The Bargaining Context of General Institution Reform

Abstract

In essence, the political behavior of human society is similar to a two-person game in a strict sense, because in most contexts, there is no third way for the generalized political behavior to choose. In view of the theoretical needs of generalization and classification, the strict two-person game actually rules out the effect of the competitive factor of the market sense, in which people have to always subject themselves to a simple choice of “yes or no” logic rather than the diversity selection.

Appendix: Definition of the Bargaining Game of General Institution Reform

Next, according to the characteristics of the two-person bargaining of political behavior and the institution expression argued above, we will give the definition of institution reform of bargaining in the sense of the book and its mathematical structure based on the context of the generalized transaction.

Institution reform refers to a social bargaining process of the two-person game. Its relationship with institution concept itself is: the status quo of the bargaining of institution reform is the initial institution, and its result is the equilibrium of the bargaining of reform and emerges as a new institution. As a dynamic description of institution, here, institution reform mainly emphasizes the process and occurrence mechanism of institution evolution. According to Definition 5.1 of Chapter 5, social institution could be taken as a set of people’s social behavior norms and expressed as a 2D topology $G_{o} \left( {n,X,\xi } \right)$. In the standard two-person game context of the chapter, it means the workers and the farmers in the social member set $i \in \left\{ {1,2} \right\}$ will obtain the generalized benefits (or punishment) $\xi_{i} (x_{i} ) \in R^{1}$ when being engaged in the social behavior $x_{i} \in X$, $(i = 1,2)$ (or we could also write it as $u_{i} = \xi_{i} \in R^{1}$, $i = 1,2$).

As a series of contracts or norms, social institution gives people’s current social behaviors corresponding social value through the inherited or realistic public rights. The economic calculation of the social values is the generalized benefit of the book. According to Definition 5.2 and Eqs. (5.1) and (5.2), the social behavior set X is a convex set in which 1 (workers) is symmetry to 2 (farmers). In association with the initial institution of the status quo, here, the concept of convex implies the social norm $G_{o} :X \to R^{1}$ has the mutual correlation of social behaviors, which can be properly understood from the concept of generalized benefit by the example of the political philosophy. For the given behavior $x_{i} \in X$ of any party, the corresponding benefits of two parties are complementary. That is to say, the sum of the generalized benefits of two parties is a certain constant. If we simply written it as the integer 1, there will be $\xi_{1} (x_{i} ) + \xi_{2} (x_{i} ) = 1$ $(\xi_{i} \in [0,1],i = 1,2)$. At this time, the social value about “absolutely equality” is essentially that $\forall \,x_{i} \in X,i = 1,2$, social institution $G_{o} :x_{i} \to \xi_{i} \in R^{1}$ will ensure $\xi_{1} = \xi_{2} = 1/2$. And the social value of the efficiency is obviously contained in the equilibrium once the bargaining deals.

According to the rational nature of generalized transaction, a bargaining of institution reform from $G_{o}$ to $G_{n}$ caused by the realistic demand can be regarded as a process of complete information within the scope of the book. It means the worker and the farmer fully understand that: if the institution $G_{o}$ is changed to be $G_{n}$, both parties will get the Pareto improvement¹² after the reform. Here, we can assume $G_{o} \cap G_{n} \ne \varnothing$, which means the bargaining reform is the reform of a limited (or single) institution rather than complete negation of overall institutions, so as to maintain the continuity of the reform process. In fact, the process of making part reform on the institution $G_{o}$ to reach the consent $G_{n}$, actually, is an intuitive and realistic description of two parties doing the bargaining around the expected benefits resulted from the reform. If the single institution identified by two sides is written as $G_{o}^{S}$, and a corresponding possible new institution is written as $G_{n}^{S}$, it is easy to understand:

Definition 6.1

The reform of a single institution $G_{o}^{S}$ is the mapping of a bargaining game $B^{S} :G_{o}^{S} \to G_{n}^{S}$, in which $G_{o}^{S} \subseteq G_{o}$, $G_{n}^{S} \subseteq G_{n}$.

Obviously, if $\overline{{G_{o} \cap G_{n} }} = \varnothing$, then $B^{S}$ means no-reform. If $G_{o} \cap G_{n} = \varnothing$ and $G_{n}^{S} = G_{n}$, $B^{S}$ means radical reform. If $G_{o} \cap G_{n} \ne \varnothing$ and $G_{n}^{S} \subset G_{n}$, $B^{S}$ means the gradual reform which may composed of a continuous several such small reform together.

If the universal set of all possible schemes of player i about institution reform is written as $S_{i} = \left\{ {\omega_{j} ,s_{j} } \right\}$, where $s_{j}$ comes from $G_{o}^{S}$ and makes up the new institution $G_{n}^{S}$ with the probability of $\omega_{j}$. The bargaining of the institution reform is written as $B^{reform} (n,d,G_{o} ,S,\Omega )$, where $n = \{ 1,2\}$; $d = (d_{1} ,d_{2} )$ is the status quo of two bargainers’ benefit under the initial institution $G_{o}$; $S = \left\{ {(s_{1} ,s_{2} );s_{i} = \sum\nolimits_{j \in J} {\omega_{j} s_{j} } \in S_{i} \subseteq G_{n} ,i \in \{ 1,2\} ,\sum\nolimits_{j = 1}^{J} {\omega_{j} } = 1} \right\}$ is the bargaining set composed of all strategic propositions of two sides about the institution reform scheme; $\Omega = U(S)$ is the joint utility function (temporarily being understood as the expected utility function of V-N-M), which is based on the preference relations of each side over S, $u_{i} :s_{j} \to {\mathbb{R}}_{ + }^{1}$ ($s_{j} \in S_{j}$, $i = 1,2$), corresponding to the bargaining set S of institution reform and can also be written as $B^{reform} :S \Rightarrow\Omega \subset {\mathbb{R}}_{ + }^{2}$ or:

$$ \Omega = \left\{ {(u_{1} ,u_{2} );u_{i} = u_{i} (s_{i} ),s_{i} \in S_{i} ,i = 1,2} \right\} $$

Or, $\Omega = \left\{ {(u_{1} ,\phi (u_{1} ));u_{1} = u_{1} (s_{1} ),s_{1} \in S} \right\}$ ($\phi$ is the boundary curve of S).

Traditionally, it can be known from the discussion of the convexity of the social behavior set S, the social association of the institution norm ${\text{G}}_{\text{o}}$ and $G_{n}$, Axiom 6.2 of the strict two-person bargaining, Axiom 6.4 of bounded rationality, and Axiom 6.5 of non-violent bargaining that the bargaining set S and its set of the joint utility function $\Omega$ meet the following properties:

• (a)
  $$ \Omega \subseteq \left\{ {(u_{1} ,u_{2} ):u_{i} > d_{i} ,i = 1,2} \right\} $$

  (6.1)
• (b)
  $$ \Omega \cap \left\{ {\left( {u_{1} ,u_{2} } \right):u_{i} > d_{i},i = 1,2} \right\} \ne \varnothing $$

  (6.2)
• (c)
  $$ (d \le v \le u) \wedge (u \in U) \Rightarrow (v \in\Omega ) $$

  (6.3)

Here, (a) means that the bargaining set is superior to the status quo; (b) means the bargaining set can be Pareto improvement; (c) means the bargaining set is a compact set; the union of three equations means the boundary curve $\phi$ of the bargaining set is a continuous and differentiable concave function; or, when the “trade objects” in the specific political negotiation are different items of institution reform, Participants 1 and 2 have the rational preference for the negotiation content of institution reform. Thus, there is a continuous convex set $\Omega$. Generally, in the case that n is clearly specified, we simply write the bargaining game of the institution reform as $B^{reform} (G_{o} ,d,S,\Omega )$, which is to emphasize that the final result or equilibrium of the reform negotiation depends on the status quo $d = (d_{1} ,d_{2} )$ of $G_{o}$, and the nature of the bargaining set S. Obviously, under the condition of Nash bargaining, the bargaining game of institution reform in the book is generalized as Nash bargaining demand solution. Thus, there is:

Definition 6.2

Nash bargaining solution $B^{reform} :G_{o} \to G_{n}$ of an institution reform meets (Fig. 6.3):

Fig. 6.3

The behavior space under the institution G₀

$$ B^{re} \left( {G_{o} ,d,S,\Omega } \right) = {\text{agr}}\mathop {\text{Max}}\limits_{{(s_{1} ,s_{2} ) \in S}} (u_{1} (s_{1} ) - d_{1} )(u_{2} (s_{2} ) - d_{2} ) $$

Furtherly, from asymmetry Axiom 6.3, we can assume that there is $\pi (\alpha ,\beta ) \in (0,1)$, here $\alpha$ and $\beta$ ($\alpha ,\beta \in R_{ + }^{1}$) are the social indexes of 1 and 2 respectively (the next chapter will discuss in more detail), so as to have:

Definition 6.3

The asymmetric Nash bargaining solution $B^{reform} :G_{o} \to G_{n}$ of an institution reform game meets:

$$ B^{re} \left( {G_{o} ,d,S,\Omega } \right) = {\text{agr}}\mathop {\text{Max}}\limits_{{s_{i} \in S,u_{i} \ge d_{i} }} (u_{1} (s_{1} ) - d_{1} )^{\pi } (u_{2} (s_{2} ) - d_{2} )^{1 - \pi } . $$

Thus, through the definitions of institution, institution reform, and relevant variables above, we complete the description of a bargaining game about the institution reform and its equilibrium concept, so that to simplify the mechanism of institution change as a bargaining process of political behavior of both sides about the institution reform profits.