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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: