How Generational Replacement Undermined the Electoral Resilience of Fianna Fáil

This chapter focuses on party switching. The civil-war cleavage that differentiated the two main Irish parties, Fianna Fáil and Fine Gael, has been gradually diminishing in importance in recent decades. This trend reached a crescendo in 2011, when the incumbent Fianna Fáil party saw a dramatic decline in voter support, with swathes of its core voters switching to the main opposition party, Fine Gael. This volatility must be seen from the perspective of a generational replacement. To understand the potential for electoral switching, as opposed to change after the fact, the chapter investigates the configuration of voters’ preferences expressed through propensity to vote questions in the INES. The general framework provides theoretical tools better to understand the scale of Fianna Fáil’s defeat, as unique commitment to that party had declined markedly from the position a generation previously and it was thus more vulnerable to punishment following the crisis.

electoral cleavages and resulting electoral losses of cleavage-based parties in many Western countries in the 1960s, 1970s, and 1980s (Franklin, Mackie, and Valen 1992Franklin 2009).
In the twenty-first century most Irish citizens have been socialized long after the conflicts that gave rise to the Irish party system (and that formed deeply held partisan identities of the generations directly involved in those conflicts), so the question arises whether the dramatic 2011 losses of Fianna Fáil could have occurred because of lack of fervour of its erstwhile supporters, particularly those of younger generations. An additional reason why younger voters are likely to have a higher potential for vote switching is that they have not yet had the opportunity to be 'immunized' against such change. Immunization is the psychological effect of the act of choosing, which, after having repeatedly voted for the same party, leads to a strong loyalty to that party (Butler and Stokes 1969;Miller and Shanks 1996;van der Eijk and Franklin 2009: 49-53). These various considerations lead us to focus in our analyses on possible generational differences.

Preferences for Multiple Parties
Election outcomes are definitive in political terms (relating to government formation and policymaking), but they are often much less definitive as reflections of voters' preferences.
As Powell (2000: 160) argues eloquently, choice does not tell us enough about voter preferences to understand electoral behaviour adequately, but the additional information that is required for that purpose cannot be derived from the ballot, not even in the single transferable vote electoral system that is used in Ireland. Voting for a party does not necessarily involve a strong preference for it, nor does not voting for a particular party imply rejection or antipathy. Additionally, some voters may have made their choice with confidence and without hesitation, while others will have been deeply uncertain and hesitant about the choice that they eventually made. In order to understand how definitive choices are, and thus also what the potential is for changes in those choices, information is required about the electoral attractiveness of each of the parties for a voter. From this we can derive which parties are held in a positive regard by a voter, or between which she hesitates. Such information can be obtained only from surveys in which the relevant questions were asked. If such data are available, then we can, at least in principle, address questions about the potential for changes in individual voters ' choices, and, in  An empirical approach to gauge the potential for electoral volatility that has acquired considerable traction since the 1990s is based on so-called non-ipsative electoral preferences (van der Eijk et al. 2006). These are absolute preferences for each of the parties and candidates that one can vote for-or in Downsian terms 'utilities' (Downs 1957)-in contrast to relative preferences that are expressed on the ballot. Such non-ipsative preferences define, at the individual level, the parties that a voter may consider supporting in an election (socalled consideration sets; on this see Pieters and Verplanken 1995; Wilson 2008) and they constitute a basis for deducing how easy or how difficult it would have been for the individuals concerned to have marked the ballot differently than they actually did.
Aggregating this over individuals yields an indication of the potential for electoral volatility at the aggregate level. This approach can be used in a generic way that just focuses on the ease with which voters could have switched from one party to each of the others (van der Eijk and Niemöller 1984;Kroh et al. 2007), or in a specific way that focuses on the aggregate electoral consequences of specific changes in the context within which voters make their choices (van der Brug et al. 2007; Walter and van der Eijk 2016).
Non-ipsative preferences for parties can be measured in different ways. The three most widely known are: (1) the so-called feeling thermometers used in the American National Election studies and subsequently in election studies in many other countries; (2) questions about how strongly one likes or dislikes each of the parties, as used in the Comparative Study of Electoral Systems (CSES); and (3) questions about how likely it is that one will 'ever' vote for each of the parties ('propensity to vote' or PTV questions). Of these three the last-the PTVs-have been demonstrated to be most strongly related to actual electoral behaviour (van der Eijk and Marsh 2011): the party supported on the ballot is almost always the one with the highest preference score. More importantly, such questions have been included in various surveys of the Irish electorate, including the Irish National Election Study (INES). 1 We therefore focus on these questions in our attempt to shed further light on the 2011 electoral collapse of Fianna Fáil.

Data and Analytical Design
1 The actual wording of the question is 'We have a number of political parties in Ireland each of which would like to get your vote. How probable is it that you will ever give your first preference vote to the following parties? Please use the numbers on this scale to indicate your views, where "1" means "not at all probable" and "10" means "very probable". ' In INES 2002 and2007 this question was asked for each of the following: Fianna Fáil, Fine Gael, Greens, Labour, Progressive Democrats (PDs), and Sinn Féin, and 'an Independent candidate'. In the 2011 INES the PDs were not included in this question, but instead the United Left Alliance (ULA) was; moreover, the response options in 2011 ranged from 0 to 10. We are interested in the changing strength over time of party preferences for Fianna Fáil, and in the possible role of generational replacement in this. Because the passing of generations is a slow and gradual process, this requires observations over as long a time period as possible.
We therefore complement the data from the INES of 2002INES of , 2007INES of , and 2011 almost a decade-with the Irish samples of the European Parliament Election Studies (EES) of 1989, 1994, 1999, 2004, 2009, and 2014 2 -which jointly provide a window of observations of twenty-five years. All these studies were designed as representative sample surveys of the Irish electorate, and all were fielded in the immediate context of an actual election. 3 Yet, at the same time, these surveys also pose comparability problems. Although both INES and EES aim to be representative surveys, they differ in sampling and fieldwork procedures, in response rates, in modes of interviewing, and so on. As a consequence, none of them presents an accurate aggregate perspective on the Irish adult population at the time of the surveys, 4 and each is subject to its own particular kind of biases.
Within a single survey this can be ameliorated in principle by weighting, but such a strategy is more problematic when dealing with multiple surveys that are not even fully comparable in terms of criteria to be used in weighting. We therefore refrain from weighting, and rely instead on within-survey comparisons between groups (such as generations), the results of which are subsequently used in over-time comparisons of the surveys. An additional complication is that the INES and EES surveys were conducted in different political contexts-namely, the Dáil elections and the European Parliament elections respectively (in 1989 these two elections were held on the same day, but on all other occasions they did not coincide). Although party choice in European Parliament elections is predominantly determined by domestic political considerations (Reif and Schmitt 1980; Oppenhuis, van der Eijk, and Franklin 1996; van der Brug and van der Eijk 2007), the relative weight of these considerations is not the same as in Dáil elections, which also invalidates direct comparisons. 5 Within each of the surveys we compare groups distinguished on the basis of generations and of age. In terms of generations we distinguish three groups-referring to them as cohorts, a more specific and therefore preferred term in social research (Glenn 2005). The oldest cohort consists of people born before 1940. This broad group reached adulthood mainly before the widespread modernization of Irish society of the 1960s. The older members of this group experienced first hand the foundation period of the Irish party system, while for most of the younger members of this cohort their parents had this direct experience. This group is therefore most likely to hold partisan preferences and identities defined by traditional electoral cleavages. In 1989, when the first of our surveys was fielded, this group was 50 years or older. By 2014 many of this group had passed away, and the survivors were aged 75 and beyond. Our second cohort consists of people born between 1940 and 1960. When this group reached adulthood, the party system was well established. In 1989 they were aged 30 and older and were in the prime of their lives; by 2014 they were aged 55 and older, with many having retired by then. The third cohort that we distinguish consists of people born in by 2014, when they were under 55, this cohort had grown considerably by incorporating all those who became eligible to vote after 1989.
We also compare four age groups: the over-70s, those aged 50-70, the 30-50-year-olds, and those under 30. Obviously, at any given moment in time, these age groups are unequivocally linked to cohorts, but across the surveys they are not. 6 When comparing information over time, three kinds of processes have to be distinguished that impinge on the phenomena under consideration: cohort effects, ageing or life-cycle effects, and period effects (Glenn 2005; Neundorf and Niemi 2014). Cohort effects refer to more or less stable characteristics of cohorts that are generally thought to be the consequence of socialization, and shared exposure to important events during people's formative years. Cohort effects are unique to cohorts. Age or life-cycle effects refer to the changes in attitudes, preferences, and behaviour that emanate from the social and physical consequences of ageing (which include the differences in expectations, opportunities, and constraints associated with different stages in the life cycle). Period effects refer to the consequences of events that affect everyone, irrespective of cohort or age. In this chapter we use these conceptual distinctions. We will not, however, perform a fully elaborated estimation of cohort, age, and period effects because of the comparability problems already referred to and the occasionally small sizes of groups that we focus on in our analyses.

How Strongly Do Fianna Fáil Voters Prefer Fianna Fáil?
We use the responses to the propensity to vote (PTV) questions (see n. 1) to gauge the strength of electoral preferences for Fianna Fáil and other parties, or, conversely, the 'softness' of these preferences. By contrasting vote choice and these electoral preferences, we can therefore assess the strength of preferences underlying choices for Fianna Fáil, or, looked upon from the opposite perspective, the plausibility that these choices could have been different, or that they may lead to vote switching at a next election.
At the level of individual voters, choices for Fianna Fáil are very certain if the voter has a very strong preference for this party and no other party. Choices are less certain, however, when Fianna Fáil is the most preferred party, but not very strongly preferred. The certainty (or vulnerability to change) of a given choice can thus be assessed on the basis of the PTV score for the party in question (or, for a group of people, by the percentage of people with a very strong preference for Fianna Fáil as expressed in their PTV scores). Using this perspective on the underlying strength of preference for Fianna Fáil yields The first row of Table 7.1 gives data for all; the remaining rows present the same kind of information for the cohorts and for the age groups that we distinguish, and yield some interesting findings. Within each of the studies we see that Fianna Fáil voters in earlier cohorts have stronger underlying preferences for Fianna Fáil than those in later cohorts.
Obviously we see the same difference when comparing older age groups with younger ones within each of the studies. There are a few exceptions to this (e.g., in 2014), but these are of minor magnitude in comparison to the general pattern. When looking at the figures over time for the various cohorts, we see that none of the cohorts is immune from the effect of the shock of the economic crisis in 2008. Moreover, we see that the differences between the age groups in each of the surveys do not seem to reflect an ageing effect of the kind in which growing older would lead to stronger underlying preference for Fianna Fáil. Were that to be the case, we should have seen that those who were 30-50 years old in 1989 would have stronger underlying preferences for Fianna Fáil twenty years later, in 2009, when they had become 50-70 years old. In 1989 67.4 per cent of Fianna Fáil voters who were then 30-50 years old awarded Fianna Fáil a PTV score of 10; but, even though they have aged twenty years, the corresponding percentage in 2009 has not increased, but has rather decreased to 45.6 per cent. Similarly, the under-30s in 1989 who aged to become the 30-50-year-olds in 2009 show not a strengthening of preferences but rather a weakening (percentage scoring Fianna Fáil at 10 on the PTV scale declining from 61.3 to 38.1). A problem with this comparison is that it stretches beyond the onset of the economic crisis of 2008, the moment of a strong period effect, but, even after accounting for this, there is no evidence of an ageing effect. 10 10 We are aware that the cohort and age distinctions used here, in conjunction with the timing of the studies, do not result in tables that are easily perused for the presence of cohort and age effects.
However, as explained in n. 4, practical considerations prevented more refined distinctions.
Nevertheless, approximate accounting for the period effect can be done by comparing the difference between the EES 2004 and EES 2009 percentages for the last two cohorts (these differences are 12.6% and 20.0% respectively), with the decline in percentages mentioned in the main text for the groups that were under 30, and 30-50 in 1989, and who had aged to 30-50 and 50-70 in 2009 (these declines are 21.8% and 23.2% respectively). In other words, the declines in percentages of cases scoring Fianna Fáil lower after having aged twenty years exceed the approximate period effect for the two cohorts in which these cases are located. As ageing effects would have resulted in increases rather than decreases of these percentages, it is clear that there is no evidence whatsoever for such ageing effects.
All in all, then, Table 7.1 provides clear evidence of a strong period effect that is located in 2008, and that is not limited to particular groups, but that has an across-the-board character. Table 7.1 also provides evidence of distinct cohort effects, with earlier cohorts being more certain of their votes for Fianna Fáil than later cohorts. The implication of this is that the gradual replacement of earlier cohorts by later ones had placed the party by the time of the economic crisis in 2008 in a more vulnerable electoral position than it had been earlier.
Finally, Table 7.1 does not provide any evidence of ageing effects, which means that the higher commitment within older as compared to younger age groups in each of the studies is predominantly driven by cohort differences.

Support
The decreasing strength of the underlying preference for Fianna Fáil-as shown in Table   7.1-is not the only factor that makes the party electorally vulnerable to shocks such as the 2008 crisis. A quite different factor is the co-occurrence of electoral preferences for Fianna Fáil and other parties. A respondent who, for example, has a very strong preference (as expressed in her PTV score) for Fianna Fáil, and a similarly strong preference for, for example, the Greens, can easily vote for either party. Even if she did vote for Fianna Fáil in a given election, her continued support for Fianna Fáil cannot be taken for granted, as it is vulnerable to switching. Empirically, this cannot be ascertained on the basis of only the PTV for Fianna Fáil, but it requires the PTV scores for all parties to be taken into consideration.
When doing so, we should consider not only the potential of some Fianna Fáil's actual voters to change and switch to another party, but also the complementary possibility of people who voted for other parties to switch to Fianna Fáil because they have a strong (but shared) electoral preference for Fianna Fáil.
Analysing co-occurring preferences for several parties is easiest done in terms of parties' so-called potential electorates and the overlap between these. This represents the share of the vote a party could obtain in a given election if in its competition for votes with other parties everything went its way while at the same time its competitors did as badly as possible. These potential electorates of parties sum to more than 100 per cent, or, in other words, they overlap, as has been demonstrated repeatedly for Ireland in previous research (Marsh 1996(Marsh , 2006aMarsh and Cunningham 2011). From PTV scores one can derive for each party estimates of the magnitude of its potential electorate, of the magnitude of the overlap of their potential electorates with each of the other parties, as well as with all other parties together.
This, in turn, makes it possible to determine the size of a party's 'unique' electorate, which is the component of a party's potential electorate that is non-overlapping with the joint potential of all other parties. This unique component is important in the sense that it reflects electoral support that is effectively uncontested by other parties, and that can therefore be interpreted as a share of the votes that a party can expect to obtain in a worst-case scenario, whereas the potential electorate reflects the share of the votes that it could obtain in a best-case scenario.
The details of these procedures are specified in the Appendix to this chapter. Neither of these two scenarios, the best case and the worst case, are likely ever to materialize, but they define useful anchors for assessing the actual electoral performance of parties. Finally, the ratio of the unique to the potential electorate reflects the extent to which parties are dependent on the outcome of electoral competition with other parties for the same voters; the higher this ratio, the less vulnerable they are. Armed with this repertoire of measures, we now turn to an empirical analysis of preferences and support for Fianna Fáil.
In Table 7.2 we report a perspective of Fianna Fáil's electoral vulnerability that is based on the co-occurrence of respondents' electoral preferences for multiple parties. The rows present a comparison of the percentage of Fianna Fáil's unique potential electorate (which is not contested by other parties) to its total potential electorate, or, in other words, the degree to which the party's vote share in each of these elections is vulnerable-that is, dependent on the success of its competition with other parties. A high ratio indicates that the actual support of a large proportion of potential voters can be taken for granted; a low ratio indicates that the party has actively to compete for the actual support of a large share of those who may potentially vote for it.

< INSERT TABLE 7.2 >
In many ways Table 7.2 shows similar patterns to Table 7.1. In each of the years the ratio of unique to total potential electorate is highest for earlier cohorts and lowest for more recent cohorts. We see, therefore, also that in each of the years the ratio is higher in older age groups and lower in younger ones. With respect to the cohorts, the earliest cohort (comprising those born before 1940) is most distinctive, and the two more recent cohorts resemble each other more than the earliest one. When looking at the development over time we see, as we did in Table 7.1, a clear period effect of 2008 that affects all cohorts, including the earliest. The two earliest cohorts (born before 1940, and born between 1940 and 1960, respectively) show no trend prior to 2008, but the most recent cohort does, which reflects the continuous expansion of this cohort by the influx of those who become eligible to vote. These patterns are clearly indicative of cohort effects. Just as in Table 7.1, we fail to see any clear signs of an ageing (or life-cycle) effect, which, as discussed earlier, would have shown itself in fewer multiple electoral preferences and thus a higher ratio of unique to total potential electorate. Finally, To some extent this reflects that the numerator of the ratio (the unique potential electorate) has declined less than the denominator (the total potential electorate), which one would expect to occur with electoral losses: those who had other attractive alternatives to Fianna Fáil switch, leaving among Fianna Fáil voters a larger proportion that does not see other parties as a viable alternative. Whether the increasing 2014 numbers reflect also something else cannot be ascertained at the time of writing, and requires data from later election surveys.
The main story that Our main conclusion from the data presented in Tables 7.1 and 7.2 is that, over the past decades, Fianna Fáil has become more vulnerable to the consequences of external shocks and to the unpredictable outcomes of the electoral tug-of-war with other parties for the votes of the same groups of people. The increased vulnerability to the consequences of external shocks is reflected in gradually weakening certainty (or commitment) of a choice for Fianna Fáil. The increased vulnerability to the outcomes of electoral competition-and thus indirectly also to external shocks that figure in election campaigns-is reflected in a gradually decreasing ratio of unique to total electoral potential. Both seem to be driven over the long run by 'demographic metabolism': the process of generational replacement with later cohorts having weaker preferences for Fianna Fáil and a larger number of other parties for which they also have preferences compared to earlier cohorts. This generational replacement has been overlaid with the period effects brought about by the 2008 economic crisis.

Compatibility or incompatibility of electoral preferences for Fianna Fáil and Fine Gael
The previous section concluded with the observation that, in more recent cohorts, voters who have a preference for Fianna Fáil also have, more than those in earlier cohorts, electoral preferences for other parties. This conclusion can be specified further by examining the overlap of such preferences with Fine Gael. This more refined focus is motivated by the contention in the extant literature that shared affections between these two parties are particularly rare. Fine Gael is also of particular importance because of its size, and because of the explicit appeal that the party made in the 2011 election campaign to Fianna Fáil supporters to 'lend' their vote to Fine Gael. 11 Indeed, the shifts recorded in the 2011 election result were possible only because significant numbers of voters switched from Fianna Fáil to Fine Gael (Marsh and Cunningham 2011: 180).
The extent of co-occurring preferences for Fianna Fáil and Fine Gael can be expressed in a coefficient of dyadic overlap between the potential electorates of these two parties (see the Appendix to this chapter for the detailed definition of this coefficient). This coefficient is 0 if there is no overlap at all between these potential electorates, and 1 in the case of complete overlap (which would imply that the potential electorate of the smaller of the two is entirely contained within that of the larger one). Table 7.3 reports the values of this coefficient for the entire sample of each of the surveys we use, as well as for subgroups defined in terms of cohorts and age groups.
The results in Table 7.3 show that for the entire period from 1989 to 2014 there has always been a substantial degree of 'shared affections', as reflected in the coefficients of dyadic overlap. They also show that, with very few and minor exceptions, in each of the studies more recent cohorts and younger age groups exhibit higher degrees of such cooccurring preferences than earlier cohorts or older age groups. It is telling, though, that by far the lowest coefficients were recorded for the oldest age group in 1989 and 1994. This group consisted at those times of people born before 1920 or 1925 respectively, or, in other words, the group whose formative political experiences are closest to and most affected by the foundational period of the party system, and to the civil war that spawned the formation of Fianna Fáil and Fine Gael. This unique group passed away at an increasing rate after 1989, which helps to explain the somewhat higher numbers in the 70+ age group since 1999. These sparse observations lend strong support to the thesis that in the more distant past support for Fianna Fáil and Fine Gael was indeed incompatible and that civil-war experience (either direct or inculcated via parental transmission) was one of the foundations of that incompatibility. But, as these traumatic events receded in the past, they became increasingly less important for new entrants who reached voting age. For obvious reasons, any mutual exclusivity of preference remained probably strongest among those who identify themselves with either of these political parties, but that is a group that has also shrunk as a consequence of generational replacement (Mair and Marsh 2004;Marsh 2006b;Marsh et al. 2008: ch. 4).
The notion of incompatible electoral preferences for Fianna Fáil and Fine Gael was evidently relevant for generations that had dominated the electorate through the 1960s and 1970s, but has increasingly become irrelevant since then. 12 The fluctuations over time that are reported in Table 7.3 for each of the cohorts reflect mainly non-systematic fluctuation, while the coefficients increase over time for each of the age groups. These patterns suggest a clear presence of cohort effects: once having acquired its political identity in its formative years, a cohort does not change much in terms of the overlap of electoral preferences for Fianna Fáil and Fine Gael. The earliest cohorts record the smallest degree of dyadic overlap of preferences for these two parties; more recent cohorts display considerably higher co-occurring preferences for them. As cohort members age over time, this pattern of trendless fluctuation is therefore also indicative of the absence of any age or life-cycle effects. The observation that the coefficients for age groups increase over time is entirely consistent with this, as these categories become over time populated with members of more recent cohorts. 13 In contrast to Tables 7.1 and 7.2; we do not see in Table 7.3 a clear period effect in the form of a step change from before to after 2008, which emphasizes by its absence even more the important role of generational replacement.

Concluding Remarks
The dramatic vote loss by Fianna Fáil in the 2011 elections is generally attributed to its handling of the 2008 economic crisis in government. The evidence for this interpretation is compelling, and this chapter does not contest it. But this interpretation does not address the underlying question why the crisis could have this spectacular electoral consequence. In this chapter we have argued that the continuous process of generational replacement has weakened the electoral resilience of the party in at least two complementary ways.
The first of these is a gradually decreasing strength of electoral preference for the party among its own voters (as illustrated in Table 7 Table 7.2), a phenomenon that includes a growth in shared affections for Fianna Fáil and Fine Gael (as illustrated in Table 7.3). Both components are clearly propelled by generational replacement of earlier cohorts (in which Fianna Fáil was less vulnerable) by later ones (in which Fianna Fáil is more vulnerable). One implication of this is that, had the 2008 economic crisis occurred some twenty-five years earlier, its electoral consequences would have been less extensive than they were in 2011.
In the 2011 election Fianna Fáil was the party that most dramatically suffered the consequences of these developments. Although we have focused in this chapter on Fianna Fáil, it has to be noted that increased vulnerability to electoral change is not restricted to this party only. Other Irish parties, among them first and foremost Fine Gael, are affected in similar ways by the same gradual generational replacement. It is, therefore, plausible that, had Fine Gael, not Fianna Fáil, been the leading government party in 2008, the fate that befell The changes in strengths and structure of electoral preferences that have been demonstrated in the previous sections also indicate that what clearly in the past has been an electoral cleavage between Fianna Fáil and Fine Gael has gradually lost most of its relevance for cohorts entering adulthood over the past few decades. As discussed in the beginning of this chapter, this seems to be an almost unavoidable evolution in political systems in which the root causes of an electoral cleavage have been pacified. The waning of what once was an electoral cleavage opens the way for more open electoral contests for votes in which no parties are sheltered from external shocks or from electoral competition. That makes all of them vulnerable in the electoral sense discussed, and implies that none of them can take for granted the support of substantial segments of the electorate. Instead, citizens increasingly perceive more than one party as a potentially worthwhile recipient of their vote, and elections will be decided on the basis of competition for the actual support of these voters. That makes parties more vulnerable to external shocks than they were in the past, but it also offers electoral opportunities

Electorates and their Overlaps
All measures and coefficients used in this chapter are derived from individual-level responses to the propensity to vote (PTV) questions (see main text, n. 1).
Let PTViJ be the score of respondent i (i = 1 … n) to PTV question for party J (J = A … K), then individual-level contributions to party J's potential electorate are obtained by: with being a monotone non-decreasing function and 0 ≤ ≤ 1.
In this chapter we defined transformation function as a linear function with 0 and 1 as values for the lowest and highest PTV scores respectively. In some of the surveys the PTV responses were to be given on a 10-point scale (from 1 to 10), and in others on an 11-point scale (from 0 to 10). In the 10-point scale the values of (the transformed PTV scores) thus progress from 0 to 1 with increments of 0.11111…, while in the 11-point scale they progress from 0 to 1 with increments of 1. This particular transformation is supported by Tillie's calibration (1995) of PTV scores with magnitude estimation procedures.
An interesting form that such non-decreasing functions can take are step functions of zeros and ones. Such functions define so-called consideration sets with the threshold for inclusion defined by the location along the PTV scores of the 0 to 1 step. In this chapter we do not use this form of , but comparison of the linear function that we use and a step function with the two or three highest PTV scores transformed to 1 and the others to 0 generally lead to similar substantive conclusions.
Aggregating over i yields the magnitude of party J's electoral potential (sometimes also referred to as potential electorate) as a proportion of the total sample: When considering a pair of parties, J and K, the individual-level contribution to their joint electoral potential is Consequently, at party level (aggregate level) the joint electoral potential (or joint potential electorate) of parties J and K is: The joint electoral potential of a set of three parties (I, J, and K) is then: which can obviously be extended to sets of parties of any magnitude in an analogous fashion.
The overlap between the potential electorate of two parties J and K, as a proportion of the total sample, is then: The overlap of the potential electorate of two parties ∩ defined above is expressed as a proportion of the entire sample, and is obviously dependent not only on the degree of shared electoral preferences, but also on the size of the parties involved. When focusing on competitive relations between parties, it may be preferable to express it in a form independent of party sizes. That leads to the following coefficient of dyadic overlap of potential electorates, as used in Table 7.3 of the main text: If ( + ) ≤ 100 : And if ( + ) > 100 : Note: Ratios expressed as percentages.