To appreciate the contribution of the probabilistic voting model, it is useful to reconsider the source of nonexistence problems with nonsingle-peaked preferences. The source of the problem is illustrated in (4.2), which links the probability of winning an election for a party to the preferences of the median voter, when preferences are single-peaked. We repeat this equation as specifying the probability that party A offering platform qA will win against party B offering policy qB:
(12.1)
where M denotes the median voter. The important feature of this equation is that the probability that party A wins is a discontinuous function of its policy; as qA varies, this probability jumps from 0 to 1/2 and then to 1. To illustrate the reason, suppose that the policy vector in question, q, is unidimensional and that the median voter M’s preferences are single-peaked, with his or her most preferred policy denoted by qM. Then, when the two parties offer the policies qA and qB such that qA = qB + ε < qM, where ε is a small positive number (in the limit, infinitesimally small). The median voter prefers party A, which is offering a policy closer to his or her preferred point. Now imagine that party B changes its policy by a small amount, increasing it by 2ε. This causes the median voter to prefer party B and because the party that attracts the median voter wins the election, this change in policy causes a discontinuous change in P(qA, qB) from 1 to 0.
To guarantee the existence of pure strategy, Nash equilibria requires continuity of payoff functions in all strategies (as well as strategy sets to be bounded, closed, and convex and the payoff functions to be quasiconcave in their own strategies; e.g., Fudenberg and Tirole 1991, Theorem 1.2, p. 34). As this discussion illustrates, the Downsian party-competition model does not satisfy these assumptions. Nevertheless, discontinuities do not necessarily lead to nonexistence, but they do imply that we cannot establish existence under general conditions. In fact, as the analysis in Chapter 4 established, with single-peaked preferences the Downsian model generates a unique equilibrium (even though the objective functions of the political parties are not continuous). This demonstrates that continuity is sufficient to guarantee the existence of an equilibrium, but it is not necessary — an equilibrium can exist even if behavior is discontinuous. However, the discontinuity of the objective functions leads to nonexistence when preferences are not single-peaked or the policy space is multidimensional.
How can we ensure the existence of an equilibrium? One way is to smooth out the discontinuities in the payoff functions - in this context, the probability that party A wins the election, P(qA, qB). This is what the probabilistic voting approach does.
The idea of the probabilistic voting approach is that an equation like (12.1) should apply at the individual level (for individual voting decisions) but because of heterogeneities at the individual level and random shocks to preferences, the probability that party A wins the election should be a smooth function of its platform. Specifically, let pi(qA, qB) be the probability that individual i votes for party A offering policy qA rather than party B offering policy qB. This is given by the following equation, similar to (12.1):
(12.2)
Why would P(qA,qB) differ from pi(qA, qB)? The most common approach in the literature is to presume that there are some nonpolicy-related reasons for uncertainty in individuals’ preferences (either related to “ideology” or to the “valance” of the politicians), so that individual voters have slightly different preferences (e.g., Lindbeck and Weibull 1987; Coughlin 1992; Persson and Tabellini 2000). As a result, when aggregated over individuals, P(qA, qB) will be a smooth function of policy platforms, and a small change in policy only gets a small response in terms of aggregate voting behavior. This is the approach we develop next. Our particular interest in this model is not only for the technical reason that an equilibrium may exist where otherwise it would not, but also because the probabilistic voting model incorporates different ideas about who has power in a democracy.
2.2 Probabilistic Voting and Swing Voters
Let the society consist of N distinct groups of voters (i.e., all voters within a group have the same economic characteristics). Examples would be the rich and the poor in the two-class model, or the rich, the middle class, and the poor in the three-class model.
There is electoral competition between two parties, A and B, and letbe the fraction of voters in group n voting for party j where j = A, B, and let λn be the share of total voters in group n and, naturally,1 Xn = 1. Then, the expected vote share of party j is
Under Downsian electoral competition, because all voters in n have the same economic preferences,is given by (12.2), and jumps discontinuously from 0 to 1 because voters in group n always vote with certainty for the party that promises the policy that they prefer more. As summarized in Proposition 4.2, this type of Downsian electoral competition leads to the policy most preferred by the median voter. We now see how different outcomes emerge when ideological differences are incorporated in voting behavior.
Instead, imagine that an individual i in group n has the following preferences:
(12.3)
when party j comes to power, where q is a vector of economic policies chosen by the party in power. Assume that q ∈ Q ⊂IIBS so that q is an S-dimensional vector. Here, Vn(q) is the indirect utility of agents in group n as before and captures their economic interests. All individuals in a particular group have the same Vn(q). In addition, the termcan be interpreted as nonpolicy-related benefits that the individual receives from party j. The most obvious source of these preferences would be ideological. So, this model allows individuals within the same economic group to have different ideological or idiosyncratic preferences.
Now defining the difference between the two parties’ ideological benefits for individual i in group n bythe voting behavior of individual i can be represented by an equation similar to (12.2):
(12.4)
Because this equation makes it clear that all that matters is the difference between the two ideological benefits, we work directly withni. Let the distribution of this differential benefitni within group n be given by the smooth cumulative distribution function Fn defined over ( — ∞, +∞), with the associated probability density function fn. Then, (12.4) immediately implies:
(12.5)
Furthermore, and somewhat differently from before, suppose that parties maximize their expected vote share.23 In this case, party A sets this policy platform qA to maximize:
(12.6)
Party B faces a symmetric problem, which can be thought of as minimizing πA. Equilibrium policies then are determined as the Nash equilibrium of a game in which both parties make simultaneous policy announcements to maximize their vote share.
We first look at the first-order condition of party A with respect to its own policy choice, qA, taking the policy choices of the other party, qB, as given. This requires:
where V V“ (qA) denotes the gradient vector of the function V” (qA); that is,
and the superscript T denotes the transpose of the vector ∇ Vn (qA) - So, in other words, the derivative of the vote share in (12.6) needs to be equal to zero with respect to each component of the policy vector q.
This first-order condition characterizes a maximum when the second-order condition is also satisfied. The second-order sufficient condition is for the matrix:
(12.7)
to be negative definite, in which ∇24 Vn (qA) denotes the Hessian of the function Vn (qA) evaluated at the policy vector, qA.
This condition is satisfied if voter utilities are concave functions of platforms, so that ∇24 Vn (qA) is negative definite and the density of ideological differences is not increasing sharply - or, specifically, if it is similar to a uniform distribution. Although ensuring that the second-order conditions hold in general is difficult, here, we follow the literature on probabilistic voting and assume that they do.
Because the problem of party B is symmetric, it also promises the same policy; hence, in equilibrium, we have policy convergence with qA = qB .24 Therefore, Vn(qA)
= Vn(qB) and equilibrium policies, announced by both parties, are given by:
(12.8)
Equation (12.8), which gives equilibrium policies, also corresponds to the solution to the maximization of the following weighted utilitarian social-welfare function:
(12.9)
where
are the weights that different groups receive in the social-welfare function. We state this result as the following proposition for future reference:
Proposition A. 1. (Probabilistic Voting Theorem): Consider a set of policy choices Q, let q EQ ⊂ RSbe a policy vector, and let preferences be given by (12.3) as a function of policy and which party is in power, with the distribution function of nibeing F”. Then, equilibrium policy if it exists is given by q* that maximizes the weighted utilitarian social-welfare function (12.9).
There are two features worth emphasizing here. First, an equilibrium exists as long as the second-order conditions in (12.7) are satisfied; we do not need single-peaked preferences and now the policy space, Q, can be a subset ofs for S > 1, no longer necessarily unidimensional. Therefore, the probabilistic voting model partially avoids the nonexistence problems associated with either the failure of single-peakedness or the multidimensionality of policy spaces. This is a result of the smoothing of the individual-level discontinuities by aggregation.
Second, and more important, this model gives us a way to parameterize the different political power of various groups. If the fn(0)‘s, the density of ideological biases between parties’ at the point where both parties’ platforms give the same utility (i.e., at Vn(qA) = Vn(qB)) are identical across groups, (12.9) becomes exactly the utilitarian social-welfare function. The actual equilibrium in this political economy game differs from the maximization of this utilitarian social welfare function because different groups have different sensitivities to policy. For example, imagine two groups n and n’ such that n is more “ideological,” meaning that there are individuals in this group with strong preferences toward party A or party B. This corresponds to the distribution function Fn having a relatively large amount of weight in the tails. In contrast, imagine that group n’ is not very ideological and the majority of the group votes for the party that gives them slightly better economic policies. This corresponds to having relatively little weight in the tails of Fn and, therefore, a significant value of fn’ (0). In this case, voters from group n’ become the “swing voters” receiving more weight in the political competition game because they are more responsive to changes in policies. Intuitively, tilting policies in favor of groups that are more likely to be responsive to policies (rather than ideological issues) is more attractive to the parties as a strategy for winning votes, so in the political equilibrium, policies are more responsive to the swing group’s preferences.
This discussion has immediate implications for our two-class workhorse model. Although the poor are more numerous, it does not follow that political parties offer a policy platform that is the ideal point of the poor because in the probabilistic voting model, it is not just “mere numbers” that count. When there is ideology, what also matters is how willing voters are to switch their allegiance from one party to the other. This typically means that political parties consider the preferences of the rich as captured by our reduced-form model in the text where the political process maximized a weighted utilitarian social-welfare function similar to (12.9). In this context, we can also think of changes in the weight of the rich χr (or with the microfoundations here fr (0)) affecting how redistributive democratic politics will be.
3. Lobbying
The models discussed so far allow only the votes of the citizens to affect policies. In practice, different groups, especially those that can organize as a lobby, make campaign contributions or pay money to politicians to induce them to adopt a policy that they prefer. In this section, we develop a simple lobbying model and investigate how this affects the determination of equilibrium policies.
With lobbying, political power comes not only from voting but also from other sources, including whether various groups are organized, how many resources they have available, and their marginal willingness to pay for changes in different policies. The most important result is that even with lobbying, equilibrium policies look like the solution to a weighted utilitarian social-welfare maximization problem.
We now develop a baseline model of lobbying from Grossman and Helpman (1996,2001). Imagine again that there are N groups of agents, each with the same economic preferences. The utility of an agent in group n , when the policy q is implemented, is equal to:
where V” (q) is the usual indirect utility function and γn (q) is the per-person lobbying contribution from group n. We allow these contributions to be a function of the policy implemented by the politician; to emphasize this, it is written with q as an explicit argument.
To obtain sharp results, we now abstract entirely from electoral politics and assume that there is already a politician in power. Suppose that this politician has a utility function of the following from:
(12.10)
where, as before, λn is the share of group n in the population. The first term in (12.10) is the monetary receipts of the politician and the second term is utilitarian aggregate welfare. Therefore, the parameter a determines how much the politician cares about aggregate welfare. When a = 0, he or she only cares about money; when a → ∞, he or she acts as a utilitarian social planner. One reason that politicians might care about aggregate welfare is because of electoral politics; for example, in the last subsection, the vote share that he or she receives might depend on the welfare of each group (Grossman and Helpman 1996).
Now consider the problem of an individual i in group n. By contributing some money, he or she might be able to sway the politician to adopt a policy more favorable to his or her group. But he or she is one of many members in his or her group, and there is the natural free-rider problem associated with any type of collective action (see Chapter 5). Consequently, he or she might let others make the contribution and simply enjoy the benefits. This is the typical outcome if groups are unorganized (e.g., there is no effective organization coordinating their lobbying activity and excluding noncontributing members from some of the benefits). On the other hand, organized groups might be able to collect contributions from their members to maximize group welfare.
We think that of the N groups of agents, L < N of those are organized as lobbies and can collect money among their members to further the interests of the group. The remaining N — L are unorganized and make no contributions. Without loss of any generality, let us rank the groups such that groups n = 1, ... , L are the organized ones.
The lobbying game takes the following form: every organized lobby n simultaneously offers a schedule γn(q) > 0, which denotes the payments they would make to the politician when policy q is adopted. After observing the schedules, the politician chooses q. The important assumption is that contributions to politicians (i.e., campaign contributions or bribes) can be conditioned on the actual policy that is implemented by the politicians. This assumption may be a good approximation to reality in some situations but in others, lobbies might simply have to make upfront contributions and hope that they help the parties that are expected to implement policies favorable to them to get elected.
This is a potentially complex game because various different agents (here, lobbies) are choosing functions (rather than scalars or vectors). Nevertheless, noticing the fact that this looks like an auction model along the lines of the work by Bernheim and Whinston (1986), it can be shown that the equilibrium has a simple form.
In particular, the following proposition can be established25:
Proposition A.2 (Grossman-Helpman Lobbying Equilibrium): In the lobbying game described above, contribution functions for groups n = 1, 2 ... L, {n (.)}n=1,2...Land policy q*constitute a subgame perfect Nash equilibrium if:
1. n(.) is feasible in the sense that 0 ≤ n(q) ≤ Vi(q).
2. The politician chooses the policy th
at maximizes his welfare; that is,
(12.11)
3. There are no profitable deviations for any lobby, n = 1, 2, .., L; that is,
(12.12)
Although this proposition at first looks complicated, it is quite intuitive. Condition 1 is simply feasibility; negative contributions are not allowed and no group would pay in amounts that would give negative utility.
Condition 2 has to hold in any subgame perfect equilibrium because the politician chooses the policy after the lobbies offer their contribution schedules. This condition simply states that given the lobbies’ contribution schedules, the politician chooses the policy that maximizes his or her objective.
Economic Origins of Dictatorship and Democracy Page 51
