Parliament Shapes and Sizes

PARLIAMENT SHAPES AND SIZES

RAPHAEL GODEFROY and NICOLAS KLEIN∗

This paper proposes a model of Parliamentary institutions in which a society makes three decisions behind the veil of ignorance: whether a Parliament should comprise one or two chambers, what the relative bargaining power of each chamber should be if the Parliament is bicameral, and how many legislators should sit in each chamber. We document empirical regularities across countries that are consistent with the predictions of our model. (JEL D71, D72)

between the population and the number of repre- I. INTRODUCTION sentatives.2 In “Federalist No. 62,” he argues in Parliamentary institutions vary widely favor of an Upper Chamber as a safeguard against 3 across countries. For instance, the Indian the errors of a large Lower Chamber. Lower Chamber—Lok Sabha—has 543 to 545 However, little is known on the institutional seats, and the Indian Upper Chamber—Rajya regularities of Parliaments across countries. Sabha—has 250. The U.S. Congress has 535 Indeed, only two stylized facts have been docu- seats: 435 in the House of Representatives and mented: a linear relationship between the log of 100 in the Senate. The Luxembourg Parliament the size of the population and the log of the size has 60 legislators, in a single chamber.1 As of of Parliament (Stigler 1976) and an increasing 2012, there were 58 bicameral and 110 unicam- relationship between the population size and eral systems recorded in the Database of Political the probability to have a bicameral Parliament Institutions (DPI; Beck et al. 2001 [updated (Massicotte 2001). in 2012]). The purpose of this paper is twofold: to pro- The main questions that arise when design- pose a model of Parliament design and to doc- ing Parliamentary institutions are fundamentally ument empirical institutional regularities across quantitative: should there be one or two cham- countries, following the lead of the predictions of bers, what should be their respective bargaining the model. One of our contributions is that our power, what should be the size of Parliament, and simple model generates predictions concerning so on. James Madison, in the “Federalist No. 10,” variables that have an unambiguous equivalent in postulates a concave and increasing relationship the data we observe.

2. “In the first place, it is to be remarked that, however ∗ The authors gratefully acknowledge financial support small the republic may be, the representatives must be raised from the Fonds de Recherche du Québec—Société et Culture. to a certain number, in order to guard against the cabals of a The second author also gratefully acknowledges financial few; and that, however large it may be, they must be limited support from the Social Sciences and Humanities Research to a certain number, in order to guard against the confusion of Council of Canada. a multitude. Hence, the number of representatives in the two Godefroy: Assistant Professor, Department of Economics, cases [will be] proportionally greater in the small republic” Université de Montréal and CIREQ, Montréal, Quebec (Madison 1788a). H3T1N8, Canada. Phone 514-343-2397, Fax 514-343- 3. “The necessity of a senate is not less indicated by the 7221, E-mail [email protected] propensity of all single and numerous assemblies, to yield Klein: Assistant Professor, Department of Economics, to the impulse of sudden and violent passions, and to be Université de Montréal and CIREQ, Montréal, Que- seduced by factious leaders into intemperate and pernicious bec H3T1N8, Canada. Phone 514-343-7908, Fax resolutions” (Madison 1788d). 514-343-7221, E-mail [email protected]

1. Throughout the paper, we use the term “legislator” to refer to a member of Parliament. If a Parliament is unicameral, ABBREVIATIONS we refer to its chamber as the House or the Lower Chamber, DPI: Database of Political Institutions indifferently. If there is a second chamber, we refer to it as the Senate or Upper Chamber, indifferently. i.i.d.: Identically and Independently Distributed

There are two stages in the model: a Constitu- from a two-stage game. In the first stage, par- tional stage and a Legislative stage. At the Con- ties may form coalitions to maximize the share of stitutional stage, society, which is made up of a seats they obtain in a proportional single-district discrete population of individuals, designs a Con- election. In the second stage, each individual stitution by writing three decisions into the social casts one vote, either for a party or for a coalition contract: whether a Parliament should comprise of parties, to maximize his expected utility, which one or two chambers, what the relative bargain- is a function of the policy adopted, under per- ing power of each chamber should be if the Par- fect information about legislators’ partisan affili- liament is bicameral, and how many legislators ations, but imperfect information about the exact should sit in each chamber. These decisions are value of their bliss points.5,6 made behind a veil of ignorance, that is, before The problem at the Constitutional stage the members of society know their own prefer- involves two trade-offs. On one hand, there is a ences. As everyone is identical behind the veil of trade-off between the unit cost of a member of ignorance, they will unanimously agree that the Parliament and the marginal benefit of having a goal should be to maximize the expected utility larger Parliament to lower the risk that the policy of the representative agent. They know that each will be decided by legislators whose preferences individual’s utility is the opposite of the quadratic are far from those of the population at large. difference between the policy adopted by the Par- One could see this trade-off in terms of external liament and the individual’s bliss point, which costs and internal or decision-making costs. are both real numbers, minus the costs associated (External costs are the costs that individuals have with the functioning of Parliament. to bear as a result of others’ decisions whenever We assume that the population is partitioned an action is chosen collectively. Internal costs into parties and that two types of issues may arise: stem from an individual’s participation in an partisan and nonpartisan. For nonpartisan issues, organized activity, such as legislative bargaining, individuals’ bliss points are identically and inde- see Buchanan and Tullock 1962, Chapter 15.) pendently distributed (i.i.d.), whereas for partisan The former decrease in expectation as the size issues, all members of a party share the same of Parliament increases; indeed, the larger the bliss point, the party’s bliss point. Parties’ bliss Parliament the more precisely it will estimate the points are i.i.d. across parties. Individuals do not average bliss point of the population. The latter know their partisan affiliations at the Constitu- costs, by contrast, will increase as Parliament tional stage. Nor do they know which type of becomes larger; indeed, the more numerous the issue may arise, the distribution of shares of par- assembly, the costlier it will be to ensure its ties’ members in the population, or the realized proper functioning. In our model, we assume that distribution of bliss points in the population or the marginal internal cost of parliamentarians is among legislators. constant. On the other hand, there is a trade-off At the Legislative stage, the Parliament between the unit cost of a chamber and the decides on a policy. To formalize society’s infor- benefit of instituting a Senate to mitigate the mation about Parliament’s decision process, we negative impact of the error term in the adopted assume that only one issue arises, and that the policy. That error term formalizes in the simplest policy adopted for that issue is the weighted aver- way possible the common point in defenses age of the policies that maximize the aggregate of bicameralism, such as Madison’s, which is utility of each chamber, subject to some error.4 that members of the Lower Chamber might not We consider two allocation systems of Parlia- mentary seats: a proportional representation and 5. Any game using a nonproportional voting system a nonproportional representation system. In the requires many more assumptions than the game in which voting is at-large and proportional. For the latter, we would need proportional representation system, the distribu- to specify the number of districts, the size of each district tion of seat shares across parties is equal to the (which may not be uniform in practice), the distribution of distribution of shares of partisans in the popula- partisans in each district, the possibility for parties to form coalitions within and across districts, and so on. We do not tion. In the nonproportional system, these distri- propose such a model; instead, we make an assumption on the butions may differ. In an extension (Appendix C), system of allocation of seats that is consistent with an empir- we show that the distribution of legislators in the ical analysis of nonproportional voting systems. proportional representation system is obtained 6. The conclusions of the model would be the same if we considered an electorate whose votes are determined by partisan loyalty, which is the main determinant of voting 4. In a unicameral system, the policy adopted is the policy behavior according to Achen and Bartels (2016), rather than proposed by the unique chamber. policy preferences. GODEFROY & KLEIN: PARLIAMENT SHAPES AND SIZES 3 adopt the best policy according to some measure in nonproportional representation systems. First, of welfare. we estimate that, on average across observations The model provides several predictions. First, at the country/party/election year level, the party the log of the size of Parliament increases linearly ranked first by decreasing share of votes obtains with the log of the size of the population, which around 10% of seats in the Lower Chamber. Sec- is consistent with one of the two stylized facts ond, we find that the rest of the seats are allo- mentioned above. cated across all parties, including the party ranked Second, the number of legislators ought not to first, so that the share of seats and the share of depend on the unicameral or bicameral structure votes of a party are the same, regardless of the of the legislature. party’s rank. Third, the relative bargaining power of a The rest of the paper is set up as follows: chamber in a bicameral system should optimally Section II reviews related literature; Section III be equal to the share of legislators belonging to presents the model, which is used to derive the this chamber. predictions exhibited in Section IV; Section V Fourth, countries with a larger population are presents the empirical analysis; Section VI con- more likely to have a bicameral legislature, the cludes. Descriptive statistics are in Appendix A. other stylized fact mentioned above. Fifth, the Appendix B presents tests of the assumptions model predicts no impact of the factors that affect used in nonproportional representation systems. the size of Parliament, except the size of the pop- Appendix C presents a two-stage voting game ulation, on whether a country has a unicameral or that substantiates the reduced-form analysis of bicameral Parliament. proportional systems. We find that the number of legislators depends on whether the system is proportional or not. Our sixth prediction is that in a proportional voting II. LITERATURE REVIEW system, the makeup of the partisan structure of Our contribution relates to the literature on a country does not affect the number of legisla- 7 tors. Our seventh prediction is that, in nonpro- endogenous political institutions. The distinc- portional systems, by contrast, we predict that as tion between a Constitutional stage, in which the partisan structure becomes more dispersed, as decision procedures are created, and a subsequent measured by the Herfindahl–Hirschman index, legislative stage, in which a policy is chosen, was the number of legislators ought to rise. To derive studied in Romer and Rosenthal (1983). While testable predictions, we compute the optimal Romer and Rosenthal (1983) find a certain una- number of seats for a specific nonproportional nimity rule to be optimal in their setting, Aghion system in which a fixed number of seats are and Bolton (2003) find that it is optimal to require an interior majority threshold for a change in the granted to the party with the biggest share of 8 partisans in the population, and the remaining status quo policy. seats are randomly distributed across all parties, The seminal theory concerning the size of leg- including the largest party, according to their islatures is the cube root formula proposed by share of partisans in the population. Taagepera (1972), which holds that the num- Empirically, we find that all these predic- ber of legislators should equal the cube root tions are consistent with the results of a series of the population of a country. This formula is of estimations that we conduct on a sample of designed to prevent excessive disproportionality 75 nonautocratic countries for which we have in representation—see Lijphart (2012). Theoret- detailed political information over the period ically, it results from the minimization of the 1975 to 2012. For the average bargaining power number of communication channels for assembly of Upper Houses across bicameral countries, we members, who need to communicate with their use the estimation provided in Bradbury and Crain (2001). 7. Stigler (1992) claimed a role for economists in the study of legal institutions, writing: “Understanding the Our model also provides specific predictions, source, structure, and evolution of a legal system is the kind of which are borne out by the data, concerning the project that requires skills that are possessed but not monopo- coefficient of the Herfindahl–Hirschman index lized by economists, for it is in good part an empirical project of partisan fractionalization as well as that of addressed to rational social policy.” 8. The crucial difference is that Aghion and Bolton other terms. (2003) assume that there is a deadweight loss associated with Finally, we provide two tests of the assump- implementing monetary transfers designed to compensate the tion we use to compute the size of Parliament losers of the reform being proposed. 4 ECONOMIC INQUIRY constituents while also communicating with their on the process of legislative bargaining and sub- fellow assembly members. Empirically, this for- sequent policy choices. Ansolabehere, Snyder, mula underpredicts the sizes of Parliaments. and Ting (2003) analyze a situation where the Auriol and Gary-Bobo (2012) adopt a House has one member per district while the Sen- mechanism-design approach to the design of ate has one member per state. As in Baron and Parliamentary institutions. In their model, the Ferejohn (1989), the object of legislative bargain- principal (the “Founding Fathers”) does not ing is to divide a fixed budget of resources. House know the distribution of preferences over a districts are equal in population size, while states one-dimensional policy choice to be made, and may encompass several House districts. Legis- knows that no agent in society will know them. lators in both chambers are responsive to their Rather, the “Founding Fathers” have a diffuse respective median voter. Both chambers have to prior over the set of possible distributions of agree to a proposed division of the resources, and preferences. Furthermore, there is an execu- both vote by majority rule, but only representative branch, made up of a randomly chosen tives can propose a bill. They show that smaller single agent who is the residual claimant of states, which are over-represented in the Senate, all decision rights not specifically delegated to do not get a higher per capita share of spend- the legislative branch. The legislative branch, ing. The reason is that a minimum winning coali- by contrast, is made up of n randomly chosen tion in the House carries a majority in the Senate agents, whose role consists of revealing their as well. However, small state bias reappears if preferences truthfully. Truthful revelation is there are supermajority rules in the Senate, Sen- effected by a Vickrey-Clarke-Groves mechanism ators have proposal power, or goods are lumpy applied to the agents making up the legislative (i.e., they cannot be targeted toward a single dis- branch. In our model, by contrast, there is no trict). Kalandrakis (2004) analyzes a variant of executive branch, and the legislators’ salaries are this model where resources can only be targeted exogenously given. This simpler model allows us at the state level, and Senators can be recognized to derive additional predictions concerning the as proposers as well. He finds that supermajorities structure of Parliaments, such as their bicameral may occur in equilibrium, but they only ever do or unicameral nature, the impact of different in one chamber at a time. Parameswaran (2018) voting systems, and the effect of preferences that adopts this model by letting the goods be targeted are homogeneous within given groups. at the House-district level. He finds that small We do not explicitly model the decision states may actually fare worse with a Senate in process in Parliament, which is the focus of which they are over-represented than in a uni- the legislative bargaining literature, starting with cameral system without malapportionment. The Baron and Ferejohn (1989), Baron and Diermeier reason is that, if a Senator from a big state is rec- (2001), and Diermeier and Merlo (2000). ognized as the proposer, he can at no cost buy As the predictions of these bargaining models off all the House members from his state. Thus, often depend on the fine details of the bargaining he needs to include fewer small state legislators protocol, which varies widely across countries, in his winning coalition, which in turn makes it we instead use a spatial model of policy pref- less likely that representatives from a small state erences and assume that each chamber adopts will be included in the winning coalition. Knight as its policy the average bliss point of its mem- (2008) investigates empirically the role of repre- bers. This policy maximizes the sum of leg- sentation for the division of funds in the United islators’ utilities, and thus corresponds to the States. He finds that small states receive a larger behavior they would optimally adopt in a set- share of appropriations originating in the Senate ting in which utilities are transferrable.9 In con- than of those originating in the House. He dis- trast to, for instance, the analysis in Tsebelis and tinguishes between the proposer-power channel Money (1997), our legislators do not anticipate resulting from a state’s representation on the rele- the impact of their decisions on the bargaining vant committees and the vote-cost channel result- process with the other chamber in a bicam- ing from the proposer’s need to build a winning eral system. coalition. Snyder, Ting, and Ansolabehere (2005) The literature on bicameralism has devoted show that when votes are weighted in a unicam- some attention to the impact of a second chamber eral setting, a legislator’s ex ante expected share of the resources equals his voting weight. Thus, the biases introduced by a malapportioned sec- 9. See also our discussion in Section III.B. ond chamber could in principle be replicated by GODEFROY & KLEIN: PARLIAMENT SHAPES AND SIZES 5 a unicameral system with weighted voting. Vespa stability (Blais 1991; Grofman and Lijphart (2016) runs a laboratory experiment testing the 1986). Aghion, Alesina, and Trebbi (2004) impact of weighted voting in a unicameral setting show that countries with greater ethno-linguistic as compared to bicameralism with a malappor- fractionalization are more likely to have plu- tioned second chamber, confirming the theoreti- rality voting systems, which are interpreted as cal predictions that there will be a small state bias a means of achieving greater insulation of the with weighted voting. In a bicameral system, this political leadership. While also minimizing the bias appears if and only if Senators have proposal role of strategic behavior in the establishment of power, as predicted by Ansolabehere, Snyder, and political parties, Lijphart (1990) finds that the Ting (2003).10 voting system has little impact on the number of Facchini and Testa (2005) study the inter- parties. Finally, a few studies link voting systems action of legislators with lobbying groups in a to voter turnout (Herrera, Morelli, and Nunnari bicameral setting. Rogers (1998) also formalizes 2016; Herrera, Morelli, and Palfrey 2014). Our an informational justification for bicameralism. paper contributes to this literature by relating the He assumes that larger chambers have lower costs structure of Parliament to the voting system and for information acquisition, giving them a larger to partisan fractionalization. first-mover advantage. Our paper complements these papers. Instead III. MODEL of looking in detail at a model that tries to formalize the U.S. Congressional system in a Collective A. Setup Bargaining framework, we use a reduced form We consider a population of M individuals. model to formalize as simply as possible the most At the Constitutional stage, they want to maxi- universal motivation in favor of a Senate, which is mize the representative agent’s expected utility to reduce the probability of legislative errors. We through their choice of the setup of Parliament. do not attempt to explain where such error comes They choose whether Parliament will be unicam- from. Instead, we formalize in greater detail what eral or bicameral, and set the number of mem- the Parliament designers’ objectives may be. For bers of the House nH. If they choose a bicameral the same reasons, our empirical analysis aims less Parliament, they also set the number of mem- at explaining the policies adopted by a specific bers of the Senate nS, and the bargaining power Parliament, such as the U.S. Congress, than at of the House and the Senate, denoted 𝛼 and explaining what may have motivated the specific 1 − 𝛼, respectively. design features (number of seats and number of Preferences. An individual i’s utility is: ( ) Chambers) of Parliaments across countries in the 2 first place. ui (x) =− x − xi − ((NC + nc) ∕M) We also do not consider issues pertaining to where x ∈ ℝ is the policy to be adopted by the Par- the acquisition or transmission of information ℝ liament, xi ∈ is i’s bliss point, N ∈ {1, 2} is the within the legislature, all legislators being per- number of chambers, C the unit cost of a cham- fectly informed of their respective bliss points. ber, n the number of members of Parliament, and By contrast, an early investigation of legislators’ c the unit cost of a member of Parliament.11 incentives to acquire information is provided by The first term represents the payoff obtained Gersbach (1992). Austen-Smith and Riker (1987) from the policy adopted, and the second term analyze legislators’ incentives to reveal or con- represents the per capita contribution to the fund- ceal private information they may have. ing of Parliament.12 Our paper is also related to the literature on electoral systems. The argument most often cited Distribution of Bliss Points. The population is in favor of proportional systems is that they lead partitioned into partisan groups or parties, and to a more faithful representation of a population’s opinions in Parliament, whereas plurality 11. The unit cost of a chamber may comprise its main- systems are more likely to obviate the need for tenance cost and the potential rent of the building where its multiparty coalitions, thus leading to greater members meet. The unit cost of a member of Parliament may comprise his salary. 10. See also Buchanan and Tullock (1962, Chapter 16), 12. While we make the assumption of quadratic util- Riker (1992), Diermeier and Myerson (1994), Diermeier and ity for the purpose of tractability, one could interpret the Myerson (1999), Rogers (1998), Bradbury and Crain (2001), quadratic utility function as a second-order Taylor approx- Tsebelis and Money (1997), and the references in these imation of a more general utility function. Indeed let ui(x) papers. be agent i’s smooth utility function depending on the policy 6 ECONOMIC INQUIRY two types of political issues may arise: partisan utilities of the members of each chamber, subject and nonpartisan. For partisan issues, all members to some error, where the weight is the bargaining i of a partisan group share the same bliss point; power conferred to each chamber at the Constitu- bliss points are drawn independently across par- tional stage. ̃ ties. For nonpartisan issues, bliss points are drawn We assume that the error term ZX is indepen- independently across individuals. dently drawn from a distribution of mean zero > We assume that the distribution from which and square mean vX 0. The error term cap- bliss points are drawn, either parties’ bliss points tures any difference between the policy adopted for partisan issues or individual bliss points for by a chamber and the cooperatively optimal pol- nonpartisan issues, is continuous and has a vari- icy for its members. It may be interpreted as ance 𝜎2 ∈ (0, ∞). the result of imperfections in the deliberation process, which may manifest themselves in var- Policy Adopted at the Legislative Stage. Legisla- ious ways: it could be the probability that a tors have the same type of preferences as the rest subgroup of members imposes its favorite pol- of the population. Once in Parliament, legislators icy, or that negotiations among members fail, adopt a policy for the unique issue that has arisen. and so on. With probability q ∈ [0, 1], the issue that arises is partisan. Information. At the time of the design of Par- If Parliament members are numbered 1 liament, it is not known which issue will arise, through nH + nS, with members of the House or what the actual distribution of bliss points numbered first and members of the Senate num- across individuals or across legislators will be. It bered second, we assume that the policy adopted is known, however, that there will be G parties. for this issue is:( ) Society also has a prior belief at the Constitu- ∑nH tional stage regarding the shares( of) these groups ∗ 𝛼 1 ̃ G (1) x = xk + ZH 𝛄 𝛾 , G n in the overall population, = g ∈ (0 1) H k=1 ∑ g=1 G 𝛾 (with g=1 g = 1). They know too how the leg- ( ) islators’ preferences map into the adopted policy nH∑+nS (Equation (1)). 𝛼 1 ̃ + (1 − ) xk + ZS nS k=nH+1 Representation in Parliament. We consider two systems of representation: a system of propor- where x is legislator k’s bliss point for the k tional representation in which the partisan distri- issue discussed, 𝛼 represents the bargaining ̃ bution of shares of seats in Parliament is equal power of the House, and ZX is an error term for to the distribution of shares of partisans in the chamber X. population, and a system of nonproportional rep- The policy adopted is the weighted average resentation. Legislators’ bliss points for nonpar- of the policies that maximize the sum of the tisan issues are i.i.d. decision x. By taking the Taylor expansion of u at agent i’s i B. Discussion of Assumptions preferred policy xi we can write ( ) 2 Society Sets Up a Parliament. Why would soci- ( ) ( ) ( ) x − xi ( ) u (x) = u x + x − x u′ x + u′′ x ety set up a Parliament in the first place? This i i i i i i 2 i i (( ) ) is a question broached by Buchanan and Tul- 2 . + o x − xi lock (1962, Chapter 15). On one hand, dele- gating decision-making powers to a Parliament If the population taken into account at the Constitutional increases the external costs, as compared to direct stage is homogeneous so that x will be close enough to xi,the democracy, because there will always be some 2 terms o((x − xi) ) will be small enough to be neglected. As, chance that Parliamentary representation might by definition, u assumes its maximum at x , u′(x ) = 0 ≥ u′′ i i i i i lead to a biased sample of population preferences. (x ). If individual i is risk averse, u′′(x ) < 0, and maximiz- i [ ] i i [ ( ) ] Yet, as Buchanan and Tullock (1962, paragraph 𝔼 𝔼 2 ing ui will be tantamount to minimizing − x − xi . 3.15.6) argue, the risk of bias has to be traded Anecdotal historical evidence suggests that Parliamentary off against the decrease in decision-making costs. institutions have often been ushered in by and for groups of like-minded people, which is consistent with the fact that the Representative (as opposed to direct) democracy degree of population homogeneity is a critical determinant of facilitates collective action. Equations (2) and (3) the size of a nation (Alesina and Spolaore 2005). can be interpreted as the formalization of this GODEFROY & KLEIN: PARLIAMENT SHAPES AND SIZES 7 very trade-off13; the benefit part in the equations The Electoral System Is Exogenously Given. In corresponds to a decrease in external costs, while our model, society should always prefer a pro- the cost part could be interpreted as an increase portional voting system over a nonproportional in decision-making costs, as the size of Parlia- system. This is because we do not integrate any ment increases. Madison also discusses the opti- offsetting benefits of nonproportional systems, mal size of the House of Representatives, inter which may for instance lead to greater stabil- alia, in the “Federalist No. 55” and “Federalist ity of Parliamentary majorities and may favor No. 58,”14 see the discussion of the size princi- stronger ties between legislators and their con- ple in Ostrom (2008, Chapter 5). Ostrom (2008) stituents, thus leading to greater accountability. In points out that the larger a legislative assembly particular, we assume that there is no relationship becomes, the less time individual representatives between the voting system and the cost of Par- will have to make their arguments, as only one liament or with the error terms that arise in the representative can speak at a time. Thus, large adopted policy of either chamber. This assump- assemblies will tend to be dominated by an oli- tion implies that the voting system is not signif- garchic leadership, as is arguably the case in the icantly correlated with the probability to have a House of Commons in the United Kingdom. At second chamber; we examine this implication in the same time, a certain minimum number of rep- Section V. resentatives is needed “to secure the benefits of In a system of proportional representation, a free consultation and discussion, and to guard party’s share of seats corresponds to its share of against too easy a combination for improper pur- the votes. Thus, we assume that voters vote for poses” (Madison 1788b). their respective parties. We show in Appendix While it is necessary for society to know the C that this is the unique subgame-perfect Nash variance in the distribution of bliss points at the equilibrium outcome if voters only know their Constitutional stage already, none of our calcu- partisan affiliations at the time of the vote. In par- lations depend on society’s already knowing the ticular, it is assumed that they do not know the mean or any other characteristics of this distri- realizations of the various parties’ bliss points bution. If the dispersion of this mean is large at the time of voting. This assumption captures enough, it will not be practical for society to set the idea that voters will be uncertain about the a policy at the Constitutional stage already. details of the question that will arise before the Legislature. Wages c Are Exogenous. We could endogenize Achen and Bartels (2016) argue that vot- wages by assuming that higher wages will lead ers choose parties and candidates on the basis to better legislators being selected, that is, vX is of social identities and partisan loyalties, even decreasing in c. We refrain from doing so here adjusting their policy views to match these loy- because we do not observe vX in the data and thus alties. Such behavior would be perfectly in line have no way of ascertaining the functional form with our assumptions.15 The observation that of the dependency of vX on c. the link between voters’ partisan loyalties and their policy preferences was tenuous would cor- The Population Considered at the Constitutional respond to a low q in our model. Stage May Differ from the Actual Population. Such a case may arise for two reasons: a fixed size The Role of the Second Chamber. While of Parliament may still apply many years after one rationale that is often used to justify the it was set, when the size of the population has existence of second chambers is to give some changed, or the preferences of parts of the popu- (over-)representation to certain minorities, our lation, for example, a disenfranchised population, model makes an implicit assumption that either may not be included in the Constitution Design- chamber equally represents any citizen. Indeed, if ers’ objective functions. the goal were to (over-)represent some minority, We need not interpret M as the actual popula- this could also be achieved by quotas or weighted tion size; in fact, our predictions would continue voting in a unicameral system (see Snyder, Ting, to hold if M were a fraction or a multiple of the and Ansolabehere 2005 for a Baron and Ferejohn actual population size. 1989 bargaining type model). By the same token,

13. We are indebted to an anonymous referee for pointing 15. Of course, Achen and Bartels (2016) focus their out this connection to us. analysis on the United States, which does not use proportional 14. See Madison (1788b, 1788c). representation. 8 ECONOMIC INQUIRY if the role of a second chamber were merely this is no doubt a somewhat naïve view of the to make it more difficult to pass legislation, legislative process, there is some evidence that the same could be achieved in a unicameral legislators will often trade their votes intertem- system, for example, by imposing supermajority porally across issues (so-called log-rolling, see, rules. Indeed, Cutrone and McCarty (2006) con- e.g., Stratmann 1992). If one takes for granted clude: “Regardless of the theoretical framework that the utility legislators derive from the enact- or the collective action problem to be solved, ment of particular policies is transferable in we find that both the positive and normative this way, legislators should always adopt the arguments in favor of bicameralism tend to be policy that maximizes the sum of their utilities, weak and underdeveloped. Most of the effects of as we assume in Equation (1). Any failure to do bicameralism are due primarily to quite distinct so would amount to an error on their part, as institutional choices, such as malapportionment it reduces the surplus to be distributed among and super-majoritarianism, which correlate them. In our opinion, this admittedly idealized empirically with bicameralism. There seems to view of the legislative bargaining process has the be no logical reason why the benefits of these merit of not depending on the fine details of the institutions—like the protection of minority Parliamentary procedures governing coalition rights and the preservation of federalism—could formation within and across legislative chambers not be obtained by a suitably engineered uni- (e.g., the choice of proposer, the navette system cameral legislature or through vigorous judicial between chambers, etc.), which vary widely review.” across countries. By contrast, we see the goal of bicameralism, which cannot easily be reproduced within IV. RESULTS a unicameral system, as providing for a second, independent, deliberative process in the law- We first analyze the case of proportional vot- making procedure.16 Indeed, we view represen- ing, before turning to nonproportional systems. tative democracy as a sampling of population We neglect integer problems throughout preferences, which are subsequently aggregated. our analysis. Each Parliamentary chamber corresponds to such an aggregation process. As aggregation processes A. Proportional Representation are subject to error, it can be useful to have a second, independent aggregation process. Consis- For Proportional Representation, we assume tent with this view, we abstract from any design that the shares of a party’s members in the popu- differences in the two legislative chambers, as it lation and in Parliament are equal. is not clear why certain peculiarities of second Unicameral Parliament. If a nonpartisan issue chambers, for instance those designed to protect arises, which happens with probability (1 − q) ∈ a minority or the federal structure of a country, (0, 1), the policy choice is could not be replicated in a unicameral setting. ∑ n x The analysis of the impact of a malapportioned ∗ d=1 d ̃ , second chamber, or malapportionment in general, x = + ZH n ( ) which motivates much of the literature on bar- n where n is the number of legislators, and xd d=1 gaining in bicameral legislatures (see our discus- are their bliss points. These are i.i.d. draws from sion in Section II), is thus left outside the scope a distribution with variance 𝜎2. of this paper. If∑ a partisan issue arises, the policy choice is: x∗ = G 𝛾 x + Z̃ . The Legislative Bargaining Process and the g=1 g g H Determination of the Adopted Policy. We assume At the Constitutional stage, society maximizes: that, at the Constitutional stage, the Constitution [ ] [( ) ] 2 C + n c designers anticipate that legislators within either (2) 𝔼 u =−𝔼 x∗ − x − H i i M Chamber act cooperatively when choosing the { ( [ ]) policy to be adopted (see Equation (1)). While ∑G 𝜎2 𝔼 𝛾2 =− q 1 − g 16. As we shall see below, more heterogeneous countries g=1 are not significantly more likely to have a bicameral Parlia- ( )} ment. One could interpret this finding as suggesting that the C + n c 1 H . primary role of a second chamber was not to enhance the rep- + (1 − q) 1 + − vH − resentation of minorities in heterogeneous countries. nH M GODEFROY & KLEIN: PARLIAMENT SHAPES AND SIZES 9

Thus, when all parties are guaranteed propor- Unicameral Parliament. First, fix a realization 𝜸 tional representation in Parliament regardless of of partisan shares .Letkg be the number of its size, only the nonpartisan issues matter for legislators who belong to party g. Given that a the optimal size of Parliament. Indeed, the only representative agent i’s utility conditional on a role of Parliament in our model consists in the nonpartisan issue arising is given by the same sampling of population preferences. In particular, expression regardless of the electoral system, we the size of Parliament will be independent of the here compute his utility conditional on a partisan partisan fractionalization of society. The optimal issue arising. We call this event 𝒬. This expected number of members of Parliament is given by: utility is given by √ [ ( ) ] 𝔼 ∗ 2 |𝒬, 𝛄 M − x − xi n∗ = n∗ = 𝜎 (1 − q) . H c ( ) ⎡∑G ∑G 2 ⎤ ⎢ kg′ ⎥ Bicameral Parliament. With two chambers, the =−𝔼 𝛾 x − x ′ |𝛄 − v ⎢ g g n g ⎥ H representative agent’s ex ante expected utility is ⎣g=1 g′=1 ⎦ given by: ( ) G ⎡ G 2 ⎤ { ( [ ]) ∑ ∑ k ′ G ⎢ g ⎥ [ ] ∑ =− 𝛾 𝔼 x − x ′ |𝛄 − v 𝔼 𝜎2 𝔼 𝛾2 g ⎢ g n g ⎥ H (3) ui =− q 1 − g g=1 ⎣ g′=1 ⎦ g=1 [ ( ) ] ( )} ∑G ∑G 2 2 kg kg′ 𝛼2 (1 − 𝛼) =−𝜎2 𝛾 𝔼 1−2 + |𝛄 − v , + (1 − q) 1 + + g n n H n n g=1 g′=1 H S( ) wherewehaveusedthatthex are i.i.d. draws ( ) 2C + nH + nS c g − 𝛼2v + (1 − 𝛼)2 v − . from a distribution with a variance of 𝜎2.Wehave H S M [ ( ) ] 𝔼 ∗ 2 |𝒬, 𝛄 It thus follows that, in the unique can- − x − xi didate for an interior optimum, the relative { ∑G bargaining power of a chamber equals its 2 [ ] ( ( )) =−𝜎2 1 − 𝛾 𝔼 k |𝛄 share of seats: 𝛼∗ = n∗ ∕ n∗ + n∗ . More- n g g √ H H S g=1 ∗ 𝜎 vS M ∗ } over, n = (1 − q) and n = G [ ] √H vH+vS c S ∑ v 1 𝔼 2|𝛄 . 𝜎 H M 𝛼* + k − vH (1 − q) , so that = (vS/(vH + vS)), 2 g vH+vS c n g=1 and the overall number of members of Parliament ∑n is equal to the number of members of Parliament We can write kg = d=1 1d,g, where 1d, g is an in a unicameral system, indicator equal to 1 if and only if the legislator √ assigned to seat d (d ∈{1,···, n}), is of party g, M and 0 otherwise. n∗ = 𝜎 (1 − q) . c Thus, [ ] ∑n [ ] Finally, the difference in welfare between a 𝔼 k |𝛄 = 𝔼 1 |𝛄 bicameral and unicameral system is: g d,g ( ( )) d=1 2 . n v ∕ vH + vS − (C∕M) ∑ ( ) H |𝛄 𝛄 , = Pr 1d,g = 1 = nλg ( ) If vH,vS, and C are uncorrelated with the pop- d=1 ulation of a country, this difference decreases as ∑n ( ) 1 the size of the population increases. This implies where λ (𝛄) ∶= Pr 1 , = 1|𝛄 is the aver- g n d g that more populous countries are more likely to d=1 have a Senate. age probability (over Parliamentary seats) that a member of party g becomes a legislator. By the same token, B. Nonproportional Representation [ ] ( ) ( [ ]) 2 𝔼 k2|𝛄 = Var k |𝛄 + 𝔼 k |𝛄 In a nonproportional representation system, g g g the shares of a party’s members in the population ( ) = Var k |𝛄 + n2λ2 (𝛄) . and among legislators may differ. g g 10 ECONOMIC INQUIRY

Now, ( ) Thus, the optimal number of members of Par- ( ) ∑n liament is given by √ (4) Var k |𝛄 = Var 1 , |𝛄 √ g d g M d=1 n∗ = 𝜎 q (1 − 𝔼 [ℋ ]) + 1 − q, c ∑n ( ) ∗ ∑G = Var 1 , |𝛄 ℋ 𝛾2 d g where we write ∶= g=1 g for the d=1 Herfindahl–Hirschman index of parti- ∑n [ ( ) ] san fractionalization. |𝛄 |𝛄 = Pr 1d,g = 1 (1 − Pr(1d,g = 1 )) d=1 (2) Bonus to the Largest Group 𝛄 , Without loss of generality, the group g = 1, = nςg ( ) ∑ [ ( ) also denoted Party 1, refers to the party with the 1 n where ς (𝛄) ∶= Pr 1 , = 1|𝛄 (1 − Pr largest number of members in the population at g ] n ∑d=1 ( d g ) 1 n the time of the allocation of seats. If the allo- (1 , = 1|𝛄)) = Var 1 , |𝛄 is the arith- d g n d=1 d g cation of each seat were determined by the out- metic mean over seats d of the variance of 1 , d, g come of an election in a unique district attached and where we have used the assumption that the to that seat, if there were no coalitions formed, partisan affiliations of legislators are indepen- if individuals voted sincerely to elect members dently drawn across seats for the equality marked of Parliament and if the distribution of parti- by *. Thus, we have san affiliations were homogeneous across dis- { tricts, Party 1 would win all seats in Parliament. [ ] G ( ) ∑ Although all these assumptions may fail in coun- 𝔼 ∗ 2 |𝒬, 𝛄 𝜎2 𝛄 𝛾 − x − xi =− 1−2 λg ( ) g tries that use nonproportional voting systems, g=1 } the party that represents the largest share of the ∑G ∑G population may still have some advantage over 1 + λ2 (𝛄) + ς (𝛄) − v . other parties. g n g H g=1 g=1 To formalize that advantage, we assume that party g = 1 automatically wins a proportion 1 − 𝜉 Therefore, the size of Parliament matters for of seats. In the remaining proportion 𝜉 of the n the representative agent’s ex ante expected utility 𝜸 seats, we make the assumption that the seat allo- unless Var(kg| ) = 0 for all g, that is, unless the composition of Parliament is deterministic, as in cations are i.i.d., with the probability that a seat a proportional voting system. is allocated to party g equal to its share of par- In order to reduce the dimensionality of the tisans in the overall population. This assumption 𝜸 is motivated by the fact that, in practice, at the problem of estimating Pr(1d, g = 1| ) for all (d, g) ∈{1, ···, n} × {1, ···, G} (which would be neces- time of the design of Parliamentary institutions, the Parliament Designers may not know perfectly sary to compute the (ς1, ···, ςG)), further assumptions on the seat-allocation process are needed. how individuals will migrate from one district We shall discuss two simple sets of assumptions to another, or may not even know how district below: (1) the case of an i.i.d. allocation of seats boundaries will be defined, etc. We examine this assumption empirically in Appendix B. and (2) a bonus for the biggest party. 𝜸 𝜉 𝛾 𝜸 Thus, we have λ1( ) = 1− (1− 1), λg( ) = 𝜉𝛾 ≠ 𝜸 𝜉 𝛾 𝜉 𝛾 (1) i.i.d. Allocation of Seats g for g 1, ς1( ) = (1− 1)(1− (1− 1)), and 𝜸 𝜉𝛾 𝜉𝛾 ≠ 𝜸 𝜸 ςg( ) = g(1− [g)forg 1. Using] this in our In this case, Pr(1d, g = 1| ) =λg( ) for all seats ( ) 𝔼 ∗ 2 |𝒬, 𝛄 d ∈{1,···, n}. We shall furthermore assume that expression for − x − xi , we find the allocation of seats is fair in the sense that [ ( ) ] λ (𝜸) = 𝛾 for all g ∈{1,···, G}. In this case, 𝔼 ∗ 2 |𝒬, 𝛄 g g (5) − x − xi = society’s objective at the Constitutional stage is { given by ( ) ∑G {[ [ ] ( [ ])] 2 2 2 ∑G ∑G −𝜎 1 + (1 − 𝜉) 1 − 2𝛾 − 𝜉 (2 − 𝜉) 𝛾 1 1 g −𝜎2 q 1 − 𝔼 𝛾2 + 1−𝔼 𝛾2 g=1 g n g [ ( )]} g=1 g=1 G ( )} 𝜉 ( ) ∑ 1 C + nc + 2 1 − 𝛾 (1 − 𝜉) −𝜉 1+ 𝛾2 − v . + (1 − q) 1 + − v − . n 1 g H n H M g=1 GODEFROY & KLEIN: PARLIAMENT SHAPES AND SIZES 11

Thus, society’s objective at the Constitutional One shows by calculations similar to those stage is given by above that

(6) { [ (8) ( [ ]) { 𝜎2 𝜉 2 𝔼 𝛾 𝜉 𝜉 [ ( ) ] ∑G − q 1 + (1 − ) 1 − 2 1 − (2 − ) 𝔼 ∗ 2 |𝒬, 𝛄 𝜎2 𝛾 − x − xi =− 1 − 2 g [ ] [ g=1 ∑G 𝜉 ( [ ] ) ( [ ] ) 𝔼 𝛾2 + 2 1 − 𝔼 𝛾 (1 − 𝜉) 𝛼 1 − 𝛼 g n 1 𝔼 kH|𝛄 + 𝔼[kS|𝛄] g=1 g g nH nS ( [ ]) ]] [ ]} ∑G G ( ) ∑ 𝛼 𝛼 2 −𝜉 1 + 𝔼 𝛾2 𝔼 H 1 − S |𝛄 g + kg + kg g=1 nH nS ( )} g=1 1 C + nc 𝛼2 𝛼 2 + (1 − q) 1 + − v − . − vH − (1 − ) vS n H M { H ∑G ( ) 𝜎2 𝛾 𝛼 H 𝛄 𝛼 S 𝛄 Optimizing over n gives us the optimal size of =− 1 − 2 g λg ( ) + (1 − ) λg ( ) Parliament, g=1 √ G ( ) M ∑ 2 n∗ = 𝜎 𝛼 H 𝛄 𝛼 S 𝛄 c + λg ( ) + (1 − ) λg ( ) √ [ ( [ ] ) ] g=1 1−q + q𝜉 2 1−𝔼 𝛾 (1−𝜉) −𝜉 (1+𝔼 [ℋ ]) . ( )} 1 ∑G 𝛼2 𝛼 2 H 𝛄 (1 − ) S 𝛄 𝜉 + ςg ( ) + ςg ( ) We note that the case = 1 corresponds to our nH nS previous i.i.d. case. g=1 𝜉 𝜸 𝛼2 𝛼 2 , For = 0 and given , there is no uncertainty − vH − (1 − ) vS concerning the partisan composition of the legislature, as party g = 1 will capture all the seats. Thus, as in the case of proportional voting, the X 𝛄 where λg ( ) denotes the average probability number of Parliamentary seats matters only when over districts that a candidate of party g is it comes to nonpartisan issues. It is therefore X 𝛄 elected to chamber X, and ςg ( ) is the arith- no surprise that, in this case, the optimal size metic mean (over districts) of the variance of of Parliament corresponds to that under propor- X 𝜸 the random variable 1d,g conditional on .The tional representation. X random variable 1d,g is 1 if a candidate of party Bicameral Parliament. We now examine the case g is elected to chamber X for the seat d, and of a bicameral system in a nonproportional sys- 0 otherwise. tem. The representative agent i’s ex ante expected To make further predictions, we again analyze utility, conditional on a partisan issue arising, is the i.i.d. case and the case of a bonus to the largest given by party, as above. [ ( ) ] 2 (1) i.i.d. Allocation of Seats (7) 𝔼 − x∗ − x |𝒬, 𝛄 i In this case, (8) simplifies to [( ∑G ∑G kH [ ] g′ ( ) ∗ 2 =− 𝛾 𝔼 x − 𝛼 x ′ 𝔼 − x − x |𝒬, 𝛄 g g n g i g=1 g′=1 H ( ) 2 2 )2 𝛼 (1 − 𝛼) ∑G kS ⎤ =−𝜎2 1 + + g′ ⎥ n n − (1 − 𝛼) x ′ |𝛄 H S n g ⎥ g′=1 S ⎦ ℋ 𝛼2 𝛼 2 , (1 − ) − vH − (1 − ) vS −𝛼2v − (1 − 𝛼)2 v , ∑ H S ℋ G 𝛾2 where ∶= g=1 g again denotes the X where kg is the number of legislators of party g Herfindahl–Hirschman index of partisan frac- in chamber X. tionalization. At the Constitutional stage, society 12 ECONOMIC INQUIRY maximizes ( )} 2 𝛼 2 ( ) 𝛼 (1 − ) 2 2 𝛼 2 [ + (1 − q) 1 + + − 𝛼 v 2 𝛼 (1 − ) H − 𝜎 1 + + q (1 − 𝔼 [ℋ ]) nH nS ( ) nH nS ] 2C + n + n c 2 2 𝛼 2 H S . +1 − q − 𝛼 v − (1 − 𝛼) v − (1 − ) vS − ( H ) S M 2C + nH + nS c Optimizing, we again find − , ( ( )) M 𝛼∗ , = vS∕ vS + vH and the optimum is given by and for the optimal total number of legislators ( ( )) √ 𝛼∗ = v ∕ v + v , S H S M (12) n∗ = 𝜎 √ c √ √ M [ ( [ ] ) ] ∗ 𝜎𝛼∗ 𝔼 ℋ , 1−q+q𝜉 2 1−𝔼 𝛾 (1−𝜉) −𝜉 (1 + 𝔼 [ℋ ]) , nH = q (1 − [ ]) + 1 − q 1 cH √ as in the unicameral setting. As before, 𝛼 = √ ( ) M n∗ ∕n∗,orn∗ = 𝛼*n* and n∗ = (1−𝛼*)n*. n∗ = 𝜎 1 − 𝛼∗ q (1 − 𝔼 [ℋ ]) + 1 − q, H H S S c Both models (1) and (2) predict that the trade- S off between a unicameral and a bicameral system implying is the same as under a proportional voting system, √ √ implying that a bicameral system is better if and ∗ M only if (9) n = 𝜎 q (1 − 𝔼 [ℋ ]) + 1 − q , ( ( )) c 2 ≥ . vH∕ vH + vS (C∕M) as in the unicameral case. Thus, a country’s choice of bicameralism (2) Bonus to the Largest Group depends on the size of its population, but does Straightforward calculations show not depend on its voting system or its level of [ ( ) ] partisan fractionalization. 𝔼 ∗ 2 |𝒬, 𝛄 (10) − x − xi = { C. Predictions ( ) ∑G −𝜎2 1 + (1 − 𝜉)2 1 − 2𝛾 − 𝜉 (2 − 𝜉) 𝛾2 This analysis yields the following predictions. 1 g Prediction g=1 1. The log of the number of legis- ( )[ lators is linearly increasing in the log of the size 𝛼2 𝛼 2 ( ) 𝜉 (1 − ) 𝛾 𝜉 of the population, with a coefficient close to 0.5. + + 2 1 − 1 (1 − ) Prediction 2. The number of legislators is nH nS ( ) ]} independent of whether a Parliament is unicam- ∑G eral or bicameral. 𝜉 𝛾2 𝛼2 𝛼 2 . − 1 + g − vH − (1 − ) vS Prediction 3. In bicameral systems, the rela- g=1 tive bargaining power of a given chamber is equal Thus, society’s objective at the Constitutional to the share of legislators sitting in this chamber. stage is given by Prediction 4. More populous countries are { [ ( [ ]) more likely to have a bicameral Parliament. 𝜎2 𝜉 2 𝔼 𝛾 Prediction 5. The factors that impact the size (11) − q 1 + (1 − ) 1 − 2 1 [ ]]} of Parliament in the model (except the size of the ∑G population) have no impact on the probability that 𝜉 𝜉 𝔼 𝛾2 − (2 − ) g a country has a bicameral Parliament. g=1 Prediction 6. In proportional systems, the level ( ) 𝛼2 (1 − 𝛼)2 ( ( [ ] ) of partisan fractionalization has no impact on +𝜉 + 2 1 − 𝔼 𝛾 (1 − 𝜉) the size of Parliament. The log of the size of n n 1 H ( S [ ]))] Parliament is: ∑G 𝜎 . . 𝜉 𝔼 𝛾2 (13) log n = log + 0 5logM − 0 5logc − 1 + g g=1 + 0.5log(1 − q) . GODEFROY & KLEIN: PARLIAMENT SHAPES AND SIZES 13

Prediction 7. In nonproportional voting sys- “Autocracies” (score between −10 and −6), and tems, for a given q and 𝜉, the log-linearization of other regimes. Equation (12) implies that the log of the size of Parliament is: Sample of Observations. The longest period cov- (14) log n = log 𝜎 + 0.5logM − 0.5logc ered by all the data sources spans 1975 to 2012 ( [ ( [ ] ) (with some missing information). Many countries . 𝜉 𝔼 𝛾 𝜉 +0 5log 1 − q + q 2 1 − 1 (1 − ) have been governed by an autocratic regime for at least some years of that period. Since we con- −𝜉 (1 + 𝔼 [ℋ ])]) . sider that our model does not explain the institu- Approximating the last term, we have: tional features of such regimes, we aim to exclude 𝜎 . . these observations from our sample. To do so, (15) log n = log + 0 5logM − 0 5logc we include only “Nonautocratic” regimes in the q Polity IV classification (i.e., we exclude countries +0.5log(1 − q) − 0.5 𝜉2𝔼 [ℋ ] 1 − q with a score between −10 and −6 in 2012). q ( [ ] ) In addition, we exclude countries that have not −0.5 2𝜉 (1 − 𝜉) 𝔼 𝛾 + 𝜉2 − 2𝜉 . had at least two legislative elections for which 1 − q 1 no party obtained all the votes and there were no reports of substantial fraud (the DPI contains a binary variable that indicates whether an election V. EMPIRICAL ANALYSIS was marred by fraud).19 A. Methodology Since observations in a country across years usually cannot be considered independent—for This section documents a certain number of instance, the size of the U.S. Congress over empirical regularities of Parliamentary institu- the whole period is 535, due to a rule set in tions across countries. The size of the sam- 1911—we run all regressions on a sample of data ple, small by nature, and the lack of exogenous with a unique observation by country.20 sources of variation in the explanatory variables limit causal inference. With this caveat in mind, Definition of Variables. To assess the value of we formulate the identification assumptions of all institutional variables unrelated to electoral out- estimations, but do not discuss their plausibility. comes (the number of chambers of Parliament, Instead, we discuss the results of the estimations in the context of the predictions of the model. the size of each chamber of Parliament) of a country, which are all nonrandom, we use the obser- Data. The estimations use data from the DPI vation for 2012. To assess the size of the popu- 2012 (Beck et al. 2001 [last update in 2012]). lation of that country, we use the observation at the time of the most recent change in those insti- These data contain information, for almost every 21 country and every year between 1975 and 2012, tutional variables before 2012. To assess the concerning the number of chambers, the number values of variables related to electoral outcomes, of members of either chamber, the voting sys- which may be random, we compute their empir- tem (proportional/nonproportional), and electoral ical means over all the elections (with no fraud outcomes in election years, namely vote shares reported and with more than one party) that took 22 and seat shares across parties, for the Lower place between 1975 and 2012. Chamber.17 Population by country comes from the World Bank Database. To assess the “degree of democratization,” we use the Polity IV score 19. This restriction excludes countries that may have had from Polity data.18 These data indicate the Polity more than one election, but for which there is no electoral IV score of most countries for every year between information. 20. Including many years of observations would artifi- 1800 and 2015. The Polity IV score ranges from cially increase the significance of the impact of any variable −10 to 10, and is used to partition regimes into that changes little between 1975 and 2012, such as the size of the Parliament, the population, the number of chambers, and so on. 17. There is no information on Upper Chamber elections. 21. In most countries, Parliament features change “regu- In fact, in many countries (e.g., in the United Kingdom or in larly.” Only a few countries have not changed the number of Canada), members of the Upper Chamber are not elected. Parliamentary seats since 1975. 18. The data are available on the Polity IV website http:// 22. All results are similar if we use 2012 population levels www.systemicpeace.org/inscrdata.html. for all countries instead. 14 ECONOMIC INQUIRY

As in the model, Party 1 refers to the party that, of votes obtained by Party 1 in country k across 𝜉 in a given election, obtained the largest share of democratic elections, k is the ratio of one minus the total number of votes in the country.23 the average share of seats obtained by Party 1 The variables related to electoral outcomes across democratic elections over one minus the are the Herfindahl–Hirschman∑ index of partisan average share of votes obtained by Party 1 across ℋ 𝛾2 fractionalization = g g , the share of total democratic elections in country k. The dependent 𝛾 𝜉 votes obtained by Party 1 1, and . The actual variable log nk is the log of the total number of ℋ 𝛾 value of and 1 for any democratic election members of Parliament in country k. that took place between 1975 and 2012 can be Equation (16) does not include variables for computed directly from the data, and we use which there is no information (c) or that have no their empirical means over democratic elections obvious empirical proxy (q and 𝜎). as proxy variables for their expected values.24 Predictions 6 and 7 imply that: 𝜉 Conversely, the actual value of cannot be • Under any voting system, the model pre- obtained directly from any variable in the data. 𝜉 dicts β1 = 0.5. In fact 1− , which we informally refer to as • Under a proportional voting system, 𝜉 is “the bonus to the largest party,” may depend equal to 1, β2 = 0. Term (3) and the intercept on many institutional features—such as redis- would be collinear in theory, yet in practice, if tricting rules—that cannot be easily quantified. 𝜉 𝜉 k = 1 + uk where uk is some random error of Instead, to assess the value of ,weusethefact mean 0—due for instance to integer problems in that, by definition, the[ ]expected[ proportion] of 𝔼 𝔼 𝜉 𝜉𝛾 the attribution of seats—we may have β3 = 0. seats won by Party 1, λ1 ,is 1 − + 1 . • Under a nonproportional voting system 1−𝔼[λ ] The proxy variable for 𝜉 is then 1 , where with an i.i.d. distribution of seats across parties, 1−𝔼[𝛾 ] q 1 𝜉 is equal to 1, β =−0.5 , which implies we use averages across elections to estimate the 2 1−q 25,26 < expectations in the equation. that β2 0. Term (3) and the intercept would be Appendix A reports the descriptive statistics collinear in theory, yet in practice, as under a of the variables used in the empirical analysis. proportional voting system, we may have β3 = 0. • Under a nonproportional voting system that Specification. The basic specification for the esti- gives a bonus of seats to Party 1, β =−0.5 q 2 1−q mations is: and β =−0.5 q , which implies that β < 0, 3 1−q 2 (16) < β3 0, and β2 =β3. 𝜉2𝔼 ℋ log nk =β0 +β1 log Mk +β2 [ ]k +β3 ( ( ) [ ] k ) 𝜉 𝜉 𝔼 𝛾 𝜉2 𝜉 B. Results of the Estimations 2 k 1 − k 1 k + k − 2 k +εk Table 1 reports the estimation of the coeffi- with one observation by country k, and where Mk 𝔼 ℋ cients of Equation (16) for the sample of nonau- is the size of the population, [ ]k is the average Herfindahl–Hirschman[ index] across democratic tocratic regimes. The identification assumptions elections in country k, 𝔼 𝛾 is the average share are: (1) The unobservable variables that also 1 k affect the size of Parliament of a country, that is, the variance of bliss points 𝜎2 and the salaries 23. The identity of Party 1 may of course change from one election to the next. of members of Parliament, are not correlated 24. In certain cases, the vote shares of some par- with the size of the population, the probability ties may be either missing or not consistent. Since pre- of bicameralism, or the terms specific to partisan cise information on vote shares is crucial to compute the Herfindahl–Hirschman index, we include only observations issues. (2) The total number of seats in Parliament such that the sum of the known shares of the parties is between has no impact on the size of the population, the 0.90 and 1.10. probability of bicameralism, or the terms specific 25. The model assumes[ that ξ is nonrandom.] [ ] If it were to partisan issues. 𝔼 𝔼 random, the relationship ξγ[1 ]+ 1 −ξ = λ1 still holds, Column 1 shows that the log-linear relation- 𝔼 but cannot be used to derive ξ in general if ξ and γ1 are not independent. In fact, there would be no general proxy variable ship from the model holds, with a coefficient for the expected values of the polynomials of ξ and the share close to 0.5, which is consistent with Prediction 1. of votes that appear in the model if we did not assume that ξ In column 2, we include as a covariate a binary and the vote share γ1 were independent. variable equal to 1 if and only if country k has a 26. We note that the maximum value of ξ is larger than bicameral system. This variable has no significant 1, which is due to the fact that, in a few cases, the party that obtained the largest share of votes got fewer seats in impact on the log of the size of Parliament, which Parliament. is consistent with Prediction 2. GODEFROY & KLEIN: PARLIAMENT SHAPES AND SIZES 15 ** ** *** 1.132 1.049 0.518 0.412 − − . There is one observation * *** 1.049 ith no party getting all the votes and − 0.427 *** 0.399 0.194 − 0.353 *** 0.345 − 0.355 Log Size of Parliament ** ** *** 0.847 0.767 (0.320)(0.333) (0.600) (1.001) (0.463) (0.482) 0.403 − − TABLE 1 ** *** 0.801 (0.367) (0.588) (0.525) 0.410 − Size of Parliaments in Nonautocratic Regimes *** (0.089) 0.407 *** (1) (2) (3) (4) (5) (6) (7) (8) 0.825 0.831 0.836 0.840 0.736 0.739 0.874 0.879 (0.023) (0.023) (0.022) (0.023) (0.040) (0.040) (0.029) (0.030) 휉 × .001. 2 < ] − p 2 ℋ 휉 [ + 𝔼 ] 1 훾 .01, *** [ < 𝔼 p × ) 휉 : DPI; Polity; World Bank. 3 This table reports the estimation of the regression of the log of the total number of members of Parliament on the log of population and other covariates ] − .05, ** 1 ( ℋ value =β [ < p 2 × 𝔼 p β 휉 * Sources Notes: × 0: × 2 2 휉 Herfindahl–Hirschman 2 Bicameral ParliamentVoting system# ObservationsR Any 0.147 74 Any 74 Any 74 Any 74 Proportional Proportional 32 Nonproportional Nonproportional 32 42 42 by country. The sample compriseswithout all reported countries fraud that between were 1975 not and autocratic 2012. in See 2012 Section in V the for Polity details. IV Standard classification errors in and parentheses. that have had at least two elections w Log population 0.417 Wald H 16 ECONOMIC INQUIRY

TABLE 2 Probability to Have a Bicameral Parliament Bicameral Parliament (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Log population 0.067** 0.062** 0.073** 0.080** 0.076** 0.090*** 0.074** 0.077** 0.068** 0.066** (0.030) (0.030) (0.030) (0.031) (0.030) (0.032) (0.030) (0.030) (0.031) (0.032) Proportional voting −0.138 (0.116) Herfindahl–Hirschman 𝔼 [ℋ ] 0.582 (0.492) 휉2 × 𝔼 [ℋ ] 0.506 (0.430) 2 × 휉 × (1 − 휉) × 𝔼 [훾1] + 휉2 − 2 × 휉 0.686 (0.448) Ethnic F [1] 0.062 (0.243) Linguistic F [1] −0.006 (0.231) Religious F [1] 0.370 (0.241) Linguistic F [2] 0.019 (0.202) Ethnic P [3] 0.335 (0.229) Religious P [3] −0.089 (0.185)

# Observations 74 74 74 74 73 70 73 73 66 66 R2 0.066 0.084 0.084 0.102 0.085 0.107 0.114 0.085 0.091 0.064

Notes: This table reports the estimation of the regression of a binary variable equal to 1 if and only if the country has a bicameral Parliament, on the log of population and other covariates. There is one observation by country. The sample comprises all countries that were not autocratic in 2012 in the Polity IV classification and that have had at least two elections withno party getting all the votes and without reported fraud between 1975 and 2012. F [1], F [2], P [3] refer respectively to the Fractionalization indices of Alesina et al. (2003), the Fractionalization index of Desmet, Ortuno-Ortin, and Wacziarg (2012), and the Polarization indices of Montalvo and Reynal-Querol (2005a, 2005b). See Section V for details. Standard errors in parentheses. *p < .05; **p < .01; ***p < .001. Sources: DPI; Polity; World Bank.

Columns 3 and 4 of Table 1 show that the index, as well as of the coefficients of the second coefficients of the Herfindahl–Hirschman index term of Equation (16), which is consistent with (column 3), or of the second and third terms of Prediction 7. Equation (16) (column 4) have the sign predicted Column 8 also shows that the coefficient of the by the model for a nonproportional voting sys- third term is negative, statistically significant, and tem. These results are consistent with the fact close to the value of the coefficient of the second that the coefficients are each some average of term. The p value of a Wald test of the hypothesis the coefficient in the proportional voting system H0: β2 =β3, reported at the bottom of column 8, (which is null) and in the nonproportional system shows that we cannot reject the possibility that (which is negative). these coefficients are equal at the 0.05 (or 0.1) In fact, if we restrict the sample to countries level of significance. These results are consistent with a Proportional voting system (columns 5 and with Prediction 7 in the case of a “bonus to the 6), we find no statistically significant impact of largest group.” the Herfindahl–Hirschman index, or of the coef- Remark. These results may be used to assess ficients of the second and third terms of Equation the probability q that a partisan issue arises, a (16), which is consistent with Prediction 6. parameter that has no obvious measurable equivalent. We have here that 0.5 q is around 1, so If we then restrict the sample to countries 1−q with a nonproportional voting system (columns that q is around 67%. 7 and 8), we find a negative and statistically To assess Prediction 3, we rely on Bradbury significant impact of the Herfindahl–Hirschman and Crain (2001) who provide the only estimation GODEFROY & KLEIN: PARLIAMENT SHAPES AND SIZES 17 of the ratio in the bargaining powers of the Lower the coefficient of the log of the population onthe and the Upper Chamber across countries, that probability to have a Senate.28 is, 훼*/(1 − 훼*) with the previous notations. They find that the ratio of bargaining powers is3.5 on average. This means that 훼* is around 0.78 VI. CONCLUSION on average across countries. With our data on We have analyzed a simple model of Parlia- the sizes of Lower and Upper Chambers, we mentary institutions, testing the consistency of estimate that the ratio of the number of members its predictions with cross-country data. We have of the Lower Chamber over the total number seen that the log of the size of a country’s Par- of members of a Parliament is equal to 0.73 liament increases in a linear manner with the on average among nonautocratic regimes. This log of the size of its population. The size of a average is very close to the estimate of Bradbury country’s Parliament does not depend on whether and Crain (2001).27 it is unicameral or bicameral. In bicameral sys- Table 2 examines the determinants of an tems, the relative weight of a chamber should Upper Chamber. The identification assumptions correspond to the share of legislators sitting in are: (1) The unobservable variables that also this chamber. Furthermore, the only observable affect the probability of bicameralism, that is, variable impacting the probability that a given square means of the error terms and the fixed cost country has a bicameral Parliament is the size of of an extra Chamber, are not correlated with the its population. size of the population, the voting system in place, A second set of results pertains to the impact or the terms specific to partisan issues. (2) The of partisan fractionalization and voting systems probability of bicameralism has no causal impact on Parliament size. We find that the mode of elec- on the size of the population, the voting system tion has no impact on the probability that a given in place, or the terms specific to partisan issues. country has a second chamber, and that greater In column 1, we regress a binary variable equal fractionalization increases the size of a country’s to 1 if and only if the country has a Bicameral Parliament if and only if it has a nonproportional Parliament on the log of the population of a coun- voting system. In proportional systems, fraction- try. We find that larger countries are significantly alization has no impact. more likely to have a Senate, which is consistent Our model could be extended in several ways. with Prediction 4. For instance, we do not model the bargaining In column 2, we include a covariate equal to process among legislators, assuming instead that 1 if and only if the country has a Proportional legislators within a given chamber act cooper- voting system. This variable is not significantly atively and do not take the bargaining process correlated with the dependent variable, which is with the other chamber into account. Whether it consistent with Prediction 5. would be possible to account for the huge cross- In columns 3 and 4, we find no impact of country differences in Parliamentary procedures the other terms considered before. In particular, in a richer model, which would focus on a smaller countries with greater partisan fractionaliza- subset of countries, and whether such a model tion are not significantly more likely to have would preserve, or possibly even increase, the a Senate. This finding might be interpreted as predictive power of our model, is an interesting somewhat disputing the view that the role of question for future research. second chambers is primarily to afford represen- Furthermore, we have made the assumption tation to otherwise under-represented minorities. that, in a bicameral system, errors are uncorre- To further examine this point, we also present lated across chambers. This is clearly a strong additional estimations that use measures of eth- assumption, made for the purpose of tractability, F nic, religious, and linguistic fractionalization ( ) as one could well imagine both chambers falling P and polarization ( ) presented in Alesina et al. under the sway of the same lobbying efforts (2003) ([1] in the table), Desmet, Ortuno-Ortin, or similar mood swings in published opinion, and Wacziarg (2012) ([2] in the table), and making for positively correlated errors. On the Montalvo and Reynal-Querol (2005a, 2005b) ([3] in the table). No indicator has any effect on 28. This result does not mean that ethnic, linguistic, or religious diversity has no impact on the setup of a country’s 27. If we restrict the sample to the sample of countries Parliament. In fact, diversity might affect the number of used in Bradbury and Crain’s (2001) estimations, the share is members of Parliament if, for instance, it had an impact on 0.74. q or σ. 18 ECONOMIC INQUIRY other hand, in our simple model, the Constitution our analysis. Indeed, sometimes, the second Designers would endeavor to ensure that the chamber is malapportioned in such a way as errors are as negatively correlated as possible. to over-represent some minority, as is the case, Indeed, in reality, many countries use different for example, with smaller states in the United modes of selection for the two chambers of States. One could view the fact that often only their Parliaments, which one could arguably one chamber is malapportioned in this way as an interpret as one way of effecting a negative attempt by the Constitution Designers to induce correlation between the errors they are prone negatively correlated errors between chambers. to. Furthermore, in many bicameral systems, We recommend for future research a more there are important design differences across detailed investigation of differences between chambers, from which we have abstracted in chambers in bicameral systems.

APPENDIX A: DESCRIPTIVE STATISTICS AND GRAPH TABLE A1 Descriptive Statistics Variable Mean Std. Dev. Min. Max. N Population in million 41.33 137.48 0.1 1,127.14 74 Size of Parliament 253.14 220.97 28 955 74 Size of Lower Chamber/House 211.84 171.63 15 650 74 Size of Upper Chamber/Senate 87.31 82.25 11 326 35 Bicameral Parliament 0.47 0.5 0 1 74 Proportional voting system 0.43 0.5 0 1 74 Share of legislators in Lower Chamber 0.73 0.09 0.54 0.91 35 Herfindahl–Hirschman index ℋ 0.36 0.12 0.17 0.83 74 휉 0.93 0.2 0.48 2 74 휉2 × ℋ 0.33 0.37 0.12 3.31 74 휉 휉 훾 휉2 휉 2 × × (1 − ) × 1 + –2 × −0.94 0.36 −3.63 −0.39 74

Notes: This table reports descriptive statistics for all nonautocratic countries in 2012. The size of the population is measured at the time of the most recent change in Parliament features before 2012. The electoral outcomes terms ℋ , 휉, 휉2 × ℋ and 휉 휉 훾 휉2 휉 2 × × (1 − ) × 1 + –2 × are the empirical averages of these terms over legislative elections that took place between 1975 and 2012. The number of members in a chamber may differ from the number of seats in that chamber if some seats are not taken up. Sources: DPI; Polity; World Bank. GODEFROY & KLEIN: PARLIAMENT SHAPES AND SIZES 19

FIGURE A1 Population and Parliament Size 7 ita fra ind ger jap uk mex bra polspa tur ind usa sou can 6 hun swe arg gre mal sou col bulswiaus ire por netaussrigha domtun fin mad bel mal norden bur chi ven bol sen 5 alb cro slo uru hon mac newpar isr per

estnam el jam pan malcab

Log # Members of Parliament ice lux bah cyp bot cos 4 van bar gam tri

bel gre st. 3

12 14 16 18 20 Log Population

Notes: Each country is represented by the first three letters of its name. The line is the linear interpolation of the relationship between the log of population and the log of Parliament size. Sources: DPI; Polity; World Bank.

give an advantage to the party with the largest share of votes in APPENDIX B: EMPIRICAL EXAMINATION OF THE the country, which amounts to 11% of House seats on average. BONUS TO THE LARGEST GROUP ASSUMPTION Our estimation method for 휉 implies that Equation (A1) In this section, we present a series of estimations that use is trivially satisfied for Party 1 empirically. It is also trivially data of the DPI at the election and party level, for any election satisfied empirically in elections with only two parties. We therefore focus on estimating the relation between the shares that took place between 1975 and 2012. > To do so, we use the data’s detailed information on shares of seats and votes for parties g 1 only in elections with three of seats and votes obtained by the largest parties across parties or more. elections to test the main implication of this assumption, The basic specification we use here is: namely that there exists 휉 such that: (A2) [ ] [ ] (A1) 𝔼 λ = 휉 × 𝔼 훾 + (1 − 휉) × 1 g g g=1 휉 Seats Sharek,g,t =δ0 +δ1 k × Votes Sharek,g,t with the notations of the model, and where 1g = 1 is a binary variable equal to 1 if and only if g = 1, and G∑−1 0 otherwise. + δr Party rank = rk,g,t +εk,g,t As mentioned in Section V, we do not observe 휉 directly, r=2 but infer its value from solving Equation (A1) with g = 1. where Seats Sharek, g, t is the share of seats and Votes Empirically, we find that, in nonproportional voting systems, Sharek, g, t is the share of total votes obtained by party g 휉 is equal to 0.89 on average (whereas it is equal to 0.98 on 휉 in country k for the election that took place in year t, k average in proportional systems). This estimation indicates is defined as before for country k,andParty rank = rk, g, t that countries that use nonproportional voting systems indeed is an indicator variable equal to 1 if and only if party g is 20 ECONOMIC INQUIRY

TABLE A2 Distribution of Seat Shares in Nonproportional Voting Systems Seats Share (1) (2) (3) (4) (5) (6) (7) 휉 × Votes Share 0.817*** 0.943*** 1.095*** 0.936*** 1.284*** 1.172*** 0.982*** (0.147) (0.146) (0.104) (0.176) (0.236) (0.071) (0.069) Party rank = 2 0.086 0.047 0.018 0.033 −0.014 −0.012 0.036 (0.065) (0.043) (0.030) (0.061) (0.038) (0.017) (0.023) Party rank = 3 −0.022 −0.034* −0.003 −0.026 −0.030 −0.016 (0.013) (0.016) (0.013) (0.022) (0.019) (0.010) Party rank = 4 0.001 −0.007 −0.043* −0.022 −0.003 (0.009) (0.009) (0.021) (0.015) (0.008) Party rank = 5 0.009 −0.027 −0.020* −0.002 (0.013) (0.016) (0.010) (0.007) Party rank = 6 −0.013 −0.006 −0.002 (0.008) (0.007) (0.006) Party rank = 7 −0.005** −0.001 (0.002) (0.005)

# Parties 34567 83to8

# Observations 72 84 208 100 84 70 666 R2 0.926 0.900 0.898 0.865 0.933 0.962 0.923

Wald p value H0: δ=1 .238 .702 .371 .723 .263 .061 .795

Notes: This table reports the estimation of the regression 34. The dependent variable is the share of seats in the House that party g in country k obtained after the election in year t. There is one observation by country, party, and election year. All estimations include a fixed effect for any country and election-year pair. The sample comprises all countries that were not autocratic in2012 in the Polity IV classification and that have had at least two elections with no party getting all the votes and without reported fraud between 1975 and 2012. See Section V for details. Standard errors clustered at the country level are in parentheses. *p < .05; **p < .01; ***p < .001. Sources: DPI; Polity; World Bank.

ranked rth in decreasing order of vote shares in country k These estimations show first that the coefficient ofthe 휉 for the election that took place in year t. We also include as term k × Votes Sharek, g, t is significantly different from 0, covariates the fixed effects of any country/election year. (The and close to 1. In fact, the p value of a Wald test indicates Party rank = G variable is excluded to avoid collinearity with that we cannot reject the hypothesis that the coefficient of 휉 the constant term. The party with the smallest share of votes k × Votes Sharek, g, t is equal to 1. This result is consistent is thus the reference group.) Standard errors are clustered by with Equation (A1) above. country to account for the fact that error terms are correlated The estimations also show that most indicator variables within countries. Party rank = rk, g, t have no significant effect at the 5% level, Table A2 reports the estimation of this regression sep- and none when all observations are included (column 7). This arately for all elections with a given number of parties result means that a party that obtains a share of votes such running—from three to eight parties—and for all elec- that it would be of rank r has no significant advantage or tions.29,30 disadvantage in terms of Parliamentary seats (with respect to the party with the largest rank, which is the reference group in these estimations). On average across countries, we thus cannot reject at the 5% level the hypothesis that the share 29. The data do not give information on more than nine of votes and the share of seats (deflated by the factor 휉), are parties, so that they may aggregate both votes and seats the same. for elections with more than eight parties. By definition, including elections with more than eight parties would create a measurement error that would bias the estimations of the coefficients. 30. The electoral system of a country may give an advan- APPENDIX C: A TWO-STAGE VOTING GAME tage or a disadvantage to the smallest party instead of a party of some given rank r. Reporting estimations on subsamples In a proportional voting system, the allocation of seats that of observations partitioned by number of parties allows to we assume here is the outcome of the subgame-perfect Nash check whether this is indeed the case. There are not enough equilibrium that is coalition proof in the following sense. observations by country to run the regression separately for A coalition is defined by a vector G every country, which would provide an even stronger test of ω=(θ1, η1, … , θG, ηG) ∈ ({0, 1} × [0, 1]) , such that the the assumption. share of seats obtained by the coalition that are attributed to GODEFROY & KLEIN: PARLIAMENT SHAPES AND SIZES 21

∑ G 31 ′ party g is θgηg, with g θgηg = 1. We say that g belongs where λg′ is the share of seats obtained by party g in Parlia- to the coalition ω if θg = 1. In other words, a coalition is ment. This term can be rewritten: defined by its members and by a contractη ( , ... , η )that 1 G [ ] ∑G ( ) specifies how the seats won in the chamber will beshared 휎2 2 휎2 2 . − 1 −λg − λg′ among them. g′=1,g′≠g Let Ω denote the set of all possible coalitions, and Ωg the set of all possible coalitions that g belongs to. This term is increasing in λg ∈ [0, 1] and decreasing in λg′ ′ We consider the following two-stage game. ∈ [0, 1] for g ≠ g. Stage 1. Suppose each party is headed by a party leader ̃ ≥ ′ Let λg′ 0 be the share of seats obtained by party g ∈{1, who may agree on behalf of the party to form a coalition ... , G} without individual i’s vote. i’s vote counts for 1/M with one or several other parties. The party leader wants to of the total share of votes, so, by definition of a proportional 32 G maximize the share of seats of his party. In the first stage, ∑ ̃ every party g’s leader chooses a unique ω∈Ω . Coalition ω voting system, λg′ = 1 − (1∕M). g ′ is running in the election if and only if all parties that belong g =1 If party g runs independently and i votes for it, i’s payoff to ω have chosen it. Otherwise, we impose that all parties that ( ) is: , , Ω picked ω run independently. Let Ω ω1 … ωG ∈ 2 be the { } ( ) G set of coalitions or parties that run in the election for a given 2 ∑ A =−휎2 1 −̃λ − (1∕M) + ̃λ2 . vector (ω1, ..., ωG). g g′ Stage 2. In the second stage, every individual observes g′=1,g′≠g parties’ choices, and casts a vote for a coalition (or a party) If party g runs independently and i votes for a coalition that is actually running. An individual i’s strategy is defined or for a party other than g, his vote will add some ε ′ to λ ′ , G → ∑ g g by a function Ω Ω which sets, for any vector of choices ≤ ≤ 33 where 0 εg′ (1∕M) and εg′ = (1∕M). Individual i’s (ω1, ..., ωG), which coalition (or party) ω∈Ω( individual)i g′∈G , , will vote for, and by the constraint that ω∈Ω ω1 … ωG , payoff is then: that is, that i cannot vote for a coalition that is not running in { } ( ) G ( ) the election. 2 ∑ 2 휎2 ̃ ̃ . Information. At the time of the election, individuals’ par- B =− 1 − λg + λg′ +εg′ tisan affiliations are publicly known, but the type of issue to g′=1,g′≠g arise and individual bliss points are unknown. This assump- The monotonicity in λg ∈ [0, 1] and λg’ ∈ [0, 1] implies tion aims to account for citizens’ imperfect information that A > B. Therefore, any party that runs independently will on the details of the issue that will come before the leg- obtain a share of seats that is at least equal to its share of islators to whom they delegate decision power. Over the partisans in the population. course of a term, legislators may have to address unexpected In the first stage, no party will choose a coalition unless issues, for instance, a domestic or international crisis, or it can obtain at least that share of seats by doing so. Since to acquire more information on an issue before deciding the sum of these shares is equal to 1 no party can obtain a on a policy. share of seats strictly larger than its share of partisans in any After the election, the type of issue is realized, individ- equilibrium. ◾ ual bliss points are known, and the policy adopted is as in Equation (1) of Section III. LEMMA 2. A situation in which all parties choose to run To simplify notations, we assume, in this section only, independently in the first stage and each individual votes for that the issue is partisan, that error terms Z are null, that X his own party in the second stage is a subgame-perfect Nash the Parliament is unicameral, and that C = c = 0. The results equilibrium. are the same if we relax all these assumptions, and use the assumptions of the general model instead. Proof. In such a situation, a deviation by any one party will not LEMMA 1. In any subgame-perfect Nash equilibrium of change the set of coalitions running, since no coalition can be this game, any party necessarily obtains a share of seats equal formed after a unique deviation. ◾ to its share of partisans in the population.

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