SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Observe that in Equation 59, the second and the third term can be canceled out. Moreover, for any two neighbor populations  $j, k \in V_i$ , the beliefs  $\mu_j, \mu_k$  about these two populations are updated separately and independently. Hence, the covariance of these beliefs are zero. We apply the moment closure approximation [64, 39] with the second order and obtain
$$
\bar {x} _ {i s _ {i}} \approx f _ {s _ {i}} (\bar {\mu}) + \frac {1}{2} \sum_ {j \in V _ {i}} \sum_ {s _ {j} \in S _ {j}} \frac {\partial^ {2} f _ {s _ {i}} (\bar {\mu})}{(\partial \mu_ {j s _ {j}}) ^ {2}} \operatorname {V a r} \left(\mu_ {j s _ {j}}\right). \tag {60}
$$
Hence, substituting  $\bar{x}_{is_i}$  in Lemma 4 with the above approximation, we have the mean belief dynamics
$$
\frac {d \bar {\mu} _ {i s _ {i}}}{d t} \approx \frac {f _ {s _ {i}} (\bar {\mathbf {u}}) - \bar {\mu} _ {i s _ {i}}}{\lambda + t + 1} + \frac {\sum_ {j \in V _ {i}} \sum_ {s _ {j} \in S _ {j}} \frac {\partial^ {2} f _ {s _ {i}} (\bar {\mathbf {u}})}{(\partial \mu_ {j s _ {j}}) ^ {2}} \operatorname {V a r} \left(\mu_ {j s _ {j}}\right)}{2 (\lambda + t + 1)}. \tag {61}
$$
E Proof of Proposition 3
It follows from Equation 2 and Equation 3 of the main paper that the change in beliefs between two successive time steps is as follows.
$$
\boldsymbol {\mu} _ {i} (t + 1) = \boldsymbol {\mu} _ {i} (t) + \frac {\mathbf {x} _ {i} (t) - \boldsymbol {\mu} _ {i} (t)}{\lambda + t + 1}. \tag {62}
$$
$$
\boldsymbol {\mu} _ {i} (t + \delta) = \boldsymbol {\mu} _ {i} (t) + \delta \frac {\mathbf {x} _ {i} (t) - \boldsymbol {\mu} _ {i} (t)}{\lambda + t + 1}. \tag {63}
$$
Move the term  $\mu_i(t)$  to the right hand side and divide both sides by  $\delta$ ,
$$
\frac {\boldsymbol {\mu} _ {i} (t + \delta) - \boldsymbol {\mu} _ {i} (t)}{\delta} = \frac {\mathbf {x} _ {i} (t) - \boldsymbol {\mu} _ {i} (t)}{\lambda + t + 1}. \tag {64}
$$
Assume that the amount of time  $\delta$  between two successive time steps goes to zero. We have
$$
\frac {d \boldsymbol {\mu} _ {i}}{d t} = \lim  _ {\delta \rightarrow 0} \frac {\boldsymbol {\mu} _ {i} (t + \delta) - \boldsymbol {\mu} _ {i} (t)}{\delta} = \frac {\mathbf {x} _ {i} (t) - \boldsymbol {\mu} _ {i} (t)}{\lambda + t + 1}. \tag {65}
$$
F Proof of Proposition 4
$$
\frac {d \boldsymbol {\mu} _ {i}}{d t} = \frac {\mathbf {x} _ {i} - \boldsymbol {\mu} _ {i}}{\lambda + t + 1} = 0 \Longrightarrow \mathbf {x} _ {i} = \boldsymbol {\mu} _ {i}. \tag {66}
$$
Denote the equilibrium points of the system dynamics, which satisfies the above equation, by  $(\mathbf{x}_i^*, \boldsymbol{\mu}_i^*)$  for each population  $i$ . By the logit choice rule, we have
$$
x _ {i s _ {i}} ^ {*} = \frac {\exp (\beta u _ {i s _ {i}})}{\sum_ {s _ {i} ^ {\prime} \in S _ {i}} \exp (\beta u _ {i s _ {i} ^ {\prime}})} = \frac {\exp (\beta \sum_ {j \in V _ {i}} \mathbf {e} _ {s _ {i}} ^ {\top} \mathbf {A} _ {i j} \boldsymbol {\mu} _ {j} ^ {*})}{\sum_ {s _ {i} ^ {\prime} \in S _ {i}} \exp (\beta \sum_ {j \in V _ {i}} \mathbf {e} _ {s _ {i} ^ {\prime}} ^ {\top} \mathbf {A} _ {i j} \boldsymbol {\mu} _ {j} ^ {*})}. \tag {67}
$$
Leveraging that  $\mathbf{x}_i^* = \boldsymbol{\mu}_i^*$ ,  $\forall i \in V$  at equilibrium, we can replace  $\boldsymbol{\mu}_j^*$  with  $\mathbf{x}_j^*$ . Q.E.D.
Consider an agent  $i$  in a classic network game. The set of neighbors is  $V_{i}$ , the set of beliefs about the neighbors is  $\{\pmb{\mu}_j\}_{j\in V_i}$ , and the choice distribution is  $\mathbf{x}_i$ . Given a classic network game, the expected payoff is given by  $\mathbf{x}_i^\top \sum_{(i,j)\in E}A_{ij}\pmb{\mu}_j$ . Define a perturbed payoff function
$$
\pi_ {i} \left(\mathbf {x} _ {i}, \left\{\boldsymbol {\mu} _ {j} \right\} _ {j \in V _ {i}}\right) := \mathbf {x} _ {i} ^ {\top} \sum_ {j \in V _ {i}} A _ {i j} \boldsymbol {\mu} _ {j} + v \left(\mathbf {x} _ {i}\right) \tag {68}
$$
where  $v(\mathbf{x}_i) = -\frac{1}{\beta}\sum_{s_i \in S_i} x_{isi}\ln (x_{isi})$ . Under this form of  $v(\mathbf{x}_i)$ , the maximization of  $\pi_i$  yields the choice distribution  $\mathbf{x}_i$  from the logit choice function [33]. Based on this, we establish the following lemma.
Lemma 5. For a choice distribution  $\mathbf{x}_i$  of SFP in a network game,
$$
\partial_ {\mathbf {x} _ {i}} \pi_ {i} \left(\mathbf {x} _ {i}, \left\{\boldsymbol {\mu} _ {j} \right\} _ {j \in V _ {i}}\right) = \mathbf {0} \quad a n d \quad \sum_ {j \in V _ {i}} \left(A _ {i j} \boldsymbol {\mu} _ {j}\right) ^ {\top} = - \partial_ {\mathbf {x} _ {i}} v (\mathbf {x} _ {i}). \tag {69}
$$
Proof. This lemma immediately follows from the fact that the maximization of  $\pi_{i}$  will yield the choice distribution  $\mathbf{x}_i$  from the logit choice function [33].
The belief dynamics of an agent can be simplified after time-reparameterization.
Lemma 6. Given  $\tau = \ln \frac{\lambda + t + 1}{\lambda + 1}$ , the belief dynamics of homogeneous systems (given in Equation 11 in the main paper) is equivalent to
$$
\frac {d \boldsymbol {\mu} _ {i}}{d \tau} = \mathbf {x} _ {i} - \boldsymbol {\mu} _ {i}. \tag {70}
$$
Proof. From  $\tau = \ln \frac{\lambda + t + 1}{\lambda + 1}$ , we have
$$
t = (\lambda + 1) \left(\exp (\tau) - 1\right). \tag {71}
$$
By the chain rule, for each dimension  $s_i$ ,
$$
\begin{array}{l} \frac {d \mu_ {i s _ {i}}}{d \tau} = \frac {d \mu_ {i s _ {i}}}{d t} \frac {d t}{d \tau} (72) \\ = \frac {x _ {i s i} - \mu_ {i s i}}{\lambda + t + 1} \frac {d ((\lambda + 1) (\exp (\tau) - 1))}{d \tau} (73) \\ = \frac {x _ {i s _ {i}} - \mu_ {i s _ {i}}}{\lambda + (\lambda + 1) (\exp (\tau) - 1) + 1} (\lambda + 1) \exp (\tau) (74) \\ = x _ {i s _ {i}} - \mu_ {i s _ {i}}. (75) \\ \end{array}
$$
Next, we define the Lyapunov function  $L$  as
$$
L := \sum_ {i \in V} \omega_ {i} L _ {i} \quad \text {s . t .} \quad L _ {i} := \pi_ {i} \left(\mathbf {x} _ {i}, \left\{\boldsymbol {\mu} _ {j} \right\} _ {j \in V _ {i}}\right) - \pi_ {i} \left(\boldsymbol {\mu} _ {i}, \left\{\boldsymbol {\mu} _ {j} \right\} _ {j \in V _ {i}}\right). \tag {76}
$$
where  $\{\omega_i\}_{i\in V}$  is the set of positive weights defined in the weighted zero-sum  $\Gamma$ . The function  $L$  is non-negative because for every  $i\in V$ ,  $\mathbf{x}_i$  maximizes the function  $\pi_{i}$ . When for every  $i\in V$ ,  $\mathbf{x}_i = \boldsymbol {\mu}_i$ , the function  $L$  reaches the minimum value 0.
Rewrite  $L$  as
$$
L = \sum_ {i \in V} \left[ \omega_ {i} \pi_ {i} \left(\mathbf {x} _ {i}, \left\{\boldsymbol {\mu} _ {j} \right\} _ {j \in V _ {i}}\right) - \omega_ {i} \boldsymbol {\mu} _ {i} ^ {\top} \sum_ {j \in V _ {i}} A _ {i j} \boldsymbol {\mu} _ {j} - \omega_ {i} v (\boldsymbol {\mu} _ {i}) \right]. \tag {77}
$$
We observe that  $\pi_i(\mathbf{x}_i, \{\pmb{\mu}_j\}_{j \in V_i})$  is convex in  $\pmb{\mu}_j, j \in V_i$  by Danskin's theorem, and  $-v(\pmb{\mu}_i)$  is strictly convex in  $\pmb{\mu}_i$ . Moreover, by the weighted zero-sum property given in Equation 2 in the main paper, we have
$$
\sum_ {i \in V} \left(\omega_ {i} \boldsymbol {\mu} _ {i} ^ {\top} \sum_ {j \in V _ {i}} A _ {i j} \boldsymbol {\mu} _ {j}\right) = 0 \tag {78}
$$
since  $\mu_i \in \Delta_i, \mu_j \in \Delta_j$  for every  $i, j \in V$ . Therefore, the function  $L$  is a strictly convex function and attains its minimum value 0 at a unique point  $\mathbf{x}_i = \boldsymbol{\mu}_i, \forall i \in V$ .
Consider the function  $L_{q}$ . Its time derivative is
$$
\begin{array}{l} \dot {L} _ {i} = \partial_ {\mathbf {x} _ {i}} \pi_ {i} (\mathbf {x} _ {i}, \{\boldsymbol {\mu} _ {j} \} _ {j \in V _ {i}}) \dot {\mathbf {x}} _ {i} + \sum_ {j \in V _ {i}} \left[ \partial_ {\boldsymbol {\mu} _ {j}} \pi_ {i} (\mathbf {x} _ {i}, \{\boldsymbol {\mu} _ {j} \} _ {j \in V _ {i}}) \dot {\boldsymbol {\mu}} _ {j} \right] \\ - \partial_ {\boldsymbol {\mu} _ {i}} \pi_ {i} (\boldsymbol {\mu} _ {i}, \{\boldsymbol {\mu} _ {j} \} _ {j \in V _ {i}}) \ddot {\boldsymbol {\mu}} _ {i} - \sum_ {j \in V _ {i}} \left[ \partial_ {\boldsymbol {\mu} _ {j}} \pi_ {i} (\boldsymbol {\mu} _ {i}, \{\boldsymbol {\mu} _ {j} \} _ {j \in V _ {i}}) \dot {\boldsymbol {\mu}} _ {j} \right]. \tag {79} \\ \end{array}
$$
Note that the partial derivative  $\partial_{\mathbf{x}_i}\pi_i$  equals 0 by Lemma 5. Thus, we can rewrite this as
$$
\begin{array}{l} \dot {L} _ {i} = \partial_ {\boldsymbol {\mu} _ {i}} \pi_ {i} \left(\boldsymbol {\mu} _ {i}, \left\{\boldsymbol {\mu} _ {j} \right\} _ {j \in V _ {i}}\right) \dot {\boldsymbol {\mu}} _ {i} + \sum_ {j \in V _ {i}} \left[ \partial_ {\boldsymbol {\mu} _ {j}} \pi_ {i} \left(\mathbf {x} _ {i}, \left\{\boldsymbol {\mu} _ {j} \right\} _ {j \in V _ {i}}\right) - \partial_ {\boldsymbol {\mu} _ {j}} \pi_ {i} \left(\boldsymbol {\mu} _ {i}, \left\{\boldsymbol {\mu} _ {j} \right\} _ {j \in V _ {i}}\right) \right] \dot {\boldsymbol {\mu}} _ {j} (80) \\ = - \left[ \sum_ {j \in V _ {i}} \left(A _ {i j} \boldsymbol {\mu} _ {j}\right) ^ {\top} + \partial_ {\mu_ {i}} v \left(\boldsymbol {\mu} _ {i}\right) \right] \left(\mathbf {x} _ {i} - \boldsymbol {\mu} _ {i}\right) + \sum_ {j \in V _ {i}} \left(\mathbf {x} _ {i} ^ {\top} A _ {i j} - \boldsymbol {\mu} _ {i} ^ {\top} A _ {i j}\right) \left(\mathbf {x} _ {j} - \boldsymbol {\mu} _ {j}\right) (81) \\ = \left[ \partial_ {\mathbf {x} _ {i}} v (\mathbf {x} _ {i}) - \partial_ {\mu_ {i}} v (\boldsymbol {\mu} _ {i}) \right] (\mathbf {x} _ {i} - \boldsymbol {\mu} _ {i}) + \sum_ {j \in V _ {i}} \left(\mathbf {x} _ {i} ^ {\top} A _ {i j} \mathbf {x} _ {j} - \boldsymbol {\mu} _ {i} ^ {\top} A _ {i j} \mathbf {x} _ {j} - \mathbf {x} _ {i} ^ {\top} A _ {i j} \boldsymbol {\mu} _ {j} + \boldsymbol {\mu} _ {i} ^ {\top} A _ {i j} \boldsymbol {\mu} _ {j}\right). (82) \\ \end{array}
$$
where from Equation 81 to 82, we apply Lemma 5 to substitute  $\sum_{j\in V_i}\left(A_{ij}\boldsymbol {\mu}_j\right)^\top$  with  $-\partial_{\mathbf{x}_i}v(\mathbf{x}_i)$ . Hence, summing over all the populations, the time derivative of  $L$  is
$$
\begin{array}{l} \dot {L} = \sum_ {i \in V} \omega_ {i} \left[ \partial_ {\mathbf {x} _ {i}} v (\mathbf {x} _ {i}) - \partial_ {\boldsymbol {\mu} _ {i}} v (\boldsymbol {\mu} _ {i}) \right] (\mathbf {x} _ {i} - \boldsymbol {\mu} _ {i}) \\ + \sum_ {i \in V} \sum_ {j \in V _ {i}} \omega_ {i} \left(\mathbf {x} _ {i} ^ {\top} A _ {i j} \mathbf {x} _ {j} - \boldsymbol {\mu} _ {i} ^ {\top} A _ {i j} \mathbf {x} _ {j} - \mathbf {x} _ {i} ^ {\top} A _ {i j} \boldsymbol {\mu} _ {j} + \boldsymbol {\mu} _ {i} ^ {\top} A _ {i j} \boldsymbol {\mu} _ {j}\right). \tag {83} \\ \end{array}
$$
The summation in the second line is equivalent to
$$
\begin{array}{l} \sum_ {(i, j) \in E} \left(\omega_ {i} \mathbf {x} _ {i} ^ {\top} A _ {i j} \mathbf {x} _ {j} + \omega_ {j} \mathbf {x} _ {j} ^ {\top} A _ {j i} \mathbf {x} _ {i}\right) - \left(\omega_ {i} \boldsymbol {\mu} _ {i} ^ {\top} A _ {i j} \mathbf {x} _ {j} + \omega_ {j} \mathbf {x} _ {j} ^ {\top} A _ {j i} \boldsymbol {\mu} _ {i}\right) (84) \\ - \left(\omega_ {i} \mathbf {x} _ {i} ^ {\top} A _ {i j} \boldsymbol {\mu} _ {j} + \omega_ {j} \boldsymbol {\mu} _ {j} ^ {\top} A _ {j i} \mathbf {x} _ {i}\right) + \left(\omega_ {i} \boldsymbol {\mu} _ {i} ^ {\top} A _ {i j} \boldsymbol {\mu} _ {j} + \omega_ {j} \boldsymbol {\mu} _ {j} ^ {\top} A _ {j i} \boldsymbol {\mu} _ {i}\right). (85) \\ \end{array}
$$
By the weighted zero-sum property given in Equation 2 in the main paper, this summation equals 0, yielding
$$
\dot {L} = \sum_ {i \in V} \omega_ {i} \left[ \partial_ {\mathbf {x} _ {i}} v \left(\mathbf {x} _ {i}\right) - \partial_ {\mu_ {i}} v \left(\boldsymbol {\mu} _ {i}\right) \right] \left(\mathbf {x} _ {i} - \boldsymbol {\mu} _ {i}\right). \tag {86}
$$
Note that the function  $v$  is strictly concave such that its second derivative is negative definite. By this property,  $\dot{L} \leq 0$  with equality only if  $\mathbf{x}_i = \boldsymbol{\mu}_i, \forall i \in V$ , which corresponds to the QRE. Therefore,  $L$  is a strict Lyapunov function, and the global asymptotic stability of the QRE follows. Q.E.D.
H Proof of Theorem 3
Consider a root agent  $j$  of a star structure. Its set of leaf (neighbor) agents is  $V_{j}$ , the set of beliefs about the leaf agents is  $\{\pmb{\mu}_i\}_{i \in V_j}$ , and the choice distribution is  $\mathbf{x}_j$ . Given the game  $\Gamma$ , the expected payoff is  $\mathbf{x}_j^\top \sum_{i \in V_j} A_{ji} \pmb{\mu}_i$ . Define a perturbed payoff function
$$
\pi_ {j} \left(\mathbf {x} _ {j}, \left\{\boldsymbol {\mu} _ {i} \right\} _ {i \in V _ {j}}\right) := \mathbf {x} _ {j} ^ {\top} \sum_ {i \in V _ {j}} A _ {j i} \boldsymbol {\mu} _ {i} + v (\mathbf {x} _ {j}) \tag {87}
$$
where  $v(\mathbf{x}_j) = -\frac{1}{\beta}\sum_{s_j \in S_j} x_{js_j}\ln(x_{js_j})$ . Under this form of  $v(\mathbf{x}_j)$ , the maximization of  $\pi_j$  yields the choice distribution  $\mathbf{x}_j$  from the logit choice function [33].
Consider a leaf agent  $i$  of the root agent  $j$ . It has only one neighbor, which is population  $j$ . Thus, given the game  $\Gamma$ , the expected payoff is  $\mathbf{x}_i^\top A_{ij} \boldsymbol{\mu}_j$ . Define a perturbed payoff function
$$
\pi_ {i} \left(\mathbf {x} _ {i}, \boldsymbol {\mu} _ {j}\right) := \mathbf {x} _ {i} ^ {\top} A _ {i j} \boldsymbol {\mu} _ {j} + v \left(\mathbf {x} _ {i}\right) \tag {88}
$$
where  $v(\mathbf{x}_i) = -\frac{1}{\beta}\sum_{s_i \in S_i} x_{is_i}\ln(x_{is_i})$ . Similarly, the maximization of  $\pi_i$  yields the choice distribution  $\mathbf{x}_i$  from the logit choice function [33]. Based on this, we establish the following lemma.
Lemma 7. For choice distributions of SFP in a network game with a star structure,
$$
\partial_ {\mathbf {x} _ {j}} \pi_ {j} \left(\mathbf {x} _ {j}, \left\{\boldsymbol {\mu} _ {i} \right\} _ {i \in V _ {j}}\right) = \mathbf {0} \quad a n d \quad \sum_ {i \in V _ {j}} \left(A _ {j i} \boldsymbol {\mu} _ {i}\right) ^ {\top} = - \partial_ {\mathbf {x} _ {j}} v (\mathbf {x} _ {j}) \quad \text {i f j i s a r o o t a g e n t ,} \tag {89}
$$
$$
\partial_ {\mathbf {x} _ {i}} \pi_ {i} (\mathbf {x} _ {i}, \boldsymbol {\mu} _ {j}) = \mathbf {0} \quad a n d \quad (A _ {i j} \boldsymbol {\mu} _ {j}) ^ {\top} = - \partial_ {\mathbf {x} _ {i}} v (\mathbf {x} _ {i}) \quad i f i \text {i s a l e a f a g e n t}. \tag {90}
$$
Proof. This lemma immediately follows from the fact that the maximization of  $\pi_j$  and  $\pi_i$ , respectively, yield the choice distributions  $\mathbf{x}_j$  and  $\mathbf{x}_i$  from the logit choice function [33].
Let  $\mathcal{R} \subset V$  be the set of all root agents. We define
$$
L := \sum_ {j \in \mathcal {R}} L _ {j} \quad \text {s . t .} \quad L _ {j} := \boldsymbol {\mu} _ {j} ^ {\top} \sum_ {i \in V _ {j}} A _ {j i} \boldsymbol {\mu} _ {i} + v (\boldsymbol {\mu} _ {j}) + \sum_ {i \in V _ {j}} v (\boldsymbol {\mu} _ {i}). \tag {91}
$$
Consider the function  $L_{j}$ . Its time derivative  $\dot{L}_j$  is
$$
\begin{array}{l} \dot {L} _ {j} = \left[ \partial_ {\boldsymbol {\mu} _ {j}} \left(\boldsymbol {\mu} _ {j} ^ {\top} \sum_ {i \in V _ {j}} A _ {j i} \boldsymbol {\mu} _ {i}\right) \dot {\boldsymbol {\mu}} _ {j} + \sum_ {i \in V _ {j}} \partial_ {\boldsymbol {\mu} _ {i}} \left(\boldsymbol {\mu} _ {j} ^ {\top} \sum_ {i \in V _ {j}} A _ {j i} \boldsymbol {\mu} _ {i}\right) \dot {\boldsymbol {\mu}} _ {i} \right] + \partial_ {\boldsymbol {\mu} _ {j}} v \left(\boldsymbol {\mu} _ {j}\right) \dot {\boldsymbol {\mu}} _ {j} + \sum_ {i \in V _ {j}} \partial_ {\boldsymbol {\mu} _ {i}} v \left(\boldsymbol {\mu} _ {i}\right) \dot {\boldsymbol {\mu}} _ {i} (92) \\ = \sum_ {i \in V _ {j}} \left(A _ {j i} \boldsymbol {\mu} _ {i}\right) ^ {\top} \left(\mathbf {x} _ {j} - \boldsymbol {\mu} _ {j}\right) + \left[ \sum_ {i \in V _ {j}} \boldsymbol {\mu} _ {j} ^ {\top} A _ {j i} \left(\mathbf {x} _ {i} - \boldsymbol {\mu} _ {i}\right) \right] + \partial_ {\boldsymbol {\mu} _ {j}} v \left(\boldsymbol {\mu} _ {j}\right) \left(\mathbf {x} _ {j} - \boldsymbol {\mu} _ {j}\right) + \sum_ {i \in V _ {j}} \partial_ {\boldsymbol {\mu} _ {i}} v \left(\boldsymbol {\mu} _ {i}\right) \left(\mathbf {x} _ {i} - \boldsymbol {\mu} _ {i}\right). (93) \\ \end{array}
$$
Since we have  $(A_{ij}\pmb{\mu}_j)^\top = \pmb{\mu}_j^\top A_{ij}^\top = \pmb{\mu}_j^\top A_{ji}$ , applying Lemma 7, we can substitute  $\sum_{i\in V_j}(A_{ji}\pmb{\mu}_i)^\top$  with  $-\partial_{\mathbf{x}_j}v(\mathbf{x}_j)$ , and  $\pmb{\mu}_j^\top A_{ji}$  with  $-\partial_{\mathbf{x}_i}v(\mathbf{x}_i)$ , yielding
$$
\begin{array}{l} \dot {L} _ {j} = - \partial_ {\mathbf {x} _ {j}} v (\mathbf {x} _ {j}) (\mathbf {x} _ {j} - \boldsymbol {\mu} _ {j}) + \left[ \sum_ {i \in V _ {j}} (- \partial_ {\mathbf {x} _ {i}} v (\mathbf {x} _ {i})) (\mathbf {x} _ {i} - \boldsymbol {\mu} _ {i}) \right] + \partial_ {\boldsymbol {\mu} _ {j}} v (\boldsymbol {\mu} _ {j}) (\mathbf {x} _ {j} - \boldsymbol {\mu} _ {j}) \\ + \sum_ {i \in V _ {j}} \partial_ {\mu_ {i}} v (\boldsymbol {\mu} _ {i}) \left(\mathbf {x} _ {i} - \boldsymbol {\mu} _ {i}\right) (94) \\ = \left(\partial_ {\mu_ {j}} v \left(\boldsymbol {\mu} _ {j}\right) - \partial_ {\mathbf {x} _ {j}} v \left(\mathbf {x} _ {j}\right)\right) \left(\mathbf {x} _ {j} - \boldsymbol {\mu} _ {j}\right) + \sum_ {i \in V _ {j}} \left(\partial_ {\mu_ {i}} v \left(\boldsymbol {\mu} _ {i}\right) - \partial_ {\mathbf {x} _ {i}} v \left(\mathbf {x} _ {i}\right)\right) \left(\mathbf {x} _ {i} - \boldsymbol {\mu} _ {i}\right) (95) \\ \end{array}
$$
Note that the function  $v$  is strictly concave such that its second derivative is negative definite. By this property,  $\dot{L}_j \geq 0$  with equality only if  $\mathbf{x}_i = \boldsymbol{\mu}_i, \forall i \in V_j$  and  $\mathbf{x}_j = \boldsymbol{\mu}_j$ . Thus, the time derivative of the function  $L$ , i.e.,  $\dot{L} = \sum_{j \in \mathcal{R}} \dot{L}_j \geq 0$  with equality only if  $\mathbf{x}_i = \boldsymbol{\mu}_i, \forall i \in V_j, \mathbf{x}_j = \boldsymbol{\mu}_j, \forall j \in \mathcal{R}$ . Q.E.D.
I Proof of Lemma 1
Definition 1. A nonautonomous system of differential equations in  $R^n$
$$
x ^ {\prime} = f (t, x) \tag {96}
$$
is said to be asymptotically autonomous with limit equation
$$
y ^ {\prime} = g (y), \tag {97}
$$
if  $f(t,x) \to g(x), t \to \infty$ , where the convergence is uniform on each compact subset of  $R^n$ . Conventionally, the solution flow of Eq. 96 is called the asymptotically autonomous semiflow (denoted by  $\phi$ ) and the solution flow of Eq. 97 is called the limit semiflow (denoted by  $\Theta$ ).
We first time-reparameterize the mean belief dynamics of heterogeneous systems. Assume  $\tau = \ln \frac{\lambda + t + 1}{\lambda + 1}$ . By the chain rule and Equation 61, for each dimension  $s_i$ ,
$$
\begin{array}{l} \frac {d \bar {\mu} _ {i s _ {i}}}{d \tau} = \frac {d \bar {\mu} _ {i s _ {i}}}{d t} \frac {d t}{d \tau} (98) \\ = \left[ \frac {f _ {s _ {i}} (\bar {\mu}) - \bar {\mu} _ {i s _ {i}}}{\lambda + t + 1} + \frac {\sum_ {j \in V _ {i}} \sum_ {s _ {j} \in S _ {j}} \frac {\partial^ {2} f _ {s _ {i}} (\bar {\mu})}{(\partial \mu_ {j s _ {j}}) ^ {2}} \operatorname {V a r} \left(\mu_ {j s _ {j}}\right)}{2 (\lambda + t + 1)} \right] \frac {d ((\lambda + 1) (\exp (\tau) - 1))}{d \tau} (99) \\ = \frac {f _ {s _ {i}} (\bar {\boldsymbol {\mu}}) - \bar {\mu} _ {i s _ {i}} + \frac {1}{2} \sum_ {j \in V _ {i}} \sum_ {s _ {j} \in S _ {j}} \frac {\partial^ {2} f _ {s _ {i}} (\bar {\boldsymbol {\mu}})}{(\partial \mu_ {j s _ {j}}) ^ {2}} \left(\frac {\lambda + 1}{\lambda + t + 1}\right) ^ {2} \sigma^ {2} (\mu_ {j s _ {j}})}{\lambda + (\lambda + 1) (\exp (\tau) - 1) + 1} (\lambda + 1) \exp (\tau) (1 0 0) \\ = f _ {s _ {i}} (\bar {\mu}) - \bar {\mu} _ {i s _ {i}} + \frac {1}{2} \sum_ {j \in V _ {i}} \sum_ {s _ {j} \in S _ {j}} \frac {\partial^ {2} f _ {s _ {i}} (\bar {\mu})}{(\partial \mu_ {j s _ {j}}) ^ {2}} \sigma^ {2} \left(\mu_ {j s _ {j}}\right) \exp (- 2 \tau). (101) \\ \end{array}
$$
Observe that  $\exp (-2\tau)$  decays to zero exponentially fast and that both  $\sigma^2 (\mu_{js_j})$  and  $\frac{\partial^2f_{s_i}(\bar{\mu})}{(\partial\mu_{js_j})^2}$  are bounded for every  $\pmb{\mu}$  in the simplex  $\prod_{j\in V_i}\Delta_j$ . Hence, Equation 101 converges locally and uniformly to the following equation:
$$
\frac {d \bar {\mu} _ {i s _ {i}}}{d \tau} = f _ {s _ {i}} (\bar {\mathbf {u}}) - \bar {\mu} _ {i s _ {i}}. \tag {102}
$$
Note that  $x_{is_i} = f_{si}(\bar{\mu})$  for a single representative agent, and thus the above equation is algebraically equivalent to the limit equation in Lemma 1 of the main paper. Q.E.D.
J Numerical Methods, Source Code, and Computing Resource
Numerical Method for the PDE model. Only limited types of PDEs allow analytic solutions. Hence, we numerically solve the PDE using the finite difference method [80]. The theoretical predictions in Figure 1 of the main paper are generated using the finite difference method given a specific initial setting (the initial sum of weights is  $\lambda = 10$ , the temperature is  $\beta = 10$ , the initial belief distribution is specified in the caption of the figures).
Source Code and Computing Resource. The source code for reproducing our main experiments is included in the supplementary material (available at https://sites.google.com/view/shuyue-hu). The Matlab script `finitedifference.m` numerically solves our continuity equation model. Figure 1 of the main paper is generated using the Origins Lab based on the numerical solution of the continuity equations. The Matlab script `regionofattraction.m` visualizes the region of attraction of different equilibria in stag hunt games, which are depicted in Figure 2. The Python scripts `simulation(staghunt.py` and `simulation(matchingpennies.py` correspond to the agent-based simulations in two-population stag hunt games and five-population asymmetric matching pennies games, respectively. We use a laptop (CPU: AMD Ryzen 75800H) to run all the experiments.