We use Kolmogorov complexity methods to give a lower bound on the effective Hausdorff dimension of the point (x, ax + b), given real numbers a, b, and x. We apply our main theorem to a problem in fractal geometry, giving an improved lower bound on the (classical) Hausdorff dimension of generalized sets of Furstenberg type.

We formulate the conditional Kolmogorov complexity of x given y at precision r, where x and y are points in Euclidean spaces and r is a natural number. We demonstrate the utility of this notion in two ways.1. We prove a point-to-set principle that enables one to use the (relativized, constructive) dimension of a single point in a set E in a Euclidean space to establish a lower bound on the (classical) Hausdorff dimension of E. We then use this principle, together with conditional Kolmogorov complexity in Euclidean spaces, to give a new proof of the known, two-dimensional case of the Kakeya conjecture. This theorem of geometric measure theory, proved by Davies in 1971, says that every plane set containing a unit line segment in every direction has Hausdorff dimension 2. 2. We use conditional Kolmogorov complexity in Euclidean spaces to develop the lower and upper conditional dimensions dim(x|y) and Dim(x|y) of x given y, where x and y are points in Euclidean spaces. Intuitively these are the lower and upper asymptotic algorithmic information densities of x conditioned on the information in y. We prove that these conditional dimensions are robust and that they have the correct information-theoretic relationships with the wellstudied dimensions dim(x) and Dim(x) and the mutual dimensions mdim(x : y) and Mdim(x : y).

When selecting locations for a set of facilities, standard clustering algorithms may place unfair burden on some individuals and neighborhoods. We formulate a fairness concept that takes local population densities into account. In particular, given k facilities to locate and a population of size n, we define the "neighborhood radius" of an individual i as the minimum radius of a ball centered at i that contains at least n/k individuals. Our objective is to ensure that each individual has a facility within at most a small constant factor of her neighborhood radius.We present several theoretical results: We show that optimizing this factor is NP-hard; we give an approximation algorithm that guarantees a factor of at most 2 in all metric spaces; and we prove matching lower bounds in some metric spaces. We apply a variant of this algorithm to real-world address data, showing that it is quite different from standard clustering algorithms and outperforms them on our objective function and balances the load between facilities more evenly.

<p>An important element of open educational resources (OER) is the permission to use the materials in new ways, including revising and remixing them. Prior research has shown that the revision and remix rates for OER are relatively low. In this study we examined the extent to which the openly licensed Flat World Knowledge textbooks were being revised and remixed. We found that the levels of revision and remix were similar to those of other OER collections. We discuss the possible significance and implication of these findings.</p><input id="gwProxy" type="hidden" /><input id="jsProxy" onclick="if(typeof(jsCall)=='function'){jsCall();}else{setTimeout('jsCall()',500);}" type="hidden" />

We use ideas from distributed computing and game theory to study dynamic and decentralized environments in which computational nodes, or decision makers, interact strategically and with limited information. In such environments, which arise in many real-world settings, the participants act as both economic and computational entities. We exhibit a general non-convergence result for a broad class of dynamics in asynchronous settings. We consider implications of our result across a wide variety of interesting and timely applications: game dynamics, circuit design, social networks, Internet routing, and congestion control. We also study the computational and communication complexity of testing the convergence of asynchronous dynamics. Our work opens a new avenue for research at the intersection of distributed computing and game theory. This is the authors' version of this work. The definitive version will be published in ACM Trans. Econ. Comput. A preliminary version of some of this work appeared in

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp(L) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim(a, b) is equal to the effective packing dimension Dim(a, b), then sp(L) contains a unit interval. We also show that, if the dimension dim(a, b) is at least one, then sp(L) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.

We consider the multi-armed bandit setting with a twist. Rather than having just one decision maker deciding which arm to pull in each round, we have n different decision makers (agents). In the simple stochastic setting, we show that a "free-riding" agent observing another "self-reliant" agent can achieve just O(1) regret, as opposed to the regret lower bound of Ω(log t) when one decision maker is playing in isolation. This result holds whenever the self-reliant agent's strategy satisfies either one of two assumptions: (1) each arm is pulled at least γ ln t times in expectation for a constant γ that we compute, or (2) the self-reliant agent achieves o(t) realized regret with high probability. Both of these assumptions are satisfied by standard zero-regret algorithms. Under the second assumption, we further show that the free rider only needs to observe the number of times each arm is pulled by the self-reliant agent, and not the rewards realized.In the linear contextual setting, each arm has a distribution over parameter vectors, each agent has a context vector, and the reward realized when an agent pulls an arm is the inner product of that agent's context vector with a parameter vector sampled from the pulled arm's distribution. We show that the free rider can achieve O(1) regret in this setting whenever the free rider's context is a small (in L 2 -norm) linear combination of other agents' contexts and all other agents pull each arm Ω(log t) times with high probability. Again, this condition on the self-reliant players is satisfied by standard zero-regret algorithms like UCB. We also prove a number of lower bounds.

Abstract. Dynamics in a distributed system are self-stabilizing if they are guaranteed to reach a stable state regardless of how the system is initialized. Game dynamics are uncoupled if each player's behavior is independent of the other players' preferences. Recognizing an equilibrium in this setting is a distributed computational task. Self-stabilizing uncoupled dynamics, then, have both resilience to arbitrary initial states and distribution of knowledge. We study these dynamics by analyzing their behavior in a bounded-recall synchronous environment. We determine, for every "size" of game, the minimum number of periods of play that stochastic (randomized) players must recall in order for uncoupled dynamics to be self-stabilizing. We also do this for the special case when the game is guaranteed to have unique best replies. For deterministic players, we demonstrate two self-stabilizing uncoupled protocols. One applies to all games and uses three steps of recall. The other uses two steps of recall and applies to games where each player has at least four available actions. For uncoupled deterministic players, we prove that a single step of recall is insufficient to achieve self-stabilization, regardless of the number of available actions.