$$
\begin{array}{l} \left\langle \pi_ {i, \alpha} - \pi_ {i} ^ {k}, \vec {r} _ {i} ^ {k} \right\rangle \\ = \frac {\sum_ {a _ {i}} \bar {r} _ {i} ^ {k} \left(a _ {i}\right) \pi_ {i} ^ {k} \left(a _ {i}\right) \exp \left\{\bar {r} _ {i} ^ {k} \left(a _ {i}\right) \alpha \right\}}{\sum_ {a _ {i}} \pi_ {i} ^ {k} \left(a _ {i}\right) \exp \left\{\bar {r} _ {i} ^ {k} \left(a _ {i}\right) \alpha \right\}} - \sum_ {a _ {i}} \pi_ {i} ^ {k} \left(a _ {i}\right) \bar {r} _ {i} ^ {k} \left(a _ {i}\right) \\ \geq \frac {\sum_ {a _ {j} \in \mathcal {A} _ {i _ {p}} ^ {k}} \bar {r} _ {i} ^ {k} (a _ {j}) \pi_ {i} ^ {k} (a _ {j}) \exp \left\{\bar {r} _ {i} ^ {k} (a _ {j}) \alpha \right\} + \sum_ {a _ {j} \in \mathcal {A} _ {i} \setminus \mathcal {A} _ {i _ {p}} ^ {k}} \bar {r} _ {i} ^ {k} (a _ {j}) \pi_ {i} ^ {k} (a _ {j}) \exp \left\{\bar {r} _ {i} ^ {k} (a _ {i _ {q}} ^ {k}) \alpha \right\}}{\sum_ {a _ {j} \in \mathcal {A} _ {i _ {p}} ^ {k}} \pi_ {i} ^ {k} (a _ {j}) \exp \left\{\bar {r} _ {i} ^ {k} (a _ {j}) \alpha \right\} + \sum_ {a _ {j} \in \mathcal {A} _ {i} \setminus \mathcal {A} _ {i _ {p}} ^ {k}} \pi_ {i} ^ {k} (a _ {j}) \exp \left\{\bar {r} _ {i} ^ {k} (a _ {i _ {q}} ^ {k}) \alpha \right\}} - \sum_ {a _ {i}} \pi_ {i} ^ {k} (a _ {i}) \bar {r} _ {i} ^ {k} (a _ {i}) \\ = \frac {\bar {r} _ {i} ^ {k} \left(a _ {i _ {p}} ^ {k}\right) \left(\sum_ {a _ {j} \in \mathcal {A} _ {i _ {p}} ^ {k}} \pi_ {i} ^ {k} (a _ {j})\right) \exp \left\{\left(\bar {r} _ {i} ^ {k} \left(a _ {i _ {p}} ^ {k}\right) - \bar {r} _ {i} ^ {k} \left(a _ {i _ {q}} ^ {k}\right)\right) \alpha \right\} + \sum_ {a _ {j} \in \mathcal {A} _ {i} \setminus \mathcal {A} _ {i _ {p}} ^ {k}} \bar {r} _ {i} ^ {k} \left(a _ {j}\right) \pi_ {i} ^ {k} (a _ {j})}{\left(\sum_ {a _ {j} \in \mathcal {A} _ {i _ {p}} ^ {k}} \pi_ {i} ^ {k} (a _ {j})\right) \exp \left\{\left(\bar {r} _ {i} ^ {k} \left(a _ {i _ {p}} ^ {k}\right) - \bar {r} _ {i} ^ {k} \bigl (a _ {i _ {q}} ^ {k}\bigr)\right) \alpha \right\} + \sum_ {a _ {j} \in \mathcal {A} _ {i} \setminus \mathcal {A} _ {i _ {p}} ^ {k}} \pi_ {i} ^ {k} (a _ {j})} - \sum_ {a _ {i}} \pi_ {i} ^ {k} (a _ {i}) \bar {r} _ {i} ^ {k} (a _ {i}) \\ = \frac {\bar {r} _ {i} ^ {k} \left(a _ {i _ {p}} ^ {k}\right) \left(\sum_ {a _ {j} \in \mathcal {A} _ {i _ {p}} ^ {k}} \pi_ {i} ^ {k} (a _ {j})\right) \exp \left\{\left(\bar {r} _ {i} ^ {k} \left(a _ {i _ {p}} ^ {k}\right) - \bar {r} _ {i} ^ {k} \left(a _ {i _ {q}} ^ {k}\right)\right) \alpha \right\} + \sum_ {a _ {j} \in \mathcal {A} _ {i} \backslash \mathcal {A} _ {i _ {p}} ^ {k}} \bar {r} _ {i} ^ {k} \left(a _ {i _ {p}} ^ {k}\right) \pi_ {i} ^ {k} (a _ {j})}{\left(\sum_ {a _ {j} \in \mathcal {A} _ {i _ {p}} ^ {k}} \pi_ {i} ^ {k} (a _ {j})\right) \exp \left\{\left(\bar {r} _ {i} ^ {k} \left(a _ {i _ {p}} ^ {k}\right) - \bar {r} _ {i} ^ {k} \bigl (a _ {i _ {q}} ^ {k}\bigr)\right) \alpha \right\} + \sum_ {a _ {j} \in \mathcal {A} _ {i} \backslash \mathcal {A} _ {i _ {p}} ^ {k}} \pi_ {i} ^ {k} (a _ {j})} \\ - \frac {\sum_ {a _ {j} \in \mathcal {A} _ {i} \setminus \mathcal {A} _ {i _ {p}} ^ {k}} (\bar {r} _ {i} ^ {k} (a _ {i _ {p}} ^ {k}) - \bar {r} _ {i} ^ {k} (a _ {j})) \pi_ {i} ^ {k} (a _ {j})}{\left(\sum_ {a _ {j} \in \mathcal {A} _ {i _ {p}} ^ {k}} \pi_ {i} ^ {k} (a _ {j})\right) \exp \left\{(\bar {r} _ {i} ^ {k} (a _ {i _ {p}} ^ {k}) - \bar {r} _ {i} ^ {k} (a _ {i _ {q}} ^ {k})) \alpha \right\} + \sum_ {a _ {j} \in \mathcal {A} _ {i} \setminus \mathcal {A} _ {i _ {p}} ^ {k}} \pi_ {i} ^ {k} (a _ {j})} - \sum_ {a _ {i}} \pi_ {i} ^ {k} (a _ {i}) \bar {r} _ {i} ^ {k} (a _ {i}) \\ = \bar {r} _ {i} ^ {k} \left(a _ {i _ {p}} ^ {k}\right) - \sum_ {a _ {i}} \pi_ {i} ^ {k} \left(a _ {i}\right) \bar {r} _ {i} ^ {k} \left(a _ {i}\right) - \frac {\sum_ {a _ {i}} \left(\bar {r} _ {i} ^ {k} \left(a _ {i _ {p}} ^ {k}\right) - \bar {r} _ {i} ^ {k} \left(a _ {i}\right)\right) \pi_ {i} ^ {k} \left(a _ {i}\right)}{\left(\sum_ {a _ {j} \in \mathcal {A} _ {i _ {p}} ^ {k}} \pi_ {i} ^ {k} \left(a _ {j}\right)\right) \left(\exp \left\{\left(\bar {r} _ {i} ^ {k} \left(a _ {i _ {p}} ^ {k}\right) - \bar {r} _ {i} ^ {k} \left(a _ {i _ {q}} ^ {k}\right)\right) \alpha \right\} - 1\right) + 1} \\ = \Bigl(\sum_{a_{i}}(\bar{r}_{i}^{k}(a_{i_{p}}^{k}) - \bar{r}_{i}^{k}(a_{i}))\pi_{i}^{k}(a_{i})\Bigr)\Biggl\[1 - \frac{1}{\bigl(\sum_{a_{j}\in\mathcal{A}_{i_{p}}^{k}}\pi_{i}^{k}(a_{j})\bigr)\bigl(\exp \bigl\{(\bar{r}_{i}^{k}(a_{i_{p}}^{k}) - \bar{r}_{i}^{k}(a_{i_{q}}^{k}))\alpha \bigr\} -1\bigr) + 1\Bigr] \\ \end{array}
$$