At the heart of competition for resources lie two simple principles: "survival of the fittest", which reflects how intrinsic fitnesses of competing agents influence their ability to gather resources; and "cumulative advantage", which reflects how accumulated resources help gather even more resources. In this letter we show that competitions embodying just the essence of these two principles exhibit a variety of surprising behaviors with respect to their duration, as measured by how long it takes for an indisputable winner to emerge, and intensity, as measured by how many times agents have equal amount of resources. We demonstrate that when the fitnesses of the agents are different, long-lasting competitions continue to occur in spite of a dramatic decrease in their intensity and the certainty that the fitter agent will win. We also unveil an unexpected phenomenon that we call "struggle of the fittest", where long-lasting competitions become more probable when one agent is slightly fitter than when agents are equally fit. These results have important implications for our understanding of competitions and various applications that leverage such models. Competition for resources is a ubiquitous phenomenon that naturally arises in many scientific disciplines. A variety of models for competition have been proposed to study phenomena ranging from proteins binding within a cell [1, 2] and companies competing for customers [3] to papers competing for citations [4, 5] . These models generally have the following two ingredients (also present in most versions of the Pólya's urn model [6] ): agent-dependent intrinsic fitness reflecting the diversity in their ability to gather resources; and the principle of cumulative advantage (CA), where accumulated resources (wealth) promote gathering even more resources. The latter appears in the literature under many variants such as Price's cumulative advantage model [4] , preferential attachment [7, 8] , "the rich get richer" [9] , and Matthew effect [10] . Such models are arguably among the most widely studied and applied stochastic models to capture the essence of chance competitions in the presence of cumulative advantage, henceforth referred to as CA competitions. Previous works on CA competitions generally focus on percentages of wealth gathered by different agents, known as market share, including convergence results and limit distributions [11] [12] [13] [14] . However, two important metrics naturally associated with competitions, duration, which measures how long it takes for a clear winner to emerge, and intensity, which measures how many times the competitors are neck and neck, have largely been neglected in the literature. Previous results, to the best of our knowledge, establish only that both duration and intensity are finite, always with the fittest agent ultimately winning [11, 15] , and that average intensity up to time t is approximately (log t) α , where α depends on the relative fitnesses of the competitors [16, 17] . In this letter we characterize the distributions of duration and intensity of CA competitions. This work shows, both analytically and through simulations, that duration and intensity of CA competitions exhibit a host of unexpected behaviors. In particular, competition duration is always heavy-tailed, regardless of fitness values. In contrast, competition intensity is heavy-tailed only when agents are equally fit and lighttailed otherwise. Hence, competitions between agents with different fitnesses can be significantly long but have very few ties. Interestingly, we show that when fitnesses are similar but not equal, the distribution of competition duration exhibits a significantly heavier tail than when agents are equally fit. In such scenarios the fitter agent, despite being destined to be the ultimate winner, may only achieve victory by staying in the competition for an exceptionally long time, a phenomenon that we call "struggle of the fittest". Our results are formally obtained by considering a competition between two agents X and Y with respective positive fitnesses f X and f Y . Let r = f X /f Y denote the fitness ratio. Without loss of generality, we assume that f X ≥ f Y and hence r ≥ 1. Wealth is measured in discrete units, and the competition starts at time t = 0 with agents X and Y having x 0 and y 0 units of initial wealth, respectively. At each time step, one unit of wealth is added to the system and given to either X or Y . More specifically, let the cumulative wealth of X and Y at time t be X t and Y t , respectively. The unit of wealth
The metrics under investigation in this paper, duration and intensity, have been studied in REF for linear Pólya urn process with fitness.
On the duration and intensity of cumulative advantage competitions
arXiv: Physics and Society
Mathematics Physics Biology