Policy Preferences and Policy Legitimacy After Referendums: Evidence from the Brexit Negotiations

How do votes in direct democratic ballots translate into policy preferences about future outcomes and affect the perceived legitimacy of those outcomes? This article examines these questions in the context of sovereignty referendums: specifically, the 2016 referendum on British membership of the European Union (EU). While the referendum result gave the British government a mandate for Britain leaving the EU, it did not provide any firm guidance as to the kind of Brexit that voters would prefer and consider legitimate. To examine the perceived desirability and legitimacy of different Brexit outcomes, we conducted a nationally representative conjoint experiment measuring attitudes towards different possible negotiation outcomes. Our findings show that ‘Leave’ and ‘Remain’ voters were highly divided over what they wanted from Brexit on salient negotiation issues, but also that most voters did not regard any possible outcome as legitimate.


Introduction
Referendums are increasingly used as a means of deciding important constitutional matters (Matsusaka 2005;Tolbert and Smith 2006;Hobolt 2009;Mendez and Germann 2018). They give citizens an opportunity to have a direct say on fundamental political questions. Yet the simple binary nature of most referendum ballots also means that complex policy problems are often reduced to stark either/or choices (Setälä 1999;Bowler and Donovan 2002;Rose 2019). This is a particular issue in sovereignty referendums, where a decision to reallocate powers between two territories may be followed by protracted negotiations about the future of the relationship. Policymakers are thus given a difficult challenge: how to translate the outcome of a dichotomous vote choice into policy which involves difficult political trade-offs. While there is a large literature on voting behavior in referendums, we know far less about how such vote choices map onto preferences for these future policy outcomes.
We examine this issue in the context of the 2016 referendum on British membership of the European Union (EU), a sovereignty referendum in which British voters took the historic decision to be the first member state to exit the EU. 1 While the narrow vote in favor of Britain leaving the EU ('Brexit') gave the British government a democratic mandate to exit, it was less obvious what this mandate meant in terms of the final political settlement. The post-Brexit landscape could have looked very close to the status quo of EU membership or could have involved a much more dramatic shift away from the previous legal, political and trading relationship with the EU (Richards et al. 2018). This is because the British government's negotiations on the relationship between the UK and the EU involved an array of complex tradeoffs on questions of immigration, border security, trade, citizens' rights and budgetary contributions to the EU. None of these issues featured on the referendum ballot paper, nor were these issues for which there was a clear policy mandate from either of the official referendum campaigns. A burgeoning body of literature has examined the determinants of voting behavior in the Brexit referendum (Goodwin and Heath 2016;Hobolt 2016;Vasilopoulou 2016;Becker et al 2017;Clarke and Goodwin 2017;Curtice 2017;Evans and Menon 2017;Ford and Goodwin 2017;Colantone and Stanig 2018;Fisher and Renwick 2018) and in earlier EU referendums elsewhere (Franklin et al 1995;Garry et al 2005;Hobolt 2009;Walter et al. 2018), but we know far less about attitudes towards policy changes after the vote. 2 From the perspective of democratic legitimacy, this raises two important questions. What do people want after an outcome in a sovereignty referendum that breaks with the status quo: what are their policy preferences? And what are people willing to accept after the vote: which outcomes do they think are legitimate? Political Behavior (2022) 44:839-858 To isolate public preferences and perceptions of legitimacy about the outcome of Brexit, we ran a survey experiment on a nationally representative sample of UK adults in late April 2017. This was a month after Britain initiated the process to exit the EU, but before any actual negotiations began. Specifically, we conducted a highdimensional factorial (conjoint) experiment. This type of design allows researchers to uncover the relative influence of different factors in how people make decisions over bundled outcomes (Jasso 2006;Hainmueller et al 2014;Auspurg and Hinz 2015;Leeper et al. 2020). Crucially, we measured both the British public's preferences towards the outcome of the Brexit process as well as their views of what outcomes were legitimate. We then tested how people's views were shaped by their own vote in the referendum. Our findings show that there were distinct differences in preferences between 'Leavers' and 'Remainers' on the key issues that were up for negotiation. We also find that neither side considered any of the available policy outcomes as adequately respecting the vote. This implies that the reality of policy making following some referendums may not live up to people's expectations, even for those on the winning side. It also means that while sovereignty referendum votes may be regarded as legitimate, the eventual policy outcome may not have the same legitimacy.
In what follows, we first motivate the question of policy preferences in the wake of the Brexit vote with a discussion of the literature on referendums and public opinion about policy. We then discuss the design of our study and present the results. Finally, we draw some conclusions about how our results should affect our understanding of the consequences of sovereignty referendums.

Referendum Votes and Policy Outcomes
Citizens are given a direct say on fundamental constitutional issues, such as the territorial contours of the nation, more often than ever before (Mendez and Germann 2018). In Europe, referendums are increasingly used to decide a country's relationship with the EU (Hobolt 2009;Rose 2019). Yet, direct democracy remains controversial. Most citizens like it: evidence from the Europe-wide European Election Study shows that 63% of people favor a direct vote on EU treaties (Van Egmond et al. 2011;Rose and Borz 2013). Scholars are often more skeptical about referendums, however. While some have emphasized the advantages of direct democracy (Setälä 1999;Torgler 2005;Smith 2009), others have focused on potentially undesirable consequences for minority rights (Bowler et al. 1998;Gerber 1999;Broder 2000;Ellis 2002;Butler and Ranney 1978;Gamble 1997) and on the inability of voters to make competent decisions given the influence of special interests (Magleby 1984;Lupia 1994;Lupia and McCubbins 1998;Hobolt 2009). Whether good or bad, referendums are clearly consequential. In the US, direct democracy has led to differences between states in tax and spending policies (Matsusaka 2004;Feld and Matsusaka 2004) as well as in other non-economic policy domains such as criminal punishment (Gerber 1999). In the EU, referendums have often had unintended and 'elite-defying' consequences for governments, resulting in the delay, and even rejection of, intergovernmental treaties (Hobolt 2009;Hooghe and Marks 2009;Walter et al. 2018).
Nonetheless, little attention has been paid to the question of how votes in referendums map onto preferences towards the ensuing policy outcomes. Instead, research has focused on the paradox that policies mandated by a direct democratic vote are often passed and implemented by representative institutions which were originally opposed to the policy outcome (Gerber 1999;Gerber et al. 2004) or on the effects of direct democracy on political support more generally (Marien and Kern 2018). However, this research does not address the question of how voters view the policy outcomes that follow referendums.
Yet this question is both normatively and empirically important, especially in cases where voters are offered only a dichotomous choice over a set of interlinked policy questions.
Of course, this choice is no less complex in representative democratic processes, when citizens are asked to cast their vote for parties, or candidates, that offer bundles of policy positions. But these processes are not fully analogous for two important reasons. First, at a general election policy proposals are set out explicitly in party and candidate manifestos. By contrast, those engaged in referendum campaigns are not required to set out any detailed policy proposals on how to implement the referendum result. This is a particular problem in "sovereignty referendums", that is direct popular votes on a reallocation of sovereignty between at least two territorial centers (Mendez and Germann 2018). Examples include referendums on EU membership (such as the Brexit vote) and secession referendums, such as the independence referendums in Quebec (narrowly rejected by the electorate in 1995), Scotland (rejected by the electorate in 2014) and Catalonia (declared unconstitutional by the Constitutional Court of Spain in 2017). Unlike more narrowly-focused single-issue referendums that amend specific constitutional articles, such as the same-sex and abortion referendums in Ireland, sovereignty referendums are often 3 "open-ended" in that they offer no detailed blueprint for the future relationship between the seceding state and the rest of the remaining territorial unit. As a consequence, the mandate for the policy discussions that follow on from such sovereignty referendums is unclear. Indeed, in the Brexit referendum, the government (which advocated a Remain vote) explicitly refused to engage in the question of what would happen if the Leave side won, and the politicians campaigning in favor of Leave were split on key questions of what a post-Brexit Britain would look like (Eleftheriadis 2017;Menon and Fowler 2016). 4 The second difference is that after a general election, those elected can be held to account if they are perceived to have broken their promises or if voters have a change of heart. But while referendums tend to decide significant constitutional issues, they do not provide the scheduled opportunity for a re-run that voters are familiar with in an ordinary electoral cycle. There is no clear point at which representatives can be held to account for policy outcomes. This is not to say that referendums are never re-run, they are, but voters do not know if, and when, that re-run will happen. At the point of the vote, referendums in most democracies appear to be one-shot opportunities to make a dichotomous choice, often with limited information about the policy change that follows. 5 It is thus critical to know how the public forms opinions about the policy implemented by the winning majority, and to what extent pre-referendum divisions drive those opinions. Equally, it is also important to know whether those policy outcomes are considered legitimate by people on both sides of the referendum divide.
The Brexit referendum provides an apposite case study of policy preferences and legitimacy perceptions following a direct democratic vote. 52% of voters voted Leave, and politicians on both sides agreed to respect the result in the immediate aftermath of the vote. However, the result provided limited guidance as to what kind of Brexit the public wanted: the option favored by a majority in the referendum may have been 'Leave', but the range of outcomes contained within 'Leave' immediately after the referendum was large. Brexit could have implied various different types of settlement (Eleftheriadis 2017;Richards et al 2018). At one extreme, Britain could have left the EU without a formal withdrawal deal or any agreement on future trade relations: a so-called 'no deal' option. At the other extreme, Britain could have stayed part of the European Single Market and Customs Union, accepted the continued jurisdiction of the European Court of Justice and allowed freedom of movement into and out of Britain: a so-called 'soft Brexit'.
Our aim is to examine not only how the referendum vote is related to preferences over the Brexit outcome, but also to address whether voters considered different policy outcomes a legitimate consequence of the referendum result. We expect that differences over the preferred Brexit outcome will reflect differences in referendum vote choice, as these differences of opinion drove citizens' vote choices in the first place. In other words, we expect that policy preferences are correlated with referendum vote choice, with Leavers preferring a harder Brexit and Remainers favoring a softer Brexit. We thus expect these differences to be related in a meaningful way to the salient issues raised during the referendum campaign. Specifically, there is considerable evidence that the policy issues of immigration and sovereignty were core elements of the Leave campaign, while the Remain campaign emphasized the economic risks of Brexit (Menon and Fowler 2016;Clarke and Goodwin 2017;Evans and Menon 2017). These factors were also key attitudinal drivers of people's vote choices (Hobolt 2016;Evans and Tilley 2017;Clarke and Goodwin 2017;Curtice 2017;Fisher and Renwick 2018). Consequently, to the extent that vote choices are correlated with preferences for specific policy trade-offs, we expect that Leavers were more likely to prefer policy outcomes that restrict immigration and strengthen Britain's legal sovereignty. In contrast, Remain voters were likely to prefer outcomes that facilitate continued free trade with the European Union.
However, we are not only interested in which policy options Leavers and Remainers prefer, but also whether they consider such policy outcomes legitimate. Legitimacy refers to the public acceptance of political authority. In a well-functioning representative democracy, citizens will consider a democratically elected government and its policies legitimate even if they did not vote for a party that forms the government (Anderson et al. 2005). In direct democracy, however, the authority of policy outcomes is not derived from elections, but from the referendum vote. But to what extent is there "losers' consent" after referendums? That is, do voters on the losing side respect the outcome of the vote and accept the resulting policy outcomes as reflecting the authority of the referendum? We know that there is a gap between winners and losers in political support following both elections and referendums (Anderson et al. 2005;Esaiasson 2011;Marien and Kern 2018), but, in this paper, we focus specifically on the legitimacy of policy outcomes, rather than general levels of political support. We expect that many voters, both Leavers and Remainers, will have had misgivings about the legitimacy of the possible Brexit outcomes when seen as sets of policies. As discussed, referendums rarely force the two sides to present bundles of policies in a manifesto. In open-ended sovereignty referendums, the inherent future policy trade-offs after the referendum are therefore less immediately visible to voters than they would be after a general election. Equally, referendums are not re-run at regular intervals, unlike regular general elections, which may lead to a further lack of perceived legitimacy. Overall, we therefore expect that most voters, on both the winning and losing sides, will not consider the outcomes they are presented with as legitimate. In the ensuing sections, we examine these expectations empirically.

Methods
Measurement of preferences over multifaceted objects of evaluation, like Brexit, is a difficult task. Traditional approaches to public opinion research tend to entail the measurement of preferences over outcomes as a whole (e.g. support for Brexit per se), measurement of often vague variations on that outcome (e.g. support for something simply labelled a 'soft Brexit' outcome), or the measurement of stated preferences over multiple, isolated features (e.g. separate questions measuring preferences over immigration and trade policy). Each of these approaches has significant limitations. The first two say little about why the public prefers particular outcomes over others. The second also relies on survey respondents making inferences about the meaning of specific terms. And while the third approach measures attitudes towards specific aspects of an outcome, it does not force people to make the tradeoffs between types of outcome that need to be made. Without being forced to choose among options, we cannot judge the relative importance to people of different features of any outcome. Political Behavior (2022) 44:839-858 To overcome these limitations, we rely on a conjoint experimental design. Borrowed from marketing research, where it is used to study purchasing decisions, this methodology has recently been used in public opinion research to study complex opinion formation processes such as support for immigration policies (Bansak et al 2016;Hainmueller and Hopkins 2015), voting for candidates (Hainmueller et al 2014) and preferences for labor market reform (Gallego and Marx 2017). In a conjoint study, participants are shown a series of pairs of vignettes that vary according to a determined set of features, with combinations of features randomly varied. Respondents then select which of each pair they prefer. Rather than asking people directly about each separate feature, their discrete choices reveal the acceptability of different features. In this case that means that people have to engage directly with the difficult trade-offs involved in the negotiations. 6 This allows us to make comparisons between respondents' evaluations of different bundles in order to detect the relative importance of individual features without asking them directly and without having to use any politicized labels such as 'hard Brexit', 'soft Brexit', or 'no deal'. Respondents were not asked to adjudicate on whether a 'free trade deal' or 'freedom of movement' were desirable, but rather make a choice between scenarios that use precise language to specify different kinds of arrangements.
In our design, we asked people 10 months after the referendum to consider pairs of Brexit outcome scenarios that varied along eight dimensions. Respondents were told: 'We are interested in your opinions about possible agreements between Britain and the EU regarding Britain's exit from the EU and future relationship.' They were then presented with a pair of alternative outcomes of the Brexit negotiations that varied along eight dimensions and asked to choose which of the two alternatives they preferred. The eight dimensions were carefully selected to cover the full breadth of the negotiations at that stage and were fully randomized. They used terminology drawn directly from statements of the negotiating parties, the UK government and the European Commission (UK Department for Exiting the European Union 2017; UK Prime Minister 2017a; UK Prime Minister 2017b; European Commission 2017). These eight dimensions were: (1) immigration controls, (2) legal sovereignty, (3) rights of EU nationals in the UK and UK nationals in the EU, (4) ongoing EU budget payments, (5) one-off 'divorce' payment, (6) trade terms, (7) status of the Ireland/Northern Ireland border, and (8) the timeline for Brexit. The different levels of the dimensions were designed in such a way as to range between the two most extreme negotiation outcomes: a very 'soft' Brexit with continued British membership of the EU's Single Market and Customs Union and a 'no deal' scenario with no agreement on a future trade deal. 7 A screenshot of an example pairing is shown in Our analysis focuses on two outcome questions, one measuring preferences and one measuring the perceived legitimacy of each bundled outcome. The first question asked respondents simply: 'Which of these two outcomes do you prefer?'. 8 To capture policy legitimacy, we ask people to consider which policy bundles 'respect' the referendum outcome: 'Which option do you think would respect the result of the referendum? [Option A, Option B, Both, Neither]'. We decompose answers to this second question into dichotomous measures of perceived legitimacy for each outcome: 'neither' means both outcomes are coded as zero and 'both' means that both outcomes are coded as 1. It is worth noting that this is a specific measure of policy legitimacy post-referendum rather than the legitimacy of the referendum process itself. In that sense, it does not directly address whether people thought the referendum was procedurally fair. Although this is, rightly, used as the key measure of perceived legitimacy (see, for example, Esaiasson et al. 2012Esaiasson et al. , 2019Marien and  Footnote 7 (continued) involve priming people with information about most criminologists' views of the effect of the death penalty on crime rates, here we do not prime people with most economists' views of the impact of different trade deals on economic growth. 8 We also asked respondents to separately rate their support for each of the two outcomes on continuous scales. The results are virtually identical and are reported in the Supplementary Materials.
Political Behavior (2022) 44:839-858 Kern 2018) of referendums, here we are interested in the legitimacy of the policy outcomes that follow the referendum. Consistent with methods proposed by Hainmueller et al (2014), we regress our first outcome measure on indicators for each possible level of each policy feature, clustering standard errors by respondent. We can then interpret our results as a set of average marginal component effect (AMCE) estimates. These indicate the marginal effect of each feature level on support for a particular outcome deal. This conveys the degree to which a given feature increases or decreases support for a bundle as a whole relative to some baseline scenario. In our case, we treat a 'no deal' exit of the EU as the baseline. 9 We interpret this to mean that there would be no UK-EU trade deal, full legal independence of Britain from EU law and the European Court of Justice, no one-off or continuing payments to the EU budget, full control over immigration with no continuing EU immigration, the loss of rights of EU citizens currently residing in the UK, and a full (customs and passport) border between Ireland and Northern Ireland (UK Department for Exiting the European Union 2017; UK Prime Minister 2017b). In general, positive AMCEs thus indicate support for softer Brexit outcomes and negative values indicate opposition to those scenarios. 10 We also present marginal means to illustrate differences between Leavers and Remainers in both their views of outcomes and the legitimacy of those outcomes. The marginal mean is simply an estimate of the average support for scenarios containing a given feature. The reason for the shift in statistic is twofold: first, marginal means allow us to convey differences in preferences between respondents for all feature levels, rather than differences in feature effects relative to a baseline feature level. Second, by doing so we avoid misinterpreting AMCEs due to the arbitrary selection of a baseline category, especially when comparing results for Leavers and Remainers (Leeper et al. 2020).
We recruited respondents through YouGov's online UK Omnibus panel. This draws respondents from an online panel of approximately 800,000 people to construct a quota sample representative of the British public with respect to age, sex, education and region which is then weighted to match the British adult population. We surveyed a total of 3,293 respondents between April 26th and April 27th in 2017. Each respondent was shown five pairs of outcomes, which translates into a dataset of 32,930 (2 × 5 × 3293) evaluations of outcomes. Figure 2 presents the preference AMCE estimates for each feature of different areas relative to the no deal baseline. The lines around the point estimates are 95% confidence intervals. A positive AMCE indicates that respondents would, on average, be more likely to support a deal that had that feature compared to the baseline. For example, the AMCE for a timeline of 2025 is − 0.03 (se = 0.01), indicating that respondents would be about 3% points less likely to support any deal with Britain leaving the EU in 2025 compared to leaving in 2019 (the baseline reference category).

Results
The figure shows three clear patterns. First, some facets of a harder Brexit were popular in 2017. On average, people liked the idea of greater control over immigration, greater British sovereignty and smaller one-off payments to settle Britain's outstanding fiscal obligations to the EU. It is interesting that some of the least preferred features overall were those that are a required part of EU membership: by far the least popular immigration and legal sovereignty options were the status quo. Second, it is also the case that some parts of a softer Brexit were popular. On average, people preferred fewer trade barriers and the continued protection of EU citizens' rights to stay in Britain. While the trade features seem less important than sovereignty and immigration, there was very strong support for a deal that allowed EU citizens to stay in Britain. Third, some features of the settlement seem less important in their impact. On average, the Irish border and the timing of the UK's exit appear to have mattered hardly at all to the public in 2017, and access to EU programs was popular only if that access cost very little.
Next, we want to test our expectation that preferences for policy outcomes mirror the referendum vote choices. Figure 3 therefore presents results of the experiment Fig. 3 Marginal means measuring support for features of Brexit outcomes, separately for Leave and Remain voters. Note The gray vertical bar represents the grand mean for all respondents (0.5) separated by Leave and Remain voters, using a measure of vote choice which was recorded immediately after the 2016 referendum. 11 Figure 3 presents these descriptive estimates as marginal means. Given the forced choice design, the overall mean is 0.5, indicated by the vertical gray line. When a marginal mean exceeds 0.5, respondents favor scenarios with that feature more often than not, and when a marginal mean is below 0.5, respondents oppose scenarios with that feature more often than not. Figure 3 reveals some similarity in preferences among Leave and Remain voters on some dimensions, but also considerable differences. For example, features near the bottom of the graph appear to have been viewed similarly by both groups, whereas those near the top of the figure-immigration, legal sovereignty, the rights of EU nationals living in the UK, and ongoing budget payments-appear to have been viewed rather differently by Leave and Remain voters. Figure 4 shows the differences in marginal means for both groups. Positive differences indicate the features that Remain voters favor more than Leave voters and negative differences indicate the features that Leave voters favor more than Remain voters. There are few, if any, differences for some features. Leave and Remain voters held fairly similar views on the timeline for Brexit to occur, trade policy, and the size of the one-off payment. Remainers were slightly more likely to favor a longer timeline, greater free trade and a bigger payment, but surprisingly these differences are relatively small. It is particularly noteworthy that frictionless trade was not a distinguishing feature to the extent that might be expected, given the prominence that the Remain campaign gave to the negative economic consequences of Brexit. However, while the Remain campaign clearly put more emphasis on the economic aspects of Brexit, both campaigns advocated loosely for continued 'free trade' with the European Union and this may explain why we do not observe great differences between Leave and Remain voters.
On other features, however, there are clear differences in people's preferences depending on how they voted in 2016. In three areas, these differences are very large. On immigration, full control was more popular with Leave voters than Remain voters, whereas no control was substantially more popular among Remain voters than Leave voters. Only 34% of Leavers favored the status quo, compared to 47% of Remainers. When it comes to legal sovereignty from the European Court of Justice, Leavers again disliked the status quo much more than Remainers although, as with immigration, it is important to note that on sovereignty Remainers still preferred a 'harder' Brexit outcome than the status quo. Finally, while Leavers were equally likely to prefer policy outcome scenarios where EU citizens are required to leave Britain (42% in favor) and scenarios where all EU citizens can stay indefinitely (43% in favor), Remainers had very distinct preferences. Only 30% of scenarios which involve EU citizens being required to leave were preferred by Remainers, but 60% of scenarios with the status quo of all being allowed to stay found favor. On two other issues the aggregate results in Fig. 2 also mask some heterogeneity. With regards to the ongoing payment to the EU and EU programs, no access is regarded fairly negatively by Remainers, but quite positively by Leavers. Similarly, Leavers' most preferred option on the Irish border (full passport and customs checks) was Remainers' least preferred option, while Remainers' most preferred option (no passport and no customs checks) was Leavers' least preferred option.
What people wanted Brexit to mean therefore depended to a large extent on how they voted in the referendum. Much of this is driven by the fact that many aspects of the negotiation appear to have mattered more to Leavers than Remainers. Differences in preferences between levels were clearly greater for Leavers when it comes to immigration and sovereignty, but also ongoing payments, the size of the one-off payment, the Irish border, and the timeline. The exception is EU citizen rights. Here, the different levels of the feature had much more effect on Remainers than Leavers.
We are not just interested in what outcomes people preferred, but also what outcomes they thought legitimate. Figure 5 compares marginal means for Remain and Leave voters with regard to their views of what outcomes would respect the referendum. Rather than being forced to choose, respondents were able to answer that neither scenario was legitimate and many took that option. The first thing to note here is that the overall mean is only just above 0.3. In fact, few scenarios, no matter how configured, were seen as legitimate by a majority of people. This is an important finding: no matter how one juggles the levels of each feature, there was little perceived legitimacy on either side. Equally importantly, perceptions of legitimate policy outcomes were less frequent among Leavers. For every possible value of every feature, Remainers were more likely than Leavers to think profiles containing that feature respected the referendum. The winners of the referendum were harder to satisfy than the losers when it comes to the legitimacy of the final policy outcome.
The second thing to note is that the features of the outcome mattered less to Remainers than Leavers, but in many cases mattered in a similar way. For example, on immigration everyone thought that greater control was a more legitimate outcome, but a deal's legitimacy was much more sensitive to the precise immigration policy involved for Leavers. This is also evident for sovereignty and to a lesser extent trade, the timeline and the one-off payment. Leavers are simply more varied in their assessments. They tended to say a Brexit package respected the referendum when it contained features that, on average, they preferred (such as full immigration control and full legal sovereignty) and thought that alternatives closer to the status quo would not respect the outcome. Leave voters saw a clear story: their own group preferences were the most legitimate interpretation of the referendum result. While Remainers appear to acknowledge that many of the features that they personally disliked may, in fact, respect the referendum result (the exception being EU citizens' rights), the precise configuration more weakly affects their views of legitimacy.
The combination of these two results is striking. One expectation might be those on the losing side would be more sensitive to the particular features of the deal, but instead they acquiesced to the position that a wide range of policy bundles-varying from 'no deal' to the status quo-were equally likely to respect the referendum result. Leavers, on the winning side of the referendum, saw all features as less legitimate, but it is their own preferences that appear to drive perceptions of legitimacy: perhaps unsurprisingly their own preferred scenario was the one that was seen to best respect the referendum.

Conclusion
One of the challenges of direct democracy is that voters are given a binary choice on issues that are often highly multifaceted. This is a particular problem in sovereignty referendums, which account for nearly half of referendums around the world (Butler and Ranney 1994; Mendez and Germann 2018), as such referendums are often open-ended, offering no clear blueprint for the complex future relationship between two territorial units. Considering the popularity of referendums in modern democracies today, it is important to know how to interpret the preferences expressed in those referendums and whether policy decisions based on direct democracy are perceived as legitimate. While there is a large literature exploring how people vote in referendums, less attention has been paid to how those vote choices translate into preferences about future policy outcomes. Are policy preferences aligned with vote choices? And are the policy outcomes based on the referendum results considered as legitimate by both winners and losers?
We examine these questions in the context of one prominent open-ended sovereignty referendum, namely the 2016 EU membership referendum in Britain. The paper makes three distinct contributions. First, we make use of a conjoint experiment to examine people's preferences over the key policy decisions involved in the Brexit settlement. Rather than asking people directly about each separate policy feature, the conjoint design allows their discrete choices to reveal the acceptability of different features. This means that people had to engage directly with the difficult trade-offs involved in the negotiations. We are therefore able to make comparisons between respondents' evaluations of different bundles and assess the relative importance of individual decisions. This approach is ideally suited to analyzing preferences towards complex policy-decisions that involve significant trade-offs, and our paper presents an approach for how to study policy preferences in the aftermath of direct democratic votes. Of course, if the ultimate policy alternatives are composed of a different set of issue dimensions than those used in the study, further research would be needed-a conjoint analysis can only tell us about decision making on a task similar to the one used in the study (see Abramson et al 2019).
Second, given the immense political importance of Brexit to both Britain and the EU, we also contribute valuable insights into the public's view of this historic decision. The 2016 referendum provided a mandate to leave the EU but did not bring a high degree of clarity to the question of what the British public might want from Brexit. Our results have revealed that during the initial negotiation period there was agreement among both Remainers and Leavers on some areas of the settlement. Most voters shared preferences for a settlement that ensured relatively frictionless trade between the UK and the EU and avoided any large one-off payments to settle Britain's outstanding fiscal obligations to the EU. 12 Nonetheless support for many aspects of any negotiated settlement varied substantially according to how people voted in the referendum. Deep-seated divisions remained over the issues of immigration, legal sovereignty and EU citizens' rights. In line with the literature on vote choices in the referendum, we find that Leave voters were much more concerned with ensuring an outcome that guaranteed immigration control and greater legal sovereignty from the EU than those who voted Remain, who in turn cared more about guaranteeing EU citizens' right to stay in the UK.
Finally, our paper addresses a question with wider normative implications for the study of direct democracy, and specifically open-ended sovereignty referendums: whether policy outcomes resulting from the referendum were seen to respect the outcome of the vote. Interestingly, we find that when facing actual policy bundles rather than simply the Leave/Remain choice, levels of perceived policy legitimacy are low among both Leavers and Remainers. It is not just that no policy bundle was seen as legitimate by both Leavers and Remainers, but that there was no policy bundle that could individually satisfy either group. Indeed, it is the winners of the referendum, Leave voters, who were particularly unlikely to perceive bundled policy outcomes as legitimate. Among Remainers, there is a degree of "losers' consent" as they were more likely to accept any outcome as legitimate (Nadeau and Blais 1993;Anderson et al 2005).
This suggests that direct democracy will not necessarily provide greater satisfaction with democracy among winners in the long-term since they may not think that the policies that emerge after referendums respect their vote. It also implies that while referendums are generally popular with voters, there is a danger that both winners and losers may end up less than satisfied with the eventual outcome. This challenge is much greater in open-ended and high-stakes sovereignty referendums, such as those on EU membership and secession, since there is often considerable uncertainty about the policy consequences of the binary choice made in the polling booth. This contrasts with narrowly focused single-issue referendums which are likely to offer much greater clarity on the policies stemming from the vote. It is thus vital that future research should do more to explore voters' preferences towards the policy outcomes emanating from different types of direct democracy and how these shape attitudes towards democracy itself.