Modeling Game Theory and Competitive-Cooperative Behaviors in Crowd Evacuation

Problem Description
In emergency evacuation scenarios, individuals may exhibit competitive or cooperative behaviors when faced with limited resources such as exits. Competitive behaviors (e.g., pushing, jostling) can lead to congestion and conflicts, reducing overall evacuation efficiency, while cooperative behaviors (e.g., taking turns, yielding) may enhance system throughput. Game theory provides a mathematical framework for analyzing individual decision-making logic under conflicting interests. This topic requires modeling competitive and cooperative behaviors during evacuation using game theory, analyzing the impact of strategy choices on crowd dynamics, and exploring mechanism design to promote cooperation.

Problem-Solving Process

1. Problem Formalization and Definition of Game Elements

   • Participants: Abstract individuals in evacuation as game players (e.g., pedestrian A, B).
   • Strategy Set: Each participant can choose a competitive strategy (e.g., accelerating to seize the exit) or a cooperative strategy (e.g., slowing down to yield).
   • Payoff Function: Quantify the utility of strategy choices, typically considering time benefit (quick passage), risk cost (collision penalty), and moral cost (psychological burden). For example:
     • If both cooperate, each receives a moderate time benefit (e.g., payoff = 3);
     • If one competes and the other cooperates, the competitor gains a high payoff (e.g., 5), while the cooperator receives a low payoff (e.g., 1);
     • If both compete, both may suffer losses (e.g., each gets -1).
   • Such a structure can be simplified to a prisoner's dilemma game, explaining why individual rationality leads to collective irrationality.

2. Game Equilibrium Analysis

   • Nash Equilibrium: An equilibrium is reached when no player can increase their payoff by unilaterally changing their strategy, given the strategies of others. In a typical prisoner's dilemma, (compete, compete) is the only Nash equilibrium but is not globally optimal.
   • Calculate equilibrium points using a payoff matrix, for example:
     • If A competes, B's best response is to compete (to avoid a payoff of 1);
     • If A cooperates, B still prefers to compete (payoff 5 > 3), making competition a dominant strategy.
   • This illustrates the inevitability of individual rational choices leading to crowd congestion.

3. Dynamic Games and Repeated Interactions

   • In repeated interactions (e.g., sequential decisions at multiple exits), introduce a "reputation" mechanism: current cooperative behavior may affect future payoffs.
   • Use repeated game models (e.g., iterated prisoner's dilemma) to analyze the long-term effects of strategies like "tit-for-tat."
   • Quantify the present value of future payoffs using a discount factor to demonstrate that long-term cooperation can become an equilibrium solution.

4. Evolutionary Game Theory and Evolution of Group Behavior

   • Treat individual strategies as heritable traits, assuming dynamic changes in the proportion of cooperative/competitive strategies within the group.
   • Construct replicator dynamics equations: the growth rate of a strategy depends on the difference between its payoff and the group's average payoff.
   • Analyze stable equilibrium points (e.g., the proportion of cooperators stabilizing at a certain level) and explore the impact of initial proportions and payoff structures on outcomes.

5. Mechanism Design and Intervention Strategies

   • Payoff Adjustment: Alter the payoff function through external interventions, such as imposing penalties on competitive behaviors (e.g., intervention by safety personnel) or rewarding cooperators (e.g., priority passage rights).
   • Information Transparency: Disclose real-time congestion information to reduce "blind competition" caused by information asymmetry.
   • Spatial Design: Optimize exit widths and set up buffer zones to reduce the intensity of resource competition.
   • Social Norm Guidance: Strengthen cooperative norms through training or signage, influencing moral cost parameters.

6. Model Validation and Simulation

   • Embed game logic into individual decision-making modules using multi-agent simulation platforms (e.g., NetLogo).
   • Adjust payoff parameters or intervention strategies to observe changes in indicators such as group evacuation time and conflict frequency.
   • Validate model robustness through sensitivity analysis and compare with empirical data (e.g., video analysis).

This modeling approach reveals the connection between micro-level decisions and macro-level phenomena, providing a theoretical basis for designing efficient evacuation systems.