Lower Bounds on Implementing Robust and Resilient Mediators

We consider games that have (k,t)-robust equilibria when played with a mediator, where an equilibrium is (k,t)-robust if it tolerates deviations by coalitions of size up to k and deviations by up to $t$ players with unknown utilities. We prove lower bounds that match upper bounds on the ability to implement such mediators using cheap talk (that is, just allowing communication among the players). The bounds depend on (a) the relationship between k, t, and n, the total number of players in the system; (b) whether players know the exact utilities of other players; (c) whether there are broadcast channels or just point-to-point channels; (d) whether cryptography is available; and (e) whether the game has a $k+t)-punishment strategy; that is, a strategy that, if used by all but at most$k+t$ players, guarantees that every player gets a worse outcome than they do with the equilibrium strategy

One more fitting (D=5) of Supernovae red shifts

Supernovae red shifts are fitted in a simple 5D model: the galaxies are assumed to be enclosed in a giant S^3-spherical shell which expands (ultra) relativistically in a (1+4)D Minkowski space. This model, as compared with the kinematical (1+3)D model of Prof Farley, goes in line with the Copernican principle: any galaxy observes the same isotropic distribution of distant supernovae, as well as the same Hubble plot of distance modulus \mu vs red shift z. A good fit is obtained (no free parameters); it coincides with Farley's fit at low z, while shows some more luminosity at high z, leading to 1% decrease in the true distance modulus (and 50% increase in luminosity) at z\sim 2. The model proposed can be also interpreted as a FLRW-like model with the scale factor a(t)=t/t_0; this could not be a solution of general relativity (5D GR is also unsuitable--it has no longitudinal polarization). However, there still exists the other theory (with D=5 and no singularities in solutions), the other game in the town, which seems to be able to do the job.Comment: 5 pages, 3 figure

On a game of chance in Marc Elsberg’s thriller “GREED”

A (possibly illegal) game of chance, which is described in Chap. 14 of Marc Elsberg’s thriller “GREED”, seems to offer an excellent chance of winning. However, as the gambling starts and evolves over several rounds, the actual experience of the vast majority of the gamblers in a pub is strikingly different. We provide an analysis of this specific game and several of its variants by elementary tools of probability. Thus we also encounter an interesting threshold phenomenon, which is related to the transition from a profit zone to a loss area. Our arguments are motivated and illustrated by numerical calculations with Python

Internal DLA and the Gaussian free field

In previous works, we showed that the internal DLA cluster on \Z^d with t particles is a.s. spherical up to a maximal error of O(\log t) if d=2 and O(\sqrt{\log t}) if d > 2. This paper addresses "average error": in a certain sense, the average deviation of internal DLA from its mean shape is of constant order when d=2 and of order r^{1-d/2} (for a radius r cluster) in general. Appropriately normalized, the fluctuations (taken over time and space) scale to a variant of the Gaussian free field.Comment: 29 pages, minor revisio