Risk diversification and vote decisions in mixed-member electoral systems

Shikano, Susumu; Herron , Erik S. Article — Published Version Risk diversification and vote decisions in mixed-member electoral systems Public Choice Provided in Cooperation with: Springer Nature Suggested Citation: Shikano, Susumu; Herron , Erik S. (2025) : Risk diversification and vote decisions in mixed-member electoral systems, Public Choice, ISSN 1573-7101, Springer US, New York, NY, Vol. 204, Iss. 1-2, pp. 203-219, https://doi.org/10.1007/s11127-025-01301-5 This Version is available at: https://hdl.handle.net/10419/330744 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) Public Choice (2025) 204:203–219 https://doi.org/10.1007/s11127-025-01301-5 Risk diversification andvote decisions inmixed‑member electoral systems SusumuShikano1 · ErikS.Herron2 Received: 30 September 2024 / Accepted: 31 May 2025 / Published online: 1 July 2025 © The Author(s) 2025 Abstract This paper builds on the literature about mixed-member electoral systems, exploring how ballot design interacts with voter behavior. We present a theoretical model for vote decision-making in mixed-member systems that takes into account the interaction between both tiers. The model is grounded in a spatial model for vote decision-making under risk and inspired by the logic of portfolio diversification under risk. Accordingly, voters are modeled as risk-averse decision-makers who may prefer diversified vote packages (i.e. split-ticket) when party and candidate uncertainties are highly correlated. The risk diversification strategy abates when voters cast their votes sequentially. This finding provides a potential explanation for the impact of vote sequence in mixed-member systems, an underinvestigated topic in the literature. It thus links the established literature on mixed-member systems with scholarship on ballot design and its effects. Additionally, the paper’s analysis explores the implications of combining the proposed model with the well-established wasted vote model. Keywords Mixed-member systems· Vote decision under uncertainty· Risk diversification· Vote sequence. 1 Introduction Mixed-member electoral systems typically allow voters to cast two ballots, one for plurality and the other for proportional representation. While many voting models exist for each tier, theoretical models that incorporate both tiers are scarce. For example, models based on voters’ expected utilities predict strategic voting in the form of split-ticket voting, but they model voting behavior in each tier independently and disregard any possible interactions between them. * Susumu Shikano [email protected] Erik S. Herron esher[email protected] 1 University ofKonstanz, Universitätsstr. 10, 78464Konstanz, Germany 2 West Virginia University, Morgantown, USA 204 Public Choice (2025) 204:203–219 This paper introduces a theoretical model that examines vote decisions in mixed-member systems, taking into account different interactions based on the system type. At the individual voter level, the model expands on conventional approaches by incorporating joint preferences for both ballots. By considering risk aversion, certain voters choose to reduce risk by splitting their ticket, even if their preference would be to cast a straight ticket if their decisions were independent. The model also explores the impact of ballot casting methods, distinguishing between simultaneous and sequential voting, and predicts varying levels of split-ticket voting. This finding offers a potential explanation for the influence of vote sequence in mixed-member systems, which remains an under-investigated topic. Additionally, while the proposed model is based on sincere voting, the analysis also explores the implications of incorporating strategic voting models, such as wasted voting. 2 Related literature There is a substantial body of literature exploring the impact of mixed-member systems on voting behavior and election outcomes (for an overview, see Herron etal. 2018). These studies can generally be categorized into two approaches: controlled comparison and contamination. The controlled comparison approach treats mixed-member systems as controlled experimental settings, where two distinct electoral systems operate within the same social, economic, and cultural contexts (e.g. Moser and Scheiner 2012). This approach assumes that the micro-level voting decision processes in each tier are independent of each other. One straightforward implication of this approach is that voting behavior in the proportional representation (PR) tier tends to be sincere, while strategic considerations, such as avoiding wasted votes, may influence voting behavior in the first-past-the-post (FPTP) tier (see e.g. Bawn 1999). Consequently, this could lead to a specific type of split-ticket voting pattern. However, it is important to note that this is just one possible implication. Alternatively, it can also be postulated that voting behavior in the PR tier is strategic with the aim of achieving certain coalition governments (Pappi and Thurner 2002; Shikano etal. 2009). Regardless of how voting behavior is modeled in each tier, models based on this approach commonly treat the voting decisions in each tier as independent of each other. In contrast to the controlled comparison approach, the contamination approach emphasizes the interdependencies between the two tiers of mixed-member systems (Ferrara etal. 2005). Early studies in this approach have demonstrated that running a candidate in a FPTP district can boost the party’s list votes in the same district, indicating a contamination effect (Herron and Nishikawa 2001). This finding suggests that voters’ decision-making processes in both tiers are not independent but rather interconnected. Subsequent studies have provided evidence for various types of contamination effects (e.g. Krauss etal. 2012), while others have expressed skepticism about the existence of such effects (e.g. Maeda 2008; Kurella 2016). It is important to note that the empirical evidence on contamination effects is often derived from aggregate-level election outcomes across different countries. This may contribute to the mixed findings. In this regard, Rheault etal. (2020) advanced our understanding of contamination effects at the individual voter level. Their study shows that the decision order between candidate and party votes, as well as the voter’s level of political information, plays a crucial role, which possibly explains the mixed empirical findings in the literature. However, while their contribution is primarily empirical, the theoretical foundations of contamination effects at the individual level remain less developed in the literature, particularly in comparison to the more established controlled comparison 205 Public Choice (2025) 204:203–219 approach. Recently, Bräuninger and Pappi (2023) advanced a formal theoretical model addressing contamination effects by explicitly introducing non-separable preferences in mixed-member voting. Yet, their work leaves open the substantive question of why and under what conditions these non-separable preferences arise. We still need a clear microfoundation for this non-separability. While studies of mixed-member systems have proliferated at least since the 1990 s, there are still several under-investigated topics within this field. One such topic is the effect of the ballot structure (Barnes et al. 2017). Mixed-member systems can adopt different types of ballot structures. For instance, in Germany, voters have a single ballot for both tiers, whereas in Japan, voters cast separate ballots for each tier. This difference in ballot structure leads to distinct vote sequences, which can potentially impact voting behavior. When voters have a single ballot, they can simultaneously choose a candidate and a party. In contrast, separate ballots require voters to make sequential decisions. These different vote sequences can influence voting behavior in various ways. In multiple survey experiments, Shikano etal. (2023); Herron and Shikano (2025) found that the Germanstyle simultaneous ballots facilitate more split-ticket voting compared to separate ballots for each tier. Their research also provides further insights into the mechanisms underlying vote decisions: • Simultaneous ballots require less overall decision time compared to separate ballots; • when faced with separate ballots, voters tend to spend significantly less time on the second ballot compared to the first one; • with simultaneous ballots, voters allocate their attention to both tiers before making their initial decision. While these results suggest the existence of an interaction between both tiers, they cannot be explained by either of the aforementioned approaches. Therefore, in the following sections, a new model will be proposed that takes these findings into account and provides a framework for understanding vote decision-making in mixed-member systems. 3 Theoretical model The proposed model incorporates the basic ideas outlined below. In this model, voters are faced with two decisions: selecting a district candidate in the FPTP tier and choosing a party list in the PR tier. Their choices are driven by the goal of maximizing their utility based on expected outcomes. There are two types of outcomes: district-specific pork and national-level politics. District candidates are responsible for the former, while parties are responsible for the latter. Candidates and parties make announcements that align with certain outcomes in the event of winning the district race or gaining governmental power. Voters are assumed to have knowledge of their own ideals regarding each type of outcome, enabling them to evaluate the proximity or distance between their positions and those announced by candidates and parties. The closer the alignment between voters and a candidate or party position, the higher the utility for the voter. Voters in this model are primarily concerned with the actual outcomes that may differ from the announced positions. This is because the realization of announced positions relies on the legislative process following the election. Consequently, there is inherent 206 Public Choice (2025) 204:203–219 uncertainty associated with the announced positions and their distance from the voters’ own positions. In this regard, voters are assumed to be risk-averse, meaning they discount utilities of outcomes with higher levels of uncertainty. Additionally, the uncertainty of candidates’ and parties’ positions may not be independent from each other. For instance, it is reasonable to assume that a party and its candidates have a positive correlation in their uncertainty, while a party and candidates from another party are less correlated or independent. Voters also take into account this covariance when casting both candidate and party list votes. Specifically, a vote combination with a higher covariance will be discounted to reduce the overall risk in the package of outcomes. As a result, some voters may choose to split their ticket, while they would have preferred to cast a straight ticket if the votes were cast independently. The basic idea above is inspired by the logic of portfolio diversification under risk, as originally formulated by Markowitz (1952). In his seminal work, Markowitz demonstrated that risk-averse investors do not evaluate assets in isolation, but instead consider the expected return and risk of an entire portfolio. Here, the total risk is shaped not only by the variance of individual assets but also by the covariance between them. Analogously, the model presented here treats vote combinations - candidate and party list votes - as a joint decision under uncertainty, where the total utility is influenced by both the expected alignment with a voter’s ideal points and the risk associated with deviations in realized policy outcomes. In line with the investment logic, voters are modeled as risk-averse decision-makers who may prefer diversified vote packages when party and candidate uncertainties are highly correlated. Here, we have to make some points about our model clear: First, regarding institutional differences between corrective mixed-member systems (e.g., Germany) and parallel systems (e.g., Japan) (Massicotte and Blais 1999), our basic setup is more consistent with the latter, where district-level pork and national-level government are determined separately by the FPTP and PR votes, respectively. Nevertheless, we argue that our model can also be applied to corrective systems, provided we assume that voters focus on the perceived consequences of their votes, rather than on the technical details of how votes are aggregated across tiers. Crucially, our model departs from the common assumption that the two votes are decided independently. Instead, we treat vote choice as a joint decision under uncertainty, driven by the voter’s concern for coherence and risk in the overall policy outcome. Here, it is important to distinguish between two types of independence: institutional independence in how votes are tallied, and perceptual independence in how voters evaluate the risk that elected representatives may diverge from their announced positions after the election. Our model is agnostic about the former, but fundamentally grounded in the latter. Second, our model is not a game-theoretic, but a decision-theoretic model. This means that individual voters are treated independently, and they make their own decisions based on their personal circumstances which are assumed to be given. Last, but not least, our model deals with one of many different possible types of motivation. Other types of motivation, such as expressive motivation (Schuessler 2000), may also play a role in split-ticket voting. However, the goal of this paper is not to model all potential motives, but rather to isolate and explore one theoretical mechanism: risk-averse utility maximization under uncertainty. By focusing on instrumental motives, we aim to provide a clear and formal account of how voters might evaluate vote combinations as a package when party and candidate behaviors are uncertain. This will never exclude the possibility of another model in which expressive and instrumental motives may interact or compete in such decisions. 207 Public Choice (2025) 204:203–219 In the following discussion, we will first introduce a simple model based on sincere voting without any strategic considerations. Then, we will incorporate the wasted vote model in the FPTP tier. 3.1 Basic set up Consider three-party competition under a mixed-member system with P={p1,p2,p3} . Each party can run one candidate in each district and denote the set of candidates in district d by Cd⊆{cd,1,cd,2,cd,3} for d∈{1, …,D} . A strategy for voter i residing in d is a pair vi=(c,p)∈P×Cd . Assume that voter i has an ideal point on the general policy dimension and the pork dimension. Since each voter can be eligible to vote only in one district, we drop the index d in the following so that: cd , j=cj . Voter i’s benefit from pj and cj is the function of the distance between their announced position and i’s ideal point, which we denote by di(cj) and di(pj) . Their announced positions have some uncertainty 𝜖cj and 𝜖pj , which concerns e.g. whether the announced positions will be implemented in legislation after the election. By using the squared loss function, we define i’s utility from candidate j’s winning the district race ( w c j ) as follows: Analogously, we also define i’s utility from party j’s winning the government power ( g p j ) as follows: The expected value of 𝜖cj and 𝜖pj is assumed to be zero and their variances are denoted by 𝜎cj and 𝜎pj , respectively. Further, we assume their independence from the other components in the utility function: The equivalent set-up of spatial voting model with uncertainty can be found in the literature (e.g. Enelow and Hinich 1984). Distinct from the previous models, however, we treat both decisions in FPTP and PR tiers simultaneously. And more crucially, we allow that random terms of both tiers can be correlated with covariance denoted by 𝜎cj , pj . 3.2 Independent choices inFPTP andPR Before we discuss simultaneous choice under mixed member systems, we discuss independent choices in FPTP and PR, respectively. In the following, we deal with each voter’s choice independently from each other in a decision-theoretic model. Therefore, we drop the index i in the following so that Ui( ⋅ )=U( ⋅ ) , vi=v , di(cj)=dcj and di(pj)=dpj . Fur ther more, without loss of w.l.o.g we assume for FPTP and PR, respectively: (1) U i(wc j )=−(di(cj)+𝜖cj) 2. (2) U i(gp j )=−(di(pj)+𝜖pj) 2. 𝜖 c j ⟂⟂ di(cj) , 𝜖p j ⟂⟂ di(pj) , 𝜖c j ⟂⟂ di(pj) , 𝜖p j ⟂⟂ di(cj) , (3) dc1<dc2<dc3 (4) dp1<dp2<dp3 208 Public Choice (2025) 204:203–219 For both FPTP and PR, our starting point is the decision-theoretic model of Black (1978) about strategic voting under multipartism. We extend it by using the above set-up with vote decision under uncertainty. This model’s most important element is the pivot probability pjj′ that a single vote can determine who of candidate cj and cj′ for j ≠ j′ wins the most votes in the election. According to his model,1 a voter would cast her vote in favor of the second closest c2 if: • p23 is large relative to p12 and p13 . • dc3−dc2 is large relative to dc2−dc1 . For the latter condition, Black did not explicitly consider uncertainty of the distance to candidate j: 𝜎2 cj . By taking it into account, we can add the following conditions: • 𝜎2 c3 −𝜎 2 c2 is large relative to 𝜎2 c2 −𝜎 2 c1 . The Black model was originally developed for FPTP elections but can also be extended to vote decision-making under PR. The key distinction lies in the assumption that the pivot probability between any pairs of parties is constant ( ppr ). However, even in PR systems, there may arise situations where a voter deviates from their closest party in favor of the candidate from their second closest party. This can occur if: In other words, we expect a PR vote in favor of the second closest party if the difference between the distance to the announced positions is compensated by the second closest party’s more certain position in comparison to the closest one’s more uncertain position. For more details see Appendix B. 3.3 Simultaneous choice If we take the controlled comparison approach, we can employ the above models for a single tier. In this subsection, in contrast, we will model simultaneous choice in both tiers with interaction. We assume that the voter seeks to have the minimum square of the sum of distances to the candidate and parties. This corresponds to the following: This function represents a joint evaluation of the overall policy package at both the local and national levels. The intuition behind summing the distances to the candidate and party before squaring is that voters form a holistic perception of risk across both tiers. They then penalize large overall deviations more, consistent with risk aversion. This structure also captures voters’ concerns about coherence or “alignment” between the candidate and the party. For example, the greater the total misalignment across both dimensions, the larger the potential for disappointment or regret in the policy outcomes – especially when voters (5) d p2 −dp1 <𝜎 2 p 1 −𝜎 2 p 2 (6) U (wc j ,gp k ) = −(dc j +𝜖c j +dp k +𝜖p k ) 2 1 See for more details Appendix A. 209 Public Choice (2025) 204:203–219 are uncertain about how closely elected representatives will adhere to their announced positions. We start with a special situation where C={c1} . That is, there is only one district candidate. Therefore, voter i has only three possible options (c1,p1) , (c1,p2) and (c1,p3) . Apparently, the voter has no need to strategically cast her vote in the FPTP tier. Now consider whether there exists a situation in which a voter would split the ticket in favor of p2 . First, we obtain the expected utility of casting the straight ticket ( vc1,p1 ) : Analogously, for a split ticket ( vc1,p2 ) The voter would split if: From here, we learn that the likelihood of asplit ticket is higher if: • the distances to the announced positions of Party 1 and 2 are similar. • the announced positions of both parties and Candidate 1 are closer to the voter. Since dp 2> dp1 and all d’s are non-negative, the left side of the last inequality is larger than zero. Therefore we obtain a further necessary condition for vc1,p2 : From this inequality, we can obtain some further insights concerning uncertainty of the announced position: • The larger the uncertainty of Party 1’s position and/or the smaller the uncertainty of Party 2’s position, the more likely is a split ticket. • The more consistent the party and its candidate in the stochastic term, the more likely there will be a split ticket. In other words, if the candidates have higher potentials to behave differently from their parties, voters are more likely to cast a straight ticket.2 (7) EU (vc1,p1)=−E[(dc1+𝜖c1+dp1+𝜖p1) 2 ] =−d2 c1−d2 p1−2dc1dp1−E[(𝜖c1+𝜖p1)2] =−d2 c1 −d2 p1 −2dc1dp1−𝜎2 c1 −𝜎2 p1 −2𝜎c1,p 1 (8) EU (vc1,p2)=−E[(dc1+𝜖c1+dp2+𝜖p2) 2 ] =−d2 c1 −d2 p2 −2dc1dp2−𝜎2 c1 −𝜎2 p2 −2𝜎c1,p 2 (9) EU(v c1,p1 )<EU(v c1,p2 ) − d2 c1−d2 p1−2dc1dp1−𝜎2 c1−𝜎2 p1−2𝜎c1,p1<−d2 c1−d2 p2−2dc1dp2−𝜎2 c1−𝜎2 p2−2𝜎c1,p 2 (dp2−dp1)(dc1+dp1+dp2)<𝜎2 p1 +2𝜎c1,p1−𝜎2 p2 −2𝜎c1,p2 (10) 𝜎2 p1 +2𝜎c1,p1−𝜎 2 p2−2𝜎c1,p2>0 𝜎2 p2 −𝜎2 p1 <2(𝜎c1,p1−𝜎c1,p2 ) 2 Note that this is one of the consequences of using the quadratic loss over the sum of distance from ideal points. If we instead use the Euclidean distance, that is U (c1,p1) = −(x c 1+𝜖 c 1−𝜃 ci ) 2 +(x p 1+𝜖 p 1−𝜃 pi ) 2 , neither 𝜎c 1, p1 nor 𝜎c 1, p2 plays any role. 210 Public Choice (2025) 204:203–219 This result can be extended to the case where we have not only one candidate, but two or three candidates. For illustrative purposes, we position parties and candidates along a one-dimensional spatial axis (e.g., {1, 2, 4} ), respectively. This setup is not meant to represent any specific party system, but rather to capture a stylized case where two options are ideologically closer and one is more distant. This is a simplified, but realistic configuration where parties are positioned in an asymmetrical way. The substantive results of our model, in particular those concerning risk diversification and vote sequence, do not depend on this specific configuration even if we change the relative distance between parties and/or the number of parties. While more complex placements such as two-dimensional structures could yield further insights, our goal here is to isolate and demonstrate the effect of correlated uncertainty in the simplest tractable setting. Both panels in Fig.1 present predicted votes for different voter positions in a certain candidate/party constellation on the pork and national policy dimensions. The grey dotted grid lines divide the electorate to different voting patterns based on dimensionby-dimension distance-based decisions. That is, both tiers are treated as independent. By contrast, the solid lines divide the electorate based on the model above which takes into account covariance between candidates’ and party’s uncertainty. The left and right panels differ in covariance, the left one is based on a higher and the right one is based on a lower covariance. If we focus on the left panel, some voters who are predicted to cast straight tickets in the dimension-by-dimension model would split their tickets in our model. These voters are very close to the grey grid lines, which means the closest and second closest candidates/parties are almost equidistant to them. Furthermore, this area becomes smaller if the distance on the other dimension increases. For example, if we look at the border area at vc1 , p1 and vc1 , p2 , its area is the largest if voters are very close to C1 ’s announced position. This area becomes smaller if voters are more distant from C1 . This corresponds to the second prediction requiring voter closeness of parties’ and candidates’ positions. If we turn to the right panel, the size of the border areas is much smaller than in the left panel. That is, if voters expect the district candidate would behave differently in the legislature, they are more likely to stick to the straight ticket. Fig. 1 Expected voting behavior depending on announced positions uncertainty. The left panel is based on a higher correlation between party and candidate announced position ( 𝜌 = 0.8 ). The right panel is based on a lower correlation ( 𝜌 = 0.2 ). The grey dotted lines divide voters depending on the closest vote combinations 217 Public Choice (2025) 204:203–219 Based on this inequality, Black identifies two conditions for voter i’s defection from her closest candidate in favor or the second-closest one: • p23 is large relative to p12 and p13 . • EU(c2)−EU(c3) is large relative to EU(c1)−EU(c2) . For the latter condition, Black did not explicitly consider uncertainty 𝜎2 . Therefore, the second condition in his version is as follows: • dc3 −dc2 is large relative to d c 2 −d c 1 . By taking uncertainty into account, we can add the following conditions: • 𝜎2 c3 −𝜎 2 c2 is large relative to 𝜎2 c 2 −𝜎 2 c 1 . B Extending Black’s model toproportional representation The set-up is analogous to the above plurality case, but the pivot-probability will be substantively different. Assume a proportional representation with a d’Hondt system. There are N voters who cast valid votes and M seats to distribute. A vote is decisive on whether one additional seat goes to Party j or j′ if the expected result of j and j′ without the vote at stake is m×N M − 1 and m� ×N M − 1 with m,m�∈{1, …M} . In other words, both parties need one additional vote to reach the next quota for another seat. Here, the pivot-probability under plurality and proportional representation is apparently different. Under plurality, both candidates should be expected to have a similar amount of votes so that the voter’s pivot probability among them becomes larger. This does not have to be the case under proportional representation since a voter can have a high pivot probability among a party being expected to obtain a large amount of votes and another party being expected to obtain a small amount of votes. Given a large number for M, we can reasonably assume p p 1 p 2 =p p 1 p 3 =p p 2 p 3 =p pr . If we apply this to the above result under plurality, we obtain: EU(c 1 )−EU(v 0 )<EU(c 2 )−EU(v 0 ) 2 p12(EU(c1)−EU(c2))+p13(EU(c1)−EU(c3)) −p 23( EU(c 2 )−EU(c 3 ) ) <0 EU(v p1 )−EU(v 0 )<EU(v p2 )−EU(v 0 ) 2 ppr(EU(gp1)−EU(gp2))+ppr(EU(gp1)−EU(gp3)) −ppr(EU(gp2)−EU(gp3))<0 EU(gp1)−EU(gp2)<0 −d2 p1 −𝜎2 p1 +d2 p2 +𝜎2 p2 <0 d2 p2 −d2 p1 <𝜎 2 p1 −𝜎2 p2 218 Public Choice (2025) 204:203–219 Acknowledgements This research was supported by a grant from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG; No.659600). An earlier version of the manuscript was presented at the annual meeting of the Working Group on Analytical Political Theory of the German Political Science Association in Hamburg in June 2023. We thank the participants for their constructive feedback. We are also grateful to Franz Urban Pappi and the three anonymous reviewers for their valuable comments, which significantly improved the manuscript. Special thanks go to Lilli Becker and Maja Stahl for their excellent research assistance. Funding Open Access funding enabled and organized by Projekt DEAL. Declarations Conflict of interest: The authors have no Conflict of interest to declare that are relevant to the content of this article. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References Barnes, T. D., C. Tchintian, and S. Alles. 2017. Assessing ballot structure and split ticket voting: Evidence from a quasi-experiment. 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