Paternalism in Mathematical Decision Models: Utility, Constraints and Welfare | 数学决策模型中的家长主义：效用、约束与福利

📚 Paternalism in Mathematical Decision Models: Utility, Constraints and Welfare | 数学决策模型中的家长主义：效用、约束与福利

In many A-Level mathematics questions, students meet optimisation, probability and utility-like functions. One interdisciplinary idea that links these tools is paternalism: a decision maker substitutes their own objective for an individual’s choice. This article formalises paternalism using the mathematical language of Edexcel A-Level units such as Pure, Statistics and Decision Mathematics.

在许多 A-Level 数学题中，学生都会遇到最优化、概率和类似效用的函数。连接这些工具的一个跨学科概念就是家长主义：决策者用自己的目标取代个体的选择。本文使用 Edexcel A-Level 单元（如纯数学、统计学和决策数学）的数学语言来形式化家长主义。

1. Defining Paternalism in Model Terms | 用模型术语定义家长主义

Paternalism in mathematics means a planner restricts an agent’s feasible set because the planner believes the agent’s own choice is suboptimal. If an individual has a utility function U(x), paternalism replaces the individual’s maximiser x* = argmax U(x) with a planner’s preferred point x_p.

在数学中，家长主义意味着计划者限制行为人的可行集，因为计划者认为行为人自己的选择不是最优的。如果个体的效用函数为 U(x)，家长主义将个体的最大化点 x* = argmax U(x) 替换为计划者偏好的点 x_p。

The key difference is not the mathematics itself, but whose objective enters the optimisation. The same calculus and constraint techniques from A-Level Pure Mathematics are used, but the objective function changes.

关键区别不在于数学本身，而在于谁的目标进入最优化。使用的仍然是与 A-Level 纯数学相同的微积分和约束技术，但目标函数发生了变化。

2. Utility Functions and Individual Choice | 效用函数与个体选择

In decision models, an individual’s preferences can be represented by a utility function U(x), where x may be consumption, study time or risk exposure. The individual chooses x to maximise U subject to a constraint such as x ≥ 0 or px ≤ m.

在决策模型中，个体的偏好可以用效用函数 U(x) 表示，其中 x 可以是消费、学习时间或风险暴露。个体在约束（如 x ≥ 0 或 px ≤ m）下选择 x 以最大化 U。

For example, if U(x) = 8x

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