The Model

The Approach

The approach I use to study polarization is adapted from the literature on political competition. Essentially, I extend the concept of equilibria to capture polarized policy. More extreme policies entering the equilibrium space represents polarization in the sense that policies which are attractive to a smaller subset of voters become tenable positions for parties to put forth. This, of course, is notably different from mass polarization. In this framework, polarization can come about without any change in individual voter preferences. Indeed, there are valuable discussions to be had about what/which way/s of studying polarization is/are most appropriate. I argue that policy equilibrium is a valuable approach to studying polarization. Although it might not explain why two neighbors won’t talk to each other due to party identification, it does get at why and which policies are possible in the political arena, including extremely partisan (and hence, polarized) ones. Indeed, the mere possibility, let alone implementation, of extreme policy may have an endogenous effect of making the citizenry more polarized.

I follow very closely John Roemer’s work on party factions and political competition, particularly his book Political Competition: Theory and Applications (2001). In this book (and earlier work), Roemer develops a new equilibrium concept called party-unanimity Nash equilibrium (PUNE). I will go into more detail on Roemer’s concept of PUNEs shortly, but Roemer developed this equilibrium concept in order to escape the nonexistence of normal Downsian or Wittman equilibrium in multiple dimensions and with uncertainty. The PUNE framework introduces sub-party factions to the model and allow for the existence of equilibria even in the multidimensional/uncertainty game. Roemer (2001, 145) states that “Researchers have responded to the nonexistence of Nash equilibrium in pure strategies in the multidimensional game in five ways: The mixed-strategy approach, the sequential game approach, the institutional approach, the uncovered set approach, [and] the cycling approach.” Roemer goes on to explain why each of these approaches is unsatisfactory for studying political competition. Indeed, I agree with his assertion that, if we believe elections are simultaneous move games, then none of the above approaches is satisfactory.

Although the PUNE concept was not designed for unidimensional games of certainty (where Downsian and Wittman equilibria do exist), it certainly still works and thus I adopt the concept. Since the game with factions better represents the reality of how parties are structured, this concept should provide outcomes more similar to what is actually seen in U.S. politics. Additionally, the unidimensional game is simply a simplification of the n-dimensional game, where n is one.

The Set-Up

The set-up of my model also closely follows a number of examples given by Roemer (2001, chaps. 1 and 8). His focus, again, is on multidimensional games (although he considers some unidimensional games) while mine is in one dimension. I leave the game quite general, although I will make a few specifications in order to make the exposition slightly clearer. Thus, while much of the model looks like Roemer’s (2001), there are a few variations for clarity. This game represents parties competing on a policy issue where the distribution of voter types is known. To make the game clearer I choose the policy issue to be a proportional tax rate and the voter types to be their wages. Hence, the game is only made less general in the sense that the policy space is bounded from 0 to 1 and voter types are bounded from 0 to infinity. I assume also that this tax rate is the only issue that voters care about (this is the unidimensionality) and that voters have single-peaked preferences over the tax policy.

Assumptions.

Altogether, I assume the following. Two parties, i=1,2, whose payoffs are a function of any number of things including their probability of winning the election, the policy put into place and/or the policy their party plays. I assume also a unidimensional policy space, over which voters have single-peaked preference orderings. Finally, I assume certainty. That is, I assume that the parties know the distribution of voter preferences (given by a probability function F) perfectly.

Policy.

Again, the policy over which the two parties are competing is a proportional tax rate. Since the tax rate can neither be lower than 0%, nor greater than 100%, t∈[0,1] is the policy space. The tax collected per person is simply a product of the tax rate, t, and that person’s wage, w. If we call the mean of the population wages, μ, then it can be shown that that the average tax revenue per person is tμ. Namely integrating over the wages gives the desired result,

∫_(w=0)^(w=∞)▒〖twdF(w)=tμ〗 (c.f. Ortuño Ortín and Roemer 2000, 8).

Actors.

The first actor in this game is the voter. Voters have a direct utility function that captures their utility from both a private good (x) and a public good (G). This utility function is

u(x,G)=x+αG^β, where α>0 and 0<β<1

(c.f. Ortuño Ortín and Roemer 2000, 8). The private good in this case, is an individual’s wage, w. Assuming that the government distributes the tax revenue equally in the form of the public good, then G=tμ. Hence, the voter’s indirect utility function is

v(t,w)=(1-t)w+α(tμ)^β

(c.f. Ortuño Ortín and Roemer 2000, 8). Now, the voter’s ideal policy (t_w^*, indexed for this voter’s wage) can be solved for by taking a partial derivative with respect to t, which yields,

t_w^*=min[(w/〖αβμ〗^β )^(1/(β-1)), 1]

(c.f. Roemer 2001, 14).

Other key groups of actors in this game are the factions. Again, these are the factions that Roemer (2001, chap. 8) lays out. The first faction is the “Opportunists.” The members of this faction are the same characters that appear in the Downs’ An Economic Theory of Democracy (1951). Their only concern is maximizing the probability that their party wins. They are solely interested in taking office and they have no concern for policy. Let (t_1,t_2) be a pair of policies proposed by party one and party two and (s_1,s_2) be another pair of different policies. Then, for Opportunists,

(t_1, t_2,)≽(s_1, s_2 )⇔π(t_1, t_2 )≥π(s_1, s_2 ),

where π(t_1,t_2 ) is the probability party one wins playing t_1 against t_2 and π(s_1,s_2 ) is the probability part one wins playing s_1 versus s_2 (and vice-versa for party two) (Roemer 2001, 148).

Again, continuing to follow Roemer, the second faction is the “Reformists.” These are the characters of Wittman’s “Parties as Utility Maximizers” (1973). The members of this faction care only about the policy ultimately implemented by the winning party; the election/holding office is simply a means to a policy end. Of course, to get to that end, the party does need to win the election. Hence, the Reformists maximize the party’s expected utility and for two different policy pairs,

(t_1, t_2,)≽(s_1, s_2 )⇔π(t_1, t_2 ) v_1 (t_1 )+(1-π(t_1, t_2 )) v_1 (t_2 )≥π(s_1, s_2 ) v_1 (s_1 )+(1-π(s_1, s_2 )) v_1 (s_2 )

(Roemer 2001, 148).

The third and final faction is that of the “Militants.” This faction will be the most important in the games and modifications presented shortly. The militant group is only interested in the policy their party proposes. That is, they do not care about winning office (as the opportunists do) or the policy actually enacted by the winning party (as the reformists do). Militants can be thought of as hardliners or purists; their utility comes entirely from a “pure” policy proposal from their party. Roemer (2001, 148) says that militants are “interested in publicity.” The party adopting their ideal policy position acts as a sort of advertisement for that ideological stance. The idea is that, perhaps, putting forth this policy can convince some voters to shift their preferences to militants’. Since militants care only about their own party’s position, the policy proposed by the other party has no bearing on their preference ordering, and hence

(t_1, t_2 )≽(s_1, s_2 )⇔v_1 (t_1 )≥v_1 (s_1)

(Roemer 2001, 148). It is hard to say what exactly is meant when a faction is referred to outside this framework. For example, is the Freedom Caucus in the House a Republican reformist or militant faction? I would argue that their behavior makes them appear closer to a militant faction than a reformist faction, but arguments could also be made that they are a reformist group. Ultimately, most examples of real-life factions will likely have strains of the different types listed above.

A party then, is simply a composition of the three factions. The party composition can be altered depending on which model is to be examined. For example, a game where the parties are made up only of opportunists is the typical Downsian game and a game where the parties are made up only of the reformists is a Wittman game. In these cases, the parties simply take on the functions given above. In the game with factions, the parties do not have a single function, but rather, take into account the different factions in order to create an ordering over policies. So, a party is said to (weakly) prefer (t_1,t_2 ) to (s_1,s_2) if and only if all factions weakly prefer (t_1,t_2 ) to (s_1,s_2) (and for strict preference, at least one faction must strictly prefer (t_1,t_2 ) to (s_1,s_2); Roemer 2001, 149). Due to this construction, the parties’ preference orderings (call ∏_i▒ where i indexes the parties) will be incomplete. There will be many policy pairs where two factions prefer one policy, while the other faction prefers the other policy (Roemer 2001, 149).

Party-Unanimity Nash Equilibrium (PUNE)

The concept of a “party-unanimity Nash equilibrium” (PUNE) will be vitally important to the games presented in the next section and so I define the concept. The definition of a PUNE for a pair of policies (t_1,t_2 ) is that the policy pair be a Nash equilibrium for the game with ∏_1▒ and ∏_2▒ and T. In other words (t_1,t_2)∏_1▒(s,t_2 ) (read: Party 1 prefers (t_1,t_2 ) to (s,t_2 )) and (t_1,t_2)∏_2▒〖(t_1,s)〗 where s∈T (Roemer 2001, 149). Essentially, a policy pair satisfies the PUNE criteria if (and only if) neither party can unanimously agree to alter its proposal, holding the other party’s proposal fixed. To deviate from a given policy pair, one party’s factions must be at least indifferent between the old policy and the deviation, and one faction must prefer the deviation (again, holding the other party’s proposal fixed). If this condition holds, then (and only then) will the party deviate, and hence the previous policy pair was not a PUNE (Roemer 2001, 149).

The Games

With the preliminaries now in place, different games with varying factions can examined using the PUNE concept. In this section I set up a few different games (where parties are arrayed in different spatial manners) and I find different equilibria. I also examine what happens when different factional groups emerge within a party in the different games.

Types of Games to be Analyzed.

For each individual game, I will solve for three different specifications. First, I will find the Downs equilibrium, which is the game where both parties are made up only of opportunists, and thus,

∏_1▒〖(t_1,t_2 )=π(t_1, t_2 ) 〗and ∏_2▒〖(t_1, t_2 〗)=1-π(t_1,t_2).

The Downs equilibrium will appear as a red diamond in the figures. The second specification I will solve for is the Wittman game. This is the game where parties are made up only of reformists, and thus,

∏_1▒〖(t_1,t_2 )=π(t_1, t_2 ) v_1 (t_1 )+(1-π(t_1, t_2 )) v_1 (t_2 ) 〗 and ∏_2▒〖(t_1, t_2 〗)=π(t_1, t_2 ) v_2 (t_1 )+(1-π(t_1, t_2 )) v_2 (t_2 ).

Wittman equilibria will appear as a green square. Where there is a continuum of Wittman equilibria, a green line will connect the endpoints, denoted with green squares. Finally, I will consider the game where there are intra-party factions. In this game I will solve for the PUNEs (recall, parties do not have preference functions, but rather incomplete preference relations). PUNEs will always be a continuum in the unidimensional case. The continuum’s endpoints will be denoted with a purple circle, and the rest of the continuum will be connected with a line.