On status quo bias and the existence of Condorcet cycles in binary voting situations

Abstract

In most real-life binary voting situations, multiple challenger alternatives can potentially contest the status quo alternative, but only one of them can be listed on the ballot. Consequently, Condorcet cycles cannot be detected by analyzing voting data alone, even when they exist. This paper considers a standard theoretical framework where Condorcet cycles cannot exist, but demonstrates that this need not be the case when voters experience status quo bias. Necessary and sufficient conditions for the existence of Condorcet cycles are provided. Monte Carlo simulations predict the probability that Condorcet cycles exist in binary voting situations with status quo bias to roughly one percent under different assumptions and scenarios.

Keywords

Binary voting situations Condorcet cycles Monte Carlo simulations

1. Introduction

Condorcet cycles have been one of the centerpieces of voting theory ever since the seminal work by Marquis de Condorcet in the late 18th century (classical references include Black, 1958; Gehrlein and Fishburn, 1976; Riker, 1982; Saari, 2001; Sen, 1970: among others). Such cycles are defined by at least three alternatives, say $x_{1}$ , $x_{2}$ , and $x_{3}$ , and situations where the electorate prefers $x_{1}$ to $x_{2}$ , $x_{2}$ to $x_{3}$ , and $x_{3}$ to $x_{1}$ . This type of cyclicity constitutes a democratic dilemma, commonly referred to as the Condorcet Paradox, wherein a majority of the electorate is dissatisfied with the outcome of the vote, regardless of which of the three alternatives is coined as the winner.

In this paper, we investigate Condorcet cycles in binary voting situations with a status quo alternative. In such situations, the democratic dilemma associated with Condorcet cycles can always be avoided if there exists a unique plausible challenger alternative (because it is impossible to close Condorcet cycles in the absence of a ‘third alternative’). For example, in the 1955 Swedish referendum, the status quo alternative ‘continue traffic on the left-hand side of the road’ was contested by the challenger alternative ‘traffic on the right-hand side of the road’. In this referendum, the challenger alternative was unique given it would be extremely impractical for the population not to always drive on the same side of the road. One can also imagine that such situations can occur in, for example, parliamentary votings where the elected assembly votes on introducing or abolishing bans and embargoes, and in candidate elections, such as mayoral and primary elections, where only one challenger decides to contest the incumbent candidate.

However, in many binary voting situations, there are multiple plausible challenger alternatives even if only one of them is allowed to contest the status quo alternative in the actual vote. For example, in a binary referendum¹ regarding whether the presidential term should be shortened or not, as in the 1994 Cook Islands referendum, the yes-alternative could, for instance, be represented by ‘yes, reduction by one year’ or ‘yes, reduction by two years’. Another example is the United Kingdom European Union membership referendum in 2016, commonly referred to as the Brexit referendum, where the electorate was asked whether the country should remain a member of, or leave, the European Union. Clearly, the status quo alternative to ‘remain’ was fixed, but there was more room to define the ‘exit alternative’. One plausible challenger alternative is to make a ‘hard exit’ and leave both the European Union and the European economic area. A more ‘soft exit’ is to leave the European Union, but stay in the European economic area. It is easy to imagine a similar type of multiplicity of plausible challenger alternatives in binary voting situations in, for example, committees, electoral districts, juries, legislative bodies, and parliaments.

In such binary voting situations, the presence of Condorcet cycles cannot be excluded because at least two plausible challenger alternatives potentially can contest the status quo alternative. Consequently, the selection of the challenger alternative in the binary voting situation may be decisive for the outcome of the vote in the usual sense.² But even if Condorcet cycles are present, they cannot be detected by studying election data alone because only two alternatives are listed on the ballot (see the discussions in Bochsler, 2010; Gehrlein, 2006). Information about plausible challenger alternatives that were not on the ballot also must be taken into consideration, but since this type of information normally is lacking, it is difficult to know if Condorcet cycles are present or not. Furthermore, in non-binary voting situations, voting methods that detect Condorcet cycles can be used, for example, various methods for ranked-choice voting, score voting (e.g. the Borda count), and pairwise voting (e.g. the Condorcet method). Even if these methods are informative about the existence of Condorcet cycles in non-binary voting situations, they are non-informative in binary voting situations. One additional factor that may complicate matters even further is the presence of a status quo alternative. It is empirically well-established that voters often experience a status quo bias, meaning that they experience some type of reluctance to abandon the status quo alternative (see, e.g. Bowler and Donovan, 1998; Grofman, 1985; Kuran, 1988; Lupia and Matsusaka, 2004; Samuelson and Zeckhauser, 1988: or the discussions in Section 2.2). It is difficult to obtain information about this type of bias, but it can affect the probability that Condorcet cycles exist. However, not much is known about the connection between status quo bias and Condorcet cycles. The purpose of this paper is to shed some light on this in binary voting situations with multiple plausible challenger alternatives by characterizing conditions under which Condorcet cycles exist and by quantifying the probability of their existence.

We consider the simplest possible theoretical framework with finite sets of voters and alternatives, but where only one of the plausible challenger alternatives is allowed to contest the status quo alternative in the binary voting situation. Voters are endowed with one-dimensional symmetric single-peaked preferences. In this framework, it is well-known that there cannot be any Condorcet cycles (see, e.g. Black, 1958; Downs, 1957: or Section 2.4). However, as demonstrated in this paper, if voters experience status quo bias, Condorcet cycles can be present even in this framework.³ This type of bias is modeled through ‘voter frictions’ that capture the idea that some voters may abandon the status quo alternative, but only if the challenger alternative deviates ‘sufficiently much’ from the status quo alternative.

Focus is directed toward binary voting situations that are interesting from an analytical perspective, so-called non-trivial binary voting situations. These are situations where the status quo alternative does not win against each of the plausible challenger alternatives or does not lose against all plausible challenger alternatives. The main theoretical contribution of this paper is to provide new necessary and sufficient conditions for the existence of Condorcet cycles in the considered framework with status quo bias. We then follow the tradition in the literature to evaluate the theoretical findings using Monte Carlo simulations (see the discussions in, e.g. Bochsler, 2010; Gehrlein, 2006). Mores specifically, the simulations are used to predict the proportion of non-trivial voting situations and the probability that Condorcet cycles exist. The numerical findings show that almost all binary voting situations are non-trivial and that the probability that Condorcet cycles exist is roughly one percent. The latter finding is robust under different assumptions on, for example, the number of voters, plausible challenger alternatives, and peak distributions.

1.1. Related literature

The democratic dilemma, associated with Condorcet cycles, has inspired a large theoretical literature that aims to characterize preference domains under which these types of majority cycles cannot exist (classical references include, e.g. Black, 1958; Downs, 1957; Gaertner, 2001; Inada, 1964; McKelvey, 1976; Sen and Pattanaik, 1969). But when they exist, for example, because preferences are not single-peaked, another strand of the literature holds that deliberation can protect against Condorcet cycles, not by inducing unanimity (which is unrealistic), but by bringing preferences closer to single-peakedness. This has been suggested by, for example, Dryzek and List (2003), Knight and Johnson (1994), and Miller (1992). Another approach in this vein is based on structure-induced equilibrium, which views restrictive rules either as devices for avoiding the chaos that would presumably occur when, for instance, Condorcet cycles are present or as part of an institutional arrangement for exchanging support across the various committee jurisdictions. This direction of research was pioneered by Shepsle (1979) and Shepsle and Weingast (1981).

The common theme in the articles cited in the preceding paragraph is that they either consider preference domains where Condorcet cycles cannot exist or, if they exist, the focus is on identifying structures and restrictions on the environment that eliminate, or at least drastically reduce, the probability of their existence or practical relevance. In this paper, we analyze the problem from a different angle. Namely, we position our analysis in the classical Downsian framework with one-dimensional symmetric single-peaked preferences where Condorcet cycles cannot exist and then demonstrate that they, in fact, can be present when accounting for the empirically grounded observation that voters often exhibit a status quo bias. This approach is most closely related to the part of the literature that analyzes and explains why election and voting outcomes often deviate from equilibrium predictions, that is, not only why off-equilibrium outcomes can often be expected but also how they can be explained (say, for example, why the outcome deviates from the position of the median voter even if preferences are single-peaked). One such explanation rests on non-policy factors that give candidates or alternatives so-called valence advantages. These advantages include factors such as name recognition, track record, and certainty about the incumbent’s position, and have been extensively analyzed in the literature (see, e.g. Groseclose, 2001; Stokes, 1963, 1992: and references therein). Off-equilibrium outcomes have also been explained using, for example, arguments that build on voter loyalty and the benefits of the doubt (Feld and Grofman, 1991; Sloss, 1973) as well as various voting-related costs (Buchanan and Tullock, 1962; Herzberg and Wilson, 1990; Hinich and Ordeshook, 1969; Tullock, 1967). But even if this paper also focuses on off-equilibrium outcomes, it deviates from previous literature because the main focus is not on the actual off-equilibrium outcome, but rather to demonstrate that status quo bias reintroduces Condorcet cycles in the considered framework with binary voting situations. The main theoretical contribution is to provide necessary and sufficient conditions for the existence of Condorcet cycles. These conditions are not presented as a representation of the probability that Condorcet’s Paradox is observed (that part of the literature is thoroughly reviewed by Gehrlein, 2006).

To obtain a general understanding of the prevalence of Condorcet cycles, researchers have attempted to calculate the probability of their existence and, when possible, to empirically investigate their occurrence. The findings are not conclusive and depend on a number of different factors, such as the number of voters and alternatives. The probability that a Condorcet cycle exists varies between 0% and 10%, both in simulated and empirical environments (see, e.g. Black, 1958; Gehrlein and Fishburn, 1976; Gehrlein, 2006; Gehrlein and Lepelley, 2011; May, 1971; Tideman, 2006; van Deemen, 2014). Researchers have also provided mathematical formulas for calculating the expected probability of the Condorcet Paradox in various information and voting environments (see, e.g. DeMeyer and Plott, 1970; Gehrlein, 1981; Niemi and Weisberg, 1968). This paper contributes also to this literature in that we estimate the probability that Condorcet cycles exist, specifically for binary voting situations with multiple plausible challenger alternatives where voters may experience status quo bias.

1.2. Outline of the paper

The remaining part of this paper is outlined as follows. In section 2, we introduce the model and some basic definitions. The theoretical findings and the Monte Carlo simulation can be found in sections 3 and 4, respectively. Some concluding remarks are provided in section 5. Appendix A provides a theoretical foundation for a utility representation of single-peaked preferences with status quo bias.

2. The model and basic definitions

In this section, we introduce the simplest possible model that captures the essential features of binary voting situations where voters may experience a status quo bias, noting that all results presented in the paper hold under more general assumptions. Throughout the paper, remarks related to generalizations are delegated to footnotes. We start by introducing the basic model.

2.1. The basic model

The voters are gathered in the finite set $N = {1, \dots, n}$ for some odd number $n$ . The alternatives are represented by integer numbers, collected in the finite set $X = {x_{0}, \dots, x_{m}}$ , where $x_{0}$ represents the status quo alternative.⁴ It is convenient to think about the index $j$ as a measure of the ‘policy distance’ between challenger alternative $x_{j}$ and the status quo alternative. Thus, the higher index $j$ an alternative has, the more it deviates from $x_{0}$ . For example, in a referendum about leaving the EU, it may be instructive to regard the status quo alternative as remain in the EU, and alternatives $x_{1}$ , $x_{2}$ , and $x_{3}$ as renegotiate a new deal within the EU, leave the EU but stay in the European economic area, and leave both the EU and the European economic area, respectively.

Each voter $i \in N$ has a weak preference relation over the finite set of alternatives $X$ denoted by $R_{i}$ , with corresponding strict and indifference relations $P_{i}$ and $I_{i}$ , respectively. The one-dimensional preference relation $R_{i}$ has a unique maximum element denoted by $x_{i}^{*}$ , henceforth referred to as the peak of voter $i$ . The peaks for each voter in $N$ are gathered in the vector $x * = (x_{1} *, \dots, x_{n} *)$ , and are distributed on the set $X$ according to the discrete function $g (x *)$ with corresponding cumulative density function $G (x *)$ . There is an ordering greater than of the alternatives in $X$ such that, for every voter $i \in N$ :

\begin{aligned} x_{k} < x_{j} & \leq x_{i}^{*} implies that x_{j} P_{i} x_{k} \\ x_{k} > x_{j} & \geq x_{i}^{*} implies that x_{j} P_{i} x_{k} \end{aligned}

That is, when voter

i

compares two distinct alternatives that both are either to ‘the right’ or to ‘the left’ of her peak

x_{i}^{*}

, she strictly prefers whichever alternative that is closest to

x_{i}^{*}

. The preference relation

R_{i}

is assumed to be symmetric around its peak, that is, when voter

i

evaluates two distinct alternatives on different sides of the peak, she strictly prefers whichever alternative is closest to her peak and can, therefore, only be indifferent between two distinct alternatives if their distances to the peak are identical. In case a voter needs to break ties, it is assumed that she always selects the alternative with the lowest index.⁵ The vector

R = (R_{1}, \dots, R_{n})

contains the preference relations of the voters. The model presented so far is essentially the standard Downsian framework (Black, 1958; Downs, 1957) with one-dimensional symmetric single-peaked preferences (see section 2.4 for some further remarks).

2.2. Voter types and status quo bias

The fact that voters often experience status quo bias is well-established in the literature.⁶ This type of bias will be modeled by assuming that all voters have ideological preferences (represented by the symmetric single-peaked preferences), but that they may experience some type of ‘cost’ when abandoning the status quo alternative. To model this, voters are partitioned into two distinct types. The type-S voters (S stands for ‘status quo’) have their peaks at the status quo alternative $x_{0}$ . Voters who do not have their peaks at alternative $x_{0}$ are referred to as type-A voters (A stands for ‘alternative’). Each type-A voter $i$ has symmetric single-peaked preferences $R_{i}$ on $X$ , but also experiences some type of cost when abandoning alternative $x_{0}$ . In this sense, the status quo alternative has a special meaning for all type-A voters because they are not willing to abandon it for ‘sufficiently small’ policy changes. In other words, if a challenger alternative is not located ‘sufficiently far away’ from the status quo alternative, the voter will find it too costly to abandon the status quo alternative.

This type of assumption can be justified in the existing literature. Samuelson and Zeckhauser (1988), Shepsle and Weingast (1981: Example 3), and Tullock (1967) all argue that it should be unfeasible to place alternatives on the agenda that are within a certain distance from the status quo alternative because they most likely only are dilatory, for example, as voters are unlikely to cast their vote on them anyway. This is exactly what we intend to capture in the considered model via so-called voter frictions. A voter friction describes all challenger alternatives that are considered to be ‘too close’ to the status quo alternative to even be considered by the voter. Note that it is possible to provide a theoretical foundation for a utility representation of single-peaked preferences with voting frictions using a cost-based approach (for more on costs and single-peakedness, see Coombs and Avrunin, 1977). However, given that we do not need such a representation to prove the theoretical results in this paper, we refer to Appendix A for the details.

The friction $f_{i}$ for a type-A voter $i$ belongs to the set ${x_{1}, \dots, x_{m}}$ . If $f_{i} = x_{l}$ , voter $i$ strictly prefers the status quo alternative $x_{0}$ to any of the plausible challenger alternatives in the set ${x_{1}, \dots, x_{l}}$ .⁷ Because all type-S voters have their peaks at the status quo alternative, their frictions are, without loss of generality, set to $x_{0}$ . Let the set $X^{0} = X ∖ {x_{0}}$ contain all plausible challenger alternatives. Note also that when a voter evaluates two plausible challenger alternatives, the friction is irrelevant because it is assumed to be ‘active’ only when an alternative is evaluated against the status quo alternative. The frictions of the voters in $N$ are gathered in the vector $F = (f_{1}, \dots, f_{n})$ . A profile $D = (R, F)$ is a complete description of voter preferences with frictions.

2.3. Binary voting situations

A binary voting situation is a majority vote between the status quo alternative and some challenger alternative in $X^{0}$ . In general, the profile $D$ determines the outcome of a majority vote between any two alternatives in $X$ , where $x_{j} D x_{k}$ means that alternative $x_{j}$ wins a majority vote against alternative $x_{k}$ .

To investigate if the voters in $N$ prefer challenger alternative $x_{j}$ or challenger alternative $x_{k}$ given that their peaks are gathered in the vector $x * = (x_{1} *, \dots, x_{n} *)$ , we first identify the alternative that is located exactly between alternatives $x_{j}$ and $x_{k}$ , that is, alternative $⌊ 0.5 \times (x_{j} + x_{k}) ⌋$ .⁸ For any two alternatives $x_{j}, x_{k} \in X^{0}$ with $j < k$ and a given cumulative peak distribution $G (x *)$ , the following identity identifies that mass of users who have their peaks (weakly) closer to alternative $x_{j}$ than to alternative $x_{k}$ :

mean (x_{j}, x_{k}) = G (⌊ 0.5 \times (x_{j} + x_{k}) ⌋)

So, if

j < k

and

mean (x_{j}, x_{k}) \geq 0.5

, then alternative

x_{j}

wins the majority vote against alternative

x_{k}

because at least 50% of the peaks are more closely located to alternative

x_{j}

than to alternative

x_{k}

(recall that preferences are symmetric).⁹

For a given profile $D$ , an alternative $x_{j}$ is a Condorcet winner if $x_{j} D x_{k}$ for any distinct alternative $x_{k} \in X$ . There exists a Condorcet cycle for a given profile $D$ if $x_{j} D x_{k_{1}}, \dots, x_{k_{l}} D x_{k_{l + 1}}$ , and $x_{k_{l + 1}} D x_{j}$ for $x_{j}, x_{k_{1}}, \dots, x_{k_{l}}, x_{k_{l + 1}} \in X$ and some integer $l \in {1, \dots, m - 1}$ .

2.4. On the relation to the Downsian framework

There are two assumptions that cause the model described in this section to deviate from the classical framework of political competition (Black, 1958; Downs, 1957): namely, that some political candidates must represent the status quo alternative and that some voters may experience a status quo bias. If both these assumptions are dropped, the model reduces to the classical frameworks of political competition. Among other things, this implies that the peak of the median voter $\bar{x}$ is the Condorcet winner and that there are no Condorcet cycles.¹⁰

3. Theoretical results

Throughout this section, the insights from the four panels of Figure 1 will be used to illustrate the main theoretical findings. Each panel in the figure describes the one-dimensional symmetric preferences of three voters.¹¹ In panel (a), there is one type-S voter and two type-A voters. The latter two voters have their peaks at $x_{1}$ and $x_{4}$ with frictions at $x_{2}$ and $x_{3}$ , respectively. The part of $R_{i}$ where the frictions are ‘active’ when the status quo alternative is evaluated is represented by a dashed line (see Appendix A for further interpretations). Note that the median peak $\bar{x}$ is ‘inactive’ in Figure 1(a) and (c) in the sense that the frictions make the median peak lose a majority vote against the status quo alternative. If the main findings would only hold under such a restrictive situation, they would not be particularly interesting. However, as will be established further down, all results presented in this paper hold independently regardless of whether the median peak is ‘inactive’ or not.

Figure 1.

(a)–(d) Illustrate the four leading examples considered in this paper.

The restriction will be directed toward profiles $D$ where the binary voting situation is non-trivial (the Monte Carlo simulations in Section 4 evaluate the frequency of such voting situations). Informally, these are the binary voting situations where the status quo alternative does not win against each of the plausible challenger alternatives or does not lose against all plausible challenger alternatives. Formally:

Definition 1.

For a given profile $D$ , a binary voting situation is non-trivial if there are alternatives $x_{j}, x_{k} \in X^{0}$ such that (i) $x_{0} D x_{j}$ and (ii) $x_{k} D x_{0}$ .

A voting situation based on the profile illustrated in Figure 1(a) is not non-trivial because the status quo alternative wins over any other challenger alternative in $X^{0}$ . This follows because the type-S voter always votes for $x_{0}$ . Furthermore, the voter with the peak at $x_{1}$ not only prefers $x_{0}$ to $x_{2}$ , and $x_{0}$ to any of the alternatives to the left of alternative $x_{2}$ because of her friction, but the voter also prefers $x_{0}$ to any of the alternatives to the right of $x_{2}$ . The profiles illustrated in Figure 1(b) to (d), however, induce non-trivial voting situations. To see this, note first that the status quo alternative $x_{0}$ wins over alternative $x_{1}$ in Figure 1(b) to (d), but the status quo alternative $x_{0}$ loses to alternatives $x_{3}$ , $x_{4}$ , and $x_{3}$ in Figure 1(b) to (d), respectively. Consequently, a binary voting situation is non-trivial only for some profiles.

The following result shows that there exists a special challenger alternative, called $x_{r}$ , that satisfies Definition 1(i) and, in addition, is most ‘closely located’ to any alternative that satisfies Definition 1(ii) without being located to ‘the right’ of any such alternative.

Lemma 1.

For any profile $D$ where the binary voting situation is non-trivial, there exists some alternative $x_{r} \in X^{0}$ with the highest index $r$ such that $x_{0} D x_{r}$ , and $r < k$ for any alternative $x_{k},$ where $x_{k} D x_{0}$ .

Proof.

The fact that there is an alternative $x_{j} \in X^{0}$ where $x_{0} D x_{j}$ follows immediately because the binary voting situation is non-trivial and voters have frictions. In particular, this is true for $j = 1$ because all type-A voters have frictions (see also footnote 7), which does not exclude that $j > 1$ . Define now $k$ as the lowest index where $x_{k} D x_{0}$ and note that $k > 1$ by the previous conclusions. But then because $x_{0} P x_{j}$ for $j = 1$ , there must be an alternative $x_{r} \in X^{0}$ with the highest index $r$ such that $x_{0} P x_{r}$ , and $r < k$ for any alternative $x_{k}$ where $x_{k} D x_{0}$ . □

In Figure 1(b) to (d), $x_{r}$ is given by $x_{2}$ , $x_{3}$ , and $x_{2}$ , respectively. Note also from Figure 1(b) and (c) that $x_{r}$ can be located either to ‘the left’ or ‘the right’ of the median peak $\bar{x}$ .

In the classical framework, where voters do not have frictions, for any two alternatives $x_{j} \neq \bar{x}$ and $x_{k}^{'} \neq \bar{x}$ , it always holds that $\bar{x} P x_{j}$ and $\bar{x} P x_{k}$ . Consequently, the peak of the median voter $\bar{x}$ is always the Condorcet winner and there are no Condorcet cycles (see section 2.4). When voter frictions are introduced, the next result reveals that the latter conclusion need not hold (even if the median peak obviously may be the Condorcet winner for some profiles $D$ even when frictions are present).

Lemma 2.

For profiles $D$ where the binary voting situation is non-trivial, the median peak $\bar{x}$ need not be a Condorcet winner. This holds independently of if $\bar{x} > x_{r}$ or $\bar{x} < x_{r}$ .

Proof.

To prove the lemma, it suffices to find one example of a non-trivial binary voting situation with an alternative $x_{k}$ such that $x_{k} D \bar{x}$ . The result is proved using Figure 1, where it has already been established that the binary voting situations in Figure 1(b) to (d) are non-trivial.

Figure 1(b) shows that the median peak $\bar{x}$ need not be a Condorcet winner in the case when $\bar{x} > x_{r}$ . To see this, note first that $\bar{x} = x_{4}$ and $x_{r} = x_{2}$ , and consider a majority vote between alternatives $x_{0}$ and $x_{4}$ . Because the voters with peaks at $x_{2}$ and $x_{6}$ , vote for the status quo alternative, it follows that $x_{0} D x_{4}$ .

Figure 1(c) shows that the median peak $\bar{x}$ need not be a Condorcet winner in the case when $\bar{x} < x_{r}$ . To see this, note first that $\bar{x} = x_{2}$ and $x_{j}^{r} = x_{3}$ , and consider a majority vote between alternatives $x_{0}$ and $x_{2}$ . Because the voters with peaks at $x_{0}$ and $x_{2}$ , vote for the status quo alternative, it follows that $x_{0} D x_{2}$ . □

Lemma 2 shows that the median peak need not play the same decisive role when voters have frictions as it does in the classical framework, previously discussed in Section 2.4. As will become clear shortly, this insight also means that Condorcet cycles may be present (which obviously does not mean that they are always present; there is, for example, no Condorcet cycle in Figure 1(b)). But even when Condorcet cycles are present, the challenger alternative need not be selected in such a way that the Condorcet cycle is problematic from a democratic perspective, in the sense that a majority prefers another alternative to the winning. To make this point clear, consider Figure 1(d) where there is a Condorcet cycle because $x_{0} D x_{2}$ , $x_{2} D x_{5}$ , and $x_{5} D x_{0}$ .¹² An agenda setter that nominates the challenger alternative $x_{5}$ will, consequently, win the vote against the status quo alternative $x_{0}$ . In other words, an agenda setter can successfully push the winning alternative away from the median peak. But as the observant reader will see, any of the alternatives $x_{2}$ , $x_{3}$ , and $x_{4}$ win a majority vote against alternative $x_{5}$ , and any of the alternatives $x_{3}$ , $x_{4}$ , and $x_{5}$ win a majority vote against the status quo alternative $x_{0}$ . Therefore, the Condorcet cycle can be ‘neutralized’ if the challenger alternative is appropriately selected. More precisely, in this specific example, alternative $x_{3}$ turns out to be the Condorcet winner, so even if there exists a Condorcet cycle, it need not be problematic from a democratic perspective if the agenda setter, for some reason, decides to nominate alternative $x_{3}$ to challenge the status quo alternative.

To neutralize Condorcet cycles (if they exist), the distribution of peaks as well as the distribution of frictions must be known. In any real-life binary voting situation, an agenda setter is unlikely to possess that type of information, so the process for the agenda setter to nominate the challenger alternative must somehow be approximated, for example, based on various investigations and polls.¹³ But there is always a positive probability that such information is misinterpreted or biased in some way, so the agenda setter enters the binary voting situation with a ‘non-optimal alternative’ from a voting maximizing perspective. This is particularly true given it has been established in Lemma 2 that the median peak is not as informative when frictions are present as in the classical framework without frictions. Thus, if the agenda setter enters the binary voting situation with a ‘non-optimal alternative’, the Condorcet cycles may in fact not be neutralized and thus become problematic from a democratic perspective. For that reason, it is important to characterize under which circumstances Condorcet cycles exist.

Theorem 1.

For any profile $D$ where the binary voting situation is non-trivial, there exists a Condorcet cycle if and only if (i) $\bar{x} \leq x_{r}$ or (ii) $\bar{x} > x_{r}$ and $mean (x_{r}, x_{s}) \geq 0.5,$ where $x_{s}$ is the alternative with the highest index $s \in {r + 1, \dots, m}$ such that $x_{s} D x_{0}$ .

Proof.

Note first that there exist profiles $D$ such that the binary voting situation is non-trivial in both cases (i) and (ii), so both cases are relevant (see Figure 1 and the analysis thereof). In the remaining part of the proof, such profile $D$ is fixed.

We start by proving that if condition (i) or (ii) holds, then there is a Condorcet cycle. Because the binary voting situation is non-trivial, there exists a challenger alternative that wins a majority vote over the status quo alternative. This observation together with Lemma 1 imply that there exists an alternative $x_{s}$ with the highest index such that $x_{s} D x_{0}$ . So, it needs only to be established that there is an alternative $\hat{x} \in X^{0}$ such that $x_{0} D \hat{x}$ and $\hat{x} D x_{s}$ . From Lemma 1, it follows that there exists some alternative $x_{r} \in X^{0}$ with the highest index $r < k$ such that $x_{0} P x_{r}$ for any alternative $x_{k}$ where $x_{k} D x_{0}$ . By setting $\hat{x} = x_{r}$ , it follows that $x_{0} P \hat{x}$ , so it remains only to prove that $\hat{x} D x_{s}$ . Note next that the latter condition holds if $mean (\hat{x}, x_{s}) \geq 0.5$ because $\hat{x}, x_{s} \in X^{0}$ (recall that the frictions do not play any role for the alternatives in $X^{0}$ , so the classical results, previously described in Section 2.4, hold). But this condition holds in case (i) because $\bar{x} \leq x_{r}$ and $r < s$ , and in case (ii) by the assumption in the statement of the theorem because $\hat{x} = x_{r}$ .

We next prove that if there is a Condorcet cycle, then condition (i) or (ii) must hold. Suppose that there is a Condorcet cycle, but that neither of the two conditions holds. The latter means that $\bar{x} > x_{r}$ and $mean (x_{r}, x_{s}) < 0.5$ . Note next that by definition of $x_{r}$ , for any alternative $x_{k}$ where $x_{k} D x_{0}$ , it must be the case that $k > r$ . Consequently, because there is a Condorcet cycle by assumption, there must be alternatives $x_{j}, x_{k} \in X^{0}$ such that $x_{0} D x_{j}$ and $x_{j} D x_{k}$ where $k > r$ . Because, $mean (x_{r}, x_{s}) < 0.5$ , it follows that $x_{s} D x_{r}$ . Hence, $x_{k} D x_{j}$ for any $k = r + 1, \dots, s$ and any $j = 1, \dots, r$ . But this means that a Condorcet cycle cannot exist, which contradicts our assumptions. □

As observed in the aforementioned, there is no Condorcet cycle in Figure 1(b), and the reason is that condition (ii) in Theorem 1 is violated. This follows because $\bar{x} = x_{4}$ , $x_{r} = x_{3}$ , and $x_{s} = x_{3}$ , so $\bar{x} > x_{r}$ and $mean (x_{r}, x_{s}) = 0.33 < 0.5$ .

4. Monte Carlo simulations

As already concluded in the Introduction section, even if Condorcet cycles may exist in binary voting situations with multiple plausible challenger alternatives, they are impossible to identify using voting data alone. To shed some further light on this potential problem, we follow the tradition in the literature to evaluate the theoretical findings using Monte Carlo simulations (see, e.g. Bochsler, 2010; Gehrlein, 2006: for further discussions). The purpose of the simulations is to provide an estimate of the proportion of non-trivial voting situations and the probability that Condorcet cycles exist. The latter can be achieved by employing Theorem 1.

The results are presented for averages over 1,000 simulations for each of the 600 considered instances (i.e. $5 \times 3 \times 4 \times 2 \times 5 = 600$ ) based on the variables in Table 1. The variables ‘number of voters’ and ‘number of alternatives’ are self-explained, but the others need to be clarified. ‘Type-S voter probability’ represents the probability that a given voter is of type-S, which is determined by a random draw based on the instance-specific probability. The distribution of peaks $g$ and the distribution of frictions $F$ for a given number of voters are determined by random draws from discrete Poisson distributions with mean $λ_{G}$ and $λ_{F}$ , respectively, based on the variables ‘mean peak percentage’ and ‘mean friction percentage’. For example, if there are 20 alternatives and both ‘mean peak percentage’ and ‘mean friction percentage’ are given by 0.5, then $λ_{G} = 20 \times 0.5 = 10$ and $λ_{F} = λ_{G} \times 0.5 = 10 \times 0.5 = 5$ .¹⁴

Table 1.
Variables and values for the Monte Carlo simulation. There are 600 instances in total, and for a given number of voters, there are 120 instances.

Variable Notation Considered values

Number of voters $n$ 11, 101, 1,001, 10,001, 100,001

Number of alternatives $m$ 6, 10, 20

Type-S voter probability $π_{S}$ 0.10, 0.20, 0.30, 0.40

Mean peak percentage $μ_{x *}$ 0.50, 0.75

Friction mean percentage $μ_{F}$ 0.25, 0.50, 0.75, 1.00, 1.25

Variable	Notation	Considered values
Number of voters	$n$	11, 101, 1,001, 10,001, 100,001
Number of alternatives	$m$	6, 10, 20
Type-S voter probability	$π_{S}$	0.10, 0.20, 0.30, 0.40
Mean peak percentage	$μ_{x *}$	0.50, 0.75
Friction mean percentage	$μ_{F}$	0.25, 0.50, 0.75, 1.00, 1.25

For a given number of voters, all 120 instances (i.e. $3 \times 4 \times 2 \times 5 = 120$ ) were evaluated. Almost all of the simulated binary voting situations turned out to be non-trivial as can be seen in the upper half of Table 2 (all entities in the table are mean values). This is also exactly what can be expected from real-life binary voting situations because there is little value in organizing them if there is wide consensus among the voters that the status quo alternative is preferred to all possible challenger alternatives. In fact, when the number of voters is 10,001 or 100,001, the simulation results reveal that all binary voting situations are non-trivial, and when the number of voters is 1,001, only five out of the 120,000 simulated voting situations turned out not to be non-trivial. With a smaller number of voters, there are naturally fewer non-trivial voting situations given random draws are more likely to be ‘skewed’ in some direction.

Table 2.

Summary statistics of non-trivial binary voting situations and Condorcet cycles based on averages for the 120 instances (1,000 simulations per instance) for a given number of voters.

Number of voters	11	101	1,001	10,001	100,001
Minimum proportion of non-trivial voting situations	0.610	0.824	0.997	1.000	1.000
Mean proportion of non-trivial voting situations	0.905	0.989	1.000	1.000	1.000
Maximum proportion of non-trivial voting situations	1.000	1.000	1.000	1.000	1.000
Minimum number of Condorcet cycles	1.182	0.373	0.000	0.000	0.000
Mean number of Condorcet cycles	7.410	7.616	6.815	6.591	6.491
Maximum number of Condorcet cycles	26.24	32.98	31.04	30.20	28.81

A more detailed visualization related to non-trivialness can be found in Figure 2. The figure considers the $2 \times 3 = 6$ basic cases that arise when there are 11 or 101 voters, and when the number of alternatives is 6, 10, or 20 (these are the cases with the lowest number of non-trivial voting situations according to Table 2). For a given panel, each ‘dot’ represents one of the $4 \times 2 \times 5 = 40$ possible instances that arise due to the variation in ‘Type-S probability’, ‘Mean peak percentage’, and ‘Friction mean percentage’. The color of the dot represents the mean percentage of non-trivial voting situations in accordance with the color bar. In Figure 2(a), for example, the color of the dot in the upper right corner furthest away from the ‘Type-S probability axis,’ that is, for the instance $(π_{S}, μ_{x *}, μ_{F}) = (0.4, 0.75, 1.25)$ , indicates that the mean percentage of non-trivial voting situations is roughly 60%. A visual inspection of all panels in Figure 2 indicates that the binary voting situations are almost always non-trivial independently of the values of the variables. Most exceptions can be found when there are many type-S voters ( $π_{S} \in {0.3, 0.4}$ ) and when the median peak is located far to ‘the right’ ( $μ_{x *} = 0.75$ ). This is not particularly unexpected because it is exactly for these cases where a majority of the voters are likely to support the status quo alternative regardless of which challenger alternative is chosen.

Figure 2.

(a)–(f) Illustrate the average proportion of non-trivial binary voting situations for the 240 instances specified in Table 1 when $n \in {11, 101}$ and $m \in {6, 10, 20}$ .

The lower half of Table 2 displays the average number of Condorcet cycles based on the 120 instances for a given number of voters. A general take-away from the table is that it consistently seems to be around seven cycles on average per 1,000 simulations, implying that the probability of a Condorcet cycle is roughly 0.7% (the largest number of Condorcet cycles found in any of the 600,000 simulations was 97). A more detailed visualization can be found in Figure 3. The figure should be interpreted exactly as Figure 2, but is here displayed for the $5 \times 3 = 15$ cases represented by all possible combinations of ‘Number of voters’ and ‘Number of alternatives’. As expected, the average number of Condorcet cycles grows with the number of alternatives (see the five panels furthest to the right in the figure), that is, the more alternatives, the more possible cycles. The more centered the median peak is (i.e. $μ_{x^{*}} = 0.5$ ) and the larger the proportion of type-S voters (i.e. $π_{S} \in {0.3, 0.4}$ ), the higher number of Condorcet cycles on average. In non-trivial voting situations, this is anticipated because this reflects situations where there is an alternative that is defeated by the status quo alternative, but is preferred to some alternative to ‘the far right’ that beats the status quo alternative.

Figure 3.

(a)–(o) illustrate the average number of Condorcet cycles for the 600 instances specified in Table 1.

5. Conclusions

The analysis in this paper has been based on the observations that the challenger alternative not necessarily is unique in binary voting situations, and depending on which alternative is selected, the voters may be differently likely to cast their votes on it. Specifically, if voters experience some type of status quo bias, for example, because of switching costs, voter loyalty, or benefits of the doubt, Condorcet cycles can be present even in the standard one-dimensional framework with symmetric single-peaked preferences.

Further work on the empirical side includes a thorough investigation of various polls and voter investigations to see if Condorcet cycles can be identified in real-life binary voting situations with multiple plausible challenger alternatives (here the technique in Eggers, 2021: may be helpful). On the theoretical side, future work includes more general models and careful modeling of various incomplete information settings. In this paper, no such information structure has been imposed on the model, simply because it is not needed to make the main points related to the existence of Condorcet cycles when voters have a status quo bias.

Footnotes

Acknowledgements

The author would like to thank the Editor and two anonymous referees for their excellent comments and remarks.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was financially supported by the Jan Wallander and Tom Hedelius Foundation (grant number P22-0087).

ORCID iD

Tommy Andersson

Appendix A: Utility representation with frictions

The purpose of this Appendix is to show that a utility representation of single-peaked preferences with voting frictions can be theoretically founded using a cost-based approach (see also the discussions in section 2.2). To achieve this, recall first that a standard utility representation of one-dimensional symmetric single-peaked preferences is: (1)

v_{i} (x_{j}) = α_{i} - β_{i} ∣ x_{j} - x_{i} * ∣ for x_{j} \in X

where

α_{i}

and

β_{i}

are two positive integers. Suppose next that a ‘switching cost’ for type-A voter

i

can be represented by a strictly decreasing continuous cost function

c_{i} : X^{0} \mapsto R_{+}

. The idea here is that if a challenger alternate

x_{j}

is ‘too close’ to the status quo alternative

x_{0}

, the costs for type-A voters to abandon the status quo alternative are ‘too high,’ so they prefer to stick to the status quo alternative. But if the distance between the alternatives is ‘sufficiently far’, the costs are zero and the type-A voters may consider abandoning the status quo alternative (see footnote 15 for some further remarks). In this sense, these costs create frictions for type-A voters for alternatives in

X^{0}

that are ‘too close’ to the status quo alternative (recall that the alternatives

x_{j}

are ordered, so that a higher index

j

indicated a larger ‘policy distance’ from alternative

x_{0}

Consider now the following utility representation of preferences $R_{i}$ with switching costs $c_{i}$ : (2)

u_{i} (x_{j}) = {\begin{matrix} α_{i} - β_{i} ∣ x_{j} - x_{i} * ∣ & \,for x_{j} = x_{0} \\ α_{i} - β_{i} ∣ x_{j} - x_{i} * ∣ - c_{i} (x_{j}) & \,for all x_{j} \in X^{0} \end{matrix}

With this specification, the voter friction

f_{i} = x_{l}

can be identified by finding the alternative

x_{l}

with the lowest index

l

such that voter

i

is indifferent between alternatives

x_{0}

and

x_{l}

, that is:

α_{i} - β_{i} ∣ x_{0} - x_{i} * ∣= α_{i} - β_{i} ∣ x_{l} - x_{i} * ∣ - c_{i} (x_{l})

Rearranging yields: (3)

c_{i} (x_{l}) + β_{i} ∣ x_{l} - x_{i} * ∣= β_{i} ∣ x_{0} - x_{i} * ∣

Under relatively mild assumptions on the cost function, there exists a voter friction

f_{i} = x_{l}

.¹⁵ To exemplify how the friction

f_{i} = x_{l}

can be identified, suppose that voter

i

has one-dimensional symmetric single-peaked preferences as in Figure 4(a), that is, that

m = 12

x_{i} * = 5

α_{i} = 7

, and

β_{i} = 1

. Suppose further that the cost function is given by:

c_{i} (x_{j}) = {\begin{matrix} 4 - x_{j} & \,for j \in {0, \dots, 4} \\ 0 & \,for j \in {4, \dots, 12} \end{matrix}

Figure 4.

(a)–(c) Illustrate a standard single-peaked utility representation, the cost function, and single-peaked preferences with voting frictions, respectively.

This cost function is illustrated in Figure 4(b). Note now that the solution to equation (3) is given by

x_{l} = 2

, meaning that the voter experiences a shift from the status quo alternative

x_{0} = 0

‘too costly’ if the challenger alternative is

x_{1} = 1

x_{2} = 2

. Formally, this follows because the utility from the status quo alternative

x_{0} = 0

2

, whereas the utility for alternatives

x_{1} = 1

and

x_{2} = x_{l} = 2

when taking costs into account is given by 0 and 2, respectively (recall that voters break ties by selecting the alternative with the lowest index).¹⁶ The single-peaked preferences with voter frictions are illustrated in Figure 4(c), where the dashed line represents that part of the single-peaked preferences that is ‘inactive’ (because the voter always prefers the status quo alternative over the challenger alternatives that correspond to the utility represented by the dashed line) when accounting also for the costs. See Figure 1 for additional illustrations.

Notes

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