这是一份liverpool利物浦大学MATH331的成功案例

For a function $f$ that is twice continuously differentiable, $f$ has increasing differences if and only if
$$
\begin{gathered}
t^{\prime} \geq t \Rightarrow \frac{\partial f}{\partial x}\left(x, t^{\prime}\right) \geq \frac{\partial f}{\partial x}(x, t), \
\Leftrightarrow \frac{\partial^{2} f}{\partial x \partial t}(x, t) \geq 0, \forall x \in \mathcal{X}, \forall t \in \mathcal{T} .
\end{gathered}
$$
Given the definition of a function with increasing differences, we can formally define a supermodular game as follows: for all $i$,

MATH331 COURSE NOTES ：

$$
I_{p}(f)=\sum_{e \in \mathcal{P}} I_{e}\left(f_{e}\right) .
$$
Note that this path latency is not a function of the corresponding path flow because it depends on the total flow on each of its edges. Consequently, the total latency $C(f)$ of a flow $f$ is defined as follows:
$$
C(f)=\sum_{p \in \mathcal{P}} I_{p}\left(f_{e}\right) f_{p}
$$
it can be easily shown that $C(f)$ depends solely on the edge flow, and is given by
$$
C(f)=\sum_{e \in \mathcal{E}} I_{e}\left(f_{e}\right) f_{e} .
$$