Players and Policies

A set of policy makers $\mathcal {N}=\left\lbrace 1,\ldots ,n,\ldots ,N\right\rbrace$ are in a relationship that lasts for infinitely many periods. Time is discrete and, in each period, they must decide which of the two policies L and R to implement, where L can be interpreted as a liberal policy such as a high tax rate, high public-spending level, or high level of protection of domestic industries, and R as its conservative counterpart.

Payoffs

Each policy maker $n\in \mathcal {N}$ maximizes her expected discounted sum of period payoffs with discount factor δ ∈ (0, 1). Her payoff from implementing policy R in some period $t\in \mathbb {N}$ is denoted by θ_n(t), and her payoff from implementing L is normalized to 0. Hence, if θ_n(t) is positive (negative), policy maker n prefers the conservative policy R (the liberal policy L) in t. One can view this specification as a stylized dynamic extension of the canonical spatial model in which θ_n(t) is the peak of policy maker n’s single-peaked utility function.Footnote ⁶ We refer to θ_n(t) as the sincere preferences of policy maker n in period t. Throughout, we use bold symbols to denote N-tuples, and use the symbol $\boldsymbol{\theta }$ and the term “state” to refer to a particular realization of the random variable $\boldsymbol{\theta }\left(t\right) \equiv \left( \theta _{1}\left( t\right) ,\ldots ,\theta _{N}\left( t\right) \right)$.

We assume that the stochastic process $\left\lbrace \boldsymbol{\theta }\left( t\right) :t\in \mathbb {N} \right\rbrace$ that governs the evolution of the state is i.i.d. across periods, with probability distribution P. We assume that P is such that for all $n\in \mathcal {N}$, θ_n(t) is integrable, has full support,Footnote ⁷ and satisfies the following properties:

Assumption 1(A) states that even though policy makers’ preferences evolve over time, their ideological rank order remains constant. Specifically, those with lower indices are more leftist than those with higher indices. Therefore, whenever two policy makers disagree, the relatively more leftist always prefers L and the relatively more rightist prefers R. This assumption is in line with Poole and Rosenthal (Reference Poole and Rosenthal1991), who find that “Congress as a whole may adapt by, for example, moving to protectionism when jobs are lost to foreign competition. But as such new items move onto the agenda, their cutting lines will typically be consistent with the preexisting, stable voting alignments.”Footnote ⁸ Assumption 1(B) means that any two legislators disagree with positive probability.

The Legislative Game

In each t, a status quo q(t) ∈ {L, R} is in place. First, $\boldsymbol{\theta }\left( t\right)$ is realized and each policy maker $n\in \mathcal {N}$ observes her own preferences θ_n(t).Footnote ⁹ Then she chooses whether to vote for the status quo or the alternative policy. If the set of policy makers who vote for the alternative policy is a winning coalition, this policy is implemented. Otherwise, the status quo stays in place. The implemented policy determines the payoffs for period t and becomes the status quo for the next period q(t + 1).

The Institutions

In our stylized environment, the institutional arrangement is summarized by the set of winning coalitions needed for a policy change. We allow this set to depend on the identity of current status quo: for all q ∈ {L, R}, Ω_q denotes the set of winning coalitions when the status quo is q, Ω denotes (Ω_L, Ω_R), and Γ(Ω) denotes the game thus defined. We impose the following conditions on Ω:

Assumption 2 is standard in the literature on voting (see, e.g., Austen-Smith and Banks Reference Austen-Smith and Banks1999). Monotonicity ensures that an additional vote in favor of an alternative can only change the outcome in favor of that alternative. Nonemptyness means that a unanimous vote for a given alternative guarantees that this alternative is implemented irrespective of the status quo. Properness requires that if a coalition can change the policy in place, those outside this coalition cannot reverse this change.

Assumption 2 allows for a large class of institutions. For instance,

(1) $$\begin{equation} \Omega _{R}=\Omega _{L}=\Omega ^{M}\equiv \left\lbrace C\subseteq \mathcal {N} :\left|C\right|\ge M\right\rbrace \end{equation}$$

corresponds to a parliamentary system with a single chamber operating under a supermajority rule with threshold M. Assumption 2 is satisfied for all M ≥ (N + 1)/2. The two polar cases M = (N + 1)/2 and M = N correspond to the simple majority rule and the unanimity rule, respectively. Assumption 2 allows also for nonanonymous institutions, such as the combination of simple majority rule and a veto player (or gate keeper) $v\in \mathcal {N}$:

(2) $$\begin{equation} \Omega _{R}=\Omega _{L}=\Omega ^{v}\equiv \left\lbrace C\subseteq \mathcal {N} :C\in \Omega ^{\left( N+1\right) /2}\text{ and }v\in C\right\rbrace. \end{equation}$$

Nonanonymous institutions are the de facto rules in many democracies in which the legislative body uses simple majority but is subject to the negative agenda control of the majority leader, or the veto power of the president.Footnote ¹⁰

Equilibrium

As standard for dynamic voting games, we look for stationary Markov perfect equilibria in stage-undominated strategies (henceforth, equilibria) as defined in Baron and Kalai (Reference Baron and Kalai1993). A stationary Markov strategy for player n in Γ(Ω) maps the realization of $\boldsymbol{ \theta }\left( t\right)$ and the status quo into a probability distribution over {L, R}. Stage undomination requires that in each period, each player votes for the policy that gives her the greater continuation payoff. This equilibrium refinement rules out pathological equilibria such as all players always voting for the status quo. We assume that when indifferent, a player votes for R, but this assumption is made only for expositional convenience and does not affect the results.

Comments on Assumptions

We abstract from many aspects of lawmaking to isolate the sole impact of introducing dynamics. In particular, our policy makers are purely policy motivated and we abstract from elections. We introduce elections in a reduced form in the Role of Expectations section.

The policy space consists of two alternatives, which is clearly an abstraction, albeit a useful one. Since there is only one alternative to the status quo, this assumption implies that in each period, the bill under consideration is exogenous and cannot be amended. As a result, we do not need to specify details of how bills and amendments are chosen, and can completely characterize the political decision process by collections of winning coalitions Ω. This allows us to focus on differences between static and dynamic models without keeping track of differences among static models.Footnote ¹¹ Note also that our results do not rely on the absence of “compromise” alternatives that lie between L and R. In a setting with two players and the unanimity rule, Dziuda and Loeper (Reference Dziuda and Loeper2016) show that under reasonable assumptions on the bargaining procedure, similar voting distortions arise when players can chose from a one-dimensional space of alternatives, and there is no reason to suspect that the same would not hold in the setting of this paper.Footnote ¹² Allowing for more alternatives, however, would come at a significant cost in terms of technical and expositional complexity.

We consider policies that are continuing in nature: policies that do not have an expiration date and remain in effect until a new agreement is reached. Many policies have this feature. In the U.S., this is the case of most tax policies (Gale and Orszag Reference Gale and Orszag2003) and most spending policies: by law, all mandatory spending is continuing and this spending category, which contains all entitlements, represents more than two-thirds of the total federal budget (Levit, Austin, and Stupak Reference Levit, Austin and Stupak2015). Similarly, labor laws and minimum wage legislation tend to be continuing in nature (Clinton Reference Clinton2012). Sunset provisions, that is, clauses that attach an expiration date to a legislation, historically have been the exception rather than the norm.

Static Model

Since most of the lawmaking literature in the spirit of pivotal politics is static, we first consider as a benchmark the game Γ⁰(Ω) in which policy makers play only the first period t = 0 of Γ(Ω). In the unique equilibrium of this game, each policy maker n votes sincerely. That is, she votes for R if θ_n(0) ≥ 0 and for L if θ_n(0) < 0, irrespective of the status quo. As argued in the pivotal politics literature (see, e.g., Krehbiel Reference Krehbiel1998), sincere voting implies the existence of two decisive players, called pivots, such that a policy change occurs if and only if these two pivots vote for it.

To see who these pivots are, consider first status quo L, and suppose that θ_n(0) ≥ 0, so legislator n votes for R. Assumption 1(A) implies that with probability 1, all more rightist policy makers k ≥ n also vote for R. So if {n, . . ., N} ∈ Ω_L, then R is implemented. Suppose now that θ_n(0) < 0, so n votes for L instead. Assumption 1(A) implies that only strictly more rightist policy makers k > n may vote for R. Hence, if {n + 1, . . ., N}∉Ω_L, then L stays in place. So if n is such that

(3) $$\begin{equation} \left\lbrace n,\ldots ,N\right\rbrace \in \Omega _{L}\text{ and }\left\lbrace n+1,\ldots ,N\right\rbrace \notin \Omega _{L}, \end{equation}$$

then under status quo L, the equilibrium outcome always coincides with n’s vote. Conditions A and B in Assumption 2 guarantee that Equation (3) characterizes a unique n, which we label l(Ω) and refer to as the left pivot.Footnote ¹³ The right pivot, which we denote by r(Ω), is defined in a symmetric fashion by n such that

(4) $$\begin{equation} \left\lbrace 1,\ldots ,n\right\rbrace \in \Omega _{R}\text{ and }\left\lbrace 1,\ldots ,n-1\right\rbrace \notin \Omega _{R}. \end{equation}$$

The following proposition summarizes the above analysis:

Since the mapping between institutions and pivots plays a crucial role in our analysis, in the example below, we show how pivots are determined for the examples of institutions introduced in the Model section.

Example 1 Consider five players and, for expositional simplicity, assume the following common-shock specification: for all n and $t\in \mathbb {N} ,$$\theta _{n}\left( t\right) =\bar{\theta }_{n}+\varepsilon \left( t\right) ,$ where $\boldsymbol{\bar{\theta }}\in \mathbb {R} ^{N}$ and the stochastic process $\left\lbrace \varepsilon \left( t\right) :t\in \mathbb {N} \right\rbrace$ is i.i.d. with mean 0. If ε(t) has full support and $\bar{\theta }_{n}$ is strictly increasing in n, this specification satisfies Assumption 1. The ideological ordering of policy makers corresponds to the ordering of their average preferences $\boldsymbol{\bar{\theta }},$ like in Figure 1. The common shock ε(t) captures the shock to the policy environment in period t. Consider first simple majority, denoted by Ω^{M = 3} as defined in Equation (1). The coalition of policy makers {3, 4, 5} is winning but {4, 5} is not, so Equation (3) implies that l(Ω^{M = 3}) = 3. Likewise, {1, 2, 3} is winning, but {1, 2} is not, so from Equation (4), r(Ω^{M = 3}) = 3. That is, policy maker 3 is both the left and the right pivot.

FIGURE 1.

Determination of the pivots for the 4/5th majority rule.

Consider now supermajority Ω^{M = 4}, that is, the rule which requires four votes for a policy change. The determination of pivots under this institution is illustrated in Figure 1. Under Ω^{M = 4}, {2, 3, 4, 5} is a winning coalition but {3, 4, 5} is not, so Equation (3) implies that l(Ω^{M = 4}) = 2. Also, {1, 2, 3, 4} is winning, but {1, 2, 3} is not, so from Equation (4), r(Ω^{M = 4}) = 4.

And finally, consider the institution Ω^{v = 2} defined in Equation (2) in which policy changes must be approved both by a simple majority and by the veto player, policy maker 2. The coalition {2, 3, 4, 5} is winning as it contains both a majority of policy makers and the veto player, but {3, 4, 5} is not, as it does not include the veto player, so from Equation (3), l(Ω^{v = 2}) = 2. Likewise, {1, 2, 3} is winning, but {1, 2} is not, because it does not form a majority. Hence, Equation (4) implies that r(Ω^{v = 2}) = 3.

In Example 1, l(Ω) ≤ r(Ω). More generally, conditions (A) and (C) in Assumption 2 imply that this inequality always holds.Footnote ¹⁴ That is, the left pivot is always weakly more leftist than the right pivot. When l(Ω_L) = r(Ω_R), we shall say that Ω is quasi-dictatorial: the equilibrium outcome of Γ⁰(Ω) is the same as if the unique pivot were the dictator. Quasi-dictatorial institutions include dictatorship, but also non-dictatorial institutions such as simple majority rule. When l(Ω_L) < r(Ω_R), we shall say that Ω is non-quasi-dictatorial. Examples of non-quasi-dictatorial institutions include supermajority requirements, bicameral governments, or unicameral governments with a presidential veto.

In the pivotal politics literature, the influence of the status quo for a given institution is captured by the gridlock interval, defined as the set of status quos that cannot be replaced by any proposal (Krehbiel Reference Krehbiel1998). In our model, any status quo may be replaced if the realization of $\boldsymbol{\theta }\left( t\right)$ makes it sufficiently unappealing. However, for any profile of votes, either the same policy is implemented irrespective of the status quo, or the current status quo stay in place.Footnote ¹⁵ Hence, it seems natural to adapt the canonical notion of gridlock as follows.

From Proposition 1, in the equilibrium of Γ⁰(Ω), gridlock occurs in state $\boldsymbol{ \theta }$ if the pivots’ sincere preferences disagree in that state. Since l(Ω) ≤ r(Ω), in that case, l(Ω) prefers L and r(Ω) prefers R. So the set of gridlock states is

(5) $$\begin{equation} G^{0}\left( \Omega \right) =\lbrace \boldsymbol{\theta }\in \mathbb {R} ^{N}:\theta _{l\left( \Omega \right) }<0\le \theta _{r\left( \Omega \right) }\rbrace. \end{equation}$$

For instance, for the environment considered in Example 1 and Ω^{M = 4}, the set of gridlock states in equilibrium is $G^{0}\left( \Omega ^{M=4}\right) =\lbrace \varepsilon \in \mathbb {R} ^{N}:\bar{\theta }_{2}<-\varepsilon \le \bar{\theta }_{4}\rbrace$. In Figure 1, this is the area between $\bar{\theta }_{2}$ and $\bar{\theta }_{4}.$

Dynamic Model

In the dynamic game Γ(Ω), policy makers’ votes determine not only today’s policy, as in the static game Γ⁰(Ω), but also tomorrow’s status quo, which in turn affects future payoffs. Let W _n(q) denote the value of the game Γ(Ω) for policy maker n given initial status quo q ∈ {L, R}. In any period t, her continuation payoff from implementing R is θ_n(t) + δW _n(R) and from implementing L is 0 + δW _n(L). So the stage-undominated action for n is to vote for R if and only if

(6) $$\begin{equation} \theta _{n}\left( t\right) +\delta \left( W_{n}\left( R\right) -W_{n}\left( L\right) \right) \ge 0. \end{equation}$$

Denote d _n ≡ δ(W _n(R) − W _n(L)). By comparison with Γ⁰(Ω), we see that in equilibrium, n behaves as if the game was static but her preferences were θ_n(t) + d _n instead of θ_n(t). We call the term θ_n(t) + d _n her strategic preferences in t, and d _n her voting distortion.Footnote ¹⁶ The voting distortions $\boldsymbol{d}$ reflect policy makers’ preferences over the next status quo.

Let us first investigate when sincere voting, that is, $\boldsymbol{d}=\left( 0,\ldots ,0\right) ,$ can be an equilibrium. Consider the environment of Example 1, and suppose first that Ω is the simple majority rule. If policy makers vote sincerely in Γ(Ω), then as argued in Example 1, voter 3 is pivotal under either status quo, so her most preferred policy is always implemented. Therefore, in any period t, the status quo does not affect the future policy path, and hence W _n(R) = W _n(L). So by Equation (6), it is optimal for policy makers to vote sincerely. Note that the same argument holds for any quasi-dictatorial rule.

Consider now the case of a non-quasi-dictatorial institution, say the 4/5th majority rule as in Example 1 and Figure 1. If policy makers vote sincerely, then policy makers 2 and 4 are the pivots. From Assumption 1(B), the sincere preferences of these pivots disagree with positive probability; hence, there are states in which the status quo stays in place. In those states, policy maker 2 prefers policy L and policy maker 4 prefers policy R, and hence W ₂(L) > W ₂(R) and W ₄(L) < W ₄(R). Therefore, by Equation (6), sincere voting cannot be an equilibrium.

How do policy makers distort their vote when Ω is non-quasi-dictatorial? For any profile of votes, either the outcome of the vote is the same irrespective of the status quo, or both status quos stay in place. So status quo R is weakly more likely to result in policy R than status quo L. Therefore, policy makers with more rightist policy preferences also have more rightist preferences over the next status quo. Given Assumption 1(A), this means that d _n = W _n(R) − W _n(L) increases in n. The latter property implies that in any equilibrium, policy makers’ strategic preferences $\boldsymbol{\theta }\left( t\right) +\boldsymbol{d}$ are rank-ordered as their sincere preferences $\boldsymbol{\theta }\left( t\right).$ That is, more rightist policy makers should vote in a more rightist way. As a result, in equilibrium, the same players as in Γ⁰(Ω) are pivotal under each status quo, and by the same argument as before, l(Ω) and r(Ω) distort their vote in favor of L and R, respectively.

The following proposition formalizes the above intuition:

Proposition 2 has three noteworthy implications. First, policy makers vote sincerely in Γ(Ω) only when Ω is quasi-dictatorial. Second, when Ω is non-quasi-dictatorial, in any equilibrium, the voting distortions $\boldsymbol{d}$ have a polarizing effect. To see this, note that from Assumption 1(A) and Proposition 2(i), for any non-quasi-dictatorial Ω, with probability 1,

(7) $$\begin{eqnarray} &&\left|\left( \theta _{n}\left( t\right) +d_{n}\right) -\left( \theta _{m}\left( t\right) +d_{m}\right) \right| = \left|\left( \theta _{n}\left(t\right)\right.\right.\nonumber\\ &&\quad\left.\left. -\theta _{m}\left( t\right) \right) +\left( d_{n}-d_{m}\right) \right| >\left|\theta _{n}\left( t\right) -\theta _{m}\left( t\right) \right|. \end{eqnarray}$$

The distance between the strategic preferences of policy makers n and m on the left-hand side of Equation (7) can be interpreted as their degree of strategic polarization. Hence, Equation (7) states that under any non-quasi-dictatorial rule, the degree of strategic polarization between any two policy makers is greater than their degree of ideological polarization |θ_n(t) − θ_m(t)|.

Finally, Proposition 2 has important implications for policy dynamics. In any equilibrium of Γ(Ω), pivots have the same decisive role as in Γ⁰(Ω), so gridlock occurs in state $\boldsymbol{\theta }$ if the pivots’ strategic preferences disagree in that state. In such states, l(Ω) votes for L and r(Ω) votes for R. Therefore, in any equilibrium with distortions $\boldsymbol{d}$, the set of gridlock states is

(8) $$\begin{equation} G\left( \Omega \right) =\lbrace \boldsymbol{\theta }\in \mathbb {R} ^{N}:\theta _{l\left( \Omega \right) }+d_{l\left( \Omega \right) }<0\le \theta _{r\left( \Omega \right) }+d_{r\left( \Omega \right) }\rbrace. \end{equation}$$

When comparing Equation (5) to Equation (8), Proposition 2(ii) implies that the set of gridlock states is greater—in the inclusion sense—in any equilibrium of Γ(Ω) than in the equilibrium of Γ⁰(Ω). Thus, the strategic polarization that arises in the dynamic game exacerbates gridlock: policy change occurs less frequently than predicted by the static model.Footnote ¹⁷

The next proposition characterizes the equilibrium distortions. Recall that d _n is defined as δ(W _n(R) − W _n(L)), that is, policy maker n′s preference over the next status quo, and the latter matters only if the next state is in the gridlock interval. In any gridlock state, either status quo stays in place, so the continuation payoff gain of having status quo R instead of L is given by θ_n(t) + δ(W _n(R) − W _n(L)). This delivers the following:

Proposition 3 For all $\boldsymbol{d}\in \mathbb {R} ^{N}$, there exists an equilibrium of Γ(Ω) whose profile of voting distortions is $\boldsymbol{d}$ if and only if for all $n\in \mathcal {N},$

(9) $$\begin{equation} d_{n}=\delta \int _{\boldsymbol{\theta }\in G\left( \Omega \right) }\left( \theta _{n}+d_{n}\right) dP\left( \boldsymbol{\theta }\right) , \end{equation}$$

where G(Ω) is given by Equation (8). In particular, in any equilibrium,

(10) $$\begin{eqnarray} \frac{1}{N}\sum \nolimits _{n\in \mathcal {N}}d_{n} &=&\frac{\delta }{1-\delta P\left( G\left( \Omega \right) \right) }\nonumber\\ &&\times\int _{\boldsymbol{\theta }\in G\left( \Omega \right) }\left( \frac{1}{N}\sum \nolimits _{n\in \mathcal {N} }\theta _{n}\right) dP\left( \boldsymbol{\theta }\right). \qquad \end{eqnarray}$$

Note that, in general, the right-hand side of Equation (10) does not need to be 0, which means that policy makers’ average strategic preferences $\frac{1}{N}\sum _{n\in \mathcal {N}}\left( \theta _{n}\left( t\right) +d_{n}\right)$ may be more rightist or leftist than their average sincere preferences $\frac{1}{N}\sum _{n\in \mathcal {N}}\theta _{n}\left( t\right)$. Hence, policy makers may exhibit an average policy bias: they may vote as if they were on average biased toward one alternative. Note that this bias depends on the preference distribution across policy makers, but also on the political institution via G(Ω). Whereas strategic polarization increases the probability of gridlock, the average bias affects the relative likelihood of liberal versus conservative policy change, and thus the relative prevalence of policies L and R. We explore the effect of the policy bias more carefully in the Biased Institutions section.

Remark 1 Proposition 3 implies that the equilibria are easy to characterize. One must first solve Equation (9) for n = l(Ω) and n = r(Ω); that is, solve a (nonlinear) system of two equations with two unknowns. Any solution (d_l(Ω), d_r(Ω)) pins down the gridlock set G(Ω). Once G(Ω) is determined, for n∉{l(Ω), r(Ω)}, Equation (9) becomes a straightforward linear equation in d_n. Moreover, in symmetric setups, d_r(Ω) = −d_l(Ω), so the behavior of the pivots is characterized by a single equation.Footnote ¹⁸ For instance, for the specification in Example 1, if F is the distribution of ε(t), this equation becomes

(11) $$\begin{eqnarray} &&\hspace*{-7pt}d_{r\left( \Omega \right) }\nonumber\\ &&\quad\hspace*{-7pt} =\frac{\delta \left( F\left( d_{r\left( \Omega \right) }+\bar{\theta }_{r\left( \Omega \right) }\right) -F\left( -d_{r\left( \Omega \right) }-\bar{\theta }_{r\left( \Omega \right) }\right) \right) }{ 1-\delta \left( F\left( d_{r\left( \Omega \right) }+\bar{\theta }_{r\left( \Omega \right) }\right) -F\left( -d_{r\left( \Omega \right) }-\bar{\theta } _{r\left( \Omega \right) }\right) \right) }\bar{\theta }_{r\left( \Omega \right) }. \nonumber\\ \end{eqnarray}$$

Propositions 2 and 3 imply a fundamental difference between static and dynamic models. At first sight, the predictions of these models seem observationally equivalent. In both, each policy maker votes for the policy closer to her ideal point, the only difference being that in the static theory, this ideal point reflects her sincere preference θ_n(t), whereas in the dynamic theory the ideal point θ_n(t) + d _n is also affected by strategic, forward looking considerations. However, this apparent equivalence overlooks an important point. First and foremost, the static approach treats policy makers’ preferences as an independent primitive. In contrast, Proposition 2 implies that preferences are equilibrium objects, and as such, are endogenous to the political decision process Ω. Hence, when considering the impact of institutional changes, one cannot take as given the preferences as revealed by the current legislative behavior, but has to understand how these strategic preferences change with Ω. Second, policy makers’ strategic preferences depend crucially on their expectations about the future gridlock. Hence, changes in expectations, even when not accompanied by observable changes in the current institution or environment, are likely to affect current behavior.

Having established that institutions and expectations about future economic and political environment shape policy makers’ behavior, it is important to better understand the mapping between the former and the latter. We do this in the Institutions and Voting Distortions and Role of Expectations sections.

Before we proceed, however, we note that the game Γ(Ω) may have multiple equilibria.Footnote ¹⁹ This is because gridlock feeds on itself: if policy makers expect more frequent future gridlock, they have greater incentives to distort their vote for the policy they prefer in case of gridlock, and this in turn increases the likelihood of gridlock. One can prove, however (see Lemma 1 in the Appendix) that equilibria can be ranked in terms of the degree of strategic polarization of the pivots. Formally, we say that an equilibrium with distortions $\boldsymbol{d}^{\prime }$ is strictly more polarized than another equilibrium with distortions $\boldsymbol{d}$ if d′_l(Ω) < d _l(Ω) and d′_r(Ω) > d _r(Ω). Then for any two distinct equilibria, one is strictly more polarized than the other. In what follows, when multiplicity is an issue, the comparative statics are derived for the least polarized equilibrium (which always exists), but the same results hold for the most polarized equilibrium.

Institutions and Strategic Polarization

In this section, we investigate how the equilibrium voting behavior depends on voting institutions. We use the following partial order on institutions.

In other words, Ω′ requires a greater consensus than Ω if changing the status quo requires the approval of a greater set of policy makers under Ω′ than under Ω. For example, rules that use a higher supermajority threshold require a greater consensus, and adding a veto player, or more generally adding checks and balances, increases the degree of consensus required. By definition of l(Ω) and r(Ω), an institution that requires a greater consensus leads to more extreme pivots.Footnote ²⁰ Formally, if Ω′ requires a greater consensus than Ω, then

(12) $$\begin{equation} l(\Omega ^{\prime })\le l\left( \Omega \right) \le r\left( \Omega \right) \le r(\Omega ^{\prime }). \end{equation}$$

Proposition 4 Let Ω and Ω′ be two institutions and let $\boldsymbol{d}\left( \Omega \right)$ and $\boldsymbol{d}\left( \Omega ^{\prime }\right)$ denote the voting distortions of the least polarized equilibrium of Γ(Ω) and Γ(Ω′), respectively. If Ω′ requires a greater consensus than Ω, then

(13) $$\begin{eqnarray} d_{l\left( \Omega ^{\prime }\right) }\left( \Omega ^{\prime }\right) &\le& d_{l\left( \Omega ^{\prime }\right) }\left( \Omega \right) \le d_{l\left( \Omega \right) }\left( \Omega \right) \le 0\le d_{r\left( \Omega \right) }\left( \Omega \right)\nonumber\\ & \le& d_{r\left( \Omega ^{\prime }\right) }\left( \Omega \right) \le d_{r\left( \Omega ^{\prime }\right) }\left( \Omega ^{\prime }\right) , \end{eqnarray}$$

and for all $n,m\in \mathcal {N}$, with probability 1,

(14) $$\begin{eqnarray} &&\left|\left( \theta _{n}\left( t\right) +d_{n}\left( \Omega ^{\prime }\right) \right) -\left( \theta _{m}\left( t\right) +d_{m}\left( \Omega ^{\prime }\right) \right) \right|\nonumber\\ &&\quad\ge \left|\left( \theta _{n}\left( t\right) +d_{n}\left( \Omega \right) \right) -\left( \theta _{m}\left( t\right) +d_{m}\left( \Omega \right) \right) \right|. \qquad \end{eqnarray}$$

If, furthermore, (l(Ω), r(Ω)) ≠ (l(Ω′), r(Ω′)), then the outer inequalities in Equation (13) and the inequality in Equation (14) are strict.

The first claim in Proposition 4 states that increasing the required consensus exacerbates the voting distortions of the left and right pivot in favor of policy L and R, respectively, and does so for two reasons. First, from Equation (12), more extreme policy makers become pivots, and from Proposition 2, for a fixed institution Ω, more extreme policy makers have greater voting distortions. This effect explains the four inner inequalities in Equation (13). Second, the voting distortions of the new pivots are greater under Ω′ than under Ω, which results in the two outer inequalities in Equation (13). The latter effect comes from the fact that as more extreme players become pivots, the probability that pivots disagree, and thus that gridlock occurs in the future increases. The greater likelihood of future gridlock in turn increases the incentives of each new pivot to distort her vote in favor of her preferred status quo.

Inequalities in Equation (13) imply that the pivots exhibit a greater degree of strategic polarization under the rule that requires a greater consensus. The inequalities in Equation (14) state that the same is true for any two policy makers.

An important consequence of Proposition 4—illustrated in Figure 2 for the common-shock specification of Example 1—is that the static and the dynamic approaches predict different impact of institutional change. To see why, note that when assessing the impact of a change from Ω to Ω′, the static approach assumes that policy makers’ past voting behavior under Ω, as given by their strategic preferences $\left( \theta _{n}\left( t\right) +d_{n}\left( \Omega \right) \right) _{n\in \mathcal {N}}\!,$ is sincere, and thus that they would continue voting the same way under Ω′. Given the new pivots l(Ω′) and r(Ω′), the static approach would then conclude that gridlock under Ω′ would occur when

$$\begin{equation*} \theta _{l\left( \Omega ^{\prime }\right) }\left( t\right) +d_{l\left( \Omega ^{\prime }\right) }\left( \Omega \right) < 0\le \theta _{r\left( \Omega ^{\prime }\right) }\left( t\right) +d_{r\left( \Omega ^{\prime }\right) }\left( \Omega \right). \end{equation*}$$

The dynamic approach implies instead that gridlock under Ω′ occurs when

$$\begin{equation*} \theta _{l\left( \Omega ^{\prime }\right) }\left( t\right) +d_{l\left( \Omega ^{\prime }\right) }\left( \Omega ^{\prime }\right) <0\le \theta _{r\left( \Omega ^{\prime }\right) }\left( t\right) +d_{r\left( \Omega ^{\prime }\right) }\left( \Omega ^{\prime }\right). \end{equation*}$$

The two outer inequalities in Equation (13) imply hence that the static approach underestimates the inertial effect of replacing Ω by Ω′, by not taking into account policy makers’ behavior change.

FIGURE 2.

The top picture illustrates static gridlock under Ω^{M = 4} for the common-shock specification. The middle picture illustrates policy makers’ strategic preferences and the gridlock under Ω^{M = 4} as well as the gridlock predicted by static models when the institution changes to Ω^{M = 5} taking the behavior of policy makers under Ω^{M = 4} as sincere. The bottom picture illustrates the actual gridlock under Ω^{M = 5}.

These results provide potentially testable implications. Consider an institutional reform that increases the degree of consensus required for policy changes, such as adding a second chamber, allowing presidential veto, or replacing simple with a qualified majority. Then any two policy makers who are in office before and after the reform takes place should exhibit a greater degree of polarization in their voting behavior after the reform is implemented.

Proposition 4 also speaks to the impact of filibuster in the U.S. Senate. Pundits often argue that the U.S. Senate should abandon the filibuster since the increased ideological polarization makes it hard to gather 60 votes on virtually any important legislation. Proposition 4 implies that indeed abandoning the institution of the filibuster should decrease gridlock in the U.S. Congress, not only because the approval of fewer senators would be required to pass a policy change, but also because the voting behavior of all U.S. congressmen should become less polarized. This logic also suggests an alternative interpretation of the correlation between the more frequent use of the filibuster and the growing polarization among the U.S. legislators. The filibuster was once an infrequently used tool reserved for the most important and controversial bills, but since the ’70s it has become a routine practice in American politics (Koger Reference Koger2010). At the same time, polarization in Congress started increasing (McCarty, Poole, and Rosenthal Reference McCarty, Poole and Rosenthal2008). Pundits tend to interpret the rise in the use of the filibuster as a tactic used by a more ideologically polarized Senate to stifle reforms. Our results suggest a possible alternative explanation. The new norm of systematically using the filibuster de facto shifted the pivotality in the U.S. Congress—not only in the U.S. Senate—to more ideologically polarized policy makers. Hence, Proposition 4 predicts that this de facto institutional change should have resulted in more polarized voting behavior and an increase in gridlock.

Biased Institutions

Some democratic institutions use different voting rules depending on the policy change under consideration. For example, in sixteen U.S. states, tax increases require the approval of a qualified majority in each house, whereas tax cuts can be approved by simple majority. Similarly, in the U.S. budget process, the Byrd Rule essentially requires a filibuster-proof majority to pass bills that raise deficit, whereas simple majority is sufficient to pass bills that lower it.Footnote ²¹ The explicit goal of these fiscally conservative institutions was to limit growth of the public sector and the public debt. Indeed, within a static framework, increasing a hurdle for a policy change in one direction decreases the probability that such policy change occurs. However, the analysis below shows that once the strategic effects are taken into account, fiscally conservative voting rules can induce a more liberal voting behavior on average, and can even lead to more liberal policies. This insight is consistent with the empirical finding that effects of such biased voting rules on the level of state taxes in the U.S. have been modest, if any.Footnote ²² It is also consistent with the observation that the Byrd rule did not prevent the federal debt from growing steadily since its implementation.

To see why the inequality in Definition 3 corresponds to an institution biased for policy R, observe that from Proposition 2, status quo L is replaced by R when all policy makers in {l(Ω), . . ., N} vote for R. Conversely, status quo R is overturned when all policy makers in {1, . . ., r(Ω)} vote for L. So the condition l(Ω) + r(Ω) > N means that a greater coalition is needed to repeal R than to repeal L. For instance, if Ω requires a supermajority with threshold M _q under status quo q, then Ω is biased in favor of R if and only if M _L < M _R.

Proposition 5 below identifies a class of environments in which the government as a whole has no ideological bias, but when operating under a rule biased in favor of R, the average voting bias makes it vote as if it was in favor of L on average.

Under the conditions of Proposition 5, when Ω is unbiased, L and R receive the same expected number of votes. Proposition 5 states that when Ω is biased in favor of R, the expected number of votes for L is greater. The intuition for that result is that policy makers anticipate that it may be hard to replace R with L in the future, and are therefore more inclined to vote for L today. Thus, in our dynamic setting, increasing the institutional hurdle to pass liberal reforms makes not only liberal reforms, but also conservative reforms less likely to be implemented.

Proposition 5 raises a question whether the increase in the expected number of votes for L may be large enough to offset the institutional bias in favor of R. Example below shows that the answer is affirmative: biasing political institution against, say, liberal policy reforms, always reduces the likelihood of both liberal and conservative reforms, but may also result in policies that are more liberal on average.

Example 2 Consider the common-shock specification like in Example 1, where ε(t) follows a standard normal distribution. Unlike in Proposition 5, however, assume that the government is ideologically biased toward L, with median policy maker’s ideological position $\bar{\theta }_{\left( N+1\right) /2}=-0.5$. Under the simple majority rule Ω^{M = (N + 1)/2}, the median voter is always pivotal, and there are no voting distortions; hence, the probability that L is implemented in any period is $\Pr \left( -0.5+\varepsilon <0\right) =\Phi \left( 0.5\right) =0.7.$ Suppose now that the hurdle to implement L increases. The median policy maker remains the left pivot, whereas a more rightist policy maker becomes the right pivot, $\bar{\theta }_{r}>-0.5.$ Hence, the probability that R is replaced by L goes down, and is equal to $\Phi \left( -\bar{\theta } _{r}-d_{r}\right) <\Phi \left( -\bar{\theta }_{r}\right) <\Phi \left( 0.5\right) $. However, the median policy maker distorts her voting behavior in favor of L, d_{(N + 1)/2} < 0, so the probability that L is replaced by R also goes down, $1-\Phi \left( -\bar{\theta }_{\left( N+1\right) /2}-d_{\left( N+1\right) /2}\right) <1-\Phi \left( 0.5\right) $. Hence, in terms of taxes, making a voting rule biased against tax increases makes both tax increases and decreases less likely.

Figure 3 depicts the probability of L according to the stationary distribution of the underlying Markov process over policies; that is, the probability that in a distant future the policy is L for the biased voting rule (solid line) and for simple majority (dashed line). Increasing the bias in favor of R, which is equivalent to increasing $\bar{\theta }_{r},$ does not monotonically increase the long-run prevalence of R, and it can even make L more likely to be implemented in the long run.Footnote ²³

FIGURE 3.

Probability that L is in place in the long run a function of the bias of the voting rule in favor of R.

Voting Distortions and Welfare

The proposition below states that the greater the voting distortions of the pivots, and thus the greater gridlock, the worse off the moderate policy makers are.

To see the intuition for this result, consider an equilibrium and suppose that the right pivot deviates by voting for R in a greater set of states (because an institutional change makes the pivot ideologically more extreme, and/or because she increases her voting distortion). This deviation is detrimental to all legislators more leftist than her because the outcome under status quo R is less likely to coincide with their preferences. The best response of the left pivot to this deviation is to become more biased in favor of L, so as to avoid status quo R, and this is detrimental to all legislator more rightist than her for the same reason.

It is worth pointing out, however, that our model abstracts from many aspects of policy making that could make checks and balances and the resulting inertia desirable. For example, our policy makers do not have dynamic inconsistency problem (Piguillem and Riboni Reference Piguillem and Riboni2013, Reference Piguillem and Riboni2015) that gridlock could solve, two policies leave no room for policy smoothing across periods (Bowen, Chen, and Eraslan Reference Bowen, Chen and Eraslan2014; Bowen et al. Reference Bowen, Chen, Eraslan and Zapal2017), and majorities have limited flexibility to design policies concentrating benefits and spreading the costs (Baron Reference Baron1991; Battaglini and Coate Reference Battaglini and Coate2007, Reference Battaglini and Coate2008), which precludes the tyranny Role of Expectations even under rules that require little consensus.

Volatility of the Environment

The realization of $\boldsymbol{\theta }\left( t\right)$ can be interpreted as the current economic situation, state of knowledge, or citizens’ sentiment. In some areas, these variables are likely to change more frequently than in others. We investigate now how strategic polarization and gridlock depend on policy maker’s expectations about the volatility of the policy environment.

To this end, we relax the assumption that the state is i.i.d. over time. Instead, we assume that in each t ≥ 1, conditional on $\boldsymbol{ \theta }\left( t-1\right) ,$$\boldsymbol{\theta }\left( t\right) = \boldsymbol{\theta }\left( t-1\right)$ with probability 1 − v, and with probability v, $\boldsymbol{\theta }\left( t\right)$ is redrawn independently of previous states, and according to the same probability distribution P as in the basic model. Thus, v ∈ [0, 1] measures the volatility of the environment.Footnote ²⁴

Policy gridlock is bound to be particularly detrimental in environments in which frequent policy changes are a desirable response to a volatile environment. In the next proposition, we show that, unfortunately, it is in such environments that strategic polarization, and hence gridlock, tend to be greater.

Proposition 7 Let Ω be a non-quasi-dictatorial institution, and let $\boldsymbol{d}$ and $\boldsymbol{d}^{\prime }$ be the voting distortions of the least polarized equilibrium of Γ(Ω) with volatility v and v′, respectively, with 0 < v < v′ ≤ 1. Then

$$\begin{equation*} d_{l\left( \Omega \right) }^{\prime }<d_{l\left( \Omega \right) }<0<d_{r\left( \Omega \right) }<d_{r\left( \Omega \right) }^{\prime }. \end{equation*}$$

When the environment is static—that is, v = 0—then in the unique equilibrium, $\boldsymbol{d}=\left( 0,..,0\right) $.

The intuition for Proposition 7 is as follows. As the volatility of the environment increases, the policy implemented today is more likely to require revision tomorrow; hence, securing a favorable status quo becomes more important relative to implementing the sincerely preferred policy today.

Proposition 7 suggests that the impact of the environment volatility on the likelihood of policy change is ambiguous. As volatility increases, more frequent shocks prompt more frequent policy changes, but policy makers become strategically more polarized, and are thus less likely to agree on policy response after shocks. Figure 4 above demonstrates that the latter effect may dominate, and policy persistence may be greater in environments that require more frequent policy reforms.

FIGURE 4.

Probability that policy changes from one period to another as a function of preference volatility.

Figure 4 considers again the common shock specification from Example 1 with Ω generating symmetric pivots, that is, $\bar{\theta }_{r\left( \Omega \right) }=-\bar{\theta }_{l\left( \Omega \right) }=0.5$. For this environment, Figure 4 depicts policy volatility—measured by the probability that the policy changes in the next period—as a function of v. Initially, these two volatilities move together. This is because for low volatility, the strategic effect is small. However, as volatility becomes large, the strategic effect dominates and the policy becomes more persistent.

Tax, welfare, or trade are prominent examples of volatile policy domains. The goal of most tax reforms oscillate between stimulating the economy (i.e., lowering tax rates) and reducing the deficit (i.e., increasing tax rates). These reforms are prompted by recurring shocks such as business cycles, changes in fiscal needs, or the vagaries of public opinion. Likewise, since WWII, welfare policies have had to adapt to the emergence of new social risks, growing inequalities, and changing demographics (Hacker Reference Hacker2004). Trade policies must also react to sectorial shocks, domestic pressures, and the international environment. Proposition 7 provides a rationale for procedural rules that limit the degree of consensus required in such areas. Examples of such rules include the reconciliation process, which limits the use of the filibuster for fiscal policies, or the fast track authority, which limits the ability of Congress to veto the decisions of the president for trade policies (Koger Reference Koger2010; Howell and Moe Reference Howell and Moe2016).

Political Turnover

The basic model assumes that policy makers stay in office indefinitely. Realistically, elections imply political turnover. To investigate how turnover shapes behavior, we extend the model as follows. Let $\mathcal {N}=\left\lbrace 1,\ldots ,N\right\rbrace$ denote the set of political positions (e.g., presidency, congressional seats), and $\mathcal {C} =\left\lbrace 1,\ldots ,C\right\rbrace$ the set of potential candidates for these positions. In any period $t\in \mathbb {N}$, the preferences of each candidate $c\in \mathcal {C}$ are denoted by $\theta _{c}\left( t\right) \in \mathbb {R}$ and the candidate elected in each position $n\in \mathcal {N}$ is denoted by $e_{n}\left( t\right) \in \mathcal {C}$. The institution Ω specifies coalitions of political positions that are winning, independently of which candidates are currently appointed to these positions, and Ω is assumed to satisfy Assumption 2.

In each period t, before voting, candidates commonly observe the preferences $\boldsymbol{\theta }\left( t\right)$ and the electoral outcome $\boldsymbol{e}\left( t\right) $. To capture policy makers’ expectation about the future political environment, we assume that they also observe a common signal $s\left( t\right) \in \mathcal {S}$ that conveys information about the next elections $\boldsymbol{e}\left( t+1\right) $. For example, s(t) may contain information that more leftist candidates are more likely to be elected next period. To retain the stationarity of the model, we assume that $\left\lbrace \left( \boldsymbol{\theta }\left( t\right) ,s\left( t\right) ,\boldsymbol{e}\left( t\right) \right) :t\in \mathbb {N} \right\rbrace$ is a stationary process. To make sure that s(t) only captures electoral expectations, we further assume that for all t ≥ 1, $\boldsymbol{\theta }\left( t\right)$ and s(t) are independent of the past and of $\boldsymbol{e}\left( t\right) ,$ whereas $\boldsymbol{e}\left( t\right)$ is correlated with the past only via s(t − 1). In line with Assumption 1, the rank order of the candidates’ preferences is fixed, but we slightly relax that assumption to allow some candidates to have the same preferences. Formally, for all t, θ₁(t) ≤ . . . ≤ θ_C(t) with probability 1. To keep this extension in line with the basic model, we assume that each candidate c cares about the implemented policy independent of whether she is in office.Footnote ²⁵

Proposition 8 is a natural extension of Proposition 2, but reveals an interesting point overlooked by the latter. Voting distortions do not depend on which candidates are currently in office, $\boldsymbol{e}\left( t\right) $. Instead, they depend on policy makers’ expectations about who will be in office tomorrow, as captured by s(t). To see why, recall that policy makers distort their votes to influence which alternative prevails in case of future gridlock, and the latter depends only on who is elected tomorrow, not today. Note, however, that the electoral outcome $\boldsymbol{e}\left( t\right)$ still affects the policy outcome in t, as it determines who is pivotal in that period.

To illustrate the applicability of this extension, consider a simple case in which candidates can be of two types, say Republicans and Democrats, and suppose that these types disagree with positive probability in the sense of Assumption 1(B). The electoral outcome $\boldsymbol{e}\left( t\right)$ can lead to one of three cases: either there exists a winning coalition of elected Democrats, or there exists a winning coalition of elected Republicans, or no winning coalition of the same type exists. The first two cases correspond to a united government, whereas the last case corresponds to a divided government.

Thus, Democrats distort their voting behavior in favor of L and Republicans in favor of R. Moreover, voting distortions depend not on whether the current government is divided, but on the probability that it is divided in the future, and are increasing in the latter. The intuition for that result is as follows. When the government is united, a winning coalition of Democrats or of Republicans can unilaterally implement any policy irrespective of the status quo. Conversely, when the government is divided, with positive probability, no policy receives a bipartisan support, in which case either status quo stays in place. Hence, if policy makers expect future government to be united, they do not care about the future status quo and vote sincerely, whereas if they expect future government to be divided, they are biased in favor of their most preferred status quo.

A careful analysis of the differences between united and divided governments in a dynamic setting is beyond the scope of this paper. It is interesting, however, that policy distortions occur even under united government. To see that, consider a united Republican government that inherited high tax rates from the previous government. Suppose further that because of a high level of public debt, both parties sincerely prefer to leave those rates unchanged. Republicans may nevertheless decrease them to assure a favorable status quo should they need Democrats’ approval to implement a tax cut in the future.

The last proposition generalizes Proposition 9 to an arbitrary number of types. For any realization of the electoral outcome $\boldsymbol{e}\left( t\right)$, one can reorder the political positions from the position held by the most leftist appointee to the position held by the most rightist appointee, keeping unchanged the role of each position in Ω. One can then define the left and right pivots as in the Static Model section. Proposition 8 implies that, almost surely, θ₁(t) + d ₁(s(t)) ≤ . . . ≤ θ_C(t) + d _C(s(t)). That is, more rightist appointees vote in a more rightist way, so for almost all realizations of $\boldsymbol{\theta }\left( t\right)$ and s(t), the outcome of the vote under status quo L (R) coincides with the vote of the left (right) pivot given $\boldsymbol{e}\left( t\right)$, as in the basic model.

Proposition 10 For all $\left( l,r\right) \in \mathcal {C}^{2}$ and s ∈ S, let $\Pr \left( l,r|s\right)$ denote the probability that $\boldsymbol{e}\left( t+1\right)$ is such that the left and right pivots in t + 1 are candidates l and r, respectively, conditional on s(t) = s. Then in any equilibrium, for all $c\in \mathcal {C}$, there exists $\left( \Delta _{c}\left( l,r\right) \right) _{1\le l<r\le C}\in \mathbb {R} ^{\frac{C\left( C+1\right) }{2}}$ such that for s ∈ S,

$$\begin{equation*} d_{c}\left( s\right) =\sum _{1\le l<r\le C}\Pr \left( l,r|s\right) \times \Delta _{c}\left( l,r\right) . \end{equation*}$$

Moreover, Δ_c(l, r) weakly decreases as l moves closer to c from either side, and weakly increases as r moves closer to c from either side.

Proposition 10 states that policy maker c’s distortion is more rightist the closer she expects to be to the next right pivot c _r and the farther she expects to be from the next left pivot c _l. The intuition for this is as follows. As in the basic model, the willingness to distort her vote in favor of R relative to L reflects her preferences over these policies conditional on pivots disagreeing. The more similar she expects to be to the right pivot and the more dissimilar she expects to be to the left pivot, the more likely she is to prefer R whenever the former votes for R and the latter votes for L.

Proposition 10 is derived in a stylized model; hence one has to exercise caution when applying it to the real world. Nevertheless, it is instructive to derive its implications in particular situations to illustrate its difference with the static approach.

Consider, for instance, presidential elections in which a Republican is expected to win with a coattail effect in the concurrent congressional elections. Since the coattail effect usually wanes in midterm elections, the expectations held before the presidential election about the ideology of the pivots right after the election are likely to be more rightist than the expectations held in the first two years of the presidency about the ideology of the pivots after the incoming midterm elections. Proposition 10 implies hence that moderate policy makers—including the legislator who becomes the left pivot during the first two years of the presidency—will vote in a more rightist way during the first two years of the presidency than during the two years before. So a government under Republican presidents may be more successful in replacing liberal status quos (i.e., L) by conservative policies (i.e., R) in her first two years not only because the ideology of the current president and Congress is more conservative, but also because their expectation of liberal shift in the next midterm elections increases their incentives to approve conservative policies today.

Finally, Proposition 10 suggests that the effects of the 1986 Supreme Court decisions on redistricting may be more nuanced than predicted by the static literature. Theoretically, redistricting increases expected ideological polarization of future delegations from the affected states. Suppose that redistricting happens in a conservative state with current representatives being to the right of the current right pivot. The expected right shift of some of its representatives will not affect who the future pivots are, but the shift of the remaining representatives to the left increases the probability that the future right pivot or even both pivots are more leftist candidates (see Shotts (Reference Shotts2003), for a discussion of how redistricting affects national legislature). Proposition 10 implies then that redistricting affects expectations in a way that makes moderate policy makers vote in a more rightist way.