This paper presents a new model for computing optimal randomized security policies in non-cooperative Stackelberg Security Games (SSGs) for multiple players. Our framework rests upon the extraproximal method and its extension to Markov chains, within which we explicitly compute the unique Stackelberg/Nash equilibrium of the game by employing the Lagrange method and introducing the Tikhonov regularization method. We also consider a game-theory realization of the problem that involves defenders and attackers performing a discrete-time random walk over a finite state space. Following the Kullback-Leibler divergence the players' actions are fixed and, then the next-state distribution is computed. The player's goal at each time step is to specify the probability distribution for the next state. We present an explicit construction of a computationally efficient strategy under mild defenders and attackers conditions and demonstrate the performance of the proposed method on a simulated target tracking problem.
Males of the small copper butterfly, Lycaena phlaeas daimio, exhibit two mate-locating tactics: patrolling and perching. Field investigations were conducted to determine the biotic and abiotic factors affecting the mate-locating behaviour of male L. phlaeas. Patrolling was often observed when light intensity was high. Perching was performed throughout the day regardless of environmental conditions, but the chasing of passing insects increased at high light intensities. The activity patterns of the males were not affected by those of the females. The thoracic temperatures of patrolling males were lower than those of perching males under cool conditions, suggesting that patrolling males lose heat more easily. In contrast, perching males may more easily regulate their body temperature to a suitable level as they fly for shorter periods and can bask while waiting for mates. These results highlight several reasons (i.e., heat loss, energetic costs) why males patrol when weather conditions are favourable.