## 3 Preliminaries
### Problem Setting  
To evaluate our method, we employ the object collection game where agents collect as many objects as possible in a grid-like environment of size \(G_{\mathrm{X}}\times G_{\mathrm{Y}}\), as shown in Figure 1a, where \(G_{\mathrm{X}}=G_{\mathrm{Y}}=25\). At the beginning of each episode, agents are placed at the initial positions indicated by numbered blue and red cells in Figure 1a and begins exploration in the environment. At each time step, agents decide their action \(a_{i}\in\mathcal{A}_{i}=\{up,down,right,left\}\), wherein each element describes the movement direction of agents. Agent \(\forall i\) receives a reward \(r_{i}(s,a)\) depending on the state \(s^{\prime}\in\mathcal{S}\) at the next time step after the joint action \(a=(a_{1},\ldots,a_{i},\ldots,a_{N})\in\mathcal{A}\); i.e., \(i\) obtains a positive reward \(r_{i}(s,a)=r_{\mathrm{obs}}>0\) if it moves to the same position as an object to collect in state \(s^{\prime}\); it receives a negative reward \(r_{i}(s,a)=r_{\mathrm{col}}<0\) if it collides against other agents or walls in \(s^{\prime}\); otherwise, it receives \(r_{i}(s,a)=0\). Despite the simple tasks in this game and the required coordination structure being straightforward, it makes it easier to reason why the individual agents analyzed the specific parts of input data to decide actions for the evaluation of the proposed method, eDA3-X.  
At each step, agents encode their observations of the environment in \(N_{\text{C}}\) channels of \(R_{\text{X}}\times R_{\text{Y}}\) binary matrices in \(\{0,1\}\) or \(\{0,-1\}\) (where \(R_{\text{X}}\times R_{\text{Y}}\) is the size of the observation matrices \(\in\mathbb{R}^{R_{\text{X}}\times R_{\text{Y}}}\)) and feed their observation tensor in shape of \(N_{\text{C}}\times R_{\text{X}}\times R_{\text{Y}}\) into their neural networks. In this study, we introduce two types of observation methods, namely, _local view_ and _relative view_ methods [71], as shown in Figure 1b,c. Note that the leftmost figure in Figure 1b,c depicts the example environment, whose size is \(\{G_{\text{X}}\times G_{\text{Y}}\}=\{10\times 10\}\), including the walls; thus, they are smaller than that in Figure 1a. In these figures, black cells are walls, black star-shaped marks are objects, and blue and red numbered cells are agents in the environment.  
Local view: Agents generate the matrices of the local view based on their local observations; thus, \(R_{\text{X}}\times R_{\text{Y}}\) is the same size as the visible range. Figure 1b shows example environment and matrices when \(\{R_{\text{X}}\times R_{\text{Y}}\}=\{7\times 7\}\), where the blue square line indicates _agent_ 0's surrounding sight (visible range). Suppose \(N_{\text{C}}=5\), _agent_ 0 obtains its local observation matrices where the first three indicate agents' position, the fourth channels indicate objects' position, and the fifth channel shows walls and invisible area. Note that the area behind the walls cannot be observed.  
Relative view: Unlike with the local view method, agents obtain their observation matrices in the shape of \(\{R_{\text{X}}\times R_{\text{Y}}\}=\{G_{\text{X}}\times G_{\text{Y}}\}\) with the relative view method. An example is shown in Figure 1c, where the red square indicates _agent_ 2's visible range, whose size is \(7\times 7\) and \(\{R_{\text{X}}\times R_{\text{Y}}\}=\{10\times 10\}\). The encoding mechanism of the relative view is similar to that of the local view, where each channel indicates the locations of agents, objects, or walls within the visible range; other elements are filled with zeros. As an advantage of the relative view method, agents observe their global positions in the environment.