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In this paper we study Nash equilibria in auctions from the kinetic theory of active particles point of view. We propose a simple learning rule for agents to update their bidding strategies based on their previous successes and failures, in first-price auctions with two bidders. Then, we formally derive the corresponding kinetic equations which describe the evolution over time of the distribution of agents on the bidding strategies. We show that the stationary solution of the equation corresponds to the symmetric Nash equilibrium of the auction, and we prove the convergence to this stationary solution when time goes to infinity. We also introduce a more general learning rule that only depends on the income of agents, and we apply to both first- and second-price auctions. We show that agents learn the Nash equilibrium in first- and second-price auctions with these rules. We present agent-based simulations of the models, and we discuss several open problems. [ABSTRACT FROM AUTHOR]

A kinetic modeling of crowd evacuation with leaders and followers is considered in this paper, in which the followers may not know the full information about the walking environment and the evacuation strategy, but they follow the leaders and learn the walking strategy to get out of the walking venue. Based on the kinetic theory of active particles, the learning dynamics are considered by introducing an activity variable u , which represents the learning level of the followers and measures how much knowledge a follower has learned about the walking strategy, the walking environment, or the geometry of the walking venue. Several fundamental factors are considered in this leader–follower learning system of crowd evacuation, including: (1) the rational motion of all pedestrians, i.e. the trend to the exit or to a preferred direction, the ability to avoid collisions with walls or obstacles, and the tendency to search for less crowded direction with minimal density gradient, (2) the irrational motion of followers to follow other pedestrians induced by panic, (3) the learning dynamics of the followers who learn the walking strategy during interaction with others, and, (4) the transition from a follower to a leader when one's activity reaches the highest level of learning. A numerical comparison of a metro platform evacuation with and without learning shows a reasonably good predictive ability of the model that the learning effect plays a significant role in the evacuation dynamics. [ABSTRACT FROM AUTHOR]

The modeling of living systems composed of many interacting entities is treated in this paper with the aim of describing their collective behaviors. The mathematical approach is developed within the general framework of the kinetic theory of active particles. The presentation is in three parts. First, we derive the mathematical tools, subsequently, we show how the method can be applied to a number of case studies related to well defined living systems, and finally, we look ahead to research perspectives. [ABSTRACT FROM AUTHOR]