# ESCAPE: Evacuation Strategy through Clustering and Autonomous Operation in Public Safety Systems
## 4 Evacuate or Not? A Minority Game Approach  
Based on the reinforcement-learning component of evacuation planning as presented in Section 3, evacuees select the evacuation route that they potentially want to follow be guided to a safe place. Thus, the reinforcement-learning component implicitly acts as a clustering mechanism of evacuees per evacuation route. At this stage, evacuees act in a selfish and competitive manner in terms of finally being able to evacuate or not through the evacuation route that they initially selected. Evacuees that finally do not evacuate the disaster area at the current time slot, due to the created bottleneck and competition, re-enter the reinforcement-learning-based decision process in the next time slot. Evacuees' competitive behavior is captured through non-co-operative game theory and, specifically, a minority game-theoretic approach is introduced. The main concept of the proposed minority game is that a number of evacuees per evacuation route repeatedly compete in order to be in the minority group by selecting a strategy among the two available ones, i.e., evacuate or not. Each evacuee makes an autonomous decision regarding the strategy that they should follow, being agnostic of the strategies that the rest of the evacuees have selected on joining a specific evacuation route through the reinforcement-learning component (Section 3), based only on the history of the game's outcome. The evacuees that belong to the minority group win and promote their winning action for the next iteration of the game. It is shown in [24] that a minority game has a nonempty set of Pure Nash Equilibria.  
Let \(G_{e}^{t}=[|M|_{e}(t),\{S_{m}\}_{m\in M},\{f_{s_{m}}(m)\}_{m\in M}]\) denote the minority game per each evacuation route \(e\) that is played at each time slot \(t\) of the reinforcement-learning component. Game \(G_{e}^{t}\) consists of the set of evacuees/players \(|M|_{e}(t)\) that have selected in time slot \(t\) that they want to evacuate through evacuation route \(e\), their set of strategies \(S_{m}=\{0,1\}\), and the payoff function \(f_{S_{m}}(m)\). The strategy of each evacuee at each iteration \(ite\) of minority game \(G_{e}^{m}\) is denoted as \(s_{m}^{(ite)},s_{m}^{(ite)}\in S_{m}\). An evacuee can finally select not to evacuate through the desired evacuation route, i.e., \(s_{m}^{(ite)}=0\), or to evacuate, i.e., \(s_{m}^{(ite)}=1\). For each strategy \(s_{m}^{(ite)}\in S_{m}\), there is a function \(f_{s_{m}}:\{1,\ldots\ m,\ldots\ |M|\}\rightarrow\mathbb{R}\), where, for each \(m\), \(m\in M\), the number \(f_{s_{m}}(m)\in\mathbb{R}\) expresses the payoff that evacuee \(m\) receives. Payoff function \(f_{s_{m}}(m)\) is formulated as follows.  
\[f_{s_{m}}(m)=\begin{cases}1,&m\in\{1,\ldots\kappa\}.\\ \\ 0,&m\in\{\kappa+1,\ldots\ |M|_{e}\}.\end{cases} \tag{3}\]  
where \(\kappa\leq C_{e}-|M|_{e}(t)\) and \(C_{e}-|M|_{e}(t)\) is the threshold number of evacuees that can physically go through evacuation route \(e\) at time slot \(t\). The distributed and low-complexity algorithm to determine the Pure Nash Equilibrium of minority game \(G_{e}^{t}\) is presented and discussed in detail in Section 5.