A Rational Choice Model of Compulsory                 VotingHow does Compulsory Voting alter Riker and Ordeshook’s (1968) ...
A Rational Choice Model of Compulsory VotingSince around 1960 steadily falling rates of voter turnout across most establis...
decisive in an election (Downs 1957; Palfrey and Rosenthal 1983). Obviously, this predictionmade by costly voting models i...
number countries the option exists for those who prefer abstention to indicate “none of theabove” on their ballot. Additio...
CVi < qiCNV + DiAnd at the apathy-limit (Di → 0) a rational citizen would vote if:                                        ...
achieve turnout of 99.99% in the Soviet Union. Even when penalties are significant somecitizens will still have high costs...
According to my sixth prediction voters under CV, the average voter under a CV systemshould have a lower sense of duty (or...
social democratic policies (McAllister and Mughan 1986; Nagel 1988; Pacek and Radcliff1995), and government spending (Hust...
So if a distinct income channel exists in determining turnout, what can be said of thepartisan effects of increased turnou...
Ordeshook‟s (1968) original approach of treating the explanatory variables in the „Calculusof Voting‟ as exogenous.Conclus...
BibliographyAbramovitz, M.         1956. “Resource and Output Trends in the United States since                       1870...
Helmke, G. and B. Meguid                     2007. “Endogenous Institutions: The Origins of Compulsory                    ...
Mackerras, M. and I. McAllister                      1999. “Compulsory Voting, Party Stability and Electoral              ...
Table 1: Compulsory Voting in 2008 (Table 2.1 in Birch 2009: 36)                              14
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A Rational Choice Model of Compulsory Voting

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A new rational choice model incorporating the institution of compulsory voting into the "Calculus of Voting" (1968). Compulsory Voting is a sure method of increasing voter turnout only if citizens
face significant sanctions and enforcement, and at the cost of those who cast a ballot being
on average more apathetic. Overtime it is likely that any potential partisan effects of compulsory voting are disappearing.

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A Rational Choice Model of Compulsory Voting

  1. 1. A Rational Choice Model of Compulsory VotingHow does Compulsory Voting alter Riker and Ordeshook’s (1968) “Calculus of Voting”? What is its impact on turnout and electoral outcomes? Jonathon Flegg London School of Economics and Political Science 10th May 2010 1
  2. 2. A Rational Choice Model of Compulsory VotingSince around 1960 steadily falling rates of voter turnout across most established democracieshave been interpreted by many as the cause of a “crisis of legitimacy” for democracy, and asone of the key trends indicative of ensuing Crouch‟s (2004) apocalyptic “post-democracy”.Arend Lijphart (1997), in his presidential address to the American Political ScienceAssociation in 1996, provoked a wave of new discussion when he argued that this“democratic dilemma” was becoming so critical that developed democracies need to considera raft of institutional changes to boost electoral turnout. The most controversial andefficacious was the introduction of compulsory voting (CV). Democratic government is apublic good, argue proponents such as Lijphart, and to avoid the free-rider problem votingshould be treated as an enforceable duty and not a voluntary right. But is compulsory votingreally the answer to society‟s lack of enthusiasm for exercising its franchise? What can wereasonably expect from compulsory voting, and does it introduce any partisan effects? Whileleaving the normative debate to one side (see Lijphart (1997) for the case in favour and Lever(2007) for the case against), this essay addresses these questions by building a rational choicemodel of compulsory voting and testing its predictions.The Calculus of VotingRational choice theories of electoral turnout begin by assuming that citizens make thedecision to vote or not vote based on rational self-interest. In the basic two-candidate modelproposed by Downs (1957), each voter considers their costs (Ci) of voting (for example, intime waiting to vote or in petrol used to drive to the polling station) and weighs that againstthe expected benefits (Bi) they might receive from the election of their preferred candidateover and above that received from the alternative candidate. This internal process, wherecitizen i will only vote if their expected utility (Ri) is positive, is often referred to as the„Calculus of Voting‟ (Downs 1957; Tullock 1967; Riker and Ordeshook 1968): Ri = pBi – CiWhile the expected benefits of a citizen‟s preferred candidate being elected can be and arelikely to be non-zero and positive, the actual utility of voting for the candidate is likely to bezero as the probability that the individual citizen‟s ballot will determine the election isessentially zero in large electorates. Hence rational voters should not vote because invariablypBi < Ci holds, where p equals the insignificant probability the individual voter will prove 2
  3. 3. decisive in an election (Downs 1957; Palfrey and Rosenthal 1983). Obviously, this predictionmade by costly voting models is an unsatisfying one for the simple observation that in alldemocratic elections citizens always do turnout in numbers significantly larger than zero.This contradiction has led some to declare it as the “paradox that ate rational choice theory”(Fiorina 1989).To save the Calculus of Voting from its own theoretical underpinnings, Riker andOrdeshook (1968) recognising that for the citizen the act of voting may include utility in andof itself. They incorporated a positive, civic duty (Di) term into the equation, where a citizenwould vote if their overall expected utility from voting (Ri) is positive: Ri = pBi – Ci + DiWhile not denying that certainly many citizens vote out of a sense of duty, the term is bothsubjective and defies measurement, making it essentially a residual „black box‟ term. Criticshave candidly overstated the implications of this point, arguing that it strips the Calculus ofVoting of its analytical value because all the important “action” is found within this term thatcannot be usefully understood through rational choice theory (Barry 1970; Mueller 2003:306). A fairer criticism, as alluded to by Aldrich (1993: 258), is that the duty term does notadd anything new of analytical value. While rational choice cannot explain where dutycomes from, the fact that it reconciles the equation to the empirics unleashes the rest of themodel to be used uninhibited in analysis. A useful analogy to macroeconomics would be therole of total factor productivity in production functions despite essentially being, asAbramovitz (1956) put it, a “measure of our ignorance”. Moreover cross-country analysis ofthe Riker and Ordeshook equation with panel data can easily deal with the duty termthrough fixed effects, if it is assumed to remain relatively constant within countries throughtime.The Institution of Compulsory VotingWhile CV is not the norm for democratic countries, it is currently used in 29 countries,around a quarter of all democracies. As can be seen from Table 1, half of the world‟s CVsystems punish abstainers with sanctions, around half have introduced the institution sinceWorld War II, and just over half of all countries with CV are found in Latin America. In onesense the term „compulsory voting‟ is a misnomer, as citizens are only required to attend thepolls. A citizen is not required to cast a preference for a valid candidate, and in fact in a 3
  4. 4. number countries the option exists for those who prefer abstention to indicate “none of theabove” on their ballot. Additionally, all current CV systems have certain exceptions from thelegal obligation to participate, that may include overseas residence or temporary absence,infirmity, literacy, or because of religious belief or practice.Multiple types of sanctions are employed in CV systems. Most common is a simple fine forthose who do not participate, ranging from three Swiss francs in the Swiss canton ofSchaffhausen to a hundred euros or more in Luxembourg for the first offence and rising upto as much as a thousand euros for repeated infringements. Some CV systems, rather thanapply monetary incentives, punish a non-voting citizen by withdrawing their franchise. InBelgium if a citizen fails to vote in four elections over a fifteen-year period, they loose theirfranchise for the following ten years. A number of Latin American countries such as Boliviaand Brazil sanction non-voters by prohibiting them from holding a position in the publicsector, or withholding access to public services. In the extreme, sanctions for those who failto vote can include imprisonment. Following the 1993 federal election in Australia, 43 non-voters who refused to pay a fine received prison sentences (Bennett 2005).How does Compulsory Voting Alter the Calculus of Voting?Making voting compulsory increases the costs of not voting, and therefore implicitly reducesthe overall opportunity cost of voting. A rational model of voting behaviour with CV mustsomehow incorporate the non-zero costs of abstention into the overall cost term (Kato 2007;Panagopolous 2008; Krasa and Polborn 2009). Crucially the assumption of costly abstentionrelies on both significant sanctions and enforcement for citizens who fail to vote(Panagopoulous 2008; Birch 2009). Specifically the cost of not voting will only be greaterthan zero if there is a positive interaction between a non-zero sanction (CNV) and theprobability or strictness of that sanction being enforced (qi). Building from the Riker andOrdeshook (1968) equation, if the true cost of voting under a CV system is Ci = CVi – qiCNV,then a citizen would choose to vote if their overall utility from voting (Ri) is positive: Ri = pBi – CVi + qiCNV + DiIf the electorate is heterogeneous, with a continuum of duty values from zero (politicallyapathetic citizens) to high (Di > CVi, citizens who value electoral participation and whowould vote even in the absence of compulsion), then rational citizens would vote if: 4
  5. 5. CVi < qiCNV + DiAnd at the apathy-limit (Di → 0) a rational citizen would vote if: CVi < qiCNVThese findings are crucial as it explains the real observation of non-universal turnout underCV systems. While CNV is a fixed sanction, CVi and qi have individual-level variation and areheterogeneous across the electorate. Citizens may still abstain under a CV system if theypossess or can acquire a large CVi or a low qi.The Effect of Compulsory Voting on TurnoutThe rational model of CV makes a number of predictions that we can examine in the realworld.Prediction 1: ceteris paribus, turnout can only be higher under CV compared with voluntary voting, asthe overall utility function of voting is increased by a positive value qiCNV.The most obvious prediction of CV systems is that turnout will increase because of the non-zero costs of abstention. This has been substantiated by a large number of empirical studiesthat find turnout to increase by between seven to sixteen percentage points in establisheddemocracies (Powell 1986; Jackman and Miller 1995; Franklin 1999; Birch 2009), and bybetween eleven and seventeen percentage points in Latin America (Fornos et al 2004). Thesize of the increase in turnout due to the introduction of CV has been shown to be mainlydetermined by how low turnout is prior to its introduction (Hirczy 1994), and of course bythe severity of the sanction and level of enforcement.Prediction 2: while turnout can only increase with CV, it will not achieve universal turnout as long asthere is individual-level variation in the values of CVi and qi.Individual-level variation in CVi and qi within the electorate also ensures that CV systems arenot a „magic bullet‟ for achieving complete turnout. The formal institution of CV operates asa legal mechanism and for enforcement purposes lacks the informational advantages ofinformal forms of coercive social or political mobilisation, such as was able to regularly 5
  6. 6. achieve turnout of 99.99% in the Soviet Union. Even when penalties are significant somecitizens will still have high costs to voting that will deter them from voting, such as beingoverseas during an election. Other citizens may reduce their probability of receiving asanction. In Australia, infringements are detected through the electoral roll, and so manywho seek to avoid detection simply never register in the first place.Prediction 3: increasing sanctions and increasing enforcement will increase turnout.Prediction 4: because of the interaction effect between qi and CNV we can predict that increasing bothsanctions and enforcement will increase turnout in excess of the effect of increasing each individually.Prediction 5: if qi or CNV or both are equal to zero then we can predict that turnout should beindistinguishable from that of a voluntary voting system.Taken together, predictions 3 to 5 would substantiate the channel by which greater turnoutis achieved. The interaction of sanctions and enforcement in CV systems has been a source ofrecent research. Panagopolous (2008) has divided all CV systems into three categories forseverity of sanctions and strictness of enforcement: no/low (-1), moderate (0), and high (+1).Using multivariate regression analysis Panagopolous found a positive and significantcorrelation between level of penalty, level of enforcement, and their interaction term withlevel of turnout, while controlling for economic growth, and electoral and legislativesystems. Moreover, CV systems with “no/low” sanctions and enforcement, such as Italy,Mexico and Thailand, showed turnout that was not statistically different to those withoutCV. In effect in these CV systems citizens have a compulsory duty to vote in an aspirationalconstitutional provision, but in effect they remain non-enforced or without penalty. In therecent volume by Birch (2009: 94) on CV, multivariate analysis failed to find any statisticallysignificant impact of CV on turnout until it was qualified as sanctioned CV, which was foundto be highly significant. The evidence to date appears to support the argument that CVincreases turnout primarily through non-trivial and enforced sanctions, and systems thatlack either sanctions or enforcement are in effect no different from voluntary systems.Prediction 6: if in a CV system a citizen will vote if Di > CVi - qiCNV, ceteris paribus, we would expectthe mean duty of a voter, D V , to be lower than under voluntary voting, where citizens will vote for allvalues that satisfy Di > CVi.  6
  7. 7. According to my sixth prediction voters under CV, the average voter under a CV systemshould have a lower sense of duty (or a greater sense of apathy) than the average voterwithout compulsion. Apathy may be manifest by a higher degree of intransitivity or protestin the ballot box. The counter argument to the prediction CV increases the average apathyamongst voters is that CV may have positive spillover effects on increasing citizen‟s dutyand political participation (Lijphart 1997). Similarly it has been suggested that CV increasesparty identification within the electorate (Mackerras and McAllister 1999). Regardless ofwhether such spillover effects occur or not, the evidence suggests that they are still notsufficient to reverse the prediction of lower duty among voters in CV systems. A number ofstudies find CV to be associated with an increase in spoiled ballots (Power and Roberts 1995;Mackerras and McAllister 1999; Reynolds and Steenberger 2006) and higher levels ofarbitrary voting, or “ballot order effects” (Power and Roberts 1995; Jackman 2001; King andLeigh 2009). Invalid voting is especially common in CV systems in Latin America, where thepercentage of spoiled ballots has exceeded forty percent in a number of elections.Partisan Effects of Compulsory VotingRiker and Ordeshook (1968) did not make any predictions about potential partisan effectsfrom their rational model. While those who fail to vote under voluntary voting could beexpected to have a lower sense of duty or a higher cost to voting, they treated both asexogenous and uncorrelated with omitted variables. However as long as the possibilityexists that the explanatory variables are correlated with omitted variables, the potential forpartisan effects exists.In actuality the model can make an obvious case for partisan effects if the cost of voting, CVi,represents the citizen‟s opportunity cost of voting, then turnout would be negativelycorrelated with income. CV therefore would have the effect of differentially inducing higher-income citizens to the polls and boosting the proportion of votes received by the party orparties they tend to support, typically regarded as right-wing parties. However thisdisinterested high-earner interpretation comes into direct conflict with a large body ofliterature that argues the exact opposite (DeNardo 1980; Pacek and Radcliff 1995; Lijphart1997). They argue that CV systems actually tend to benefit left-of-centre parties because ithas the effect of differentially induce lower-income citizens to the polls, the exact part of theincome spectrum who are most likely to abstain under voluntary systems. Corollary theoriesof this disinterested low-earner argument predict the introduction of CV to increase realised 7
  8. 8. social democratic policies (McAllister and Mughan 1986; Nagel 1988; Pacek and Radcliff1995), and government spending (Husted and Kenny 1997; Brookie 2008).What is the Evidence for Partisan Effects?While the disinterested low-earner argument suggests CV is an institution that benefits left-of-centre parties, the actual record of how the institution was introduced seems to supportthe former theory that higher-income citizens have a greater need for compulsoryinducement to the polls. In a multivariate regression analysis of 33 countries that adoptedCV between 1862 and 1998, Helmske and Meguid (2007) argued that the institution wasadopted because the contemporaneous expansion of suffrage and the burgeoning ranks of theworking class through industrialisation presented an acute electoral threat to incumbentright-wing governments. A perception existed that organised labour were succeeding inmobilising their core constituency to the polls through social coercion, a perceptioncrystallised in the growing strength of left-wing Labour parties within democraticlegislatures. To defend their incumbency, right-wing governments were compelled to inducetheir “rich and content” middle-class supporters to the polls through introducing CV.Helmske and Meguid (2007) substantiate their argument through showing the strongexplanatory power of right-wing incumbency, increasing vote shares of left-wing parties,and the percentage of workers in industry (as a negative measure of the right‟s naturalelectorate) in explaining episodes where and when CV was introduced. The history of CVsuggests that the existence of competing informal methods of achieving turnout, such associal coercion, can significantly influence the direction of possible partisan effects.Both the disinterested high- and low-earner arguments agree that CV impacts partisanadvantage specifically through an income channel. The veracity of the income channelappears to be supported by empirical analysis. Firstly, party competition must be based onsocio-economic cleavages (Lipset 1960; Pacek and Radcliff 1995), or more specifically incomemust have a determining effect on party support. There is a well-documented generalcorrelation between income and support for right-of-centre parties throughout all westerndemocracies, exception France where the Communists and Socialists receive higher supportamong higher income voters (McCarthy et al 2009). Secondly, there must be a relationshipbetween income and turnout. The empirical support between income and turnout is evenmore strongly documented (Blais et al 1996; Klingemann et al 1994), substantiating theargument that voluntary voting is generally skewed towards representing a higher incomemedian voter than that of the median citizen. 8
  9. 9. So if a distinct income channel exists in determining turnout, what can be said of thepartisan effects of increased turnout and introducing CV? The most often quoted finding isPacek and Radcliff‟s (1995) study that found turnout and turnout interacted with Powell‟s(1980) index of class-politics are both positively correlated with the proportion of votes wonby left-wing parties. Hence increasing turnout increases the proportion of left-wing electionvictories, and the effect is even more pronounced in countries where party competition isdivided along differences in class. This result however has come under fire from a largenumber of more recent studies finding little or no partisan effect of turnout (van der Eijk andvan Egmond 2007; Fisher 2007; Pettersen and Rose 2007). Birch (2009: 128) finds theintroduction of sanctioned CV does not significantly affect left-wing votes when controllingfor overall level of political and economic development. On government spending Brookie‟s(2008) regression analysis found that while a 1.0 percentage point increase in turnout wasassociated with a 0.15 point increase in government spending, the effect of CV wasinsignificant and had the wrong sign.While Lijphart (1997) disinterested low-earner argument, supported by the empirics ofstudies such as Pacek and Radcliff‟s (1995), remain the conventional wisdom with regards topartisan effects of turnout, its critics have become increasingly strident in recent years.There does seem to be a general trend towards particularly scant evidence for partisaneffects within countries with weaker socio-economic cleavages in party competition, such asthe United States and Canada. As Pacek and Radcliff (1995) themselves admitted, class-consciousness is eroding overtime as a determinant of voting in most developed democracies,and hence rendering any possible partisan advantage as increasingly impotent.The evidence from partisan effects from introducing CV, to the extent that it might beseparate from simply increased turnout, is even less compelling. There is absolutely noconclusive evidence to date that CV introduces any partisan effects. One explanation mayhave to do with the relatively large standard errors generated by the limited datasetavailable, for example Birch‟s (2009) study found sanctioned CV increased the share of left-wing party support by 3.75 percentage points, but the associated standard error was a hefty6.25. At a minimum the historical record shows that the introduction of CV systems was tiedto the perceived existence of partisan advantage, which is in itself a form of argument infavour of their existence. But overall the evidence is scant at best. The fact that establishingnot just the direction of partisan effects but their very existence validates Riker and 9
  10. 10. Ordeshook‟s (1968) original approach of treating the explanatory variables in the „Calculusof Voting‟ as exogenous.ConclusionThe introduction of CV remains a hotly debated topic within many developed democracies.This essay establishes a rational choice approach to understanding the institution of CV. Thenature of CV – a sanction applied to individuals as a simple incentive to vote – lends itselfperfectly to rational choice. CV is a sure method of increasing voter turnout only if citizensface significant sanctions and enforcement, and at the cost of those who cast a ballot beingon average more apathetic. Following Riker and Ordeshook (1968), a rational choiceexplanation considers the citizen‟s duty and costs as exogenous and uncorrelated withomitted variables such as income. While the possibility that partisan effects certainly exist,the direction and magnitude of the effect could only be determined by complex externalfactors such as the degree of socio-economic cleavages within party competition and theexistence of other informal methods of achieving turnout, such as social coercion. Overtimeit is likely that any potential partisan effects of introducing CV are disappearing. 10
  11. 11. BibliographyAbramovitz, M. 1956. “Resource and Output Trends in the United States since 1870”. American Economic Review 46:5–23.Aldrich, J. H. 1993. “Rational Choice and Turnout”. American Journal of Political Science 37(1): 246-78.Barry, B. 1970. Sociologists, Economists, and Democracy. London: Collier- Macmillan.Bennett, S. 2005. “Compulsory Voting in Australian National Elections”. Research No. 6 2005-06, Parliament of Australia Library.Birch, S. 2009. Full Participation: A Comparative Study of Compulsory Voting. Manchester: Manchester University Press.Blais, A., D. Blake, and S. Dion 1996. “Do Parties Make a Difference? A Reappraisal”. American Journal of Political Science 34: 514-20.Brookie, J. H. 2008. The Effect of Compulsory Voting on Government Spending. Unpublished dissertation. Available at: http://harri son brookie.weebly.com/uploads/3/7/4/5/374517/thesis.pdf.Crouch, C. 2004. Post-Democracy. Cambridge: Polity Press.DeNardo, J. 1980. “Turnout and the Vote: The Joke‟s on the Democrats”. American Political Science Review 74: 406-20.Downs, A. 1957. An Economic Theory of Democracy. New York: Harper.van der Eijk, C. and M. van Egmond 2007. “Political Effects of Low Turnout in National and European Elections”. Electoral Studies: 26(3): 561-73.Fiorina, M. 1989. “Is Negative Voting an Artifact?”. American Journal of Political Science 33: 423-39.Fisher, S. D. 2007. “(Change in) Turnout and (Change in) the Left Share of the Vote”. Electoral Studies 26(3): 598-611.Fornos, C. A., T. J. Power, and J. G. Garand 2004. “Explaining Voter Turnout in Latin America, 1980-2000”. Comparative Political Science 29: 205-24.Franklin, M. 1999. “Electoral Engineering and Cross-National Turnout Differences: What Role for Compulsory Voting?”. British Journal of Political Science 29: 205-24. 11
  12. 12. Helmke, G. and B. Meguid 2007. “Endogenous Institutions: The Origins of Compulsory Voting Laws”. Chicago: American Political Science Association.Hirczy, W. 1994. “The Impact of Mandatory Voting Laws on Turnout: A Quasi-Experimental Approach”. Electoral Studies 13: 64-76.Husted, T. and L. Kenny 1997. “The Effect of the Expansion of the Voting Franchise on the Size of Government”. Journal of Political Economy 105(1): 54-82.Jackman, S. 2001. “Compulsory Voting”. International Encyclopedia of the Social and Behavioural Sciences. Oxford: Elsevier.Jackman, R. W. and R. A. Miller 1995. “Voter Turnout in the Industrial Democracies in 1980s”. Comparative Political Science 27: 467-92.Kato, K. 2007. “How Compulsory Voting Can Improve Turnout: Understanding the Mechanisms”. In Compulsory Voting: Principles and Practice. Helsinki: European Consortium for Political Research.King, A. and A. Leigh 2009. “Are Ballot Order Effects Heterogeneous?”. Social Science Quarterly 90(1): 71-87.Klingemann H.-D., R. I. Hofferbert, and I. Budge 1994. Parties, Policies, and Democracy. Boulder, Colorado: Westview.Krasa, S. and M. K. Polborn 2009. “Is Mandatory Voting Better than Voluntary Voting?”. Games and Economic Behaviour 66: 275-91.Lever, A. 2007. “Compulsory Voting: A Critical Perspective”. In Compulsory Voting: Principles and Practice. Helsinki: European Consortium for Political Research.Lijphart, A. 1997. “Unequal Participation: Democracy‟s Unresolved Dilemma”. American Poetical Science Review 91(1): 1-14.Lipset, S. M. 1960. Political Man: The Social Bases of Politics. Garden City, NY: Doubleday.McAllister, I. and A. Mughan 1986. “Differential Turnout and Party Advantage in British General Elections”. San Francisco: Political Science Association.McCarthy, N., J. Pontusson and T. Yamamoto 2009. “Voting, Income Inequality, and Polarization”. Dublin: Annual ECPR Political Economy Workshop. 12
  13. 13. Mackerras, M. and I. McAllister 1999. “Compulsory Voting, Party Stability and Electoral Advantage in Australia”. Electoral Studies 18: 217-33.Mueller, D. 2003. Public Choice III. Cambridge: Cambridge University Press.Nagel, J. H. 1988. “Voter Turnout in New Zealand General Election, 1928- 1988”. Political Science 40: 16-38.Pacek, A. and B. Radcliff 1995. “Turnout and the Vote for Left-of-Centre Parties: A Cross- National Analysis”. British Journal of Political Science 25: 137-143.Palfrey, T. R. and H. Rosenthal 1983. “A Strategic Calculus of Voting”. Public Choice 41(1): 7-53.Panagopolous, C. 2008. “The Calculus of Voting in Compulsory Voting Systems”. Political Behavior 30: 455-67.Pettersen, P. A and L. E. Rose 2007. “The Dog that Didn‟t Bark: Would Increased Turnout Make a Difference?”. Electoral Studies 26(3): 574-88.Powell, G. B. 1980. “Voting Turnout in Thirty Democracies”. In R. Rose (ed.), Electoral Participation: A Comparative Handbook: 5-30. Los Angeles: Sage. 1986. “American Voter Turnout in Comparative Perspective”. American Political Science Review 80: 17-43.Power, T. and J. T. Roberts 1995. “Compulsory Voting, Invalid Ballots, and Abstention in Brazil”. Political Research Quarterly 48: 795-826.Reynolds, A. and M. Steenbergen 2006. “How the World Votes: The Political Consequences of Ballot Design, Innovation and Manipulation”. Electoral Studies 25(3): 570- 98.Riker, W. H. and P. C. Ordeshook 1968. “A Theory of the Calculus of Voting”. American Political Science Review 62: 25-42.Tullock, G. 1967. Towards a Mathematics of Politics. Ann Arbor: University of Michigan Press. 13
  14. 14. Table 1: Compulsory Voting in 2008 (Table 2.1 in Birch 2009: 36) 14

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