# cs2

**Repository Path**: baiwenchao/cs2

## Basic Information

- **Project Name**: cs2
- **Description**: Impl. for CS-EV equilibrium algo. under 2CSs case
- **Primary Language**: Python
- **License**: Apache-2.0
- **Default Branch**: master
- **Homepage**: None
- **GVP Project**: No

## Statistics

- **Stars**: 0
- **Forks**: 0
- **Created**: 2021-12-09
- **Last Updated**: 2022-01-06

## Categories & Tags

**Categories**: Uncategorized

**Tags**: None

## README

# cs2

#### 介绍
Impl. for CS-EV equilibrium algo. under 2CSs case

#### 公式备忘

* $\omega_{p}p_{1} + \omega_{d}d_{i1} + \frac{\omega_{q}}{N_{1}^{C}}(Q_{1} + f_{i1}) -\mu_{i1} = \omega_{p}p_{2} + \omega_{d}d_{i2} + \frac{\omega_{q}}{N_{2}^{C}}(Q_{2} + f_{i2}) -\mu_{i2} = \lambda_{i}$

* $$
  \begin{cases}
   \omega_{p}p_{1} + \omega_{q}(\frac{1}{N_{1}^{C}}+\frac{1}{N_{2}^{C}})Q_{1}^{*} \le T_{i} - \frac{\omega_{q}}{N_{1}^{C}} & f_{i1}^{*}=1 & iff \quad u_{i} \in \mathcal{I} \\
  \omega_{p}p_{1} + \omega_{q}(\frac{1}{N_{1}^{C}}+\frac{1}{N_{2}^{C}})Q_{1}^{*} = T_{i} + \frac{\omega_{q}}{N_{2}^{C}} - \omega_{q}(\frac{1}{N_{1}^{C}}+\frac{1}{N_{2}^{C}})f_{i1}^{*} & f_{i1}^{*}\in(0,1) & iff \quad u_{i} \in \mathcal{J} \\
  \omega_{p}p_{1} + \omega_{q}(\frac{1}{N_{1}^{C}}+\frac{1}{N_{2}^{C}})Q_{1}^{*} \ge T_{i} + \frac{\omega_{q}}{N_{2}^{C}} & f_{i1}^{*}=0 & iff \quad u_{i} \in \mathcal{K} \\
  \end{cases}
  $$

* $T_{i} = \omega_{p}p_{2} + \frac{\omega_{q}}{N_{2}^{C}}|U| + w_{d}(d_{i2}-d_{i1})$

* $T^{k} = \sum_{u_{i}\in\mathcal{J}^{k}}(T_{i}+\frac{w_{q}}{N_{2}^{C}})$

* $Q_{1}^{*} = \frac{-\omega_{p}|\mathcal{J}|}{w_{q}(\frac{1}{N_{1}^{C}} + \frac{1}{N_{2}^{C}})(|\mathcal{J}|+1)}p_{1} + \frac{\sum_{u_{i}\in\mathcal{J}}(T_{i} + \frac{\omega_{q}}{N_{2}^{C}}) + w_{q}(\frac{1}{N_{1}^{C}} + \frac{1}{N_{2}^{C}})|\mathcal{I}|}{w_{q}(\frac{1}{N_{1}^{C}} + \frac{1}{N_{2}^{C}})(|\mathcal{J}|+1)}$

* $g(x) = \omega_{p}x + \omega_{q}(\frac{1}{N_{1}^{C}}+\frac{1}{N_{2}^{C}})Q_{1}^{*}(x) = \frac{\omega_{p}}{|\mathcal{J}|+1}x + \frac{\sum_{u_{i}\in\mathcal{J}}(T_{i} + \frac{\omega_{q}}{N_{2}^{C}}) + w_{q}(\frac{1}{N_{1}^{C}} + \frac{1}{N_{2}^{C}})|\mathcal{I}|}{|\mathcal{J}|+1}$

* $p_{1}^{k} = \frac{1}{w_{p}}[(|\mathcal{J}^{k}|+1)b_{1}^{k} - T^{k} - w_{q}(\frac{1}{N_{1}^{C}}+\frac{1}{N_{2}^{C}})|\mathcal{I}^{k}|]$

  * $Case1: b_{1}^{k}=T_{l}-\frac{w_q}{N_{1}^{C}}$
    * $Q_{1}^{k+1}(p_{1}^{k}) = \frac{-\omega_{p}(|\mathcal{J}|^{k}+x)}{w_{q}(\frac{1}{N_{1}^{C}} + \frac{1}{N_{2}^{C}})(|\mathcal{J}|^{k}+x+1)}p_{1}^{k} + \frac{T^{k} + xb_{1}^{k} + w_{q}(\frac{1}{N_{1}^{C}} + \frac{1}{N_{2}^{C}})|\mathcal{I}|^{k}}{w_{q}(\frac{1}{N_{1}^{C}} + \frac{1}{N_{2}^{C}})(|\mathcal{J}|^{k}+x+1)}$

  * $Case2: b_{1}^{k}=T_{l}+\frac{w_q}{N_{2}^{C}}$
    * $Q_{1}^{k+1}(p_{1}^{k}) = \frac{-\omega_{p}(|\mathcal{J}|^{k}-x)}{w_{q}(\frac{1}{N_{1}^{C}} + \frac{1}{N_{2}^{C}})(|\mathcal{J}|^{k}-x+1)}p_{1}^{k} + \frac{T^{k} - xb_{1}^{k} + w_{q}(\frac{1}{N_{1}^{C}} + \frac{1}{N_{2}^{C}})|\mathcal{I}|^{k}}{w_{q}(\frac{1}{N_{1}^{C}} + \frac{1}{N_{2}^{C}})(|\mathcal{J}|^{k}-x+1)}$