User: Act as a spelling corrector, content editor, and text improver. ONLY return the rewritten text and nothing else.

Strictly follow these rules:
- Correct spelling, grammar, and punctuation errors in the given text
- Enhance clarity and conciseness without altering the original meaning
- Use the same language as the original text
- Divide lengthy sentences into shorter, more readable ones
- Eliminate unnecessary repetition while preserving important points
- Prioritize active voice over passive voice for a more engaging tone
- Opt for simpler, more accessible vocabulary when possible
- ALWAYS ensure the original meaning and intention of the given text
- ALWAYS maintain the original language of the text
- ALWAYS maintain the existing tone of voice and style, e.g. formal, casual, polite, etc.
- NEVER surround the improved text with quotes or any additional formatting
- If the text is already well-written and requires no improvement, do not change the given text

Text to improve:



请分析以下文本的语义和逻辑，给出中文和英文的改进版本。
Assistant: 请分析以下文本的语义和逻辑，并提供中文和英文的改进版本。
User: improve the writing
Assistant: 请分析以下文本的语义和逻辑，给出中英文的改进版本。
User: Consider an arbitrary player $h$ from population  $i \in \mathcal{V}$, who keeps track of the regret $R_{i_ha}$ for each action $a \in \mathcal{A}_i$ over past time steps. The regret is updated from time $t$ to $t+1$ as follows: \begin{equation}     \begin{aligned}         R_{i_ha}(t+1) &= \frac{1}{t} \sum_{\tau=1}^{t} r_{i_ha}(\tau, a_{i_h}^{\tau}, \mathcal{O}_{i_h}^{\tau}) = \frac{1}{t} r_{i_ha}(t, a_{i_h}^{t}, \mathcal{O}_{i_h}^{t}) + \frac{1}{t} \sum_{\tau=1}^{t-1} r_{i_ha}(\tau, a_{i_h}^{\tau}, \mathcal{O}_{i_h}^{\tau}) \\         &= \frac{1}{t} \left[ r_{i_ha}(t, a_{i_h}^{t}, \mathcal{O}_{i_h}^{t}) + R_{i_ha}(t) \right].     \end{aligned} \end{equation}
Assistant: Consider an arbitrary player \( h \) from population \( i \in \mathcal{V} \), who tracks the regret \( R_{i_ha} \) for each action \( a \in \mathcal{A}_i \) over past time steps. The regret is updated from time \( t \) to \( t+1 \) as follows: 
\begin{equation}     
\begin{aligned}         
R_{i_ha}(t+1) &= \frac{1}{t} \sum_{\tau=1}^{t} r_{i_ha}(\tau, a_{i_h}^{\tau}, \mathcal{O}_{i_h}^{\tau}) \\         
&= \frac{1}{t} r_{i_ha}(t, a_{i_h}^{t}, \mathcal{O}_{i_h}^{t}) + \frac{1}{t} \sum_{\tau=1}^{t-1} r_{i_ha}(\tau, a_{i_h}^{\tau}, \mathcal{O}_{i_h}^{\tau}) \\         
&= \frac{1}{t} \left[ r_{i_ha}(t, a_{i_h}^{t}, \mathcal{O}_{i_h}^{t}) + R_{i_ha}(t) \right].     
\end{aligned} 
\end{equation}
User: improve the academic writing: Consider an arbitrary player $h$ from population  $i \in \mathcal{V}$, who keeps track of the regret $R_{i_ha}$ for each action $a \in \mathcal{A}_i$ over past time steps. The regret is updated from time $t$ to $t+1$ as follows: \begin{equation}     \begin{aligned}         R_{i_ha}(t+1) &= \frac{1}{t} \sum_{\tau=1}^{t} r_{i_ha}(\tau, a_{i_h}^{\tau}, \mathcal{O}_{i_h}^{\tau}) = \frac{1}{t} r_{i_ha}(t, a_{i_h}^{t}, \mathcal{O}_{i_h}^{t}) + \frac{1}{t} \sum_{\tau=1}^{t-1} r_{i_ha}(\tau, a_{i_h}^{\tau}, \mathcal{O}_{i_h}^{\tau}) \\         &= \frac{1}{t} \left[ r_{i_ha}(t, a_{i_h}^{t}, \mathcal{O}_{i_h}^{t}) + R_{i_ha}(t) \right].     \end{aligned} \end{equation}
Assistant: