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Text to improve:



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}
Note that the instantaneous regret at each time step $t$ is computed based on the action $a_{i_h}^{t}$ and observation $\mathcal{O}_{i_h}^{t}$, which implies that update rule exhibits a dependency between consecutive time steps. In other words, the expected change between two successive time steps can be expressed as a conditional expectation:
\begin{equation}
    \begin{aligned}
        \mathbb{E}[R_{i_ha}(t+1) - R_{i_ha}(t)]
        &= \mathbb{E} \left[ \frac{1}{t} \left( r_{i_ha}(t, a_i, \mathcal{O}_{i}) + R_{i_ha}^{t} \right) \mid a_i = a_{i_h}^{t},\mathcal{O}_{i} = \mathcal{O}_{i_h}^{t} \right] \\
        &= \frac{1}{t} \mathbb{E} \left[ r_{i_ha}(t, a_i, \mathcal{O}_{i}) \mid a_i=a_{i_h}^{t},\mathcal{O}_{i}=\mathcal{O}_{i_h}^{t} \right] - \frac{1}{t} R_{i_ha}(t).
    \end{aligned}
\end{equation}
Moreover, as the action $a_{i_h}^{t}$ is chosen according to the its strategy $\mathbf{x}_{i_h}$, the expected instantaneous regret is given by the weighted sum of the instantaneous regret for each action:
\begin{equation}
    \begin{aligned}
        \mathbb{E} \left[ r_{i_ha}(t, a_i, \mathcal{O}_{i}) \mid \mathcal{O}_{i}=\mathcal{O}_{i_h}^{t} \right]
        &= \sum_{a_i \in \mathcal{A}_i} x_{i_ha_i} \mathbb{E} \left[ r_{i_ha}(t, a_i, \mathcal{O}_{i}) \mid \mathcal{O}_{i}=\mathcal{O}_{i_h}^{t} \right].
    \end{aligned}
\end{equation}

Now, we examine the remaining condition on the observation $\mathcal{O}_{i_h}^{t}$, which represents the action-state profile of all interactions involving player $h$ at time  $t$. Specifically, it consists of the interaction states and the actions of neighboring players, given by $\mathcal{O}_{i_h}^{t} = \bigcup_{j \in N_i} \{(a_{j_k},s_{i_hj_k}) \mid k \in V_j\} \subseteq \mathbb{R}^{|V_j| \times |\mathcal{S}_{ij}|}$. 
To analyze the regret dynamics under a population-level interaction framework, we introduce the following assumptions on the observation $\mathcal{O}_{i_h}^{t}$.
\blue{Modify to be applied for all population}
\begin{assumption}
    The action-state pairs $(a_{j_k}, s_{i_hj_k})$  from different neighbors $j_k$ are independent and identically distributed (i.i.d.) according to a joint probability distribution $P_{i_hj}(a_j, s_{ij})$, i.e., $(a_{j_k}, s_{i_hj_k}) \sim P_{i_hj}(a_j, s_{ij})$ for all $k \in V_j, j \in N_i$.
\end{assumption}
This assumption is reasonable in large-scale systems where individual neighbors make decisions independently based on their own local information, with interactions limited to their direct relationships with player $h$.
Since each neighbor set $V_j$ contains a large number of individuals and the interaction state $\mathcal{S}_{ij}$ is finite, the expectation of the instantaneous regret can be approximated by the expected value of the observation:
\begin{equation}
    \begin{aligned}
        \mathbb{E}[r_{i_ha}(t, a_i, \mathcal{O}_{i}) \mid \mathcal{O}_{i}=\mathcal{O}_{i_h}^{t}]&=\sum_{(i,j)\in\mathcal{E}}\mathbb{E}_{(a_j, s_{ij}) \sim P_{i_hj}(a_j,s_{ij})}[r_{i_ha}(t, a_i, a_j, s_{ij})]\\
        &\approx \sum_{j \in N_i} \sum_{a_j \in \mathcal{A}_j} \sum_{s_{ij} \in \mathcal{S}_{ij}} P_{i_hj}(a_j, s_{ij}) r_{i_ha}(t, a_i, a_j, s_{ij}).
    \end{aligned}
\end{equation}

Suppose that the amount of time that passes between two successive time steps is $\delta \in (0, 1]$. Then, we rewrite the difference equation to obtain the continuous-time differential equation as:
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 updates 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}
Note that the instantaneous regret at each time step \( t \) is computed based on the action \( a_{i_h}^{t} \) and the observation \( \mathcal{O}_{i_h}^{t} \). This implies that the update rule shows a dependency between consecutive time steps. The expected change between two successive time steps can be expressed as a conditional expectation:
\begin{equation}
    \begin{aligned}
        \mathbb{E}[R_{i_ha}(t+1) - R_{i_ha}(t)]
        &= \mathbb{E} \left[ \frac{1}{t} \left( r_{i_ha}(t, a_i, \mathcal{O}_{i}) + R_{i_ha}^{t} \right) \mid a_i = a_{i_h}^{t}, \mathcal{O}_{i} = \mathcal{O}_{i_h}^{t} \right] \\
        &= \frac{1}{t} \mathbb{E} \left[ r_{i_ha}(t, a_i, \mathcal{O}_{i}) \mid a_i = a_{i_h}^{t}, \mathcal{O}_{i} = \mathcal{O}_{i_h}^{t} \right] - \frac{1}{t} R_{i_ha}(t).
    \end{aligned}
\end{equation}
Moreover, since the action \( a_{i_h}^{t} \) is chosen according to its strategy \( \mathbf{x}_{i_h} \), the expected instantaneous regret is given by the weighted sum of the instantaneous regret for each action:
\begin{equation}
    \begin{aligned}
        \mathbb{E} \left[ r_{i_ha}(t, a_i, \mathcal{O}_{i}) \mid \mathcal{O}_{i} = \mathcal{O}_{i_h}^{t} \right]
        &= \sum_{a_i \in \mathcal{A}_i} x_{i_ha_i} \mathbb{E} \left[ r_{i_ha}(t, a_i, \mathcal{O}_{i}) \mid \mathcal{O}_{i} = \mathcal{O}_{i_h}^{t} \right].
    \end{aligned}
\end{equation}

Next, we examine the condition regarding the observation \( \mathcal{O}_{i_h}^{t} \), which represents the action-state profile of all interactions involving player \( h \) at time \( t \). It consists of the interaction states and actions of neighboring players, defined as \( \mathcal{O}_{i_h}^{t} = \bigcup_{j \in N_i} \{(a_{j_k}, s_{i_hj_k}) \mid k \in V_j\} \subseteq \mathbb{R}^{|V_j| \times |\mathcal{S}_{ij}|} \). To analyze the regret dynamics within a population-level interaction framework, we introduce the following assumptions regarding the observation \( \mathcal{O}_{i_h}^{t} \).
\begin{assumption}
    The action-state pairs \( (a_{j_k}, s_{i_hj_k}) \) from different neighbors \( j_k \) are independent and identically distributed (i.i.d.) according to a joint probability distribution \( P_{i_hj}(a_j, s_{ij}) \). Specifically, \( (a_{j_k}, s_{i_hj_k}) \sim P_{i_hj}(a_j, s_{ij}) \) for all \( k \in V_j, j \in N_i \).
\end{assumption}
This assumption is reasonable in large-scale systems, where individual neighbors make independent decisions based on their local information, with interactions limited to their direct relationships with player \( h \). Given that each neighbor set \( V_j \) contains many individuals and that the interaction state \( \mathcal{S}_{ij} \) is finite, the expectation of the instantaneous regret can be approximated by the expected value of the observation:
\begin{equation}
    \begin{aligned}
        \mathbb{E}[r_{i_ha}(t, a_i, \mathcal{O}_{i}) \mid \mathcal{O}_{i} = \mathcal{O}_{i_h}^{t}] &= \sum_{(i,j) \in \mathcal{E}} \mathbb{E}_{(a_j, s_{ij}) \sim P_{i_hj}(a_j, s_{ij})}[r_{i_ha}(t, a_i, a_j, s_{ij})] \\
        &\approx \sum_{j \in N_i} \sum_{a_j \in \mathcal{A}_j} \sum_{s_{ij} \in \mathcal{S}_{ij}} P_{i_hj}(a_j, s_{ij}) r_{i_ha}(t, a_i, a_j, s_{ij}).
    \end{aligned}
\end{equation}

Suppose the amount of time that passes between two successive time steps is \( \delta \in (0, 1] \). Then, we can rewrite the difference equation to obtain the continuous-time differential equation as:
User: imrpove
Assistant:  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 updates 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}
Note that the instantaneous regret at each time step \( t \) is computed based on the action \( a_{i_h}^{t} \) and the observation \( \mathcal{O}_{i_h}^{t} \). This implies a dependency in the update rule between consecutive time steps. The expected change between two successive time steps can be expressed as a conditional expectation:
\begin{equation}
    \begin{aligned}
        \mathbb{E}[R_{i_ha}(t+1) - R_{i_ha}(t)]
        &= \mathbb{E} \left[ \frac{1}{t} \left( r_{i_ha}(t, a_i, \mathcal{O}_{i}) + R_{i_ha}^{t} \right) \mid a_i = a_{i_h}^{t}, \mathcal{O}_{i} = \mathcal{O}_{i_h}^{t} \right] \\
        &= \frac{1}{t} \mathbb{E} \left[ r_{i_ha}(t, a_i, \mathcal{O}_{i}) \mid a_i = a_{i_h}^{t}, \mathcal{O}_{i} = \mathcal{O}_{i_h}^{t} \right] - \frac{1}{t} R_{i_ha}(t).
    \end{aligned}
\end{equation}
Furthermore, since the action \( a_{i_h}^{t} \) is chosen according to its strategy \( \mathbf{x}_{i_h} \), the expected instantaneous regret is the weighted sum of the instantaneous regret for each action:
\begin{equation}
    \begin{aligned}
        \mathbb{E} \left[ r_{i_ha}(t, a_i, \mathcal{O}_{i}) \mid \mathcal{O}_{i} = \mathcal{O}_{i_h}^{t} \right]
        &= \sum_{a_i \in \mathcal{A}_i} x_{i_ha_i} \mathbb{E} \left[ r_{i_ha}(t, a_i, \mathcal{O}_{i}) \mid \mathcal{O}_{i} = \mathcal{O}_{i_h}^{t} \right].
    \end{aligned}
\end{equation}

Next, we examine the observation \( \mathcal{O}_{i_h}^{t} \), which represents the action-state profile of all interactions involving player \( h \) at time \( t \). It consists of the interaction states and actions of neighboring players, defined as \( \mathcal{O}_{i_h}^{t} = \bigcup_{j \in N_i} \{(a_{j_k}, s_{i_hj_k}) \mid k \in V_j\} \subseteq \mathbb{R}^{|V_j| \times |\mathcal{S}_{ij}|} \). To analyze the regret dynamics within a population-level interaction framework, we introduce the following assumptions regarding the observation \( \mathcal{O}_{i_h}^{t} \).
\begin{assumption}
    The action-state pairs \( (a_{j_k}, s_{i_hj_k}) \) from different neighbors \( j_k \) are independent and identically distributed (i.i.d.) according to a joint probability distribution \( P_{i_hj}(a_j, s_{ij}) \). In other words, \( (a_{j_k}, s_{i_hj_k}) \sim P_{i_hj}(a_j, s_{ij}) \) for all \( k \in V_j, j \in N_i \).
\end{assumption}
This assumption is reasonable in large-scale systems, where individual neighbors make independent decisions based on their local information, while interactions are limited to their direct relationships with player \( h \). Since each neighbor set \( V_j \) contains many individuals and the interaction state \( \mathcal{S}_{ij} \) is finite, the expectation of the instantaneous regret can be approximated by the expected value of the observation:
\begin{equation}
    \begin{aligned}
        \mathbb{E}[r_{i_ha}(t, a_i, \mathcal{O}_{i}) \mid \mathcal{O}_{i} = \mathcal{O}_{i_h}^{t}] &= \sum_{(i,j) \in \mathcal{E}} \mathbb{E}_{(a_j, s_{ij}) \sim P_{i_hj}(a_j, s_{ij})}[r_{i_ha}(t, a_i, a_j, s_{ij})] \\
        &\approx \sum_{j \in N_i} \sum_{a_j \in \mathcal{A}_j} \sum_{s_{ij} \in \mathcal{S}_{ij}} P_{i_hj}(a_j, s_{ij}) r_{i_ha}(t, a_i, a_j, s_{ij}).
    \end{aligned}
\end{equation}

Suppose the time that passes between two successive time steps is \( \delta \in (0, 1] \). Then, we can rewrite the difference equation to obtain the continuous-time differential equation as:
User: Summerize it
Assistant:  a player \( h \) from population \( i \in \mathcal{V} \) who tracks the regret \( R_{i_ha} \) for each action \( a \in \mathcal{A}_i \) over time. The regret updates from time \( t \) to \( t+1 \) as follows:

\[
R_{i_ha}(t+1) = \frac{1}{t} \left[ r_{i_ha}(t, a_{i_h}^{t}, \mathcal{O}_{i_h}^{t}) + R_{i_ha}(t) \right].
\]

The instantaneous regret at time \( t \) is based on the chosen action \( a_{i_h}^{t} \) and the observation \( \mathcal{O}_{i_h}^{t} \), indicating a dependency between consecutive time steps. The expected change between time steps can be expressed as:

\[
\mathbb{E}[R_{i_ha}(t+1) - R_{i_ha}(t)] = \frac{1}{t} \mathbb{E} \left[ r_{i_ha}(t, a_i, \mathcal{O}_{i}) \mid a_i=a_{i_h}^{t},\mathcal{O}_{i}=\mathcal{O}_{i_h}^{t} \right] - \frac{1}{t} R_{i_ha}(t).
\]

The expected instantaneous regret, given that action \( a_{i_h}^{t} \) is chosen according to its strategy \( \mathbf{x}_{i_h} \), is calculated as a weighted sum over all actions:

\[
\mathbb{E} \left[ r_{i_ha}(t, a_i, \mathcal{O}_{i}) \mid \mathcal{O}_{i}=\mathcal{O}_{i_h}^{t} \right] = \sum_{a_i \in \mathcal{A}_i} x_{i_ha_i} \mathbb{E} \left[ r_{i_ha}(t, a_i, \mathcal{O}_{i}) \mid \mathcal{O}_{i}=\mathcal{O}_{i_h}^{t} \right].
\]

We analyze the observation \( \mathcal{O}_{i_h}^{t} \), which includes interactions involving player \( h \). This is defined as:

\[
\mathcal{O}_{i_h}^{t} = \bigcup_{j \in N_i} \{(a_{j_k}, s_{i_hj_k}) \mid k \in V_j\}.
\]

An assumption is made that the action-state pairs \( (a_{j_k}, s_{i_hj_k}) \) from different neighbors \( j_k \) are independent and identically distributed, allowing for the approximation of expectations of instantaneous regret based on observational data.

Assuming the time interval between steps is \( \delta \in (0, 1] \), we can rewrite the difference equation as a continuous-time differential equation.
User: summerize in 100 words
Assistant: