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1 4. Voter Turnout

2 Paradox of Voting So far we have assumed that all individuals will participate in the election and vote for their most preferred option irrespective of: the probability of being pivotal and the cost associated with voting. This appears, however, rather peculiar from a cost-benefit perspective.

3 Paradox of Voting Following the hypothesis of utility-maximizing individuals, a rational voter i will participate in election e if with p e i B e i > C e i (3.1) p e i : probability that voter i s vote is pivotal B e i : benefit from implementing voter i s favorite policy C e i : cost of voting If we assume that pi e is small, a rational voter will most likely abstain form voting.

4 Paradox of Voting Wahlbeteiligung nach Bundesländern In Prozent der Wahlberechtigten, Deutschland Prozent Bayern Baden-Württemberg Niedersachsen Schleswig-Holstein Nordrhein-Westfalen Hessen Rheinland-Pfalz Saarland Hamburg Deutschland Gesamt Bremen Berlin Thüringen Brandenburg Sachsen Mecklenburg-Vorpom. Sachsen-Anhalt Quelle: Bundeswahlleiter Stand: Jahr 2005 Bundeszentrale für politische Bildung Figure 4.1 Figure 15: Voter turnout in Germany by state

5 Paradox of Voting Wahlbeteiligung nach Altersgruppen In Prozent der Wahlberechtigten bei Bundestagswahlen, Deutschland Prozent Gesamt und älter unter * 1957* Jahr** Quelle: Bundeswahlleiter * 1957: ohne Saarland; 1953: ohne Rheinland-Pfalz, Bayern, Saarland 2005 Bundeszentrale Stand: ** 1994 und 1998: keine Angaben wegen Aussetzung der Wahlstatistik für politische Bildung Figure 4.2 Figure 16: Voter turnout in Germany by age

6 Paradox of Voting The voter turnout in many elections is in contrast to this prediction, however, significant. Downs (1957) names this contradiction the Paradox of Voting.

7 Paradox of Voting Ways to solve the paradox Introduce a direct consumption benefit of voting D e i Compliance with the civic duty to vote yields direct benefit The explanatory power of the exogenously imposed benefit is, however, rather limited (no testable prediction) The puzzle why people vote remains. Remark: Expressive Voting: Voting itself yields an direct consumption benefit (D e i ). Instrumental Voting: Voting as an instrument to implement the most preferred policy.

8 Paradox of Voting In the sequel, we discuss three different groups of models which try to explain why people vote. 1 Pivotal Voter Models: give a micro-foundation of the probability of being pivotal. 2 Ethical Voter Models: endogenize the direct benefit component D e i. 3 Uncertain Voter Models: endogenize the cost of voting C e i.

9 Pivotal Voter Model Idea: Probability of being pivotal may still be significant and thus the turnout may be positive. In particular, the model predicts that turnout is higher: the smaller the electorate, and the more close the election outcome is. The subsequently presented formal reasoning is based on the review article by Merlo (2005). The complete model and derivation of the respective equations can be found in the article.

10 Pivotal Voter Model Assumptions and model ingredients: A1 Two policy platforms A and B which divide the electorate in two groups, favoring either A or B. A2 Each individual knows his/her most preferred option, but can only infer that any other individual will favor A over B (or vice versa) with probability 0.5. A3 The size of the electorate is N which is finite (an infinite number of voters would reduce p e i to 0). A4 The utility when the most preferred option wins is 1, while it is -1 when the other option wins. A5 The cost of voting Ci e is independently and uniformly distributed on the interval [0,1].

11 Pivotal Voter Model Refereing to eq. (3.1) we can define a cut-off level C. All individuals with a cost below this level will participate in the election while the remaining individuals will abstain. To characterize individual voting behavior, we proceed as follows: 1 We determine the probability that s individuals indeed find it optimal to vote. 2 Then we derive the conditional probability that i is pivotal given that s individuals vote. 3 Combining the first two items, we compute the unconditional probability that individual i will be pivotal. 4 Finally, we characterize the induced cut-off level C.

12 Ad 1: The probability that s individuals vote Each individual will vote, if his/her individual cost of voting are below the threshold C. So, the ex-ante probability that with which any individual votes is given by ϑ = PrC e i C (3.2) Since we assume that voting costs are independently and identically distributed according to a uniform distribution on the support [0;1], we know that ϑ = PrC e i C = C (3.2 )

13 Ad 1: The probability that s individuals vote Therefore, the probability that s individuals vote is given by: ( ) N 1 pi e (s) = (C ) s (1 C ) N 1 s. (3.3) s The citizen s decision to vote or not to vote follows a binomial distribution as the decision is discrete with two realizations. Moreover, the cost draws are independent and the "success" probability (that citizen vote) is constant and equal to C.

14 Binomial Distribution The binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Example: Assume 5% of a very large population to be green-eyed. You pick 100 people randomly. The number of green-eyed people you pick is a random variable K which follows a binomial distribution with n = 100 and p = The probability of getting exactly K successes in n trials is: ( ) n p(k) = (p) k (1 p) n ks. k

15 Ad 2: The conditional probability of i being pivotal Voter i is only pivotal, if (s)he is able to alter the outcome of the election. This is the case if the preferred alternative of i is behind by one vote the pivotal voter can induce a tie both alternatives have gained the same number of votes the pivotal voter can induce his/her preferred alternative to win. In either case the citizens i s vote increases the expected utility by 1. if s 1 and s is odd, citizens i is pivotal if s 1 2 citizen vote for the preferred alternative.

16 So, we have: two discrete events: alternative A or B s independent draws each of which yields "success" with probability 0.5 the event that out of s voters s 1 2 will support option A follows a binomial distribution and the conditional probability that voter i is pivotal, given that s citizens vote, states: π e i (s) = ( ) s s 1 (0.5) s 1 2 (0.5) s s 1 2. (3.4) 2

17 Ad 3: The probability of i being pivotal If two events are independent, their joint probability is the product of the prior probabilities of each event occurring by itself: P(A B) = P(A) P(B). Thus, the unconditional probability of being pivotal is: pi e (s) = ( ) N 1 s = 0 N 1 (C ) s (1 C ) N 1 s πi e (s). (3.5) s p e i is lower the higher the cut-off level C. Intuition: A rise in C increases the probability that many voters are around; an event which is unfavorable for the prospect that voter i is pivotal.

18 The induced cut-off level The last issue which needs to be checked is whether there exists a C [0, 1], which induces individuals to vote if C e i < C. For an individual with cost realization C it must hold that p e i B e i C = 0 p e i C = 0 since the benefit of being pivotal is B e i = 1. As we know, that p e i = 1 if C = 0 and p e i = 0 if C = 1, we know by the intermediate value theorem that there exists a value C which is interior to [0,1].

19 Empirical Relevance: Pivotal Voter Model The theory can in principle explain a positive turnout (when C > 0), but the questions is whether the predicted value conforms with values observed in the data. Coate et al. (2004) take the model to a test: They use data on local referenda in the state of Texas. In doing so, they divide the data set in referenda held in large and small jurisdictions and estimate voting turnout for the sub-sample of small jurisdictions (for which the model is most suited). The result is that turnout can be quite well predicted. However, the model yields an upward bias in the predicted closeness of the election outcome.

20 Empirical Relevance: Pivotal Voter Model Figure 17: Average Turnout as Percentage of Eligible Voters

21 Empirical Relevance: Pivotal Voter Model Figure 18: Distribution over Closeness Voter Turnout in Germany by Age

22 Ethical Voter Model Idea: Elections typically divide the electorate in two groups (supporters and opposers) which triggers a kind of contest between the two groups. Members of both groups may thus base their voting decision on what they consider best for promoting the success of their group; i.e. they want to do their part to help their group win. They are inclined to follow a voting rule that, provided everybody adheres to it, maximizes the chance that the group wins. Group solidarity may be due to a common ethnic identity or people feel similarly strongly about the political issues involved (e.g. abortion policy).

23 Ethical Voter Model To formally illustrate the reasoning, we present a simple model based on Merlo (2005). The assumptions are the same as for the pivotal voter model except of: A2 The fraction of the population which supports option A is µ which is uniformly distributed on [0,1]. A3 There is a continuum of individuals. The size of the population is normalized at unity. The assumption excludes the possibility that individuals are pivotal (no instrumental voting).

24 Ethical Voter Model The cut-off levels for group A and B are denoted by CA and CB. They are chosen such as to maximize the overall expected utility of the group members. For a given fraction of individuals in favor of A, group A wins, if the number of supporters with a cost below CA (being equal to CA times the fraction of supporters µ) exceeds the corresponding value for the other group; µca > (1 µ)c B µ > C B /(C A + C B ). (3.6)

25 Ethical Voter Model The assumed cost of voting for any group A member is C A 0 CdC = 0.5(C A )2. (3.6) For a given µ, the conditional expected overall cost of voting is 0.5µ(CA )2 and the unconditional expected overall cost reads: 1 0.5µ(CA )2 dµ. (3.7) 0 In contrast, the conditional group benefit is dependent on the outcome of the election. For group A, it is µ(1) in case of victory and µ( 1) otherwise.

26 Ethical Voter Model Taking expectations over µ, the overall expected benefit for group A becomes: 0 C B C A +C B Thus, the overall expected payoff is: 0 C B C A +C B 1 µdµ + µdµ. (3.8) C B C A +C B 1 µdµ + µdµ C B C A +C B (C A )2 µdµ =

27 Empirical Relevance: Ethical Voter Model = 0 C B C A +C B 1 µ[ 1 0.5(CA )2 ]dµ + µ[1 0.5(C C B A )2 ]dµ C A +C B Differentiating the term for group A with respect to C A and solving the two FOCs for the values yields: CA = C B 0.71(Merlo, 2005). Here turnout is positive and might be significant. Note, the cost realizations are independent of the assignment to either group.

28 Empirical Relevance: Ethical Voter Model The theory relies on the assumption that each individual follows the voting rule which maximizes group welfare. The theory thus disregards free-riding behavior! Besides evaluating the plausibility of the model, another more important evaluation of the theory is to analyze the extent to which the theory can explain turnout observed in the data. Coate and Conlin (2004) use the same data set as for evaluating the pivotal voter model in Coate et al. (2004). As shown in figure 4.5, they find a significant share of voter turnout to be explained by the model. However, there is a tendency to over-predict turnout with low actual turnout rates and vice versa.

29 Empirical Relevance: Pivotal Voter Model Figure 19: Actual versus Predicted Turnout

30 Uncertain Voter Model Idea: The uncertain voter model assumes that the cost of voting is zero (Ci e = 0), but that individuals have different information about the nature of the optimal option. The lack of perfect information may lead them to make mistakes which are costly from each voter s perspective. It is in this way that voting becomes costly. To shed some light on the economics of voter participation, we start with a simple example.

31 Uncertain Voter Model Example 1: Assume there are two options, A and B, on the table and three individuals. In state A of the world all individuals prefer A to B. The state A occurs with probability µ = 0.9. In state B of the world all voters prefer B to A. The benefit is 1 when the most preferred option wins and -1 otherwise. One individual is perfectly informed about the true state of the world; A fact, which is known by the two other, uninformed individuals. In this situation the (perfect Bayesian-Nash) equilibrium involves that the informed individual will always vote, while the uninformed individuals will abstain.

32 Uncertain Voter Model Given that the uninformed individuals will abstain, it is optimal for informed individual to vote and to induce his/her most preferred outcome (assume that if nobody votes, either option is implemented with probability 0.5). Given that the informed individual votes, an uninformed individual cannot induce a higher utility by voting. When voting for A, (s)he wouldn t change the outcome in state A, but would induce a tie in state B and, thereby, produce the "wrong" outcome with probability 0.5. Equivalently, when voting for B, (s)he wouldn t change the outcome in state B, but would induce a tie in state A (costly!). Thus, abstention by uninformed individuals and voting by the informed individual is an equilibrium.

33 Uncertain Voter Model Conclusion of Example 1 The finding that all uninformed voters "delegate" the voting decision to the informed voter presupposes that the voters preferences are aligned. Delegation does not incur a cost on the uninformed voters. Voting incentives might be different if there are voters around which would vote, but not for the option the uninformed citizens prefer. To see this more clearly, we modify the example as follows.

34 Uncertain Voter Model Example 2: There is a fourth voter who has a partisan preference, i.e. (s)he will get a positive benefit from option A in both states of the world and no benefit from option B. In this situation it becomes optimal that one uninformed individual votes for option B, while the other abstains. Both the partisan and the informed individual will vote as well. Why? The informed individual will induce a tie in both states of the world when abstaining, while guaranteeing the "correct" outcome with probability 1 when voting. When the uninformed voter opts for B, (s)he will not change the election outcome in state B, but will induce a victory of B in state B.

35 Uncertain Voter Model Differently, when the uninformed non-voter opts for A, then (s)he will not change the outcome in state A, but will induce a tie relative to a victory for option B in state B (costly!!). Similarly, when opting for B, (s)he will induce a tie in state A (rather than a victory for option A) and no change in state B (also costly). The partisan individual can move the outcome in his/her most preferred direction in state A and is not influential in state B.

36 Uncertain Voter Model Conclusion of Example 2 The last example shows that uninformed, independent (i.e. non-partisan) individuals may vote and, if yes, they do so to neutralize the partisan bias. They may vote for option B, although it is less likely to be the "correct" voting decision (µ = 0.9). The result extends to more general models; i.e. the higher the fraction of partisan voters favoring option A, the higher the fraction of independent individuals who vote, and the more inclined are independent voters to opt for alternative B.

37 Uncertain Voter Model What is the prediction for voter turnout? The pivotal and ethical voter model predict a higher turnout when the race becomes more close. Such a prediction may not arise in the current model. It may be just in the situation when (independent and uninformed) voters are pivotal that they will abstain. They rationally anticipate that they may decide the election in the "wrong" direction. The phenomenon is named Swing voter s curse (in analogy to the winner s curse in auctions) - see Feddersen and Pesendorfer (2004): The latter says that if bidders get independent signals about the value of the item which is auctioned off (with the possibility that the signal is wrong as it gives a too high value), the winner may be worse off (relative to not winning).

38 Empirical Relevance: Uncertain Voter Model The model emphasizes the role of information for the decision to vote. This can rarely be observed in field data (unless there is a natural experiment where one group of voters is randomly chosen and gets more information about the alternatives under consideration). A different strategy is to resort to experimental analysis where the information level of voters and the number of partisan voters can be more accurately controlled.

39 Empirical Relevance: Uncertain Voter Model In fact, Battaglini et al. (2005) conduct a laboratory experiment. Therein, each lab session lasts 30 periods. After 10 periods the number of partisan voters, m, is changed and announced to the participants - see Figure 4.6 for a description of the different sessions, where p is the probability that state a occurs.

40 Empirical Relevance: Uncertain Voter Model Figure 4.7 gives a summary of how the uninformed voters respond to the adjustment in the number of partisan voters. In line with the theory, less individuals abstain and more vote for option b (not preferred by the partisan), the higher the number of partisans.

41 Figure 22: Experimental Outcome 2 Empirical Relevance: Uncertain Voter Model The "time-profile" of the voting decisions of uninformed individuals is depicted in figure 4.8 where the solid line gives the theoretical prediction w.r.t. the probability of voting for a and b. Following the change in m, the probability to vote for b is "correctly" adjusted, while the probability to vote for a remains at a low level.

42 References Battaglini, M, R. Morton, and T. Palfrey (2005), The Swing Voter s Curse in the Laboratory, CEPR Discussion Paper No Borgers, T. (2004), Costly Voting, American Economic Review, 94, Coate, S. and M. Conlin (2004), A Group Rule-Utilitarian Approach to Voter Turnout: Theory and Evidence, American Economic Review, 94, Coate, S, M. Conlin and A. Moro (2004), The Performance of the Pivotal-Voter Model in Small-Scale Elections: Evidence from Texas Liquor Referenda, mimeo. Downs, A. (1957), An Economic Theory of Democracy, New York: Harper Collins. Feddersen, T. and W. Pesendorfer (1996), The Swing Voter s Curse, American Economic Review, 86, Merlo, A. (2005), Whither Political Economy? Theories, Facts and Issues, PIER Working Paper No

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