A large political science and political economy literature focuses on such single-peaked preferences. This is because single-peaked preferences generate the famous and powerful MVT, which constitutes a simple way of determining equilibrium policies from the set of individual preferences. In this book, we either follow this practice of assuming single-peaked preferences making use of the MVT or simply focus on a polity that consists of a few different groups (e.g., the rich and the poor) in which it is easy to determine the social choice (see Subsection 4.2). This is because our focus is not on specific democratic institutions that could aggregate preferences in the absence of nonsingle-peaked preferences but rather some general implications of democratic politics.
3.2 The Median Voter Theorem
Let’s now move to an analysis of the MVT, originated by Black (1948). We can use the restrictions on preferences to show that individual preferences can be aggregated into a social choice. The MVT tells us not only that such a choice exists but also that the outcome of majority voting in a situation with single-peaked preferences will be the ideal point of the “median voter.” There are various ways to state the MVT. We do this first in a simple model of direct democracy with an open agenda. In a direct democracy, individuals vote directly on pairs of alternatives (some q, q’ ∈ Q); the alternative that gets the most votes is the winner. When there is an open agenda, any individual can propose a new pairwise vote pitting any alternative against the winner from the previous vote.
Proposition 4.1 (The Median Voter Theorem): Consider a set of policy choices Q ⊂ R; let q ∈ Q be a policy and let M be the median voter with ideal point qM. If all individuals have single-peaked preferences over Q, then (1) qMalways defeats any other alternative q’ ∈ Q with q’ ≠ qMin a pairwise vote; (2) qMis the winner in a direct democracy with an open agenda.
To see the argument behind this theorem, imagine the individuals are voting in a contest between qM and some policy q > qM. Because preferences are single-peaked, all individuals who have ideal points less than qM strictly prefer qM to. This follows because indirect utility functions fall monotonically as we move away from the ideal points of individuals. In this case, because the median voter prefers qM to, this individual plus all the people with ideal points smaller than qM constitute a majority, so qM defeats q in a pairwise vote. This argument is easily applied to show that any q where q < qM is defeated by qM (now all individuals with ideal points greater than qM vote against). Using this type of reasoning, we can see that the policy that wins in a direct democracy must be qM - this is the ideal point of the median voter who clearly has an incentive to propose this policy.
Why does this work? When citizens have single-peaked preferences and the collective choice is one-dimensional, despite the fact that individuals’ preferences differ, a determinate collective choice arises. Intuitively, this is because people can be separated into those who want more q and those who want less, and these groups are just balanced by the median voter. Preferences can be aggregated into a decision because people who prefer levels of q less than qM have nothing in common with people who prefer levels of q greater than qM. Therefore, no subset of people who prefer low q can ever get together with a subset of those who prefer high q to constitute an alternative majority. It is these “peripheral” majorities that prevent determinate social choices in general, and they cannot form with single-peaked preferences.
The MVT, therefore, makes sharp predictions about which policies win when preferences are single-peaked, and society is a direct democracy with an open agenda.
It is useful at this point to think of the model underlying Proposition 4.1 as an extensive form game. There are three elements in such a game (Osborne and Rubinstein 1994, pp. 89-90): (1) the set of players - here, the n individuals; (2) the description of the game tree that determines which players play when and what actions are available to them at each node of the tree when they have to make a choice; and (3) the preferences of individuals here captured by Vi(q). (In game theory, preferences and utility functions are often called payoffs and payoff functions; we use this terminology interchangeably.) A player chooses a strategy to maximize this function where a strategy is a function that determines which action to take at every node in which a player has to make a decision.12 A strategy here is simply how to vote in different pairwise comparisons. The basic solution concept for such a game is a Nash equilibrium, which is a set of n strategies, one for each player, such that no player can increase his payoff by unilaterally changing strategy. Another way to say this is that players’ strategies have to be mutual best responses. We also extensively use a refinement of Nash equilibrium - the concept of subgame perfect Nash equilibrium - in which players’ strategies have to be mutual best responses on every proper subgame, not just the whole game. (The relationship between these two concepts is discussed in Chapter 5.) Nevertheless, compared to the models we now discuss, the assumption of open agenda makes it difficult to write down the game more carefully. To do this, we would have to be more specific about who could propose which alternatives and when and how they make those decisions.
3.3 Downsian Party Competition and Policy Convergence
The previous example was based on a direct democracy, an institutional setting in which individuals directly vote over policies. In practice, most democratic societies are better approximated by representative democracy, where individuals vote for parties in elections and the winner of the election then implements policies. What does the MVT imply for party platforms?
To answer this question, imagine a society with two parties competing for an election by offering one-dimensional policies. Individuals vote for parties, and the policy promised by the winning party is implemented. The two parties care only about coming to office. This is essentially the model considered in the seminal study by Downs (1957), although his argument was anticipated to a large degree by Hotelling (1929).
How will the voters vote? They anticipate that whichever party comes to power, their promised policy will be implemented. So, imagine a situation in which two parties, A and B, are offering two alternative policies (e.g., tax rates) qA ∈ Q and qB ∈ Q - in the sense that they have made a credible commitment to implementing the tax rates qA and qB, respectively. Let P(qA, qB) be the probability that party A wins power when the parties offer the policy platform (qA, qB). Party B, naturally, wins with probability 1 - P(qA, qB). We can now introduce a simple objective function for the parties: each party gets a rent or benefit R > 0 when it comes to power and 0 otherwise. Neither party cares about anything else. More formally, parties choose policy platforms to solve the following pair of maximization problems:
(4.1)
If the majority of the population prefer qA to qB, they will vote for party A and we will have P(qA, qB) = 1. If they prefer qB to qA, they will choose party B and we will have P(qA, qB) = 0. Finally, if the same number of voters prefer one policy to the other, we might think either party is elected with probability 1/2, so that P(qA, qB) = 1/2 (although the exact value of P(qA, qB) in this case is not important for the outcomes that the model predicts).
Because preferences are single-peaked, from Proposition 4.1 we know that whether a majority of voters will prefer tax rate qA or qB depends on the preferences of the median voter. More specifically, let the median voter again be denoted by superscript M; then, Proposition 4.1 immediately implies that if VM(qA) > VM(qB), we will have a majority for party A over party B. The opposite obtains when VM(qA) < VM(qB). Finally, if VM(qA) = VM(qB), one of the parties will come to power with probability 1/2. Therefore, we have
(4.2)
The model we have developed can be analyzed as a game more explicitly than the direct-democracy model of the previous section. This game consists of the following three stages:
1. The two political parties noncooperatively choose their platforms (qA, qB).
2. Individuals vote for the party they prefer.
3. Whichever party wins the election comes to power and implements
the policy it promised at the first stage.
There are n + 2 players in this game: the n citizens with payoff functions Vi(q) and the two political parties with payoff functions given in (4.1). Individual voters do not propose policy platforms, only parties do so simultaneously at the first stage of the game. Parties have to choose an action qj ∈ Q for j = A, B, and citizens again have to vote. Thus, in this model, a subgame perfect Nash equilibrium would be a set of n + 2 strategies, one for each of the political parties and one for each of the n voters, which would determine which policies the parties offered and how individuals would vote. If such a set of strategies constituted an equilibrium, then it would have the property that neither party and no voters could improve their payoff by changing their strategy (e.g., by offering a different policy for parties or voting differently for citizens).
In the present model, however, we can simplify the description of a subgame perfect Nash equilibrium because, given a policy vector (qA, qB) ∈ Q x Q, voters simply vote for the party offering the policy closest to their ideal point and, because preferences are single-peaked, the MVT implies that the winner of such an election is determined by (4.2). Hence, the only interesting strategic interaction is between the parties. More formally, we can solve the game by backward induction. To do this, we begin at the end of the game and work backward. Parties are committed to platforms, so whichever party wins implements the policy it offered in the election. Then (4.2) determines which party wins and, considering this at the initial stage of the game, parties choose policies to maximize (4.1).
This implies that a subgame perfect Nash equilibrium in this game reduces to a pair of policies (,)such thatmaximizes P(qA, R, taking the equilibrium choice of party B as given, and simultaneouslymaximizes (1 - P (, qB))R, taking the equilibrium choice of party A as given. In this case, neither party can improve its payoff by choosing an alternative policy (or, in the language of game theory, by “deviating”).
Formally, the following theorem characterizes the unique subgame perfect Nash equilibrium of this game:
Proposition 4.2 (Downsian Policy Convergence Theorem): Consider a vector of policy choices (qA, qB) EQ x Q where Q ⊂ R, and two parties A and B that care only about coming to office, and can commit to policy platforms. Let M be the median voter, with ideal point qM. If all individuals have single-peaked preferences over Q, then in the unique subgame perfect Nash equilibrium, both parties will choose the platforms == qM.
Stated differently, both parties converge to offer exactly the ideal point of the median voter. To see why there is this type of policy convergence, imagine a configuration in which the two parties offered policies qA and qB such that qA < qB≤ qM. In this case, we have VM(qA) < VM(qB) by the fact that the median voters’ preferences are single-peaked. There will therefore be a clear majority in favor of the policy of party B over party A; hence, P(qA, qB) = 0, and party B will win the election. Clearly, A has an incentive to increase qA to some q E (qB, qM) if qB< qM to win the election, and to q = qM if qB = qM to have the chance of winning the election with probability 1/2. Therefore, a configuration of platforms such that qA < qB≤ qM cannot be an equilibrium. The same argument applies: if qB < qA ≤ qM or if qA > qB ≥ qM, and so forth.
Next, consider a configuration where qA = qB < qM. Could this be an equilibrium? The answer is no: if both parties offer the same policy, then P(qA, qB) = 1/2 (hence, 1 - P(qA, qB) = 1/2 also). But, then, if A increases qA slightly so that qB < qA < qM, then P(qA, qB) = 1. Clearly, the only equilibrium involves qA= qB= qM with P(qA = qM, qB = qM) = ½ (hence, 1 - P(qA = qM, qB = qM) = 1/2). This is an equilibrium because no party can propose an alternative policy (i.e., make a deviation) and increase its probability of winning. For instance, if qA = qB = qM and A changes its policy holding the policy of B fixed, we have P(qA, qB) = 0 < ½ for qA > qM or qA< qM. Therefore, qA = qM is a best response to qB = qM. A similar argument establishes that qB= qM is a best response to qA = qM.
As noted, the MVT does not simply entail the stipulation that people’s preferences are single-peaked. We require that the policy space be unidimensional. In the conditions of Proposition 4.1, we stated that policies must lie in a subset of the real numbers (Q ⊂ R). This is because although the idea of single-peaked preferences extends naturally to higher dimensions of policy, the MVT does not.
Nevertheless, there are various ways to proceed if we want to model situations where collective choices are multidimensional. First, despite Arrow’s theorem, it may be the case that the type of balance of power between conflicting interests that we saw in the MVT occurs also in higher dimensions. For this to be true in general, we need not simply state that preferences be single-peaked but also that the ideal points of voters be distributed in particular ways. Important theorems of this type are the work of Plott (1967) and McKelvey and Schofield (1987) (see Austen-Smith and Banks 1999, Chapter 5, for detailed treatment). There are also ideas related to single-peaked preferences, particularly the idea of value-restricted preferences, that extend to multidimensional policy spaces (e.g., Grandmont 1978). Restrictions of this type allow the sort of “balance of power” that emerges with the MVT to exist with a multidimensional policy space.
Second, once we introduce uncertainty into the model, equilibria often exist even if the policy space is multidimensional. This is the so-called probabilistic voting model (Lindbeck and Weibull 1987; Coughlin 1992; Dixit and Londregan 1996, 1998) analyzed in the appendix to this chapter.
Third, following Osborne and Slivinski (1996) and Besley and Coate (1997), once one assumes that politicians cannot commit to policies, one can establish the existence of equilibrium with many dimensions of policy. Intuitively, when politicians cannot commit to arbitrary policies to build majorities, many possibilities for cycling coalitions are removed.
We refer to the type of political competition in this subsection as Downsian political competition. The key result of this subsection, Proposition 4.2, resulting from this type of competition contains two important implications: (1) policy convergence - that is, both parties choose the same policy platform; and (2) this policy platform coincides with the most preferred policy of the median voter. As we show in the appendix, in non-Downsian models of political competition - for example, with ideological voters or ideological parties - there may still be policy convergence, but this convergence may not be to the most preferred policy of the median voter. There may also be nonconvergence, in which the equilibrium policy is partially determined by the preferences of political parties.
4. Our Workhorse Models
In this section, we introduce some basic models that are used throughout the book. As already explained, our theory of democracy and democratization is based on political and distributional conflict and, in an effort to isolate the major interactions, we use models of pure redistribution, where the proceeds of proportional taxation are redistributed lump sum to the citizens. In addition, the major conflict is between those who lose from redistribution and those who benefit from redistribution-two groups that we often conceptualize as the rich and the poor. Hence, a two-class model consisting of only the rich and the poor is a natural starting point. This model is discussed in the next three subsections. Another advantage of a two-class model is that something analogous to the MVT will hold even if the policy space is multidimensional. This is because the poor are the majority and we restrict the policy space so that no intra-poor conflict can ever emerge. As a consequence, no subset of the poor ever finds it advantageous to form a “peripheral” coalition with the rich. In this case, the policies preferred by the poor win over policies preferred by the rich. In Chapter 8, we extend this model by introducing another group, the middle class, and show how it changes a range of the predictions of the model, including the relationship between inequality and redistribution.
In addition to a model in which political conflict is between the rich and the poor, we want to examine what happens when conflict is based on other political identities. We introduce such a
model in Subsection 4.4.
4.1 The Median Voter Model of Redistributive Politics
We consider a society consisting of an odd number of n citizens (the model we develop builds on the seminal papers of Romer 1975, Roberts 1977, and Meltzer and Richard 1981). Person i = 1, 2, .., n has income yi. Let us order people from poorest to richest and think of the median person as the person with median income, denoted yM. Then, given that we are indexing people according to their incomes, the person with the median income is exactly individual M = (n + 1)/2. Letdenote average income in this society; thus,
(4.3)
The political system determines a nonnegative tax rate τ ≥ 0 proportional to income, the proceeds of which are redistributed lump sum to all citizens. Moreover, this tax rate has to be bounded above by 100 percent - that is, r ≤ 1. Let the resulting lump-sum transfer be T.
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