Steady Marginality: A Uniform Approach to Shapley Value for Games with Externalities

The Shapley value is one of the most important solution concepts in cooperative game theory. In coalitional games without externalities, it allows to compute a unique payoff division that meets certain desirable fairness axioms. However, in many realistic applications where externalities are present, Shapley's axioms fail to indicate such a unique division. Consequently, there are many extensions of Shapley value to the environment with externalities proposed in the literature built upon additional axioms. Two important such extensions are "externality-free" value by Pham Do and Norde and value that "absorbed all externalities" by McQuillin. They are good reference points in a space of potential payoff divisions for coalitional games with externalities as they limit the space at two opposite extremes. In a recent, important publication, De Clippel and Serrano presented a marginality-based axiomatization of the value by Pham Do Norde. In this paper, we propose a dual approach to marginality which allows us to derive the value of McQuillin. Thus, we close the picture outlined by De Clippel and Serrano

Using Virtual Addresses with Communication Channels

While for single processor and SMP machines, memory is the allocatable quantity, for machines made up of large amounts of parallel computing units, each with its own local memory, the allocatable quantity is a single computing unit. Where virtual address management is used to keep memory coherent and allow allocation of more than physical memory is actually available, virtual communication channel references can be used to make computing units stay connected across allocation and swapping.Comment: 5 pages, 4 figure

The noisy edge of traveling waves

Traveling waves are ubiquitous in nature and control the speed of many important dynamical processes, including chemical reactions, epidemic outbreaks, and biological evolution. Despite their fundamental role in complex systems, traveling waves remain elusive because they are often dominated by rare fluctuations in the wave tip, which have defied any rigorous analysis so far. Here, we show that by adjusting nonlinear model details, noisy traveling waves can be solved exactly. The moment equations of these tuned models are closed and have a simple analytical structure resembling the deterministic approximation supplemented by a nonlocal cutoff term. The peculiar form of the cutoff shapes the noisy edge of traveling waves and is critical for the correct prediction of the wave speed and its fluctuations. Our approach is illustrated and benchmarked using the example of fitness waves arising in simple models of microbial evolution, which are highly sensitive to number fluctuations. We demonstrate explicitly how these models can be tuned to account for finite population sizes and determine how quickly populations adapt as a function of population size and mutation rates. More generally, our method is shown to apply to a broad class of models, in which number fluctuations are generated by branching processes. Because of this versatility, the method of model tuning may serve as a promising route toward unraveling universal properties of complex discrete particle systems.Comment: For supplementary material and published open access article, see http://www.pnas.org/content/108/5/1783.abstract?sid=693e63f3-fd1a-407a-983e-c521efc6c8c5 See also Commentary Article by D. S. Fisher, http://www.pnas.org/content/108/7/2633.extrac

Neighbor selection and hitting probability in small-world graphs

Small-world graphs, which combine randomized and structured elements, are seen as prevalent in nature. Jon Kleinberg showed that in some graphs of this type it is possible to route, or navigate, between vertices in few steps even with very little knowledge of the graph itself. In an attempt to understand how such graphs arise we introduce a different criterion for graphs to be navigable in this sense, relating the neighbor selection of a vertex to the hitting probability of routed walks. In several models starting from both discrete and continuous settings, this can be shown to lead to graphs with the desired properties. It also leads directly to an evolutionary model for the creation of similar graphs by the stepwise rewiring of the edges, and we conjecture, supported by simulations, that these too are navigable.Comment: Published in at http://dx.doi.org/10.1214/07-AAP499 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org