这个实际是一个单递增函数，不过，不能保证连续。定义如下：
- F(x), a Lyapunov function
- A1: has a maximum (or minimum) value
- A2: there is a > 0 such that .if $x_{t+1} \neq x_t$, maximum, $F(x_{t+1}) \gt F(x_t) + a$ , or for a minimum, $F(x_{t+1}) \lt F(x_t) - a$
- Claim: at some point $x_{t+1} = x_t$
举个现实的例子，就是路径折半的算法，无限接近0。有一个k步骤之后到达极限值的速度评估值。前提是能算出来，折半那个么，呵呵了。
这个简单的模型有什么用？场景如下：
城市景点的游客自调节。游客通过观察，尽量避免拥挤。
The Lyapunov Function would be the total number of people who meet, with K=2 , just like in our original example. Any time a person changed her route to meet more people, the total number of people who meet would increase. So this is an example of a Lyapunov Function with a maximum value as opposed to a minimum value.
自由交换市场是否符合Lyaponuv？
这个很难说，自由交换如果引入外部性（Externality），就比较难评估，比如，某国卖给流氓国家核武器，以这两个国家的满意度来评估，或许符合。但是，周边的国家肯定不干了。如果这个幸福感的sum是所有的国家，那么，存在外部性的行为（除非全都是正面评价），评估就比较难了。
- Exchange markets with happiness as a Lyapunov metric satisfy A1 and A2. However, when externalities are involved, a Lyapunov function may not exist, particularly for negative externalities. Positive externalities are reinforcing and do not negate A1 and A2.
- Langdon’s lambda (the binary representation of the rule) from the simple cellular automata model essentially says that a system whose behavior isn’t influenced by others, tend to go to equilibrium. Conversely, where actions and behaviors are influenced by others tend to be more likely to be complex or random
- Externalities materially affect other people either negatively or positively. Negative externalities tend to cause ongoing changes
在优化的路上，存在局部最优解的陷阱，比如这图，局部最优后无法到达实际最优解。
相较之Markov模型，Lyapunov 的区别在
- Could be highly path dependent.
- Depends on the initial conditions
- Possibly many equilibrium
- Not a stochastic equilibrium, it’s fixed.
好吧，虽然Lyapunov 也是一种预测模型，和动态平衡下的Markov模型是不一样的，同时，建模时，要时刻提防着反向外部性negative externality这个幽灵。