POLITICAL APPLICATIONS

Exponential models have been used to describe the survival of political systems or entities. Relevant to this article is a model that examines the life spans of empires (Arbesman Reference Arbesman2011); another is the dissolution of cabinets in Italy (Cioffi-Revilla Reference Cioffi-Revilla1984). These studies empirically estimate the functional distribution of the duration of the system under examination.

We similarly construct an exponential decay model to inquire into the rate of reelection of the incumbent political parties, those of the governor or the premier, in state or provincial governments in Australia, Canada, and the United States. These data are suited for our purposes for several reasons. Electorally, these units of government behave in ways similar to their counterparts at the national level (Cuzán Reference Cuzán2022, 37, 52 59), with the advantage of including a much larger number of observations. All three countries share a common British heritage; arguably, this limits the range of exogenous cultural influences on the comparisons of their political systems relative to studies with multiple countries but few observations per country.

We seek to answer a simple question: “The incumbent party, having won n elections, what is its propensity to win the next?” A literature search revealed no previous inquiry into the topic.

DATA AND METHOD

The data consist in almost 1,200 state or provincial elections in the United States, Canada, and Australia (see table 1).Footnote ¹ As explained in this article, the series for each state or province begins with an “old” incumbent loss and, correlatively, the vote of the winning opposition party—that is, the party that now becomes the new incumbent. All party spells or reigns subsequent to that first election made it into our dataset. (Heggen and Cuzán Reference Heggen and Cuzán2022). We entered the name of the state or province; the year of the election; the new incumbent party vote from the first election when, in opposition, it defeated the old incumbent to gain control of the executive office; the opposition party vote; and whether the incumbent was reelected or defeated.Footnote ²

Table 1

Descriptive Statistics: State, Provincial, or Territorial Elections in Three Countries

Notes: *Australia is divided into six states and three territories and Canada is divided into 10 provinces and three territories. This article treats Australia’s Capital Territory and Northern Territory and Canada’s Yukon Territory as the equivalent of a state or province because, unlike the other territories in each country, they are governed by an assembly composed of political-party members.

Generally, in all three countries, elections are held at the end of a constitutionally or legally specified term: two to four years in the United StatesFootnote ³ and three to four years in Canada and Australia. In the latter two countries, a premature election may be called when the previous one yielded a minority government, the government fell because of a no-confidence vote, or the incumbents think it advantageous to do so.

In the US states, the governorship—a single-occupancy office analogous to the presidency at the national level—is filled by the winner of a statewide plurality vote. In Canada and Australia, the choice of head of government is made by Parliament, the members of which are elected from single-member, “first-past-the-post” districts. Australia uses a preferential ballot; however, in the interest of comparability with Canada, we entered the “primary” vote. This did not affect the rate of reelection—which is our outcome of interest—only the proportion of party vote entered.

Again, in this study, it is the party of the governor or premier, not the individual officeholder, that constitutes the incumbent. In Australia or Canada, that is true whether it forms a government alone or in coalition with other parties. Electoral choice in the United States generally is constrained to one of two parties: Democrats or Republicans. In the rare case when a third party wins the office, it is rarer still for it to secure more than one term. This is less rare in Canada and Australia: several of their states or provinces exhibit something closer to a three-party system. US state governors, although not their party, are subject to term limits. These restrictions on reelection do not apply to parliamentary premiers.

We partitioned the election history of every state or province into “strings.” Every string consists in a series of consecutive elections that the incumbent party—the one whose spell in office the string describes—contested. The string begins with Election 0, which the party won thereby gaining control of the executive office, and ends with Election n, which the party lost. Thus, every election results in either the renewal of the incumbent party’s lease on the executive office or its defeat and the beginning of a new string. We used only “closed strings”—those that begin and end within our dataset. “Open strings,” in which the incumbent party remains undefeated as of 2020, thus are not included. No string originated earlier than the 1930s, a time when many incumbents—buffeted by the Great Depression—went down to defeat. Other strings, particularly in the US South, started as late as the 1960s or 1970s, when Democrats lost their first election since Reconstruction.

The string of a party serving n consecutive terms brackets n+1 elections: the initial election, n-1 reelections, and the last election, a loss. For example, figure 1 displays a string from New Hampshire in which Republicans assumed the governorship in 1982, defeating the incumbent Democrats with 51.5% of the vote. Subsequently, they were reelected six times until their defeat in 1996, with less than 40% of the vote. The Republican Party thus enjoyed seven terms in office, with the loss marking the conclusion of an eight-election string.

Figure 1

New Hampshire’s Seven-Term Republican String, 1982–1996

FINDINGS

Table 1 lists the descriptive statistics of our data. Most of the observations are from the United States. The ratio of elections to strings and the incumbent win rate is highest in Australia and lowest in the United States. Note that the two variables are related: the higher the win rate, the longer and, therefore, the fewer the number of strings and, therefore, the lower the ratio of elections to strings. Note also that the incumbent party is reelected approximately 60% to 75% of the time.

Figure 2 displays the number of incumbent survivors by election within all of the United States strings. The bar at the far left represents Election 0, the one that catapulted a new party into the governorship and initiated a new string. All 306 members of this cohort began with victory at Election 0. In their first bid for reelection, 209 incumbents were successful. At Election 2, 98 incumbents remained to appeal to voters for a third term, and so on until the two remaining incumbents who did not win past Election 10.

Figure 2

Frequency of Incumbent Wins and Losses, US States

Figure 2, a discrete counting, converts into figure 3, which is a continuous curve matching the bar heights at each integer term. The transformation is appropriate when the sample size is large and the behavior of concern (i.e., the sustainability of an incumbent, in this case) is not at the tail. Our samples are large and our interest focuses on the more commonly experienced lefthand side and mid-domain of the distribution—less so, for example, than on the particular odds of eight-termers making it to nine. A K of 0.60 (see Equation 1) is estimated by curve fitting.

Figure 3

Incumbent Survival Curve, US States

Figures 4 and 5 exhibit the results for Canada and Australia, respectively, which were obtained using the same method.

Figure 4

Incumbent Survival Curve, Canadian Provinces

Figure 5

Incumbent Survival Curve, Australian States

Figure 6 compares all three survival curves in the same graph. In all three countries, the half-life (i.e., 0.50 on the vertical axis) occurs between Elections 1 and 2—that is, between the first and second reelection: 1.3 terms for the United States, 1.7 terms for Canada, and 2.1 terms for Australia.

Figure 6

Exponential Survival Curves: Australia, Canada, and the United States

In an electoral democracy, survival depends on winning more popular votes or capturing more seats in Parliament, singly or in combination with partners or allies. In the following example, we examine only the United States.

Figure 7 shows the scatter of the incumbent vote per election, beginning with Election 1, at the end of their first term in office when all 306 incumbents made a bid for reelection. Each observation consists of two or more incumbent votes from a single string. For example, the New Hampshire string shown in figure 1 contributes one point to each of Elections 0 through 7 (remember that the string did not make it past Election 7).

Figure 7

Incumbent Vote, US States

The overall slope of the scatter is statistically indistinguishable from 0 (p<0.01). That is, election after election, the incumbent vote share does not trend. As those defeated drop out, the winners maintain more or less their share until the electoral equivalent of energy is depleted. All incumbencies eventually end—an outcome that is guaranteed by the exponentially declining odds of remaining in office.