too unequal in the sense that some informed voters talk to many uninformed voters while the other informed voters have only relatively small audiences. Hence, in highly unequal networks, e.g., in a network with a star component, the uninformed voters would want to abstain or vote the opposite rather than follow a voting recommendation of an overly powerful opinion leader. This is due to a new type of the swing voter s curse: Following the recommendation to vote for a certain alternative is reasonable (because the recommendation is more likely to be true than the opposite), but Department of Economics, University of Hamburg, von-melle-park 5, D-20146-Hamburg, Germany. Phone: +49 40 42838 5573. Email: berno.buechel@uni-hamburg.de. Web: www.berno.info Department of Economics, University of Hamburg, von-melle-park 5, D-20146-Hamburg, Germany. Phone: +49 40 42838 9484. Email: Lydia.Mechtenberg@wiso.uni-hamburg.de 1

suboptimal when the own vote makes the difference. Consider an opinion leader who is very powerful such that a large part of the voting population follows her recommendation. Being pivotal with a vote that follows her recommendation implies that many voters from the rest of the population voted for the opposite, which implies, in turn, that they had information contradicting the opinion leader s recommendation. Hence, conditioning on pivotality, it is more likely that the voting recommendation of the opinion leader is wrong rather than correct. More generally, in highly unequal networks following the voting recommendation is neither informationally efficient nor equilibrium behavior. We find theoretically that voting equilibria are characterized by informational efficiency if the communication network is sufficiently equal. For mildly unequal communication networks there are informationally inefficient voting equilibria with information transmission. Testing our theoretical predictions in a lab experiment, we find that uninformed voters are indeed more inclined to abstain when they listen to an overly powerful opinion leader, but that abstention still occurs too rarely to prevent a loss in informational efficiency induced by unbalanced communication. In the experiment, the loss in informational efficiency is the larger, the more unequal the communication network becomes. Intuitively, the more unequal the network structure, the less balanced is power such that the final outcome is determined by the message of a few agents, in contrast to Condorcet s original idea of aggregating the signals in a large population. Related Literature Models on strategic voting are rooted in Condorcet s setting of two states of the world and independent private information about the true state. He assumed that voters are sincere (i.e. they vote their signal) and showed that the probability of a correct decision under majority voting converges to one as the number of voters grows (De Caritat, 1785). Given homogeneous preferences for finding the truth and equal signal precision, sincere voting is also equilibrium behavior. Strategic voting that does not coincide with sincere voting is obtained when some agents have different interests ( partisans ) or if some individuals are informed ( experts ) while others are not (Feddersen and Pesendorfer, 1996, 1997, 1998). The central argument is that by being pivotal an agent can deduce that he is affecting the election into an undesirable direction and thus should abstain. Laboratory experiments confirm the existence of strategic voting for both the presence of partisans (Battaglini, Morton, and Palfrey, 2010) and in the case of experts (Morton and Tyran, 2011), without only finding equilibrium behavior. We add to this literature a third type of strategic voting (and of the swing voter s curse), which we study 2

both theoretically and empirically. Preplay communication in the form of public communication is studied by Gerardi and Yariv (2007) theoretically, and by Goeree and Yariv (2011) empirically. These contributions show that public communication enables efficient information aggregation because voters can publicly reveal their own signal (deliberation). In contrast, we focus on private communication (in a social network) and find that unequal communication networks undermine efficient information aggregation. The Model There are two sets of agents: members of N who might be interpreted as ordinary voters (the people) and members of M who might be interpreted as the political class. A bipartite network g, consisting of links (i, j) N M, represents the communication structure between the two groups. Nature draws one out of two states of the world, say A or B. All voters in N are independent in the sense that they try to match the true state of the world. Some voters in M are independent too, but in addition they receive a signal about the true state of the world (we call them informed). The other voters in M are partisans who try to induce a certain outcome whatever the state of the world, e.g. due to the expectation of personal perquisites. The sequence of actions is as illustrated in the left panel of Figure 1. First, nature draws a state of the world, an allocation of informed players and partisans in M, and signals for the informed players. Then agents in M choose a message in {A, B, }, which is communicated to all i N who are linked with them. Third, all agents vote or abstain and the outcome is determined by the majority rule. The states of the world A and B have an ex ante equal probability. Conditional on the state, signals (for the informed agents) are drawn independently and are correct with probability p > 1. 2 Our model allows us to study the problem that a message may contain information (if sender is informed) or may only be propaganda (if sender is a partisan). This issue seems crucial when relying on the information of some sort of insider before participating in a collective decision. Some Theory Focusing on the case that the number of A-partisans equals the number of B-partisans, we study perfect Bayesian equilibria in this game. A first observation is that under mild conditions, informational efficiency of a strategy profile implies that it is also an equilibrium. It follows that there is an efficient equilibrium for every network structure. In this equilibrium all voters in N (who do not receive a signal) abstain and 3

Figure 1: Illustration of the model (left) and an unequal network (right). the informed vote their signal. This corresponds to the let the experts decide equilibria analyzed in Morton and Tyran (2011). While this is an equilibrium without information transmission ( babbling ), our focus lies on the following strategy profile with full information transmission: all informed communicate and vote their signal, all partisans communicate and vote their preferred outcome, and all agents in N follow their message (or abstain if they do not have a sender). This strategy profile is sincere in the sense that each agent communicates and votes the alternative that yields the highest expected utility for her (given the available information, but without conditioning on pivotality). This sincere strategy profile is an equilibrium in networks which feature a highly equal distribution of links, which we call almost regular. Almost regular networks are characterized by the following two properties which are both equivalent to an almost regular degree distribution: the sincere strategy profile is informationally efficient; the outcome of the election coincides with the outcome that would be attained if only votes among members of M counted. For networks which are not almost regular, but also exhibit quite an equal distribution of links, the sincere strategy profile is still an equilibrium but not informationally efficient. In a more unequal network structure (such as the one illustrated in the right panel of Figure 1), however, this strategy profile is not an equilibrium because voters who receive a voting recommendation by a powerful agent (i.e. who has a high degree) prefer to abstain or to vote the opposite. This argument leads to a necessary 4

Figure 2: The four treatments. condition for the sincere strategy profile to be an equilibrium, while almost regularity is a sufficient condition. The Experiment The experimental sessions took place in the experimental laboratory of the University of Hamburg in November 2014. In total 140 people (mostly students, with various subjects) participated in five sessions. Participants became either informed types (in M) or uninformed types (in N) and stayed in this role for course of the experiment. Partisans were not played by real people but replaced by the computer (using the sincere strategies). In every round all participants were randomly matched into groups of seven. In each group, we have n = 4 uninformed, m I = 3 informed, two A-partisans, and two B-partisans. The quality of the signal is set to p = 0.8. We study the four network structures illustrated in Figure 2 which we call empty, equal, unequal, and star. While there are no links in the empty network, the density, i.e. the number of links, is held constant in all other networks. These networks differ by their inequality in the degree distribution. Note that the network that we call equal does not have a fully equal degree distribution since some agents in M have no links. (Only the empty network has a fully equal degree distribution and thus satisfies almost regularity discussed above.) Each participant played every treatment ten times. The sincere strategy profile defined above all informed communicate and vote their signal, all partisans communicate and vote their preferred outcome, and all agents in N follow their message or abstain if they do not have a sender is an equilibrium in the empty and the equal network. In the empty network, the informed agents determine the outcome since the votes of the partisans cancel out and the uninformed abstain. In 5

the equal network, the uninformed have an incentive to follow their message in order to counterbalance the fact that others follow their messages in equilibrium. However, this equilibrium is informationally inefficient since it is possible that a signal that is only drawn once gets a majority by having two partisans and the informed with this signal spreading this recommendation. In the unequal network and the star network, the sincere strategy profile is inefficient and not an equilibrium. These networks violate the sufficient condition mentioned above. followed. There is a powerful agent whose message should not be We test in the laboratory whether agents take this strategic consideration into account and how efficiency is driven by the network structure. One has to keep in mind that, besides the sincere strategy profile whose informational efficiency is decreasing with the inequality of the network structure, there is always the informationally efficient let the experts decide equilibrium, which differs from the sincere profile in that all voters who receive a message simply abstain. For the empty network both strategy profiles coincide. First Results More than 90% of the informed indeed vote and communicate their signals. This holds irrespective of the treatment. The behavior of the receivers depends on the network structure: While 70% follow the recommendation in the equal network, only 56% follow it in the unequal network and only 48% do so in the star network. Thus, the participants seem to indeed take the strategic aspect into account and follow less, the more unequal the network structure. (Consistently, we find within the unequal treatment that the message of the powerful agent is less often followed than the message of the agent with only one recipient.) Most of those who do not follow a voting recommendation choose abstention. This behavior might potentially mitigate the informational inefficiency that is implied by sincere voting. Information aggregation is efficient when the signal that is obtained by a majority of informed is implemented. Within the inefficient cases we can distinguish between ties which have an expected utility of 1 and cases where the signal that is obtained by a 2 minority of informed in a group is implemented. The following graph illustrates the results. When the signals are uniform, i.e. all of the three informed agents have received the same signal, then it is simpler to reach this outcome (which maximizes expected utility) in majority voting than in the non-uniform case, where one informed agent receives a different signal than the two others. While there are no significant differences between the empty network and the equal network, there is a loss of informational efficiency in the unequal network and even more so in the star network. In the star network the majority signal, i.e. the outcome maximizing expected 6

101. replace unequal_position4 = 1 if T_unequal==1 & Type==2 & NetworkPosition ==4 (220 real changes made) 102. replace unequal_position4 = 0 if T_unequal==1 & Type==2 & NetworkPosition <4 (660 real changes made) 103. 104. ********************************************************************* 105. ********************************************************************* 106. end of do-file 107. do "C:\Users\BERNO~1.BUE\AppData\Local\Temp\STD00000000.tmp" 108. *Main (less clusters than parameters mean that joint significance (Wald test) cannot be tested) 109. ologit efficiency uniform T_equal T_unequal T_star if dup==1 & Period > 0, vce(cluster session) Iteration 0: log pseudolikelihood = -580.90936 Iteration 1: log pseudolikelihood = -517.64226 Iteration 2: log pseudolikelihood = -513.27752 Iteration 3: log pseudolikelihood = -513.2616 Iteration 4: log pseudolikelihood = -513.2616 Ordered logistic regression Number of obs = 800 Figure 3: Informational efficiency by treatment for uniform Wald signals chi2(2) (left) = and non-uniform. Prob > chi2 =. signals (right). Log pseudolikelihood = -513.2616 Pseudo R2 = 0.1165 (Std. Err. adjusted for 5 clusters in session) Robust efficiency Coef. Std. Err. z P> z [95% Conf. Interval] uniform_signal 2.026811.1351358 15.00 0.000 1.761949 2.291672 T_equal.0587647.1397429 0.42 0.674 -.2151264.3326559 T_unequal -.2761981.1640986-1.68 0.092 -.5978254.0454292 T_star -.7114327.3190904-2.23 0.026-1.336838 -.0860269 /cut1-1.610897.2076426-2.017869-1.203925 /cut2 -.5718534.1224386 -.8118287 -.3318782 110. end of do-file Table 1: An ordered logit regression with standard errors clustered on the session level. Reference treatment is the empty network. 111. do "C:\Users\BERNO~1.BUE\AppData\Local\Temp\STD00000000.tmp" 112. *Robustness w.r.t. estimation technique 113. oprobit efficiency uniform T_equal T_unequal T_star if dup==1 & Period > 0, vce(cluster session) Iteration 0: log pseudolikelihood = -580.90936 Iteration 1: log pseudolikelihood = -513.95055 Iteration 2: log pseudolikelihood = -512.77128 Iteration 3: log pseudolikelihood = -512.77107 Iteration 4: log pseudolikelihood = -512.77107 utility, is only reached in 84% of the cases when signals are uniform and in 52% of the cases when signals are non-uniform, compared to 94% and 64% in the equal network. The following ordered logit regression shows that compared to the empty treatment, Ordered probit regression Number of obs = 800 Wald chi2(2) =. Prob > chi2 =. Log pseudolikelihood = -512.77107 Pseudo R2 = 0.1173 there are significant treatment effects for the star network and effects at the edge of being significant for the unequal network. Performing several econometric robustness checks we consistently find that the star network leads to significantly lower informational efficiency than both the equal network and the empty network. These are two separate findings: First, informational efficiency is decreasing with the inequality of the network structure. Indeed, the unequal network always ranks between the equal and the star network when efficiency is measured. Second, informational efficiency is lower in an unequal communication structure than in the benchmark setting where communication is impossible. 7

References Battaglini, M., R. B. Morton, and T. R. Palfrey (2010): The swing voter s curse in the laboratory, The Review of Economic Studies, 77(1), 61 89. De Caritat, Marie Jean Antoine Nicolas, M. d. C. (1785): Essai sur l application de l analyse à la probabilité des décisions rendues à la pluralité des voix. L imprimerie royale. Feddersen, T., and W. Pesendorfer (1996): The swing voter s curse, American Economic Review, pp. 408 424. (1997): Voting behavior and information aggregation in elections with private information, Econometrica, pp. 1029 1058. (1998): Convicting the innocent: The inferiority of unanimous jury verdicts under strategic voting, American Political Science Review, pp. 23 35. Gerardi, D., and L. Yariv (2007): Deliberative voting, Journal of Economic Theory, 134(1), 317 338. Goeree, J. K., and L. Yariv (2011): An experimental study of collective deliberation, Econometrica, 79(3), 893 921. Morton, R. B., and J.-R. Tyran (2011): Let the experts decide? Asymmetric information, abstention, and coordination in standing committees, Games and Economic Behavior, 72(2), 485 509. 8