Modeling Ming Dynasty Politics: Game Theory, Optimization, and System Dynamics

"The Ming Dynasty 1566" is a historical drama that vividly portrays the political ecology of the Jiajing era; this article applies mathematical modeling to quantify its core conflicts such as the "change rice to mulberry" policy, bureaucratic games, corruption networks, and fiscal crises.

1. Economic Decision Model of Changing Rice to Mulberry

The policy of converting rice fields to mulberry trees in Zhejiang is modeled as a multi‑objective optimization problem.

1.1 Model Assumptions and Variables

Let the total cultivated land be A, the area planted with rice be R, and the area planted with mulberry be M, subject to the constraint R + M \le A. The per‑unit output of rice is y_r and that of mulberry is y_m. A discount factor \delta accounts for the delayed revenue from mulberry.

1.2 Multi‑objective Function Construction

Economic profit objective: maximize total economic return from both crops.

Food security objective: ensure a minimum rice planting area R_{min} to guarantee basic grain supply.

Political assessment objective (tax revenue): maximize tax revenue, where tax rates for rice and mulberry are \tau_r and \tau_m respectively.

1.3 Pareto Optimal Analysis

Using a weighted‑sum method, the overall utility function is

U = \alpha \cdot \text{profit} + \beta \cdot \text{food security} + \gamma \cdot \text{tax revenue}

. When the weight for the political objective is large, the optimal solution favors extensive mulberry planting; when the weight for food security dominates, more rice area is retained.

2. Nash Equilibrium Model of Bureaucratic Game

The confrontation between Hai Rui and the Yan faction is analyzed with a non‑cooperative game framework.

2.1 Game Setup

Players: Hai Rui (strategy set {rigid, principled, compromise, retreat}) and Yan Shifan (strategy set {press, pull, coerce, collaborate}).

2.2 Nash Equilibrium Solution

Assuming mixed strategies, let p be the probability that Hai Rui chooses the principled stance, and q the probability that Yan Shifan chooses the press strategy. Solving the expected payoff equations yields a mixed‑strategy Nash equilibrium where p = 2/3 and q = 1/2.

2.3 Repeated Game and Reputation Mechanism

In an infinitely repeated game with discount factor \delta, the Grim Trigger strategy can sustain cooperation when the condition R > \delta T holds (R = cooperative payoff, T = temptation payoff). Hai Rui’s reputation capital is modeled as C_t = \theta \cdot I_t + (1-\theta) \cdot D_t, where I_t denotes honest behavior and D_t denotes deviation.

3. Corruption Network Diffusion Dynamics

The corruption network depicted in the drama is analyzed with an improved epidemiological SIS model.

3.1 Improved SIS Model

Susceptible: honest officials

Infected: corrupt officials

Removed: officials who have been investigated and dismissed

The dynamics are governed by a set of differential equations (omitted for brevity) with parameters such as transmission rate \beta, investigation rate \gamma, reinstatement rate, natural exit rate, and recruitment rate of honest officials.

3.2 Basic Reproduction Number

The basic reproduction number R_0 = \beta / \gamma determines whether corruption spreads ( R_0 > 1) or can be contained ( R_0 < 1).

3.3 Network Topology

The corruption network follows a scale‑free topology with a power‑law degree distribution. Core nodes such as Yan Song and Yan Shifan act as hubs; random attacks have little effect, whereas targeted removal of hubs can dismantle the network.

4. Fiscal Crisis Dynamic System Model

The fiscal crisis of the Jiajing court is expressed with discrete‑time difference equations.

4.1 Fiscal Balance Model

Let S_t be the treasury at year t, R_t the fiscal revenue, and E_t the expenditure. The evolution follows S_{t+1} = S_t + R_t - E_t.

4.2 Corruption Impact on Tax Revenue

The effective tax rate is reduced by a corruption cost function C(Corruption) = \alpha \cdot Corruption^2, where higher corruption levels dramatically lower actual tax collection.

4.3 Stability Analysis

Linearizing the system around its equilibrium yields a Jacobian matrix; the sign of its eigenvalues determines stability. When the eigenvalues are negative, the system is stable; otherwise, a spiral‑type fiscal collapse ensues.

5. Optimal Control Problem of Hai Rui’s Reforms

Hai Rui’s reform agenda is framed as an optimal control problem aiming to minimize total social loss.

5.1 Problem Statement

The objective function combines a livelihood index, a bureaucratic integrity index, and a control variable representing reform intensity, subject to dynamic state equations that include resistance from entrenched interest groups.

5.2 Pontryagin Maximum Principle

Constructing the Hamiltonian and applying the Pontryagin Maximum Principle yields co‑state (shadow price) equations and a characterization of the optimal control law. The analysis shows that rigid reform intensity without feedback leads to failure, whereas adaptive control based on current state and shadow prices can improve outcomes.

In summary, the mathematical models reveal that policy conflicts arise from weighted multi‑objective optimization, bureaucratic interactions exhibit Nash equilibrium characteristics, corruption spreads like an epidemic on a scale‑free network, fiscal crises are dynamic system instabilities amplified by corruption, and reform attempts correspond to optimal control problems hindered by excessive external resistance.