On the Extent of Strategic Voting Chapter 2

6.2. Reduced Form Results Focusing on candidates of the five major parties, Table 5 shows results from estimating equation (4) by ordinary least squares. To allow for arbitrary forms of autocorrelation in the residuals as well as for correlation within and across districts, standard errors are clustered by state. Going from the left to the right the set of fixed e ects grows steadily. The most inclusive specification contains candidate-year fixed e ects, thus controlling nonparametrically for the appeal of individual candidates, as well as that of their competitors. The structural break in this specification is estimated to occur at 6.4%, implying that voters treat “close” third ranked contestants as if they are in the race.25 More to the point, the first row of Table 5 presents estimates of the share of voters who stick with their preferred candidate despite her having no chance of winning. Estimates of  range from .657 to .724, and are fairly precise. Taken at face value, these numbers indicate that about two thirds of all voters are sincere, whereas the remaining third abandons their favorite contestant whenever she is “out of the race.” The crucial question, however, is whether voters do so for strategic reasons. Strictly speaking, any model would be consistent with the evidence in Table 5 as long as it predicts the candidate-list vote gradient for non-contenders to be smaller than one. For instance, some individuals may simply vote for whichever candidate advertises the most, and advertising expenditures might be highly correlated with who remains in contention for victory. It would, therefore, appear as if voters abandon weak candidates, despite the fact that no agent behaves tactically. In order rule out mechanical explanations of this kind, Table 6 compares the extent of strategic voting, i.e. 1  , across a number of dierent settings. The first set of results demonstrates that individuals’ tendency to desert weak contestants depends on who remains in contention for victory. Voters are more than three times as likely to abandon their favorite candidate when that of an allied party is still “in the race” than when faced with the choice among two evils.26 Moreover, distinguishing between races that were “close” and those that were not, strategic voting appears to have been somewhat more prevalent in the former–although the dierence is not statistically significant; and disaggregating the data by election year shows that desertion of non-contenders was more common in 2005 than in 2009. The 2005 election followed 25 Bootstrapping suggests that  is statistically dierent from zero. The following parties are defined as close substitutes: CDU and FDP, SPD and Green Party, The Left and SPD. The dierence in Table 6 is even starker when supporters of The Left are assumed to have no substitute available. 26 17 a failed motion of confidence that triggered the dissolution of the Bundestag and was widely perceived to be a “critical election” (Korte 2009).27 It is, therefore, hardly surprising that a greater number of voters behaved tactically when the stakes were particularly high.28 Taken together these findings suggest that it might be more appropriate to consider strategic behavior a conscious decision as opposed to an agent’s type. That is, all agents may be capable of voting tactically, but only for a subset of them do the subjective benefits outweigh the (psychic) costs. More importantly, the evidence in Table 6 is at odds with many alternative theories for why voters abandon candidates who are “out of the race.” Any model in which voters defect candidates for non-strategic reasons would not only have to predict a correlation between desertion rates and a contestant’s chance of winning, but it would also have to explain why defection is more common when the stakes are higher, and why it depends on who remains in contention for victory. The patterns in Table 6 as well as the fact that voters who do cast split-tickets substitute toward candidates of a potential coalition partner (cf. Table 1) suggest that the observed behavior is, in fact, driven by strategic considerations.29 In order to strengthen this claim, Table 7 provides an explicit test of the pivotal voter model against the alternative that individuals receive utility from voting for the eventual winner of the election. The entries correspond to the candidate-list vote gradient among second ranked candidates, by distance to the first ranked one. If agents did, indeed, have a taste for supporting the likely winner, then one would expect runner-ups to be abandoned as well, especially those that trail far behind. By contrast, the model in Section 3 predicts that voters do not abandon the runner-up, even if her chances of winning are very small. This is because if a race were to be tied–however unlikely that may be–the tie would almost certainly involve the second ranked candidate, in which case voting for her would change the outcome of the election. Although counterintuitive, the evidence in Table 7 does support the 27 Campaigning to stay in oce, Chancellor Schröder and his SPD-Green coalition promised to undo some of their unpopular labor market and welfare reforms, while raising taxes on the rich. In stark contrast, led by Angela Merkel, the conservative-libertarian bloc sought to further increase the pace and scope of deregulation, slashing income taxes and public spending in the process. 28 Ocial statistics analogous to those in Table 1 show a substantially higher fraction of split tickets in 2005 (cf. Bundeswahlleiter 2006, 2010). Although abstention was approximately 7 percentage points lower than in 2009, the dierence in turnout alone is too small to account for the finding in Table 6. 29 Ancillary results (available from the author upon request) demonstrate that these substitution patters are, in fact, driven by expextations regarding post-election coalition formation. The empirical approach exploits the small, libertarian FDP’s position as “kingmaker” in much of the post-World War II period. From the 1960s until the advent of the Green Party in the 1980s the FDP was the only other major party represented in parliament and participated in every government from 1969 to 1998. Crucially, it switched allegiances halfway through this era. Although parties’ positions in policy space remained almost unchanged, after the end of the SPD-FDP coalition in 1982, a substantial number of FDP adherents switched from voting for SPD candidates to supporting those of its new coalition partner, the CDU. 18 pivotal voter model. 6.3. Sensitivity and Robustness Checks For the results above to correctly identify the shares of each type of voter it must be the case that the regressors are uncorrelated with the error term. One obvious source of bias may be systematic misclassification of contenders. To ameliorate this concern, Table 8 presents estimates of the share of strategic agents employing a number of alternative definitions (listed in the column on the left). For each definition, the table shows two estimates: one based on candidate fixed eects, and another one using candidate-year fixed eects. For comparison, the top row displays the main results from Table 5. Although individual point estimates do, of course, vary, the majority of them are fairly close to their baseline values. In all cases overlap the 95%-confidence intervals substantially with those in Table 5. On the whole, the evidence suggests that misclassification of contestants is not a serious problem. However, there are at least two other threats to identification. First, the argument laid out in the introduction assumes that the share of agents who desert non-contenders does not depend on precincts’ ideological composition (i.e. the position on the x-axis in Figure 1). This assumption is not innocuous. For instance, it rules out peer eects in voting, certain forms of sorting across precincts, or that the perception of candidate quality varies systematically with voters’ political views. Fortunately, the assumption is testable. To see this, note that all of the threats mentioned above imply a non-linear relationship between list and candidate votes for non-contenders. That is, if social interactions or dierences in candidate perception were important, then one would expect desertion rates (and hence the slope of the line in Figure 1) to vary systematically with the position on the x-axis. Yet, estimating  separately for each candidate and regressing it on her own party’s vote share in the same district yields no indication that voters are more or less likely to abandon a contestant when they live around others with similar political views. In fact, for noncontenders the correlation between k and v Lk,d,t is almost exactly zero. Another, perhaps more powerful test would be to estimate the relationship between list and candidate votes nonparametrically. As shown in Figure 5, there is little reason to believe that the linearity assumption is violated. The other concern relates to the behavior of political parties. For instance, whether parties nominate a particularly “good” or “bad” candidate might depend on the anticipated likelihood of winning the district. Since the empirical strategy relies on within-candidate variation, this sort of behavior poses a problem (only) if candidate quality interacts with the share of voters who behave sincerely–say, because voters might be reluctant to abandon 19 very charismatic contestants.30 Although plausible, the data do not suggest that “good” candidates (as measured by k,t ) are less likely to be deserted when they are “out of the race.” If anything, estimates of the covariance between k,t and a non-contender’s k are slightly negative. Table 9 performs a series of additional robustness checks. Again, the results do not depend on the weighting scheme or on whether one also includes candidates of “micro parties.” However, two additional pieces of evidence ought to be pointed out explicitly. As discussed in Section 4, a small numbers of voters might anticipate overhang mandates, and thus cast tactical list votes. Although one would surely want to classify these individuals as being strategic, it useful to probe the robustness of the results with respect to this potential confound. The fourth row in Table 9 restricts attention to states in which overhang mandates had never occurred before 2005. Reassuringly, the resulting point estimates are very similar to their baseline values. Since list votes are hardly a perfect proxy for preferences over candidates, one may also worry about measurement error. The resulting attenuation bias would lead to estimates of  that are smaller than one and, therefore, to incorrect inference about the prevalence of tactical behavior. Note, however, measurement error should not only aect the estimated candidatelist vote gradient among non-contenders, but also that for contenders. Thus, looking at the dierence in slopes for both sets of contestants, i.e.   , provides a way to address this concern. As the fifth row in Table 9 demonstrates, attenuation bias cannot the explain the results. In fact, the estimated extent of strategic voting remains almost unchanged. 6.4. Correlates of Strategic Voting Taken together the results above show that a non-trivial fraction of agents behaves strategically. Yet, simple averages may conceal considerable heterogeneity across individuals. Since strategic voters are more likely to be pivotal and thus exert a disproportionate influence on the positioning of political candidates, it is also important to understand who votes tactically. In order to infer whether  varies with the characteristics of voters, the present paper relies on ocial statistics for the universe of German cities and villages, published by the Federal Statistical Oce and the statistical oces of the Länder (Statistische Ämter des Bundes und der Länder 2007, 2011).31 After aggregating election results to the village level and focusing on the set of municipalities that are fully contained within an electoral district, 30 If this were the case, then the estimates above would simply describe voters’ behavior in the current equilibrium, conditional on the actions of parties and candidates. 31 Unfortunately, comparable data for polling precincts do not exist. Polling precincts are too small to produce reliable estimates from existing data sets. 20 it is straightforward to estimate specifications that allow for  to increase or decrease in some village characteristic. Table 10 displays the results. The first column demonstrates that aggregation to the municipality level does not aect the estimated share of sincere voters. The remaining four columns examine how  changes with population density, income tax revenue per capita, as well as the gender and age composition of the electorate. For ease of interpretation covariates have been demeaned, so that the estimates in the second row refer to the share of sincere voters at the sample average.32 Interestingly, urban voters are not more strategic than rural ones; nor is there a significant gender gap–although lack of variation in the composition of villages makes the latter conclusion somewhat more speculative. There are, however, important dierences with respect to socio-economic status (as proxied by income tax revenue per capita) as well as age. Since the income tax variable captures only revenues that accrue to the respective municipalities and given that the German tax system is highly non-linear, it is easiest to judge the magnitude of the coecient by an example. Consider two villages: one’s per capita income tax revenue is a standard deviation below the mean, while that of the other village is one standard deviation above the sample mean.33 The share of sincere voters is estimated to be almost 6 percentage points higher in the latter. Disparities by age are even larger. Taken at face value the coecient in column (5) suggests that the fraction of strategic agents is 24.4 percentage points lower among voters below the age of 30.34 Of course, this estimate is based on limited variation, and therefore not very precise. But, together with the results in column (4), it suggests that sophistication and learning aect tactical behavior. 6.5. Strategic Voting and Democratic Experience In order to further investigate the hypothesis of learning, this subsection uses the German Reunification as a natural experiment. Although the German Democratic Republic (GDR) held regular, formal elections to the Volkskammer (People’s Chamber), they were eectively meaningless. East Germans could only choose from candidates on a single list controlled by the Socialist Unity Party (SED), and it was customary to cast one’s ballot in public, simply accepting all nominated candidates. Not surprisingly, ocial approval rates often exceeded 99%.35 Free, democratic elec32 Estimating models that are fully interacted yields qualitatively very similar results. On the average municipalities receive about 13% of all income tax revenues. Thus, the per capita sample mean is 260 EUR and the standard deviation equals 110 EUR. 34 The legal voting age on the federal level is 18. 35 Accusations of election fraud by the SED regime are widespread. For actual evidence see Herz and Rothe 33 21 tions were only held on March 18, 1990–after months of peaceful political protest. The newly elected government then negotiated the end of the GDR. In stark contrast, citizens of the Federal Republic of Germany (FRG) have had the opportunity to participate in free elections since 1949; and from 1953 on under a two-ballot system almost identical to the current one. Thus, they had more than 40 years of democratic experience by the time the GDR joined the West. The first parliamentary elections in unified Germany were held on December 2, 1990, and were subject to (essentially) the same rules that had previously been used in the West and that continue to be in place today.36 If learning and familiarity with the electoral system do indeed matter, then one would expect large initial dierences in the share of strategic voters, which should disappear over time. This prediction corresponds exactly to the findings in Figure 6. For each election since 1990, the figure plots the estimated dierence in the share of tactical voters between East and West Germany. Negative values indicate fewer strategic agents among residents of the former GDR.37 Just two months after reunification, East Germans were almost 18 percentage points less likely to vote strategically than their Western counterparts. By 2005, however, the gap had practically vanished. Although none of the point estimates is very precise, one can nevertheless reject the null hypothesis of a constant dierence at the 1%-confidence level. Moreover, both the initial gap as well as the speed of convergence are in line with the “age eect” in Table 9. Thus, as in many laboratory experiments, sophistication and learning appear to be important determinants of tactical behavior.38 7. Structural Analysis and Counterfactual Seat Distributions Although the reduced form results provide strong evidence of strategic voting, they are subject to a number of limitations. For instance, the assumption that candidate quality enters (3) linearly might be overly restrictive. Taken literally, linearity could lead to predicted vote (2004). 36 The most important exception was that the 5%-threshold applied separately to East and West Germany. Thus, in 1990 a party had to gain more than 5% of the list vote in only one of the two regions to enter the Bundestag. 37 The specification on which the estimates are based is analogous to equation (4), but allows for dierent slopes and cuto values in East and West Germany. A qualitatively similar picture emerges if one were to restrict the cuto to be the same in both regions. 38 While the evidence is fully consistent with this interpretation, the findings in Figure 6 should be taken with a grain of salt. During this time period, East Germans gained not only familiarity with the electoral system, but other economic factors changed as well. If these changes had an independent eect on the propensity to cast tactical ballots, then the estimates in Figure 6 need not capture the true impact of learning. The convergence in income per capita, however, is almost an order of magnitude too small to explain the results. 22 shares that are greater than one or even negative. Moreover, by themselves the ordinary least squares estimates do not shed any light on which candidates strategic agents actually vote for (after deserting non-contenders). But, knowledge of substitution patterns is essential to construct appropriate counterfactuals by which to judge whether misrepresentation of preferences is of first order importance for electoral outcomes. To properly account for the drawbacks of the reduced form analysis and to assess the impact of strategic behavior, this section estimates a structural model of voting decisions in the 2009 federal election. Again, list votes provide a crucial source of identifying variation.39 7.1. Adding Structure The theoretical model in Section 3 shows that voters will generally group contestants into two categories: candidates who are “in the race” and those who are not. It is, therefore, natural to model agents’ decisions as a discrete choice problem in which sincere and strategic voters face dierent equilibrium choice sets. The former choose among all contestant in a particular district, whereas the latter consider only candidates who are in contention for victory. When it comes to the list vote, however, all voters pick from the set of major parties. In order to represent agents’ preference profiles in a tractable yet flexible fashion, assume that individual i receives utility (5) uLi,p = p,m + i,p + i,p from voting for party p’s list. Here, p,m denotes the average utility that agents living in municipality m derive from voting for p, and i,p are individual specific deviations from the mean. i,p is an i.i.d. type-I extreme value (T1EV) taste shock. Moreover, define the underlying utility from casting one’s candidate vote for the nominee of party p to equal (6) uC i,k = p,m + i,p + k + "i,k , where k indexes candidates, and k is voters’ assessment of k relative to that of her party. "i,k denotes another i.i.d. T1EV shock.40 It is critical to note that p,m and i,p appear in both (5) and (6), implying that ocial party positions influence not only voters’ perceptions of the respective organizations, but 39 Results are qualitatively and quantitatively similar when looking at the 2005 election instead. The mean utility from abstaining is normalized to zero. Since the available data do not allow turnout to be calculated for individual precincts, the analysis in this section is conducted at the municipality level instead. 40 23 also that with respect to their candidates. This assumption captures the fact that German politicians campaign heavily on their own party’s platform and it introduces the correlation between list and candidate votes that has been the identifying source of variation above. To allow individuals’ preferences to systematically deviate from the average in their municipality, i,p is assumed to follow a multivariate normal distribution with an unrestricted covariance matrix. That is, (i,p )p2P  N (0, ).41 Hence, supporters of the conservative CDU may, for example, also have a taste for the FDP, while holding more negative views of the communist Left. While p,m and i,p model commonalities in voters’ assessments of parties and the respective contestants, i,p and "i,k admit dierences in tastes that go beyond the common perception of candidate quality, i.e. k . The T1EV assumption is convenient because it results in a smooth closed form representation of individual choice probabilities (see, e.g., McFadden 1974 or Train 2009 for derivations). Given the structure of preferences, party p’s expected share of the list vote in municipality m equals (7) L vbp,m = Z exp (p,m + i,p ) d () , 1 + p0 2P exp (p0 ,m + i,p0 ) P L where P denotes the set of electable parties. Note that vbp,m does not depend on the share of strategic voters. After all, even tactical agents have an ex ante incentive to cast truthful list votes. This is not true when it comes to the candidate vote. The candidate vote is a mixture of sincere and strategic ballots: C,S C,T C vbk,m = vbk,m + (1  ) vbk,m . C,S C,T Here, vbk,m denotes candidate k’s share among sincere voters, and vbk,m that among tactical ones. As before,  is the fraction of agents who are sincere. C,S Since sincere voters consider every candidate, vbk,m is given by (8) C,S vbk,m = Z 1+ exp (p,m + i,p + k ) d () , k0 2K(d) exp (p0 ,m + i,p0 + k0 ) P where K(d) marks the set of all contestants in district d. Tactical agents, however, behave as if they are choosing only among the set of serious contenders, C(d). That is, irrespective 41 As the variance of i,p is determined by the distribution of the logit error, it is not necessary to impose a normalization. 24 of the underlying utility in (6), strategic voters disregard all candidates that are not “in the race.” Consequently, k’s share among strategic individuals equals C,T vbk,m = (9) 8 Z > < > : 1+ P exp(p,m +i,p +k ) d () exp(p0 ,m +i,p0 +k0 ) k0 2C(d) 0 if k 2 C(d) . otherwise A seemingly natural way to estimate (, , , ) would be to find the parameter combination that produces the best fit between predicted vote shares and the data. This, however, entails that preferences would be partially identified from candidate votes, which may confound strategic desertion with simple distaste. In order to avoid this problem, electorates’ average tastes should be inferred solely from list votes. Accordingly, with data on C(d) and actual vote shares in hand, estimates of (, , , ) could be obtained by minimizing the objective function: (10)   SSR , , , |v C , v L = X X  X d2D m2M (d) k2K(d) C C vbk,m  vk,m 2 subject to the set of constraints (11) L L vbp,m = vp,m 8p, m, d. Yet, as C(d) is not observed, it needs to be estimated as well. Based on the evidence of focal equilibria in Section 6.1, a candidate is assumed to be a contender if, and only if, her party trails the district’s second most popular one by less than  percentage points. Thus, estimating C(d) adds the following set of equilibrium constraints (12) v L,2 d nd (13) v L,2 d nd  v Lk,d   8k 2 C (d) , 8d  v Lk,d >  8k 2 / C (d) , 8d as well as the additional parameter .42 Given the granularity of the data, the optimization problem defined by equations (10)— (13) is extremely large. Finding the solution involves optimizing over more than 63,000 parameters, solving about 61,500 non-linear constraints, and approximating roughly 120,000 dierent five dimensional integrals. To keep the computational burden manageable without compromising the quality of the solution, this paper uses state-of-the-art numerical methods, such as integration on sparse-grids (Heiss and Winschel 2008) and mathematical program42 Experimentation with a subset of the contender classifcations in Table 8 yielded very similar results. 25 ming with equality constraints (Dube et al. 2012; Su and Judd 2012). For a description of these methods see Appendix D. Before proceeding to the results it is useful to provide some intuition on how the parameters are identified. Identification of p,m is straightforward. From Berry (1994) it follows that, for every , there exists a unique vector  which solves (11). Economically, this means that the list vote pins down the average taste in dierent markets, as desired. Akin to the analysis in the previous section, identification of  is based on the intuition in Figure 1. That is, the share of sincere voters can be inferred from the ratio of non-contenders’ observed vote shares (depicted on the y-axis) to those they would receive if all agents acted solely based on their preferences (proxied by the position on the x-axis). As the list vote determines mean preferences, structural and reduced form methods rely ultimately on very similar variation. Candidate quality, i.e. k , can be gleaned by comparing contestant’s actual performance in dierent municipalities with predictions thereof based on party preferences and . k will be positive for candidates whose vote shares systematically exceed their predicted values, and negative for those who underperform. Lastly,  is identified from the empirical covariance between non-contenders’ list and contenders’ candidate votes. Take, for instance, a district in which the FDP candidate is out of the race, while the nominee of the CDU is a contender. If the latter receives, on average, more votes in villages that have a greater taste for the FDP, then the respective parameter in the covariance matrix will be positive. Analogous arguments apply to the remaining elements of . 7.2. Results and Counterfactual Experiments With 26.3% (and a standard error of 7.8%) the estimated share of strategic voters, i.e. 1  , is strikingly close to the reduced form results in Tables 5 and 6. Unfortunately, few of the model’s other parameters are easily interpretable by themselves. Thus, instead of listing parameter estimates, the following discussion presents results in a way that relates straightforwardly to common intuition.43 In order to judge the model’s fit consider Figure 7. The upper two panels contrast the true marginal distributions of candidate and list votes (dark bars) with those predicted by the model (light bars). Given that (, , , , ) have been chosen to mimic these data, there are practically no discernible dierences. The lower panel depicts the frequency of valid list and candidate vote combinations. It is important to note that information on the joint distribution of votes come from an indepen43 A list of all estimates is available from the author upon request. 26 dent source (Bundeswahlleiter 2009a, 2010) and were not used to fit the model. Thus, the lower panel of Figure 7 provides a strong quasi-out-of-sample test of whether the estimation results are reasonable. Although there do exist dierences, on the whole the predicted distribution matches the qualitative features of its real world counterpart fairly well, lending credibility to the structural estimates. Table 11 compares actual and simulated outcomes of district level races. As can be seen from the entries on the diagonal, the model does an excellent job at ranking candidates. In particular, it predicts almost 95% of winners correctly. While Figure 7 and Table 11 are useful in evaluating the goodness of fit, a more interesting question might be for whom supporters of dierent parties would vote if their preferred candidate was out of the race. In order to shed light on the ordering of preferences, Table 12 shows the frequency with which voters would substitute toward the candidate of any other party, assuming that all but their preferred contestant were still in the race. Thus, the entries correspond to the probability of some other party’s candidate being “the next best choice.” The model predicts FDP adherents to substitute toward candidates of the CDU, whereas most supporters of the Green Party and The Left would choose SPD contenders instead. Given parties’ ideological positions, these patterns conform exactly to what one would expect. Based on the structural estimates, Figure 8 presents several counterfactual election results by which to judge the impact of strategic voting.44 The top left panel shows the actual distribution of seats in the Bundestag, whereas the panel on the right displays the distribution that would prevail if mandates were awarded based solely on a single vote counted under proportionality rule with a 5%-threshold, i.e. the list vote. Evidently, the current Bundestag mirrors a parliament formed under proportional representation fairly closely: all five major parties are represented, with more than 60% of seats accruing to the CDU and the SPD. In the current equilibrium, distortions introduced through strategic candidate votes are very small. The remaining two panels assume a single vote counted under plurality rule on the district level (akin to the candidate vote, or elections to the House of Representatives in the U.S.) The counterfactual on the bottom left shows the model’s predictions for such a first-pastthe-post scheme with 26.3% of voters behaving strategically. In the panel on the bottom right all voters are sincere. In line with common intuition, relative to proportional representation a “winner-take-all” system would result in dramatic losses for small parties. However, as comparing the panels on the right shows, these losses are due to the way dierent electoral rules map vote shares 44 For details on the computation of these counterfactuals see Appendix D. 27 into mandates and not to tactical behavior. The impact of strategic voting can be gleaned from comparing the two counterfactuals on the bottom. Given its estimated extent, tactical voting has only a modest eect on the overall allocation of seats. Not a single party’s share of seats would change by more than 5 percentage points, often substantially less. Although these results depend on the distribution of preferences, at first glance it would appear that lack of strategyproofness has only a modest eect on electoral outcomes. Yet, looking only at seat totals misses an important point. As the evidence in Table 13 indicates, compared to the current equilibrium, about one in ten districts would change hands if all voters were to cast truthful ballots. Thus, even moderate degrees of tactical voting may have a non-negligible impact on a given race–as predicted by theory. But as the number of heterogenous districts becomes large, distortions tend to average out. In fact, the eect of tactical behavior on aggregate seat totals will always be (weakly) lower than that on individual races. 8. Concluding Remarks Whether individuals vote strategically is one of the most important questions at the intersection of economics and political science. Although tactical voting has interested social scientists for decades, it has proven very dicult to estimate its extent. The fundamental problem is that voters’ preferences are generally unobserved, which makes it nearly impossible to compare them to actual votes without imposing strong assumptions. The present paper resolves this problem by exploiting the structure of parliamentary elections in Germany. The German system allows for preferences to be held fixed, while contrasting the behavior of the same individuals under dierent electoral regimes. Reduced form as well as structural estimates indicate that approximately 30% of voters behave as predicted by rational choice theory. That is, almost one third of voters abandons their most preferred candidate if she is believed to have no chance of winning. Interestingly, strategic voting is more common among individuals of high socio-economic status and those more familiar with the electoral rules, suggesting that sophistication and learning play an important role. Although strategic voting has a non-trivial impact on individual races, its eect on the overall distribution of seats is considerably more modest. No party’s seat total would change by more than a few percentage points if all voters behaved sincerely. Depending on the application it may, thus, be important to distinguish between the “micro” and “macro” eects of tactical behavior. With more than a dozen countries using electoral systems similar to the German one future research may follow the approach in this paper and construct comparable estimates for a 28 number of dierent settings. The findings from such a cross-country analysis could further enhance our understanding of tactical voting and shed light on the vulnerability of elections around the world. Appendix A: Proofs Proof of Proposition 1: To demonstrate existence of voting equilibria this proof adapts the fixed point argument in Myerson (1998, 2000). First, let ex ante beliefs about pivot probabilities be accurate and express expected utility as a function of voters’ strategies: bı (, ) = u X k2K 2 4 (k|ı) X  2Z(K) 0 @ Y k0 2K   0   1 3  k |µ k 0 | A p (k,  |ı)5 , with payos given by p (k,  |u, s) = uk and (14) p (k,  |u, t) = 8 > > < > > : 1 2 1 2 (uk  uk0 ) if 9k 0 2 K s.t.  (k 0 ) =  (k) ^  (k)   (k 00 ) 8k 00 2 K\ {k 0 } (uk  uk0 ) if 9k 0 2 K s.t.  (k 0 ) =  (k) + 1 ^  (k)   (k 00 ) 8k 00 2 K\ {k 0 } . 0 otherwise bı denotes the expected utility of a type ı 2 I voter who pursues strategy  2  (K) Here u I when everybody else plays  2  (K) , µ (k|)  n P u2U [ (k|u, s) + (1  )  (k|u, t)] f (u), and Z (K) is the set of all possible action profiles, i.e. the number of other players who choose I I actions ( (k))k2K .45 Next, define the best response correspondence B :  (K)   (K) such bı (, )  u bı ( 0 , ) 8 0 2  (K)}. With these dethat, for all , B = (Bı ())ı2I , where Bı ()  {|u finitions in hand it suces to show that there exists a set of mixed strategies for which  2 B (). The remainder of the proof verifies that the conditions for Kakutani’s Fixed Point Theorem are satisfied. I By definition of  (K) as k-dimensional simplex,  (K) is a non-empty, compact, and convex subset of a finite dimensional Euclidean space. I b = (u bı )ı2I is continuous, it follows from the definition of B, compactness of  (K) , and the As u Weierstrass Extreme Value Theorem that B () is non-empty. 45 Note that (14) could have alternatively been expressed as 8 uk if  (k)   (k 0 ) + 1 8k 0 2 K\ {k} > > < 1 0 0 00 00 0 0 2 (uk  uk ) if 9k 2 K s.t.  (k ) =  (k) ^  (k)   (k ) 8k 2 K\ {k } p (k,  |u, t) = . 1 0 0 00 00 (uk  uk0 ) if 9k 2 K s.t.  (k ) =  (k) + 1 ^  (k)   (k ) 8k 2 K\ {k 0 } > > : 2 0 0 0 00 00 uk 0 if 9k 2 K s.t.  (k ) >  (k) + 1 ^  (k )   (k ) 8k 2 K In (14) strategic voters care only about influencing the outcome of the election, whereas in the formulation above they derive utility from whichever candidate wins the elections. Since payos dier only by a constant, the two specifications are equivalent. 29 Moreover, B () is convex-valued. To see this note that if  0 ,  00 2 Bı (), then, for all  2 [0, 1], bı ( 0 , ) + (1  ) u bı ( 00 , )  u bı ( 000 , ) 8 000 2  (K). Also, u bı ( 0 , ) + (1  ) u bı ( 00 , ) = u bı ( 0 + (1  )  00 , ) because u bı is linear in . Together with the previous statement this implies u  0 + (1  )  00 2 Bı (), as required. By way of contradiction suppose that B () does not have a closed graph. Then there exists a sequence (n , n ) ! (, ) such that n 2 B (n ) but  2 / B (). Hence, 9 0 2  (K) and some bı and since n ! , it follows  > 0 for which uı ( 0 , ) > uı (, ) + 3. From continuity of u that for n large enough, uı ( 0 , n ) > uı ( 0 , )  . Combining the two preceding inequalities gives bı implies that uı (, ) + 2 > uı (n , n ) + . But uı ( 0 , n ) > uı (, ) + 2, and continuity of u uı ( 0 , n ) > uı (n , n ) +  contradicts n 2 B (n ). Hence, B () must have a closed graph. I From Kakutani’s Fixed Point Theorem it then follows that there exists at least one  2  (K) for which  2 B (), as desired. Q.E.D. Appendix B: Calculating a Party’s Number of Seats Following Spenkuch (2012), this appendix explains the algorithm that is currently used to calculate a party’s number of seats in the Bundestag. Let dp,s denote the number of direct mandates accruing to party p in state s. vp,s is the number of list votes that p received in s, with the equivalent number on the national level given by v p = calculated in three steps: P s vp,s . With this notation in hand, party p’s seat total is Step 1: Proportional Allocation of List Mandates to Parties. Absent overhang mandates, there are 598 seats in the Bundestag. These are allocated by proportionality rule to the set of parties clearing the 5%-threshold or winning at least three direct mandates. That is, the number of list mandates of party p equals lp  =  8 < 598 P vp e v p0 p 0 2P : 0 v where Pe = p| P pv p0 p0  .05 _ if p 2 Pe , otherwise  P s dp,s  3 and  = represents equality after rounding according to the Sainte-Laguë method, which ensures that P p lp = 598.46 Step 2: Proportional Allocation of Mandates to State Lists. German electoral law requires parties to compete with dierent lists in each state. Therefore, list mandates need to be allocated to the respective state lists. In practice, the number of mandates awarded to a party’s state list is proportional to the list’s contribution to the party’s vote total. More precisely, for all s and all p, lp,s 46 8 < l vp,s p vp  = : 0 if p 2 Pe otherwise , In 2005 the method of Hare-Niemeyer was used instead. 30 where  = is defined as above. Step 3: Determination of the Actual Number of Seats. However, the actual number of seats that party p receives in state s is given by np,s = max {d p,s , lp,s } . If d p,s < lp,s then, in addition to the district winners, the first lp,s  d p,s candidates on p’s list in s are elected to the Bundestag as well. Otherwise, only holders of direct mandates receive a seat. Note that only if d p,s  lp,s for all s, will party p’s seat total, np = P s np,s , be equal to the number of seats it would be assigned under proportional representation, i.e. lp . Appendix C: Variable Definitions This appendix provides a description of all data used in the paper, as well as precise definitions together with the sources of all variables. C.1. Election Results Data containing the ocial results of the 1980, 1983, 1987, 1990, 1994, and 1998 federal elections by municipality (Gemeinde) as well as the 2002, 2005, and 2009 elections by polling precinct (Wahlbezirk) have been purchased from the Federal Returning Ocer. These data include information on the number of list and candidate votes for each party and each candidate, the number of eligible voters, as well as the number of valid and invalid votes. In 2009 there were approximately 89,000 precincts. Whenever necessary precinct level numbers are aggregated using the municipality identifiers contained in the raw data. Municipalities spanning multiple districts are discarded. Throughout the analysis the following variables are used: Number of Eligible Voters is defined as the number of residents of each precinct that were allowed to vote in the particular year. In general this encompasses all German citizens over the age of 18, who have not been declared mentally unfit, or whose voting rights have not been suspended due to criminal behavior. Turnout is defined as the number of actual voters over the number of eligible voters. This number cannot be calculated for absentee precincts, as absentee voters are included in the number of eligible voters in their district of residence. Hence, in-person turnout in each district needs to be adjusted for absentee voters. In practice, this is done by multiplying the number of issued absentee ballots by .95 (which corresponds to the empirical frequency with which they are cast) and adding them to the ballots that are cast in person. Share of List Vote is defined as the portion of all valid list votes (in %) that are cast for a particular party. “Micro parties”, i.e. those not clearing the 5%-threshold, are grouped together. 31 Share of Candidate Vote is defined as the portion of all valid candidate votes (in %) that are cast for the candidate of a particular party. Votes for candidates of “micro parties” are pooled. Absentee Precinct is an indicator variable equal to one if a given precinct handles only absentee ballots. C.2. Candidate Characteristics Prior to every election to the Bundestag the Federal Returning Ocer publishes information on certain characteristics of all ocial list and direct candidates. This paper focuses only on the latter. The data have been compiled from Bundeswahlleiter (2005c, 2009b). Throughout the analysis the following variables are used: Age at the time of the election is defined as election year minus year of birth. Female is an indicator variable equal to one if a candidate is female, and zero otherwise. Doctorate is an indicator variable equal to one if a candidate holds a doctoral degree and/or a professorship, and zero otherwise. As doctoral degrees are part of Germans’ ocial names, this variable has been created using a text search for “Dr.” and “Prof.”. Currently Member of Parliament is an indicator variable equal to one if the candidate holds a list or direct mandate, and zero otherwise. Holds Direct Mandate is an indicator variable equal to one if the candidate holds a direct mandate, and zero otherwise. Also on Part List is an indicator variable equal to one if the candidate does not only run in the district race, but is also on her party’s state list (and could thus enter the Bundestag either way). Position on Party List denotes the candidate’s rank on her party’s state list (conditional on having been placed on the list). C.3. Municipality Characteristics Information on municipalities’ demographic and socio-economic characteristics is taken from Statistik lokal 2007 and Statistik lokal 2011 (Statistische Ämter des Bundes und der Länder 2007, 2011). Statistik lokal is an annual publication of the German Federal Statistical Oce and the statistical oces of the Länder containing data on various characteristics of approximately 12,000 municipalities and administrative units in Germany as of about 2 years before to the publication date. These data have been linked with the election results described above using the municipality identifier (Allgemeiner Gemeindeschlüssel) contained in both data sets. Below follows a brief description of all municipality level variables used throughout the analysis. Population Density is defined as a municipality’s total average population (in thousands) per square kilometer during the respective calendar year. 32 Fraction of Voters Female is defined as the share of women among a municipality’s population over the age of 18. Income Tax Revenue Per Capita is defined as the total income tax receipts (in 1,000 EUR) accruing to the respective municipality divided by its population during the same calendar year. Fraction of Voters under Age 30 is defined as the fraction of individuals aged 18—30 among those over the age of 18. Appendix D: Numerical Methods This appendix describes the numerical methods used to solve the optimization problem defined by equations (10)—(13) as well as the construction of counterfactual election results in Section 7.2. D.1. Mathematical Programming with Equality Constraints Typically, to recover mean utilities in models of discrete choice (i.e. p,m ) researchers turn to inverting the system of non-linear markets share equations via the nested fixed point (NFP) algorithm in Berry (1994) and Berry et al. (1995). Recently, however, Su and Judd (2012) and Dube et al. (2012) have shown how to recast extremum estimators in general, and the one in the Berry et al. (1995) in particular, as a mathematical programming problem with equality constraints (MPEC). Key to the MPEC approach is the insight that the inner loop can be eliminated entirely by recasting the estimator as an optimization problem subject to a set of non-linear constraints, i.e. (11), which require predicted market shares to equal observed ones. Since objective function and market share equations are usually smooth, one can rely on stateof-art optimization software, such as KNITRO or SNOPT, to find candidate solutions. Moreover, dispensing with the inner loops avoids numerical problems associated with loose inner loop error tolerances (see Dube et al. 2012 for a discussion of the NFP algorithm’s numerical properties), and it may significantly increase computational speed because the system of market share equations does not have to be solved exactly at each iteration. (The constraints have to be satisfied only at the solution.) Importantly, Su and Judd (2012) prove that MPEC and NFP solve the same problem, yielding the same estimates with the same statistical properties. The implementation of MPEC in this paper is based on the MATLAB code of Dube et al. (2012), using both of the KNITRO solver’s interior-point and active set algorithms (Byrd et al. 1999, 2004, 2006). To improve numerical accuracy as well as computational performance, KNITRO is provided with hand-coded first-order analytical derivatives of the objective function and the constraints, second order derivatives, as well as the sparsity patterns of the constraint Jacobian and the Hessian. Since the Hessian contains almost 4  109 elements of which only about 1.8  106 are non-zero, supplying the solver with the sparsity pattern is critical in order to economize on memory usage and time. To increase the likelihood of finding the global optimum five dierent starting points are used. Relative optimality and feasibility error tolerances, i.e. the maximum violation of the first 33 order conditions and the constraints, have each been set to 106 . Reported standard errors are based on the block-bootstrap with 100 iterations. In order to provide the solver with a completely smooth optimization problem, the constraints in (12)—(13) have been rewritten as an indicator function for each candidate, ck (), and are numerically approximated by the hyperbolic tangent. That is, ck () = for  = 5,000. Thus, equation (9) becomes C,T vbk,m = Z 1+ P 1 2   L,2 + 12 tanh   + v L k,d  v d ck exp(p,m +i,p +k ) d (). ck exp(p0 ,m +i,p0 +k0 ) nd  k0 2K D.2. Sparse Grid Integration Instead of solving the approximately 120,000 five dimensional integrals in equations (7), (8), and (9) using simulation methods, the present paper relies on sparse grid integration (SGI), introduced into economics by Heiss and Winschel (2008). SGI provides a way to approximate integrals numerically avoiding the curse of dimensionality associated with conventional quadrature rules (see Judd 1998). Monte Carlo evidence by Skrainka and Judd (2011) indicates that SGI imposes a significantly lower computational burden than simulation methods achieving the same level of accuracy. SGI is closely related to conventional Gaussian quadrature rules, but by exploiting symmetry properties it relies only on a small subset of nodes and (appropriately rescaled) weights. This paper uses a Konrad-Patterson rule with Gaussian kernel for choosing nodes, as explained in Heiss and Winschel (2008). This particular rule has only 151 nodes; yet it exactly integrates (over five dimensions) all complete polynomials of total order less than 7. Experimentation with more accurate rules yielded essentially the same point estimates, but required significantly more CPU time. D.3. Construction of Counterfactuals The counterfactual election results in Section 7.2 of this paper have been constructed by simulation. More specifically, for each municipality in the data 100 times its actual number of voters have been simulated by randomly drawing , , and " from the respective (estimated) distributions. A fraction  of simulated voters (rounded to the nearest integer) are designated to behave tactically. Next, each voter’s candidate and party specific utilities are calculated and his (partial) preference orderings for the list and candidate vote are determined. Naturally, sincere voters consider all candidates, whereas tactical voters choose only among those contestants who are estimated to be contenders. Election results are then constructed by aggregating votes to the appropriate level, and applying the specified electoral rule. 34