Simple centrifugal incentives in spatial competition

Abstract

This paper studies the effects of introducing centrifugal incentives in an otherwise standard Downsian model of electoral competition. First, we demonstrate that a symmetric equilibrium is guaranteed to exist when centrifugal incentives are induced by any kind of partial voter participation (such as abstention due to indifference, abstention due to alienation, etc.) and, then, we argue that: (a) this symmetric equilibrium is in pure strategies, and it is hence convergent, only when centrifugal incentives are sufficiently weak on both sides; (b) when centrifugal incentives are strong on both sides (when, for example, a lot of voters abstain when they are sufficiently indifferent between the two candidates) players use mixed strategies—the stronger the centrifugal incentives, the larger the probability weight that players assign to locations near the extremes; and (c) when centrifugal incentives are strong on one side only—say for example only on the right—the support of players’ mixed strategies contain all policies except from those that are sufficiently close to the left extreme.

Appendix

In this appendix we provide some discussion of pure strategy equilibria when the distribution of voters in nonuniform and some microfoundations of the constant functions’ quadruple $(\alpha ,\beta ,\gamma ,\delta )$ that was extensively used throughout the paper.

1.1 A.1: Nonuniform distributions

In the case of a uniform distribution, when $\hat{\alpha }=\hat{\beta }$, the game admits Nash equilibria in pure strategies: $(\frac{1}{2},\frac{1}{2})$, no differentiation, when $\hat{\alpha }>\frac{1}{2}$ and $\left( 0,1\right)$ , maximal differentiation, when $\hat{\alpha }<\frac{1}{2}$. The existence of a local ²¹ Nash equilibrium exhibiting no differentiation when $\hat{\alpha }>\frac{1}{2}$ and differentiation when $\hat{\alpha }<\frac{1}{2}$ continues to hold for a large class of distributions. Consider a distribution described by the density f which will be assumed differentiable, symmetric with respect to $\frac{1}{2}$ (that is, $f(x)=f(1-x)$ for all $x\in \left[ 0,\frac{1}{2}\right]$) and strictly increasing on $\left[ 0,\frac{1}{2}\right]$ (and so strictly decreasing on $\left[ \frac{1}{2},1\right]$). Let (x, y) be a profile of locations such that: $x<y$ . Without loss of generality²² assume that $\frac{x+y}{2}\le \frac{1}{2}$. For the player located on the left, moving on the right leads to a gain of $\frac{\gamma }{2}f(\frac{x+y}{2})$ and to a loss of $\gamma (1- \hat{\alpha })f(x)$. If $\hat{\alpha }\ge \frac{1}{2}$, then the gain is always larger than the cost and the equilibrium is defined by $x=y=\frac{1}{2 }$.²³ If otherwise $\hat{\alpha }<\frac{1}{2}$, a local equilibrium is obtained when the marginal rate of substitution $\frac{\frac{1 }{2}f(\frac{x+y}{2})}{\left( 1-\hat{\alpha })\right) f(x)}$ is equal to 1, that is,

$$\begin{aligned} \frac{1}{2}f\left( \frac{x+y}{2}\right) =\left( 1-\hat{\alpha }\right) f(x) \end{aligned}$$

or

$$\begin{aligned} f(x)=\frac{1}{2\left( 1-\hat{\alpha }\right) }f\left( \frac{x+y}{2}\right) \end{aligned}$$

Let us test when the symmetric profile $\left( x,1-x\right)$ with $x<\frac{1 }{2}$ is a local Nash equilibrium.²⁴ From above, the first order condition writes:

$$\begin{aligned} f(x)=\frac{1}{2\left( 1-\hat{\alpha }\right) }f\left( \frac{1}{2}\right) \end{aligned}$$

If $f(0)=0$, then the above equation has a unique solution $x^{*}$. We may also check that the second order condition is satisfied. Indeed the second derivative at $x^{*}$

$$\begin{aligned} \frac{1}{4}f^{\prime }\left( \frac{1}{2}\right) +(\hat{\alpha }-1)f^{\prime }(x^{*})=( \hat{\alpha }-1)f^{\prime }(x^{*}) \end{aligned}$$

is negative as $\hat{\alpha }<1$ and $f^{\prime }(\frac{1}{2})=0$. For the sake of illustration, consider the (symmetric) Beta distribution:²⁵

$$\begin{aligned} f(x)=\frac{\Gamma (2\xi )}{\Gamma (\xi )^{2}}\left( x\left( 1-x\right) \right) ^{\xi -1}\text { over }\left[ 0,1\right] \end{aligned}$$

with $\xi >1$. In such case, $x^{*}$ is the solution of the equation:

$$\begin{aligned} x(1-x)=\frac{1}{4}\left( \frac{1}{2\left( 1-\hat{\alpha }\right) }\right) ^{ \frac{1}{\xi -1}} \end{aligned}$$

We obtain:

$$\begin{aligned} x^{*}=\frac{1}{2}-\frac{\sqrt{1-\left( \frac{1}{2\left( 1-\hat{\alpha } \right) }\right) ^{\frac{1}{\xi -1}}}}{2} \end{aligned}$$

As we observe differentiation (defined by $1-2x^{*}\left( \hat{\alpha } ,\xi \right)$) is reduced when F is more concentrated around the center (large values of $\xi$) and when the centrifugal incentives are less intense (large values of $\hat{\alpha }$). This is illustrated on Table 1 for $\hat{\alpha }=0$ and on Table 2 for $\hat{\alpha }=\frac{1}{4}$.

Table 1 Concentration of voters around the center and location of the first player, when centrifugal incentives are intense

Table 2 Concentration of voters around the center and location of the first player, when centrifugal incentives are moderate

1.2 A.2: An application

In this section we develop a model of electoral competition among four parties, two instrumental and two parametric ones, and we demonstrate that it is a particular case of the general model studied above. Let $\{l,1,2,r\}$ be the set of political parties and [0, 1] the policy space. The policy platforms of the extremist parties, l and r, are fixed exogenously and are 0 and 1 respectively.²⁶ Parties 1 and 2—which we call mainstream—strategically decide their policy platforms, x and y, in order to maximize their vote-share and voters’ ideal policies are distributed uniformly on the interval [0, 1].

We consider a two-stage extended form game of perfect information and, naturally, the solution concept that we apply is subgame perfection. In the first stage, the two mainstream parties decide their policy platforms and then the voters observe $(x,y)\in \left[ 0,1\right] ^{2}$ and they vote. We allow both pure and mixed strategies for any of our players. Let $(\sigma ^{1},\sigma ^{2})$ denote a pair of mixed strategies for the two mainstream candidates (probability distributions such that their support is a subset of [0, 1]) and $\hat{\sigma }^{i}(x,y)$ denote a mixed strategy of a voter with ideal policy $i\in [0,1]$ (probability distribution such that its support is a subset of $\{l,1,2,r\}$) when the voter observes that the policy platforms of the two mainstream parties are given by $(x,y)\in \left[ 0,1\right] ^{2}$. Then a strategy profile in our game is given by $\sigma =\{(\sigma ^{1},\sigma ^{2}),\hat{\sigma }$ such that $\hat{\sigma }(x,y)\in \hat{\sigma }$ for all $(x,y)\in \left[ 0,1\right] ^{2}\}$ where $\hat{\sigma } (x,y)$ is such that $\hat{\sigma }^{i}(x,y)\in \hat{\sigma }(x,y)$ for every $i\in [0,1]$ and each possible subgame $(x,y)\in \left[ 0,1\right] ^{2}$.

Each extremist/niche party is assumed to promote a cause that is independent of the main policy issue and which voters may find attractive or repulsive.²⁷ Voters’ valuation of a cause promoted by a party is captured by an attraction/repulsion parameter $\phi _{k}\in \mathbb {R}$ for $k\in \{l,1,2,r\}.$ If $\phi _{k}>0$ then the promoted goal is attractive and if $\phi _{k}<0$ it is repulsive (we assume that mainstream parties do not promote such causes and hence $\phi _{1}=\phi _{2}=0$).

Our voters are expressive and sophisticated at the same time; they know that they cannot individually influence the outcome of elections—hence they derive utility only from voting for the party that they prefer—but they are able to take in account expectations about the voting behavior of their fellow citizens whenever this is relevant. We consider that the payoff of a voter with ideal policy $i\in [0,1]$ who votes for a party $k\in \{l,1,2,r\}$ with policy platform $\psi _{k}\in [0,1]$ when we are in subgame $(x,y)\in \left[ 0,1\right] ^{2}$ and all other voters are expected to behave according to $\hat{\sigma }^{-i}(x,y)=\hat{\sigma }(x,y)-\{ \hat{ \sigma }^{i}(x,y)\}$ is given by:

$$\begin{aligned} u_{i}(\psi _{k},\phi _{k},\hat{\sigma }^{-i}(x,y))=-\left| \psi _{k}-i\right| +\phi _{k}\times v_{k}(\hat{\sigma }^{-i}(x,y)) \end{aligned}$$

where $v_{k}(\hat{\sigma }^{-i}(x,y))$ is the expected vote-share of party $k\in \{l,1,2,r\}$ when all other voters are expected to vote according to $\hat{\sigma }^{-i}(x,y)$.

Notice that the component of the above utility function which does not relate to the position of the party on the main political issue is not fixed²⁸ as in Groseclose (2001) or Aragonès and Palfrey (2002) but it also depends on the “electoral power” of each party. This is so because, if the cause that a niche party $k\in \{l,r\}$ promotes is attractive for the voters then the impact of this niche party on government policy outputs—and hence on the voters welfare—will be increasing in the niche party’s vote-share and vice versa.

Consider that we are in subgame $(x,y)\in \left[ 0,1\right] ^{2}$ with $x<y$ and that $\hat{\sigma }(x,y)$ is such that both mainstream parties get a strictly positive vote-share. Then, if $\hat{\sigma }(x,y)$ is a Nash equilibrium of the $(x,y)\in \left[ 0,1\right] ^{2}$ subgame there should exist $i_{l1}\in [0,x]$ such that:²⁹

$$\begin{aligned} u_{i_{l1}}(0,\phi _{l},\hat{\sigma }^{-i_{l1}}(x,y))=u_{i_{l1}}(x,\phi _{1}, \hat{\sigma }^{-i_{l1}}(x,y)) \end{aligned}$$

which is equivalent to

$$\begin{aligned} i_{l1}=\frac{1}{2-\phi _{l}}x \end{aligned}$$

because when F is uniform, $x<y$ and $\hat{\sigma }(x,y)$ is a Nash equilibrium of the $(x,y)\in \left[ 0,1\right] ^{2}$ subgame such that all parties get a strictly positive vote-share, it must be the case that $v_{l}( \hat{\sigma }^{-i_{l1}}(x,y))=i_{l1}$.

Equivalently, one can show that in a Nash equilibrium of the $(x,y)\in \left[ 0,1\right] ^{2}$ subgame in which all parties get positive vote-shares there should also exist $i_{12}\in [x,y]$ and $i_{2r}\in [y,1]$ such that:

$$\begin{aligned} u_{i_{12}}(x,\phi _{1},\hat{\sigma }^{-i_{12}}(x,y))=u_{i_{12}}(y,\phi _{2}, \hat{\sigma }^{-i_{12}}(x,y)) \end{aligned}$$

and

$$\begin{aligned} u_{i_{2r}}(y,\phi _{2},\hat{\sigma }^{-i_{2r}}(x,y))=u_{i_{2r}}(1,\phi _{r}, \hat{\sigma }^{-i_{2r}}(x,y)) \end{aligned}$$

which are equivalent to

$$\begin{aligned} i_{12}=\frac{x+y}{2} \end{aligned}$$

and

$$\begin{aligned} i_{2r}=\frac{1+y-\phi _{r}}{2-\phi _{r}} \end{aligned}$$

It is straightforward that if the attraction parameter of the leftist niche party is sufficiently large ($\phi _{l}\ge 1$) then in any Nash equilibrium of any subgame $(x,y)\in \left[ 0,1\right] ^{2}$, it should be the case that no mainstream party gets any votes and, hence, our game becomes trivial: the mainstream parties are indifferent among all available strategies. This obviously holds for the attraction parameter of the rightist niche party too. Therefore, we focus on attraction parameters which make our game non-trivial: $\phi _{l},\phi _{r}<1$.

If the attraction parameters of the niche parties are small enough, $\phi _{l},\phi _{r}<1$, then for every subgame $(x,y)\in \left[ 0,1\right] ^{2}$ there exists a unique Nash equilibrium such that both mainstream parties get a positive vote-share. The payoff of mainstream party 1 in this Nash equilibrium of any $(x,y)\in \left[ 0,1\right] ^{2}$ subgame is given by:

$$\begin{aligned} \pi _{1}(x,y)=\left\{ \begin{array}{l} x(\frac{1-\phi _{l}}{2-\phi _{l}})+\frac{y-x}{2}\quad \text { if }0\le x<y\le 1 \\ \frac{1}{2}\left( x\frac{1-\phi _{l}}{2-\phi _{l}}+(1-x)\frac{1-\phi _{r}}{2-\phi _{r}}\right) \quad \text { if }x=y \\ (1-x)\frac{1-\phi _{r}}{2-\phi _{r}}+\frac{x-y}{2}\quad \text { if }0\le y<x\le 1 \end{array} \right. \end{aligned}$$

and the payoff of mainstream party 2 is given by:

$$\begin{aligned} \pi _{2}(x,y)=\left\{ \begin{array}{l} y(\frac{1-\phi _{l}}{2-\phi _{l}})+\frac{x-y}{2}\quad \text { if }0\le y<x\le 1 \\ \frac{1}{2}\left( y\frac{1-\phi _{l}}{2-\phi _{l}}+(1-y)\frac{1-\phi _{r}}{2-\phi _{r}}\right) \quad \text { if }y=x \\ (1-y)\frac{1-\phi _{r}}{2-\phi _{r}}+\frac{y-x}{2}\quad \text { if }0\le x<y\le 1 \end{array} \right. \end{aligned}$$

Hence, this two player game is identical to the one we extensively analyzed in the previous section for $\alpha =\frac{1-\phi _{l}}{2-\phi _{l}}$, $\beta =\frac{1-\phi _{r}}{2-\phi _{r}}$ and $\gamma =\delta =1$. When both extremist parties promote repulsive causes $\phi _{l},\phi _{r}<1$ (strong centripetal incentives from both sides) we have a unique convergent equilibrium to the side of the more repulsive extremist party, when one extremist party promotes a repulsive cause and the other extremist party promotes an attractive cause (mixed incentives) then we have a symmetric equilibrium in mixed strategies such that mainstream parties locate (in expected terms) to the side of the extremist party which promotes the repulsive cause and when both extremist parties promote attractive causes, there is a symmetric equilibrium in mixed strategies such that the mainstream parties locate (in expected terms) in the center of the policy space.