The Adaptive Dynamics of Turnout - CiteSeerX

The Adaptive Dynamics of Turnout∗ Nathan A. Collins

Sunil Kumar

Department of Political Science

Graduate School of Business

Stanford University

Stanford University

[email protected]

kumar [email protected] Jonathan Bendor

Graduate School of Business Stanford University bendor [email protected] June 12, 2008

∗

We thank Jeremy Bulow, Simon Jackman, Ken Shotts and seminar participants at the Department of Economics,

Stanford University, for helpful comments.

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Abstract We present a dynamic model of turnout in which voters behavior in one election depends only on whether they voted in the last election and whether their party won. This assumption may be justified by assuming citizens satisfice or by assuming they adjust subjective beliefs about being pivotal in ways that depend on whether they voted and on the outcome of the election. Regardless of the individual-level mechanism, this assumption (prior participation and prior electoral outcome affect current turnout) has empirical support, and we show that it implies turnout dynamics that are in accord with observed dynamics in several countries. Thus, this assumption or something very much like it is a necessary feature of any model of turnout. The ensuing model meets a basic requirement of any turnout model, substantial steady-state turnout, and correctly predicts some counterintuitive dynamical features of aggregate turnout — including declining turnout despite close elections and nonmonotonic changes — following significant political events such as the (re)establishment of democracy or major wars.

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Much theoretical research in political economy has focused on why people vote, that is, on why turnout reaches substantial numbers; relatively little has focused on how turnout changes over time. Presently, several models predict that turnout reaches substantial levels. Given that these models all pass this test, the next step is to add new hurdles by deriving additional predictions and comparing them to data. A natural set of tests involve turnout dynamics rather than statics. Dynamics are particularly important because they have a number of surprising and unexplained features, among them declining turnout despite close, competitive elections, as is the case in some new democracies and large, nonmonotonic changes in turnout that take place over the course of decades.1 We develop a model of turnout based on the idea that voters condition their behavior in the current election on whether they voted in the last election and whether their party won that election. Specifically, we examine four groups — winning voters, winning shirkers, losing voters, and losing shirkers — and hypothesize that the probability of switching actions (from voting to shirking or vice-versa) depends only on the group a citizen belongs to. There are several different ways to justify this approach at the individual level. One is based on satisficing (Simon 1955): an agent switches her action only when it turns out to be unsatisfactory. In the context of voting, our model arises from this satisficing theory if the probability of satisfaction depends on which of the four groups a citizen belongs to.2 Another individual-level approach is based on subjective utility maximization: if a citizen’s subjective belief about the probability of being pivotal depends on whether she votes or stays home and on whether her party won the last election, the model again arises naturally. (Such a model might, however, require that belief-revision violate Bayes’ law in some circumstances.) While we are agnostic about the particular justification, we note that there is compelling evidence that citizens condition their turnout decisions on past voting behavior (e.g. Gerber, Green, and Shachar 2003), and there is some evidence that citizens also condition their behavior on whether their party wins or loses (Kanazawa 1998). This paper is concerned with deriving and testing the implications of these observations. In either case, interesting aggregate-level dynamics result because there are several groups and in general each adjusts its behavior at a different rate. One party’s turnout — we assume that citizens either vote for a particular party or stay home, as in Palfrey and Rosenthal (1985), though our essential results do not depend on that assumption — might go up over time while another goes down. As a result, we can explain a range of dynamical features of turnout, including 3

declining turnout despite close elections and nonmonotonic changes in turnout. A side benefit of our approach, consistent with turnout in safe electoral districts in the United States, is that it predicts that adherents of the losing party will vote persistently even when their side is certain to lose. Finally, because there are always some citizens who stay home in one election and vote in the next, our model also meets a basic requirement of any turnout model, positive steady-state turnout; under plausible parametric conditions, this steady-state level is substantial. Our approach differs in fundamental ways from other models of turnout, including those based on other kinds of adaptation. Most basically, our model is inherently dynamic, since citizens’ decisions depend on what happened in the previous election, while most others are static. Feddersen and Sandroni’s (2006) model of ethical voting, for example, is a model of turnout in a single election; likewise Levine and Palfrey’s (2007) model, based in part on quantal response equilibrium, lacks dynamics. Other static models do not even explain positive turnout in a satisfactory way, either because they predict negligible turnout or they predict asymmetric equilibria (Palfrey and Rosenthal 1983, 1985; Myerson 1998). These models were designed to explain essentially static features of turnout, such as why turnout and margins of victory are inversely related, while also explaining substantial turnout. They were not designed to answer the questions we study here. Further, our model identifies the simultaneous importance of voting or staying home and winning or losing an election. In fact, interesting dynamics arise in the model only because citizens condition their turnout decisions on both their actions in and the outcome of the previous election. If turnout depended only whether citizens voted or only on whether their party won, aggregate turnout would follow first-order dynamics — i.e., geometric approach to an equilibrium. To see the importance of this idea, consider constructing a dynamic model of turnout based on the decision-theoretic approach of Riker and Ordeshook (1968). Suppose citizens vote based on their evaluation of the last election, e.g., a person who likes to vote and whose party wins evaluates the decision to vote positively. Citizens maximize a utility function that depends largely on the payoffs of the act of voting itself, since there is little chance a single person can change the electoral outcome. Therefore, except for voters with exceedingly small costs or benefits of voting, whether one’s party won or lost is irrelevant. As will become apparent, this produces dynamics that are in general too simple to explain observed aggregate turnout dynamics. A final set of models in the literature are adaptive and therefore could explain turnout dynamics in principle but are either not appropriate for studying long-term dynamics or are substantially 4

less tractable than ours. In Diermeier and Van Mieghem’s (2005) poll-based approach, each citizen in sequence updates her turnout decision after observing noisy information about other citizens’ intended voting decisions; the authors show that this process converges and yields high expected turnout. The process is dynamic and adaptive, but since one citizen at a time reevaluates her decision and since the basis of the reevaluation is essentially a new estimate of the probability of being pivotal, this model is best applied to single elections. Bendor, Diermeier, and Ting’s (2003) aspiration-based adaptive model is the closest in spirit to our paper. In that model, voters adjust their probability of voting based on whether the outcome of an election meets their aspirations or not. However, the inclusion of explicit aspirations in that model renders intractable explicit calculations of steady-state turnout, let alone turnout dynamics. We must add to these important considerations the following simple but crucial fact about our model: it is the only formal model of turnout dynamics. Furthermore, as we show below, our model’s essential properties are necessary features of any model of turnout dynamics. In particular, we will show through empirical analysis that any stationary model of long-term turnout dynamics must assume that a society comprises at least two groups that adjust their turnout levels at different rates.3 Available empirical research (e.g. Gerber et al. 2003; Kanazawa 1998) supports our assumption that the probability of switching actions depends on whether a citizen votes and on whether her party wins, which in turn implies that there are (at least) two groups that adjust their participation at different rates. We wish to address some likely concerns about the model up front. First, we perform much of our analysis in continuous time, though of course elections do not occur continuously. It is therefore vital to understand that there are no qualitative differences and few quantitative differences between our discrete-time and continuous-time models. We employ continuous time only to make the analysis of a series of close elections more tractable, and we show computationally that the discrete-time model exhibits nearly identical behavior as the continuous-time version. Second, our model is stationary, i.e. the parameters of the model do not vary over time. This may arouse suspicion as we will apply the model to turnout changes over the course of several decades. We do not pretend that our model explains all features of turnout dynamics. In particular, we will not attempt to explain short-term fluctuations in turnout, nor (importantly) will we try to explain the correlation between turnout and the winning party’s margin of victory. Instead, we explain broad trends in turnout that can be thought of as recovery from electoral shocks accompanying 5

significant political events. We present turnout data in the United States, United Kingdom, and other countries between World War II and the 1973 oil crisis, a period of time of relative calm between globally significant political and economic events. Thus, the model would fruitfully be interpreted as explaining the baseline level of turnout about which various other effects — e.g. closeness, intensity of preference, individual differences — create election-to-election fluctuations. Finally, our model assumes only two parties, so applying the model to parliamentary systems that are dominated by more than two parties might seem inappropriate. However, the essential dynamics of the model are unaffected by the number of parties. That a party’s turnout adjusts over time and the manner in which it adjusts depends on the decision rule we employ, not the number of parties. Likewise the driving force behind turnout in our model is that some abstainers vote in the next election, either because they are dissatisfied (under the satisficing mechanism) or because they believe the probability of being pivotal is sufficiently high (under the maximizing mechanism) — not because they ignore costs, as one might be tempted to think. A similar point applies to concerns that our voters can only change their turnout decisions; we could, for example, allow them to change parties instead without changing our basic results. The rest of the paper is organized as follows. First we present the discrete-time model and its continuous-time limit, which we use to study competitive elections. As part of this presentation, we derive steady-state turnout levels and basic results about dynamics. Following that we discuss dynamics in more detail and derive solutions of the continuous-time model differential equations. Having derived the main features of the model, we compare the model’s predicted aggregate turnout patterns with actual turnout from several nations around the world and then conclude.

The Model Following Palfrey and Rosenthal (1985), we study turnout in a simple strategic setting. There is a sequence of elections at dates t = 0, 1, 2, 3, . . .. The electorate comprises two fixed factions A and B.4 Each citizen is a member of exactly one of these factions. At each election, each citizen either votes for her faction or stays home, and whichever faction turns out to vote in greater numbers wins. If both sides mobilize the same number of voters, we assume that some party wins, but we do not specify how to break ties. (This choice will not affect the analysis in any significant way.) Note that parties are not active players here, and citizens who turn out to vote have already made

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up their minds about whom to vote for. We assume that a citizen’s decision to vote at t + 1 depends on whether she voted at t and whether her party won that election. We use f to denote the probability of switching actions. A winning voter’s switching probability is fwv , a losing shirker’s switching probability is fls , and so on. As we discussed in the introduction, there are different individual-level ways to justify this model. In the satisficing interpretation, f is simply the probability of being unsatisfied.5 Then, for example, winning voters are unsatisfied with probability fwv and therefore stay home in the next election. In the subjective utility interpretation, a citizen i’s utility difference between voting and staying home is pi b + ci , where p is a subjective estimate of the probability of being pivotal, b is the utility gained if i’s party wins, and ci is the utility associated with voting, which may be positive or negative. If there is some distribution of ci in the population and the distribution of estimates of pi depends on whether a citizen voted in the last election and whether her party won, then fwv (for example) is the probability that for an individual chosen at random, pi,wv b + ci < 0. Similarly, fls = P(pi,ls b + ci > 0). Regardless of the interpretation, we can represent the idea that voting is costly by assuming fws < fwv and fls < flv ; likewise, we can represent the (rather banal) hypothesis that winning is valuable by assuming fwv < flv and fws < fls . However, making such assumptions is not necessary: the key predictions of the model do not depend on whether voting is costly or even whether winning is valuable, though of course it would be odd if the latter didn’t hold at least in expectation. Beyond the assumptions we have already made, we need only assume that f ∈ [0, 1], i.e. that the chances of switching are in fact probabilities. Our model simplifies considerably in the large population limit, in which the model becomes deterministic. Because individuals do not change parties, they can only change whether they vote. Thus, citizens exiting the voting set enter the shirking set, and vice-versa. This system will be in equilibrium when these flows exactly offset each other, i.e., when the fraction of citizens exiting the voting set equals the fraction of citizens entering that set. Figure 1 presents a graphical representation of the process for one party. Note how simple this system is: the state of the system is characterized by two variables, the fraction of each party that votes. This simplification — which is the source of our tractability advantage over the Bendor et al. (2003) model, in which a state consists of an aspiration and propensity to vote for each citizen — makes it possible to derive both statics and dynamics analytically. 7

[Figure 1 about here.]

The Discrete-Time Model The discrete-time model is most useful when applied to cases in which one party wins each election for some finite set of elections t = 0, 1, . . . , M . Real political behavior in such cases is somewhat counterintuitive from the perspective of full-rationality models. For example, since 1968 all of Wyoming’s electoral college votes have gone to a Republican presidential candidate, yet in 2000 about 30 percent of Wyoming’s 210,000 presidential votes went to Gore. Lest we think that 60,000 people came to the polls concerned about some other issue on the ballot and voted for Gore only because they already happened to be there, consider voters in Big Horn County. Roughly 5,000 citizens cast their votes in the county; of those about 1000 voted for Gore. No Democrat came even close to winning in any other partisan election, and the fates of judges and constitutional amendments on the ballot were decided by substantial margins. Barring fairly widespread electoral fervor regarding elections for such things as city council and school board positions, rational voters would have to have a taste for voting to explain this effect. As we show, our model predicts the effect even when voting is costly, i.e. when voters are more likely than shirkers to get a low payoff, so that fwv > fws and flv > fls .6 We now state the discrete-time model in the form of difference equations that are fundamental for the rest of the paper. Let vA,t be the voting rate for party A at election date t, i.e., the fraction of the electorate that votes for party A. The fraction who identify with party A is SA . Similarly, vB,t is the voting rate for party B and SB is the size of this party. Assume, without loss of generality, that A wins at time t. There are vA,t voters and SA − vA,t shirkers in party A. A fraction fws of the A-shirkers switch to voting at time t + 1. A fraction 1 − fwv of the A-voters keep voting. The same reasoning applies to party B, except that this party loses. Then, the changes in A and B’s turnout from t to t + 1 are vA,t+1 − vA,t = fws SA − (fws + fwv ) vA,t , and

(1)

vB,t+1 − vB,t = fls SB − (fls + flv ) vB,t . These equations hold if A wins election t, either because vA,t > vB,t or because vA,t = vB,t and the

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tie breaks in favor of A. If, on the other hand, B wins, then vA,t+1 − vA,t = fls SA − (fls + flv ) vA,t , and

(2)

vB,t+1 − vB,t = fws SB − (fws + fwv ) vB,t . We emphasize again that these equations may be derived from the satisficing model, in which citizens switch actions when they are dissatisfied, or from a subjective maximization model in which citizens switch actions because their estimates of the probability of being pivotal change. The key feature that we exploit is that, whatever the behavioral reason, the probability a citizens changes her action depends on whether she voted and whether her party won. Because our model is based on an adaptive rule, dynamics are a natural part of the model. In the model, turnout changes in proportion to the difference between current and steady-state participation levels, which implies that turnout changes geometrically with time if one party always wins. Proposition 1 In discrete time, if A always wins and the turnout levels at election 0 are vA,0 and vB,0 , then the turnout levels at election t are   fws SA fws SA + vA,0 − [1 − (fws + fwv )]t , and vA,t = fws + fwv fws + fwv   fls SB fls SB vB,t = + vB,0 − [1 − (fls + flv )]t , fls + flv fls + flv

(3)

Proof. Rewrite Equation (1) as   fws SA fws SA vA,t+1 − = vA,t − [1 − (fws + fwv )] and fws + fwv fws + fwv   fls SB fls SB = vB,t − [1 − (fls + flv )] . vB,t+1 − fls + flv fls + flv

(4)

Applying this equation t times beginning with vA,0 and vB,0 yields Equation (4). Deriving steady-state turnout levels is straightforward if one party always wins: simply set vt+1 = vt in Equation (1) and solve. If A always wins, fws , and fws + fwv fls = SB . fls + flv

vw,A = SA vl,B

(5)

Naturally, we can obtain the steady states when B wins by exchanging the A and B labels. Figure 2 presents some examples for the case SA = SB . The top left presents a new-democracy case with high initial turnout that declines over time. The top right provides a second example that 9

parallels the Utah Democrats example: one party loses forever yet always has positive turnout. We include in each figure aggregate turnout, which we will examine further below. [Figure 2 about here.] Of course, one can find many examples of elections where one party is not guaranteed to win all the time. The discrete-time model handles these cases as well. The figures in the bottom row of Figure 2 illustrate two examples in which one party does not always win. For some period of time in both examples, the turnout levels “chatter,” that is, the parties alternate as winner and loser, and their turnout levels adjust accordingly. A special feature of the first example on the bottom left is that the turnout levels remain in this chattering state forever. (This property holds because in this case the winner asymptotically approaches a turnout level below that of the loser; see the supplementary material for details.) Unfortunately, writing down the steady-state turnout levels, dynamics, or comparative statics is difficult when each party frequently switches from winner to loser and back again in discrete time. Hence we now consider the model’s continuous-time limit.

The Continuous-Time Model We study the continuous-time limit of the model so that we can understand the steady states, dynamics, and comparative statics of the model when elections are competitive — i.e., when the winner frequently changes from election to election, as in the bottom row of Figure 2. U.S. presidential elections and some House and Senate races often fall in this category. The continuous-time model enables us to study such situations quantitatively. Happily, this model is qualitatively and quantitatively very similar to its discrete-time counterpart, so little is lost by using the more tractable version.7 This section establishes three main results. First, we present the continuous-time limit of the discrete-time model. Second, we derive steady-state turnout levels. Third, we derive conditions under which competitive elections can persist in the continuous-time model. A competitive election in the present model is, as we will argue, essentially a tie in the continuous-time limit. Competitive elections are not unusual in politics, and for this reason we want to emphasize that ties are not unusual in our continuous-time model. In continuous time, ties happen for a wide range of parameter values, and for some values they are inevitable.

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The Continuous-Time Limit. We begin by developing some intuition for the continuous-time limit. If one party always won, the continuous-time limit of the model would be easy to derive. Roughly, we replace vt+1 − vt = f (vt ) (Equation (1), say) with vt+∆t − vt = ∆t f (vt ), where heretofore ∆t was one. Divide by ∆t, and take the limit ∆t ↓ 0; Equation (1) becomes dvA (t) = fws SA − (fws + fwv ) vA (t), and dt dvB (t) = fls SB − (fls + flv ) vB (t). dt

(6)

The case of interest, however, is the chattering (rapidly alternating winners and losers) depicted in the bottom row of Figure 2. First, note that the continuous-time limit of a series of close races is tied turnout. In the updating rule vt+∆t − vt = ∆t f (vt ), the difference vt+∆t − vt scales with ∆t. There are two consequences. First, vA and vB cannot cross until they are close to each other — roughly on the order of ∆t apart. Second, and more importantly, as long as they continue switching, they remain within roughly ∆t of each other. As ∆t → 0, then vA = vB as long as the alternating behavior continues. In other words, the two parties tie for some interval of time. The question then reduces to how we handle ties in continuous time. Intuitively, since some party wins and some party loses at each time t, the updating rule must be some convex combination of the rules for winners and losers. When the parties are not tied the combination is simple: the party with the larger turnout obeys the winner’s update rule and the other one obeys the loser’s. The tied case is more involved. For convenience, we define λw = fws + fwv and λl = fls + flv . Proposition 2 The continuous-time limit of the model given in Equation (1) is a pair of differential equations, dvA = 1vA >vB (SA fws − λw vA ) + 1vA vB (SB fls − λl vB ) + 1vA vB or vA < vB in the above equations, the levels are the same as those given in Equation (5) for the discrete-time model. Calculating steady-state turnout in the case of competitive elections is trickier in discrete time, and a desire to simplify this calculation helped motivate the continuous-time limit of the model. With the aid of that limit, the problem is again straightforward. Under the assumption turnout is tied, 1vA =vB = 1 in Equation (7) and the other indicator functions are zero. Since vA (t) = vB (t) by assumption, we set vA (t) = vB (t) = v(t). Doing so results in a quadratic equation because of the presence of the function α(t). With equal party sizes, Equation (8) implies α(t) = 1/2. Then, setting dv/dt = 0 yields a linear equation in v, and the steady states are again simple to calculate. Proposition 3 If SA = SB and vA (t) = vB (t) in the steady state, then the steady state of the continuous-time limit are v∞ =

fls + fws 1 . 2 fls + flv + fws + fwv

(9)

When SA 6= SB , α(t) is linear in v, so the equation dv/dt = 0 is quadratic in v; thus there are two possible solutions. For convenience we define  = SA − SB , λw = fws + fwv , and λl = fls + flv .

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Proposition 4 If SA 6= SB and vA (t) = vB (t) in the steady state, then the steady state of the continuous-time limit is v∞

" # q

1 1 2 2 λl fls − λw fws ± (λw fls − λl fws ) + 2 λ2l − λ2w fls2 − fws . = 2 λ2l − λ2w

(10)

The restriction that these expressions reduce to Equation (9) when SA = SB implies that the positive root holds when fws /(fws + fwv ) > fls /(fls + flv ) and the negative root holds when fws /(fws + fwv ) < fls /(fls + flv ). When fws /(fws + fwv ) = fls /(fls + flv ), the two expressions are equal. An important implication of these results is substantial turnout even when voting is costly, i.e. when voters’ switching rates are larger than shirkers’. The intuition is the same in the discrete-time and continuous-time cases. For example, in the satisficing interpretation, since some shirkers will be dissatisfied in every election, some of them vote in subsequent elections. Dissatisfied shirkers — who are always present — in one election are always a source of voters in the next, whether or not voting is costly. Finally, the comparative statics of the steady-state turnout levels are generally in line with observed patterns and with static models of turnout. For example, increasing the switching rate of voters relative to shirkers — roughly equivalent to making voting more costly — decreases steadystate turnout. Likewise, decreasing winners’ switching rates relative to losers — roughly equivalent to increasing the benefit of winning or the probability of being pivotal — increases turnout. Ties and Tie Persistence. If the parties are of equal size, solving Equations (7) and (8) is straightforward.8 One can see that if SA = SB , vw,A = vw,B and vl,A = vl,B , Equation (8) implies α(t) = 1/2 — i.e. each party wins half of the elections, which we should expect from the symmetry of the problem. Since α(t) does not depend on turnout or on time in this case, Equation (7) has the same form as the non-tied case: linear differential equations replace linear difference equations. Turnout does not in general remain tied forever. The intuition is simple: ties persist if the turnout equations drive the winner to become the loser and vice versa, precisely the phenomenon depicted in discrete time in the bottom row of Figure 2. Roughly, for the case SA = SB we set vA (t) = vB (t) = v(t) and assert that the winner’s turnout level must decrease or at least not change relative to the loser’s: 1 1 fws − (fws + fwv ) v(t) < fls − (fls + flv ) v(t). 2 2 13

(11)

This assertion leads to the following proposition. Proposition 5 Let v(t) = vA (t) = vB (t). If SA = SB , tied turnout persists if and only if v(t) ≤ v ∗ ≡

fls − fws 1 . 2 fls + flv − fws − fwv

(12)

(This result follows from Proposition 7 in the supplementary material.) Several other useful facts follow. First, turnout levels must be below v ∗ in order to tie in the first place; if vA , vB > v ∗ the difference |vA − vB | can not decrease. Second, ties can persist forever if and only if v∞ < v ∗ . If turnout is tied, v < v ∗ , and if v∞ < v ∗ , Equation (7) and Proposition 5 imply v must remain less than v ∗ . If, on the other hand, v∞ > v ∗ , turnout will eventually be greater than v ∗ , so elections can not remain tied. We note that our results about the persistence of competitive elections are in principle testable implications of the model. In particular, given estimates of the switching rates f — obtainable, for example, from ANES panel data — the model predicts that close elections will not begin or persist if turnout is above v ∗ .

Turnout Dynamics An important feature of our model is that it allows us to derive plausible dynamics. As we noted earlier, the discrete-time version of our model predicts that turnout changes geometrically with time (see Proposition 1). We considered the continuous-time limit to understand what happens when elections are often close, as in the bottom row of Figure 2. In this limit, the chattering behavior we depicted in those figures becomes a tie. Equations (7) include this case. When the parties are of equal size α(t) = 1/2, the equations are linear differential equations, so the solutions are exponentials of time. Proposition 6 If vA (t) = vB (t) for all t ∈ [T, T 0 ] and vA (T ) = vB (T ) = v(T ), then vA (t) = vB (t) = v∞ + (v(T ) − v∞ ) e−θ(λw +λl )(t−T )/2 ,

(13)

where we defined the tied-turnout steady state v∞ in Equation (9). For completeness, we note that in continuous time when turnout is not tied for some period of time, the solutions are just exponential functions. We could obtain this result as a limit of Equation (3) or by integrating Equation (6) under the assumption that vA (t) > vB (t) or vB (t) > vA (t). 14

As we pointed out earlier, although we can integrate dv/dt to find t as a function of v when the parties have different sizes, we can not analytically invert the result. We could do so numerically, but the possibility of calculating discrete-time turnout evolution exactly (given initial conditions) makes such an approach somewhat less relevant.

Continuous-Time Turnout Trajectories Combining Proposition 6 with our earlier observations about tie persistence at the end of the section on the continuous-time model, we can derive solutions of Equation (7), which for present purposes defines the model. We have already stated much of the technical justification for these solutions. In this section we focus on describing the solutions and the intuitions behind them. Because we do not have a closed-form solution in the case of tied parties of different sizes, we specialize to equal party sizes. In this case, we can derive all possible solutions of Equation (7). Note that the dynamics of the continuous-time model are a good approximation of the discretetime dynamics. In particular, if v(t+1)−v(t) is small compared to one, then v(t) for integer t closely approximates an exponential function. Even for extreme cases, the magnitude of v(t + 1) − v(t) must be less than about 1/2 (for equal-size parties the inequality is exact) and must decrease as time goes on, so the approximation is rarely poor and gets better as time passes. The first case to consider is when no election is ever tied. If vw = fws /2(fws + fwv ) is larger than vl = fls /2(fls + flv ) and if vA (t) is far enough away from vB (t), the two turnout rates will never meet, so ties never occur. We call these solutions No-Tie solutions. One party starts as the winner and remains the winner forever, and since the parties never tie, their turnout rates simply approach their steady-state values at an exponentially decaying rate. We present some examples of No-Tie solutions in the top row of Figure 3. (We include aggregate turnout in these plots, which we examine empirically below.) Assuming A is the winning party, No-Tie solutions look like this: vA (t) = vw + (vA (0) − vw )e−θλw t , and −θλl t

vB (t) = vl + (vA (0) − vl )e

(14)

,

where vw and vl are the steady-state levels given in Equation (5) with SA = SB = 1/2 and vA (0) and vB (0) are (mathematically arbitrary) initial conditions. Note that No-Tie solutions can exist when SA 6= SB : we would replace vw and vl with vw,A and vl,B , but otherwise the form is exactly the same. 15

[Figure 3 about here.] There are two cases in which the turnout levels meet, so that Equation (14) can not describe subsequent turnout. In the first case, the steady-state turnout level for the losing party is greater than that for the winning party, so the levels must cross at some point. If the loser’s turnout kept increasing and the winner’s kept decreasing, the loser would become the winner and vice-versa. In that case, the new winner’s turnout would decrease and the new loser’s turnout would increase toward the steady-state levels, so that turnout levels meet again. In the discrete-time model, this behavior forces the turnout levels to remain near each other, so we get the chattering depicted in the bottom row of Figure 2; in the continuous-time model it forces them to remain equal. Following our earlier tie-persistence argument, this equality lasts forever. We call this situation a Tie solution:

vA (t) =

vB (t) =

  vw + (vA (0) − vw ) e−θλw t

if t ≤ T ,

 v + (v (T ) − v ) e−θ(λw +λl )(t−T )/2 ∞ ∞ A   vl + (vB (0) − vl ) e−θλl t

if t > T ;

 v + (v (T ) − v ) e−θ(λw +λl )(t−T )/2 ∞ ∞ A

if t > T ,

(15) if t ≤ T ,

where we defined v∞ in Equation (9). We present an example of a Tie solution in the bottom-left panel of Figure 3. Though Tie solutions are possible when SA 6= SB , they do not take exactly this form. In fact, because α(t) depends on the value of turnout, solving exactly for the tied turnout dynamics is not possible. (See the supplementary material for details.) Ties can also occur if vw > vl but both turnout levels are less than v ∗ . In this case, tied turnout asymptotically approaches v∞ , but v∞ > v ∗ , so at the time when v(t) = v ∗ , one party becomes the definite winner and the turnout levels deviate. Once this happens, since v ∗ < vl < vw , turnout cannot tie again, and the winner remains victorious forever. In such a circumstance, Fork solutions result. After the turnout levels separate, the solutions look just like No-Tie solutions, and turnout evolves as if there had never been a tie. We present an example of a Fork solution in the bottom

16

right panel of Figure 3. Fork solutions look like this:   vw + (vA (0) − vw ) e−θλw t     vA (t) = v∞ + (vA (T1 ) − v∞ ) e−θ(λw +λl )(t−T1 )      v + (v (T ) − v ) e−θλl t A 2 l l   vl + (vA (0) − vl ) e−θλl t     vB (t) = v∞ + (vA (T1 ) − v∞ ) e−θ(λw +λl )(t−T1 )      v + (v (T ) − v ) e−θλw t w w A 2

if t < T1 , if T1 ≤ t < T2 , if T2 ≤ t; (16) if t < T1 , if T1 ≤ t < T2 , if T2 ≤ t.

Note that T1 is the first t when vA (t) = vB (t), and T2 is the date when turnout crosses v ∗ . Similar solutions are possible when SA 6= SB , but the tied portion of the solution, for t ∈ (T1 , T2 ), lacks a closed-form expression. T1 and T2 can be computed numerically given current turnout and switching rates, so that one could predict the times when competitive elections begin and end. Thus, these dates are in principle testable predictions of the model, although sufficiently accurate measurement may be difficult.

Comparison With Empirical Dynamics In this section, we present evidence that our model predicts nontrivial features of long-term turnout trends better than other plausible models. While we will not present an exhaustive analysis, we will present a rigorous statistical comparison between our model and several other candidate models. We first introduce the set of data we analyze and describe the criteria we used to select these data. We then introduce the set of candidate empirical models and discuss the criteria by which we will judge them, including the method of estimation and the specific tests we perform to discriminate between the models. Finally, we discuss the outcomes of our tests. As noted below, the tests provide substantial evidence in favor of the most important component of our model, the existence of (at least) two groups that adjust their turnout levels at different rates.

Data Although our model also makes predictions about the turnout of individual parties, the most readily available data concerns aggregate turnout. We use data from the International Institute for

17

Democracy and Electoral Assistance Voter Turnout website, which lists turnout in national elections between 1945 and 2006 for countries that have had some meaningfully free elections during that period.9 Data are available for absolute turnout and as a fractions of the registered and voting age populations; we used data on turnout as a fraction of the voting-age population. Because our model and others can only be expected to be stationary between two significant electoral shocks, we examined data between the end of World War II and the 1973 oil crisis.10 Both events had political and economic consequences on a global scale and therefore satisfy the requirements of a significant electoral shock. Our goal, then, was to find states with enough reliable turnout data during this period to carry out a useful statistical analysis. We restricted attention to countries that met several criteria. Most basically, we required a voting-age population that was never below 100,000 for the time period of interest, so that largepopulation approximations would be justified, and we required that there be at least eight elections between 1945 and the end of 1973. Requiring eight elections during this period strikes a balance between having enough data in any one case to perform a meaningful analysis and having enough cases to justify any conclusions about which model is best. Next, we required that the freedom of an election was beyond doubt, since it is not obvious our model applies to those that are not free, and moreover data from unfree contests may have been tampered with. The IDEA data set lists political rights and civil liberties scores compiled from the Freedom House website, http://www.freedomhouse.org, as well as a summary measure. We analyzed turnout only from states in which IDEA listed each election as free, which corresponds to high freedom ratings on both the political rights and civil liberties scales. We also excluded data for states in which the government underwent a major reorganization between 1945 and 1973; in particular, we excluded France, which created a new constitution in 1958. Finally, we required that the data contained no irregularities in turnout as a fraction of votingage population (VAP turnout) versus turnout as a fraction of the population registered to vote (registered turnout). In several instances the IDEA data record the voting-age population as smaller than the registered population and therefore record VAP turnout as higher than registered turnout.11 We exclude data from countries where this occurs. The elections that satisfy our requirements are the parliamentary elections of Australia, Belgium, Canada, Sweden, and the United Kingdom. We decided to analyze presidential elections in the United States as well. Although there were only seven elections between 1945 and 1973, 18

we were interested in how the model plays out in our own country and felt the addition rounded out the set of countries well. We note that each of the countries we have selected except Sweden was actively involved in World War II, and Sweden was impacted in important ways. Similarly, although the 1973 oil embargo was not explicitly directed at all of the countries we examine, it had global economic impact and had effects beyond the states targeted by the embargo.

Candidate Models We now describe the four candidate models that we estimated. The models are to varying degrees theoretically motivated. We first describe the empirical model derived from our formal model. We then consider a special case that is consistent with our formal model, and two others that might be empirically useful but that to our knowledge have no theoretical justification. From a purely empirical standpoint, our formal model belongs to a class that might be termed relaxation models, i.e. those in which aggregate turnout is a sum of different component turnouts. Each component is an exponential function with a different time constant which relaxes to some steady state. For example, in a two-party system such as the United States, citizens who favor the winning party in a presidential election may have different switching rates — hence different time constants — than those who favor the losing party. Formally, v(t) = x1 + x2 e−(t−t0 )/x3 + x4 e−(t−t0)/x5 ,

(17)

where xi are parameters to be estimated.12 A special case of the relaxation model is the following, which we call simply the exponential model, v(t) = x1 + x2 e−(t−t0 )/x3 ,

(18)

where xi are the parameters to be estimated. We set t0 = 1945 for all the estimations we report below. The relaxation model for aggregate turnout in Equation (17) follows immediately from Equation (14) for No-Tie solutions and is a reasonable approximation to Tie and Pitchfork solutions, Equations (15) and (16), since in these solutions the largest turnout changes and any nonmonotonic turnout changes occur when parties’ turnouts are not too close to each other. Likewise, the exponential model in Equation (18) follows from our formal model when elections are consistently competitive, i.e. vA (t) ≈ vB (t), when the parameters λw and λl that control turnout dynamics are approximately equal, or when all but one party’s initial turnout is near its steady-state. 19

While both the full relaxation and the exponential models follow from our formal model, an important prediction of the theory is that we will at times require two (or more) factions with different switching rates to adequately describe real turnout dynamics. The comparison between the relaxation and exponential models is thus the key component of our investigations. A third model is a simple polynomial, probably political scientists’ default model of nonlinear time dynamics. We estimate a five-parameter polynomial model: v(t) = x1 + x2 (t − t0 ) + x3 (t − t0 )2 + x4 (t − t0 )3 + x5 (t − t0 )4 ,

(19)

where xi are parameters to be estimated. We set t0 on a case-by-case basis.13 The polynomial model has no theoretical justification, formal or otherwise, that we know of. A fourth alternative is a sum-of-sines or Fourier model.14 We estimated a five-parameter model: v(t) = x1 + x2 sin

2(t − t0 ) 3(t − t0 ) 4(t − t0 ) t − t0 + x3 sin + x4 sin + x5 sin , T T T T

(20)

where xi are parameters to be estimated, t0 is the year of the first election for the country in question, and T is the time between the first and last elections for each country. In principle, the Fourier model could arise if turnout in one election depends on turnout in the two previous contests, i.e. if one could describe turnout with a second-order differential equation. We are unaware of any substantive theory that generates such a model. We assume that errors are independently and identically normally distributed; hence, we can write the log-likelihood of the data as l(x, σ 2 ) = −n ln(2Πσ 2 )/2 −

X

(v(t; x) − vt )2 /2/σ 2 .

(21)

t

Here, n is the number of elections, v(t) is one of the functions described above, vt is the observed turnout as a fraction of the voting-age population, and the sum over t is over the years in which an election occurs. The variance σ 2 is a parameter to be estimated. Note that the maximum-likelihood estimates of x will be the same as least-squares estimates, but the maximum-likelihood approach will allow us to make statistical comparisons between models — in particular, between non-nested models — more easily. Our main concern is whether the relaxation model fits the data better than other models. It is not necessary that our model does better in each case. Rather, we seek evidence (1) that the relaxation model fares better than the simple exponential model in some cases, (2) that the relaxation 20

model fares better than other models in some cases, and (3) that in each case either the exponential model, the relaxation model, or both does better than the other models. The first condition ensures that multiple factions with different switching rates are necessary to adequately describe turnout in general, the second condition ensures the empirical utility of the relaxation model more generally, and the third condition ensures the overall utility of relaxation models (possibly including the exponential model as a special case). Combined with the theoretical motivation for the relaxation model and the lack of theoretical motivation for the other alternatives, this collection of evidence would favor some form of relaxation model. To compare the full relaxation and exponential models, we perform likelihood-ratio tests. Such a test is sensible only if one model is nested within the other. This condition obviously does not hold for comparisons between the relaxation or exponential models and the polynomial or Fourier models. For these comparisons we perform Vuong tests. The Vuong test compares a difference in log-likelihoods divided by a standard error for this difference, which one computes using the sample variance of the individual data points’ contributions to the log-likelihood. Vuong (1989) shows that this statistic has an asymptotically standard normal distribution. The Vuong test has been applied in a variety of political settings, from international relations (e.g. Clarke 2001) to political behavior (e.g. Mebane and Sekhon 2002). For clarity, we will form the Vuong statistic so that positive values indicate the full relaxation model is a better fit to the data than other models, while negative values indicate it is a worse fit than other models. We will say that one model fares better than another if the likelihood-ratio or Vuong test is significant at the p = 0.1 level or better. Note that since the Vuong test statistic is normally distributed with mean zero, we may encounter cases with p > 0.9. Given the way we form Vuong tests, such values indicate that a model other than full model is a better fit. A second question is whether the estimated parameters of the relaxation models are in line with our assumptions about switching rates. The most important of these concerns the constants λl = fls + flv and λw = fws + fwv . Since the switching rates are probabilities, λ must be in the interval [0, 2]. Note that λw and λl are measured in inverse elections (i.e., the fraction switching per election), since each time step corresponds to an election; x3 and x5 in Equation (17) are the inverses of these parameters and should therefore be greater than 0.5 elections. With elections typically spaced two to four years apart, we expect that the time constants are at least 1 or 2 years. Since we have little data to work with, the standard errors on the estimated time constants are 21

large, and we will not be able to say much about this question. We therefore dispense with it here: nine out of the twelve estimated time constants for the multifaction relaxation model are greater than 3 years, in line with our predictions. Each of the other time constants is about 1/4 year, but the standard error on each is at least an order of magnitude larger — owing, we believe, to substantial correlation between the model parameters and to the relatively small amount of data. Thus the parameter estimates provide some additional suggestive evidence in favor of our model; however, due to large standard errors, the evidence is weak. Finally, since we are using time-series data, there is a danger of serially correlated errors which may inflate likelihoods. Such errors may also violate the assumptions underpinning the Vuong test. We therefore computed Durbin-Watson statistics, which measure serial correlation, for the estimated models and found that they were all between 1 and 3 and typically between 1.5 and 2.5. (The Durbin-Watson statistic is always between 0 and 4, with 2 corresponding to no serial correlation.) While these values suggest some serial correlation, with the number of elections we consider — about 10 — normally distributed, uncorrelated errors will generate Durbin-Watson statistics outside the interval (1, 3) about twenty percent of the time and outside the interval (1.5, 2.5) about fifty percent of the time. Thus, we do not find reason to be particularly concerned about serially correlated errors.

Results Overall, we find support for the relaxation model and therefore for the class of models to which ours belongs. We first compare the full relaxation model to the other three alternatives using, as appropriate, likelihood ratio tests and Vuong tests; we report the results of these tests in Table 1. First, in four cases the relaxation model is a better fit than the exponential. This result shows that in general, relaxation models of turnout require multiple groups with differing relaxation rates—in the context of our model, at least two parties with differing switching rates. Second, the relaxation model performs better than the polynomial in half the cases, and they are statistically indistinguishable in the other half. Third, the relaxation and the Fourier models are statistically indistinguishable in three cases, and the relaxation model is better in two cases while the Fourier is better in one. [Table 1 about here.]

22

However, in that last case (Canada) the relaxation and exponential models are statistically indistinguishable, indicating the full relaxation model has extra parameters that are doing no work. We therefore compared the exponential model with the others using Akaike’s Information Criteria, which includes a penalty for extra parameters, in place of the log-likelihood. Note that this does not change the distribution of the Vuong statistic (Vuong 1989). The exponential is not statistically distinguishable from the polynomial model (Vuong statistic 1.00, p = 0.56) or the Fourier model (0.56, p = 0.29). The exponential model is also indistinguishable from three- and four- parameter versions of polynomial and Fourier models; using the uncorrected Vuong test we find significances of roughly 0.6 to 0.7. Since our model also makes predictions regarding party-level turnout, we briefly analyze these predictions here. To the extent we have been able to find appropriate data, the data are generally in line with our model’s predictions. Recall that the model predicts that party-level turnout evolves exponentially with time, though during competitive periods the exponential will explain at most a general trend (see Figure 2), i.e., we should not necessarily expect very large R2 . Using least-squares fits of the U.S. party-level data to an exponential function, we find R2 = 0.2 for Democrats and 0.3 for Republicans for the period between 1948 and 1968 (inclusive). In the United Kingdom between 1945 and 1972, we find R2 = 0.36 for the Conservative party and 0.39 for the Labour party. For the four largest parties in the Belgian parliament during this period, R2 values are between 0.20 and 0.62.15 The U.S. data during this period are roughly consistent with the behavior depicted in Figure 2, bottom right panel, until about t = 10, though with higher initial and steady-state turnout levels; the U.K. data are roughly consistent with Figure 2, bottom left panel, though with initial turnout levels rather closer to each other than in the figure. Finally, we briefly analyze turnout post-1973. In the U.S. and U.K. we have useful party-level data post-1973 and can use the least-squares approach as above. For the period between 1972 and 1992, we find R2 = 0.24 for Democrats, 0.43 for Republicans, 0.70 for the Conservative party, and 0.68 for the Labour party. We do not have useful post-1973 party-level data for the other countries we analyzed, but we do have aggregate-level data. In Australia turnout levels are monotonically declining between 1974 and 1998 the last year for which we have Australian data. This suggests that the exponential model, Equation (18), will fit best (see Figure 2, top left panel and bottom left panel after about t = 2). We therefore fit the exponential model to the aggregate data and found R2 = 0.44. The situation in Belgium is similar, although the period between 1973 and 1980 23

appears to be a continuation of the pre-1973 pattern and turnout peaks in 1981. We therefore fit the exponential model to the aggregate data from 1981 to 1999, the last year for which we have Belgian data, and found R2 = 0.97. Canadian turnout post-1973 is roughly flat until 1988 and then falls off 15 percent by 2000, suggesting the pattern in Figure 2, bottom left panel, i.e., a short period in which one party wins, followed by a period of competitive elections. We therefore fit the full relaxation model Equation (17) and found R2 = 0.90. Swedish turnout follows a similar pattern between 1973 and 1998, so again we fit the full relaxation model and found R2 = 0.85. We note that, while these results are quite favorable to our model, we are not able to evaluate the party-level turnout predictions that correspond to our assumptions — e.g., we do not know whether Canadian party-level turnout actually corresponds to Figure 2, bottom left panel.

Discussion To summarize, we identified a time period — between the end of World War II and the 1973 oil crisis — that should be relatively free of significant political or economic shocks, justifying the application of stationary turnout models during the period. We then identified countries in which there was reliable turnout data and enough elections between 1945 and 1973 to allow meaningful statistical analysis. We defined several candidate models: two were theoretically motivated by our formal model, and the other two lack theoretical motivation we are aware of. (We sought other formal models that analyze this problem but found none.) We presented statistical tests which taken together support our model: some relaxation model performs at least as well as any other equally (or less) parsimonious model we could think of and generally better, and the full relaxation model, Equation (17), often performs better than the simple exponential. Furthermore, the rates of relaxation are generally consistent with our additional assumption that voters switch their behaviors on an election-by-election basis, which led to the conclusion that the estimated time constants should be (roughly) larger than 2 years. Finally, to the extent we have been able to analyze party-level turnout and turnout after 1973, the model fits the data fairly well. Thus, we find substantial evidence in favor of the multifaction relaxation model. While we feel this analysis is thorough, we have left alone a few issues. First, serially correlated errors may be a more significant problem than we assume. Although there is no statistical reason to be worried about serially correlated errors here, an exhaustive analysis would take them into 24

account. Second, there are plausibly non-stationary explanations for turnout dynamics, notably the post-World War II rise in fertility (in the United States, at least). Since younger citizens vote less (e.g. Rosenstone and Hansen 1993) the fertility rise could affect turnout starting in the mid to late 1960s as the population’s age profile changes. Both of these issues are worth investigating, but they are beyond the scope of this paper. We hope that others with greater empirical skill take up these issues in the future.

Conclusion Classical decision-theoretic and strategic models of turnout fail to explain why people vote at all. Current rational and behavioral models do much better at this static task, but they do not explain important dynamic features of turnout. Typically such models do not fail to do so; they don’t even try. In this paper we have taken a step toward explaining major aspects of turnout dynamics. We have shown in particular how a simple, adaptive model of turnout explains qualitative features of turnout dynamics that we observe in aggregate data and, to the extent we were able to analyze it, party-level data. Among these are roughly geometric or exponential changes in turnout over time and nonmonotonic changes in aggregate turnout over time. It is important to understand that these properties do not depend on the assumption of two parties or citizens’ allegiance to one or the other party. Our results do not depend on the continuous time limit, which we used solely for analytical convenience; it makes no qualitative and little quantitative difference. Nor do they depend on the apparently linear, first-order nature of the model — which, in fact, is not linear at all. Instead, our key results depend on the presence of multiple factions that adjust their turnout levels at different rates (i.e. with different time constants). Likewise, the prediction that turnout can decline despite close elections—observed in some new democracies and counterintuitive, since close elections are often thought to increase voting — holds when steady-state turnout is lower than initial turnout. Again, this has nothing to do with the restriction to two parties or the restriction that citizens either vote for a particular party or stay home. It arises simply when turnout is initially high; then at least some voters will be disappointed and so switch to staying home, whether or not elections are close. Our model also sheds some light on qualitative features of turnout such as persistent voting among citizens of a party sure to lose. (For example, see Figure 2, top right, and the Wyoming example discussed earlier.)

25

Finally, our model has some additional implications that are at least in principle testable. For example, the model predicts that close elections cannot persist above a certain level. This level depends on the switching rates, so if these can be measured reliably, it is possible to test this prediction directly. Such a study would probably also shed light on the validity of our assumption of stationary dynamics, i.e., whether the switching rates remain constant over extended periods of time. While we have derived the most basic results, there are several questions we have left unanswered. We did not attempt to explain phenomena such as margin-of-victory effects. Perhaps more significantly we have not allowed our actors do something real voters do: change parties. While relaxing assumptions about the number of parties or citizens’ allegiance to them makes no qualitative difference to the turnout dynamics, relaxing the latter in particular allows us to study party and turnout choices simultaneously. In so doing we afford our citizens a greater measure of realism. We hope to model these aspects of electoral participation in the future.

Notes 1

We discuss these observations further in the section on empirical dynamics.

2

The full justification goes like this: suppose that citizens receive either a high h or low payoff l < h. Then, their

aspirations must be between l and h (see Bendor, Kumar, and Siegel 2004, for details). If the probability that a voter receives the high payoff depends only on whether she voted and on whether her party won, then her probability of being satisfied likewise depends only on whether she voted and whether she won. 3

Of course, we cannot rule out nonstationary models, i.e., models with parameters that depend explicitly on time.

On the other hand, nonstationary models are not necessarily falsifiable, while our model is falsifiable at the aggregate and individual levels. 4

This is primarily a simplifying assumption. Our results are not qualitatively different if we allow more parties or

if we allow citizens to change their choice of party. 5

More generally, if some voters are inertial, i.e., stay with their current action despite dissatisfaction, f is the

probability of being unsatisfied and switching. See Erev and Haruvy (2005) for a discussion of the importance of inertia in models of low rationality. 6

Note that this is a feature of any model of dynamics in which there are positive flows between the shirking and

voting states. 7

An alternative interpretation, approximately consistent with our model, is that elections occur in discrete time,

but voters continuously update, i.e., they continuously switch the action they would take depending on their action in and the outcome of the previous election. There are some differences, though. For example, any electoral ties, discussed below, will persist until the next discrete election after turnout reaches a critical level rather than when it

26

reaches that level. 8

Solving the equations when SA 6= SB is more complicated since α(t) is not constant. We can integrate the

equation for dv/dt to find t as a function of v, but we can not invert the result exactly since it contains non-integer powers of v. 9 10

http://www.idea.int/vt/. Accessed September 27, 2007. In one case, Australia, we restricted attention to turnout from 1949 — the year that Australia granted aboriginal

citizens the right to vote — to 1973. 11

It appears this results from various lags in recording. For example, Swiss women were first granted the right

to vote in federal elections in 1971. The registered population size for the 1971 election reflects this fact, but the voting-age population does not: it stays near two million, while in the next election it jumps to about four million. 12

Note that t0 must be set, as given x3 and x5 , it will not be possible to identify x2 , x4 , and t0 simultaneously.

13

As with the relaxation models, it is not technically possible to identify all the xi as well as t0 . However, setting

t0 to some fixed value for all cases and plotting the estimated functions results in fits that are obviously not as good as they could be. The issue is that while the xi and t0 are not jointly identified, the relationship between the different parameters is sufficiently complex to make estimation with a fixed value of t0 difficult. We therefore ran initial, least-squares estimations with t0 as a free parameter. We fixed t0 at its estimated value and used the estimated xi as starting values for the maximum likelihood estimations we report below. While this may be a bit of an unusual approach, we found that we were able to achieve much better fits for the polynomial model than with other estimation methods. We note that we have gone to greater lengths to fit the polynomial than we have our own model. 14

We use this name since the model is a truncated Fourier series.

15

We obtained the U.S. data from the Web site of the Clerk of the U.S. House of Representatives and the U.K.

data from the Web site of the U.K. House of Commons Library. Belgian data are from the Centre d’etude de la vie politique at the Free University of Brussels, http://dev.ulb.ac.be/cevipol/en/elections.html.

References Jonathan Bendor, Daniel Diermeier, and Michael M. Ting. A behavioral model of turnout. American Political Science Review, 97:261–280, 2003. Jonathan Bendor, Sunil Kumar, and David A. Siegel. Satisficing: A pretty good heuristic. Paper presented at the annual meeting of the Midwest Political Science Association, Chicago, 2004. Kevin A. Clarke. Testing nonnested models of international relations: Reevaluating realism. American Journal of Political Science, 45:725–744, 2001. Daniel Diermeier and Jan A. Van Mieghem. Coordination and turnout in large electorates. Typescript, 2005.

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Ido Erev and Ernan Haruvy. Generality, repitition, and the role of descriptive learning models. Journal of Mathematical Psychology, 49:357–71, 2005. Timothy Feddersen and Alvaro Sandroni. A theory of participation in elections. American Economic Review, 96:1271–1282, 2006. Alan S. Gerber, Donald P. Green, and Ron Shachar. Voting may be habit-forming: Evidence from a randomized field experiment. American Journal of Political Science, 47:540–550, 2003. Satoshi Kanazawa. A possible solution to the paradox of voter turnout. Journal of Politics, 60: 974–995, 1998. Daniel K. Levine and Thomas R. Palfrey. The paradox of voter participation? a laboratory study. American Political Science Review, 101:143–158, 2007. Walter R. Mebane and Jasjeet S. Sekhon. Coordination and policy moderation at midterm. American Political Science Review, 96:141–157, 2002. Roger Myerson. Population uncertainty and poisson games. International Journal of Game Theory, 27:375–392, 1998. Thomas Palfrey and Howard Rosenthal. A strategic calculus of voting. Public Choice, 41:7–53, 1983. Thomas Palfrey and Howard Rosenthal. Voter participation and strategic uncertainty. American Political Science Review, 79:62–78, 1985. William Riker and Peter Ordeshook. A theory of the calculus of voting. American Political Science Review, 62:25–42, 1968. Steven J. Rosenstone and John Mark Hansen. Mobilization, Participation, and Democracy in America. Macmillan Publishing Company, New York, 1993. Herbert A. Simon. A behavioral model of rational choice. Quarterly Journal of Economics, 69: 99–118, 1955. Quang H. Vuong. Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica, 57(2):307–333, 1989. 28

Country Australia Belgium Canada** Sweden UK US

Relaxation vs. Exponential (LLR) 3.19* (0.04) 0.00 (1.00) 0.14 (0.87) 4.29* (0.01) 8.46* (0.00) 2.83* (0.06)

Relaxation vs. Polynomial (Vuong) 2.37* (0.01) 5.57* (0.00) -0.92 (0.82) -0.35 (0.64) 4.20* (0.00) -0.13 (0.55)

Relaxation vs. Fourier (Vuong) 0.63 (0.26) 5.87* (0.00) 1.68 (0.95) 0.19 (0.42) 4.89* (0.00) -0.39 (0.65)

Table 1: Log-likelihood ratio statistics and Vuong statistics for the full relaxation model compared to the exponential model, the polynomial model, and the Fourier model, by country; p-values are in parentheses. We indicate tests statistically significant at the p = 0.05 level or better with asterisks. Note that the log-likelihood ratios must be positive, while the Vuong statistics may be positive or negative. Thus a Vuong statistic with a p = 0.95 or above indicates that the relaxation model is statistically worse than the other model in question. A similar value for LLR statistics indicates the two models are indistinguishable. * = p < 0.05. ** See text for a discussion of the comparison between the relaxation model and the Fourier model in Canadian turnout data.

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voters fail; some remain voters

shirkers succeed and remain shirkers some failed voters shirk

θ Voters some failed shirkers vote voters succeed and remain voters

Shirkers

θ

shirkers fail; some remain shirkers

Figure 1: We model a party’s population as two subsets, voters and shirkers. With probability θ, individuals move from the voting (shirking) set to the shirking (voting) set when they get a low payoff, that is, when they fail. When voters or shirkers succeed, they stay in their present subsets, that is, they do not change their actions. The system reaches a fixed point when the number of voters who switch to shirking equals the number of shirkers who switch to voting.

30

Figure 2: Turnout evolution in the discrete-time model. (Top left) The new democracy case. (Top right) One party turns out to vote even though it never wins. (Bottom left) For some parameters, the losing party is drawn toward a steady-state larger than the winning party’s steady state. In this situation, each party alternates as winner and loser. (Bottom right) The electoral back-and-forth can happen even when the loser’s steady-state is below the winner’s.

31

Figure 3: Continuous-time solutions. (Top row) Two examples of No-Tie solutions. (Bottom left) An example of a Tie solution. (Bottom right) An example of a Fork solution.

32