Patient and impatient pedestrians in a spatial game for egress congestion

Large crowds evacuating through narrow bottlenecks may create clogging and jams that slow down the egress flow. Especially if people try to push towards the exit, the so-called faster-is-slower effect may occur. We propose a spatial game to model the interaction of agents in such situations. Each agent has two possible modes of play that lead to either patient or impatient behavior. The payoffs of the game are derived from simple assumptions and correspond to a hawk-dove game, where the game parameters depend on the agent's location in the crowd and on external conditions. Equilibrium configurations are computed with a myopic best-response rule and studied in both a continuous space and a discrete lattice. We apply the game model to a continuous-time egress simulation, where the patient and impatient agents are given different individual parameter values, which are updated according to the local conditions in the crowd. The model shows how threatening conditions can increase the proportion of impatient agents, which leads to clogging and reduced flows through bottlenecks, even when smooth flows would be possible.

Figure 1

Illustration of the parameters of the cost function. The function in the figure has the parameter values $T_{\mathrm{ASET}}=100$, $T_0=50$, and $C=2$.

Figure 2

The black squares show the average fraction of hawks (impatient agents) in equilibrium configurations as a function of the parameter $\Delta u\left(T_{ij}\right)/C$. The values are averages of 20 simulations in a $50\times 50$-cell lattice with periodic boundary conditions. The dashed line describes the fraction of hawks in the game's mixed strategy equilibrium, which is achieved in a nonspatial well-mixed setting.

Figure 3

An equilibrium configuration for 3180 agents with parameter values $T_{\mathrm{ASET}}=T_0=2800$ s. Black squares represent impatient agents and white squares represent patient agents. The value of the parameter $C/\Delta u\left(T_{ij}\right)$ depends on the distance to the exit and the areas with different black-white patterns correspond to the eight different levels of hawks in Fig 2.

Figure 4

Equilibrium configurations with 628 agents for different values of the parameters $T_{\mathrm{ASET}}$ and $T_0$. The black squares represent impatient agents and the white squares represent patient agents.

Figure 5

Equilibrium configurations in a continuous setting with 628 agents. The black agents are impatient and the gray ones are patient.

Figure 6

The egress of 50 patient and 50 impatient agents with fixed strategies from a room with one 1.2-m-wide exit. The graph displays the average number of patient and impatient agents inside the room over time.

Figure 7

Average flow for 200 agents through a 0.8-m-wide exit with different proportions of impatient agents in the population.

Figure 8

Results of simulations with a time-dependent available safe egress time $T_{\mathrm{ASET}}$. Two hundred agents were set to evacuate through a 0.8-m-wide exit and, in different scenarios, the value of $T_{\mathrm{ASET}}$ changed over time. The results for each scenario are averages over 50 simulations. The figures display the cumulative number of evacuated agents (the curves starting from the origin) and the percentage of impatient agents among the agents inside the room. Figure 8a shows three scenarios where the value of $T_{\mathrm{ASET}}$ linearly decreases over time. Figure 8b shows two scenarios with linearly increasing $T_{\mathrm{ASET}}$. The scenarios are presented in Table . For reference, both figures display the average cumulative exit counts with populations consisting of only impatient and only patient agents.

Figure 9

The cumulative exit counts for each of the 50 simulations of (a) scenario 1 and (b) scenario 5.