# Artificial Empathy and Imprecise Communication in a Multi-Agent System
## 3 Proposed Model
### Agent Decision Making  
At any given moment, an agent in state \((\mathbf{X},\mathbf{Y})\) faces the task of selecting its next action. This decision is guided by the reward function \(R:\mathbb{R}\times\mathbb{R}\to[-1,1]\), which evaluates the potential effectiveness of the current state for future action. The rationale behind \(R\) is that it considers the current state \((\mathbf{X},\mathbf{Y})\) as the initial condition in a prospective state sequence. Within this sequence, while the agent's cognitive state (\(\mathbf{Y}\)) remains constant, its emotional state (\(\mathbf{X}\)) can change. Each potential sequence of states is assessed using the goal operator \(G^{\mathcal{A}}(S)\), to identify the sequence that maximizes this value, hence defining the reward. It is important to note that the goal operator \(G^{\mathcal{A}}(S)\), through its prefix evaluation mechanism, is also good at assessing actions that have just started. Thus, the reward function is formalized as:  
\[R^{\mathcal{A}}(\mathbf{X},\mathbf{Y})=\sup_{ \begin{subarray}{c}S=((\mathbf{X },\mathbf{Y}),(\mathbf{X}_{i},\mathbf{Y}),\dots,\\ S\text{ is a feasible state sequence}\end{subarray} }G^{\mathcal{A}}(S)\,. \tag{11}\]  
This formulation encapsulates the process of maximizing the goal operator over all feasible state sequences starting from the current state, thereby providing a metric for the short-term strategic value of a current state in the context of the agent's long-term objectives predefined by \(G^{\mathcal{A}}(S)\). To demonstrate empathetic behavior, an agent must have a target for its empathy. This target has to communicate its state, perhaps through a distress signal or a self-descriptive status update. The agent exhibiting empathy (\(\mathcal{A}\)) needs to process, visualize, or assess the state of the target (\(\mathcal{B}\)) using its knowledge base. This process forms a fundamental aspect of cognitive empathy. In the framework of our empathetic model, agent \(\mathcal{A}\) evaluates the target \(\mathcal{B}\)'s communicated state by contrasting it with its knowledge base. The reward for an empathetic response is computed using the same principles as the agent's rewards, as defined in (11). These rewards, calculated for both the empathizing agent and the target of empathy, are crucial in guiding \(\mathcal{A}\)'s subsequent actions. The agent then faces a choice between two behavioral strategies--either pursuing self-interest (egoistic behavior) or engaging in empathetic cooperation--depending on which strategy offers a higher reward. For each agent in its neighborhood:  
\[\mathcal{N}^{\mathcal{A}}=\Big{\{}\mathcal{B}_{i}=\Big{(}\mathbf{X}^{ \mathcal{B}_{i}},\mathbf{Y}^{\mathcal{B}_{i}}\Big{)},\quad i=1,\dots,\
u\Big{\}} \tag{12}\]  
\(\mathcal{A}\) compares three key reward values:  
\[R^{\mathcal{A}}(X^{\mathcal{A}},Y^{\mathcal{A}})-\text{potential reward for $\mathcal{A}$'s egoistic action}, \tag{13}\] \[R^{\mathcal{A}}(X^{\mathcal{B}_{i}},Y^{\mathcal{B}_{i}})- \mathcal{A}\text{'s projection of $\mathcal{B}_{i}$ current reward},\] (14) \[R^{\mathcal{A}}(X^{\mathcal{A}},Y^{\mathcal{B}_{i}})- \mathcal{A}\text{'s projection of reward for helping $\mathcal{B}_{i}$}. \tag{15}\]  
The first value reflects \(\mathcal{A}\)'s current prospects independent of its neighbors, while the subsequent two represent \(\mathcal{A}\)'s projections for \(\mathcal{B}_{i}\) situation. The first of these projections is an estimate of \(\mathcal{B}_{i}\)'s reward based on \(\mathcal{A}\)'s knowledge, as \(\mathcal{A}\) only knows \(\mathcal{B}_{i}\)'s state, not its knowledge. The last value predicts the reward \(\mathcal{A}\) might receive for assisting \(\mathcal{B}_{i}\), using its emotional state and assuming the cognitive state of \(\mathcal{B}_{i}\).