Effect of shape asymmetry on percolation of aligned and overlapping objects on lattices

We investigate the percolation transition of aligned, overlapping, non-symmetrical shapes on lattices. Using the recently proposed lattice version of excluded volume theory, we show that shape-asymmetry leads to some intriguing consequences regarding the percolation behavior of asymmetric shapes. We consider a prototypical asymmetric shape - rectangle - on a square lattice and show that for rectangles of width unity (sticks), the percolation threshold is a monotonically decreasing function of the stick length, whereas, for rectangles of width greater than two, it is a monotonically increasing function. Interestingly, for rectangles of width two, the percolation threshold is independent of its length. The limiting case of the length of the rectangles going to infinity shows that the limiting threshold value is finite and depends upon the width of the rectangle. Unlike the case of symmetrical shapes like squares, there seems to be no continuum percolation problem that corresponds to this limit. We show that similar results hold for other asymmetric shapes and lattices. The critical properties of the aligned and overlapping rectangles are evaluated using Monte Carlo simulations. We find that the threshold values given by the lattice-excluded volume theory are in good agreement with the simulation results, especially for larger rectangles. We verify the isotropy of the percolation threshold and also compare our results with models where rectangles of mixed orientation are allowed. Our simulation results show that alignment increases the percolation threshold. The calculation of critical exponents places the model in the standard percolation universality class. Our results show that shape-anisotropy of the aligned, overlapping percolating units has a marked influence on the percolation properties, especially when a subset of the dimensions of the percolation units are made to diverge.Comment: 12 pages, 10 figures, 3 table

Cooperation amongst competing agents in minority games

We study a variation of the minority game. There are N agents. Each has to choose between one of two alternatives everyday, and there is reward to each member of the smaller group. The agents cannot communicate with each other, but try to guess the choice others will make, based only the past history of number of people choosing the two alternatives. We describe a simple probabilistic strategy using which the agents acting independently, can still maximize the average number of people benefitting every day. The strategy leads to a very efficient utilization of resources, and the average deviation from the maximum possible can be made of order $(N^{\epsilon})$, for any $\epsilon >0$. We also show that a single agent does not expect to gain by not following the strategy.Comment: 7 pages, 5 eps figure

Emergence of Cooperation as a Non-equilibrium Transition in Noisy Spatial Games

The emergence of cooperation among selfish agents that have no incentive to cooperate is a non-trivial phenomenon that has long intrigued biologists, social scientists and physicists. The iterated Prisoner's Dilemma (IPD) game provides a natural framework for investigating this phenomenon. Here, agents repeatedly interact with their opponents, and their choice to either cooperate or defect is determined at each round by knowledge of the previous outcomes. The spatial version of IPD, where each agent interacts only with their nearest neighbors on a specified connection topology, has been used to study the evolution of cooperation under conditions of bounded rationality. In this paper we study how the collective behavior that arises from the simultaneous actions of the agents (implemented by synchronous update) is affected by (i) uncertainty, measured as noise intensity K, (ii) the payoff b, quantifying the temptation to defect, and (iii) the nature of the underlying connection topology. In particular, we study the phase transitions between states characterized by distinct collective dynamics as the connection topology is gradually altered from a two-dimensional lattice to a random network. This is achieved by rewiring links between agents with a probability p following the small-world network construction paradigm. On crossing a specified threshold value of b, the game switches from being Prisoner's Dilemma, characterized by a unique equilibrium, to Stag Hunt, a well-known coordination game having multiple equilibria. We observe that the system can exhibit three collective states corresponding to a pair of absorbing states (viz., all agents cooperating or defecting) and a fluctuating state characterized by agents switching intermittently between cooperation and defection. As noise and temptation can be interpreted as temperature and an external field respectively, a strong analogy can be drawn between the phase diagrams of such games with that of interacting spin systems. Considering the 3-dimensional p − K − b parameter space allows us to investigate the different phase transitions that occur between these collective states and characterize them using finite-size scaling. We find that the values of the critical exponents depend on the connection topology and are different from the Directed Percolation (DP) universality class