Theoretical Extensions: Unbundling Authoritarian Institutions and Their Changes under Xi Jinping

Abstract

This chapter provides an analytic narrative to help us understand both theoretically and empirically the authoritarian institutional changes Xi Jinping has brought to the Chinese regime. Theoretically, the chapter develops a framework for unbundling the authoritarian institutions—power-sharing and co-opting institutions—and understanding their interactions. Its theoretical predictions are then examined through narratives on Xi’s institutional endeavors to tame the growth curse and marshal social support for the regime through co-optation. In a broader sense, the findings also tell the students of comparative authoritarianism how authoritarian regimes evolve. Combining both analyses on Xi’s restructuring of China’s elite politics and the new initiative in social co-optation completes my depiction of the institutional trajectory Xi’s political maneuvering has taken.

Appendix: Derivations and Proofs

1.1 Proof for Lemma 8.1.1

The result in Lemma 8.1.1 is directly adopted from Boix and Svolik (2013). The following proof is provided to clarify the relationship between the result and the global games approach.

To prove that Eq. (8.6) is a Bayesian Nash equilibrium, we now introduce a global coordination game between two political elites. Suppose now there are two allies who try to decide independently if they should accept the offer from the challenger and rebel, or stay with the dictator. However, while both of them might be equally interested, as we have argued above, the dictator’s allies face great uncertainties in the payoffs and therefore they don’t necessarily share the same perceptions of the payoff structure. As a result, the nature of the issue at hand implies that the standard coordination game with complete information (about the payoffs) does not apply here. To incorporate the feature of uncertain payoffs, we specify the payoff structure for both allies in the following way (for simplicity, the subscript $t$ is removed here):

Table 8.1 Payoff structure for a rebellion

where ${f}^C$ is the counter-offer made by the challenger. Given the payoff structure in Table 8.1, there could be three sets of equilibrium scenarios as shown in Fig. 8.5.

Fig. 8.5

Two-sided limit dominance

There are two ranges of values for $\omega {f}^C$ in which unique dominant strategies exist. When $\omega {f}^C<0$, (Not Rebel, Not Rebel) is the unique dominant strategy. On the contrary, (Rebel, Rebel) becomes the unique dominant strategy when $\omega {f}^C>1$. What is worth noting here is that, in the language of global games, the structure of equilibria in Fig. 8.5 exhibits the two-sided limit dominance, and this property is sometimes replaced by its variant, the one-sided limit dominance, in some applications. Specifically, as Bueno de Mesquita (2010) argues in his analysis of revolutionary entrepreneurs, there is no value of $\omega {f}^C$ that can make joining a revolution a dominant strategy. In other words, in the world of revolution, there is no situation in which an actor would want to participate in a revolution no matter what decisions other players make, while there could be a range of values for $\omega {f}^C$ that make inaction a unique dominant strategy. This is why the limit dominance has only “one side.” While this chapter also deals with contentious politics, we stick to the convention in the global games literature with a game structure exhibiting two-sided limit dominance. The main justification is that the barrier for a coup to take place is far lower than that for a revolution, and therefore, when the benefit from joining a rebellion is high enough (taking into account the cost of failing), one would take initiative regardless of whatever action the other player might take. When $0<\omega {f}^C<1$, however, the game of Table 8.1 becomes a standard coordination game that has multiple equilibria (i.e., (Not Rebel, Not Rebel) and (Rebel, Rebel)). This is the case where the global games approach can help us find an equilibrium.

When both players decide to rebel, they both receive the payoff of $\omega {f}^C,$ which is not directly observable to both of them since the regime strength is opaque. While players do not directly observe $\omega,$ they receive private signals about it:

$$ {x}^i=\omega +{\varepsilon}^i\kern1.75em i=1,2 $$

(8.28)

where $\varepsilon \sim \mathcal{N}\left(0,\kern0.5em {\sigma}^2\right)$. Ally $i$ holds an improper prior that $\omega$ is uniformly distributed on the real line. We can therefore compute player $i$’s posterior belief about $\omega$ after she receives the private signal, $\omega \mid {x}^i\sim \mathcal{N}\left({x}^i,\kern0.5em {\sigma}^2\right)$. We can then figure out player $i$’s higher-order beliefs–that is, her belief about the other player’s belief about $\omega,$ the other player’s belief about hers, and so on. Player $i$’s higher-order belief about player $j$’s private signal given hers is simply ${x}^j\mid {x}^i\sim \mathcal{N}\left({x}^j,2{\sigma}^2\right)$.

Now suppose that player $i$ who receives a private signal ${x}_i$ believes that player $j$ uses a switching strategy around ${x}^i,$

$$ s(x)=\Big\{{\displaystyle \begin{array}{ll}\mathbf{Rebel}& \mathbf{if}\ {x}^i>{x}_t^{\ast}\\ {}\mathbf{Not}\ \mathbf{Rebel}& \mathbf{if}\ {x}^i\le {x}_t^{\ast}\end{array}} $$

(8.29)

Since we have derived the distribution of $\left({x}_j|{x}_i\right)$ above, we know that the probability for player $j$ not to rebel is $\mathit{\Pr}\left({x}_j\le k\right)=\varPhi \left(\frac{k-{x}_1}{\sqrt{2}\sigma}\right).$

We then proceed to compute the expected payoffs for an elite $i$ given her own private signal and another elite $j$’s choices. Elite $i$’s expected payoff of “Rebel” is

$$ {\displaystyle \begin{array}{c}E{U}_i\left[\mathbf{Rebel}|{x}_i,\kern0.5em {s}_j\right]=E\left[\theta Pr\left({x}_j>k\right)+\left(\theta -1\right)\mathit{\Pr}\left({x}_j\le k\right)\right]\\ {}={x}_i-\varPhi \left(\frac{k-{x}_i}{\sqrt{2}\sigma}\right).\end{array}} $$

(8.30)

Her expected payoff of “Not Rebel” is as follows:

$$ E{U}_i\left[\mathbf{Not}\ \mathbf{Rebel}|{x}_i,{s}_j\right]=E\left[0\mathit{\Pr}\left({x}_j>k\right)+0\mathit{\Pr}\left({x}_j\le k\right)\right]=0 $$

(8.31)

Now since we have obtained player $i$’s expected payoffs for both actions given $j$’s choices, we are able to write $i$’s best responses as follows:

$$ \Big\{{\displaystyle \begin{array}{ll}\mathbf{Rebel}& \mathbf{if}\ x-\varPhi \left(\frac{k-x}{\sqrt{2}\sigma}\right)>0\\ {}\mathbf{Not}\ \mathbf{Rebel}& \mathbf{if}\ x-\varPhi \left(\frac{k-x}{\sqrt{2}\sigma}\right)\le 0\end{array}} $$

(8.32)

$x-\varPhi \left(\frac{k-x}{\sqrt{2}\sigma}\right)$ is strictly increasing in $x.$ This implies that we can solve for a unique solution of $x$ when this term is set to zero. With two unknowns ($x$ and $k$) in this equation, the solution $x$ will be a function $k$ and is denoted by $b(k).$

We can then write player $i$’s best response as follows:

$$ \Big\{{\displaystyle \begin{array}{ll}\mathbf{Rebel}& \mathbf{if}\ x>b(k)\\ {}\mathbf{Not}\ \mathbf{Rebel}& \mathbf{if}\ x\le b(k)\end{array}} $$

(8.33)

In other words, a best response to a switching strategy around $k$ is equivalent to a switching strategy around $b(k).$ The key idea here is to derive an expression for player $i$’s best response to the switching strategy played by the other player. Based on the distributional assumption, the derivation above gives us player $i$’s best response as a solution to his payoff function conditioned on the other’s actions.

1.2 A More General Case for the Public Goods Provision

For a more general case,

$$ {g}_t=\Big\{{\displaystyle \begin{array}{l}\begin{array}{ll}\varOmega +{\chi}_t\left[\frac{\sum_Jy\left({\kappa}_t^J\right)}{2}\right]-{m}_t& \mathbf{if}\ {\lambda}_t{H}_g\left(\varOmega +{\chi}_t\left[\frac{\sum_Jy\left({\kappa}_t^J\right)}{2}\right]-{m}_t\right)\ge 2\left(1-\beta \right)\\ {}0& \mathbf{if}\ {\lambda}_t{H}_g(0)<2\left(1-b\right)\\ {}g\left({\lambda}_t,2\left(1-b\right)\right)& \mathbf{otherwise}\end{array}\\ {}\end{array}} $$

(8.34)

The $2\left(1-b\right)$ in the interior solution is obtained by setting the first-order condition Eq. (8.15) to zero and is the marginal value of public goods modified by ${\lambda}_t.$ When ${H}_g\left({g}_t\right)=1,$ both corner solutions have exhausted all possible values ${g}_t$ can take.