On the Convergence Time of the Best Response Dynamics in Player-specific Congestion Games

We study the convergence time of the best response dynamics in player-specific singleton congestion games. It is well known that this dynamics can cycle, although from every state a short sequence of best responses to a Nash equilibrium exists. Thus, the random best response dynamics, which selects the next player to play a best response uniformly at random, terminates in a Nash equilibrium with probability one. In this paper, we are interested in the expected number of best responses until the random best response dynamics terminates. As a first step towards this goal, we consider games in which each player can choose between only two resources. These games have a natural representation as (multi-)graphs by identifying nodes with resources and edges with players. For the class of games that can be represented as trees, we show that the best-response dynamics cannot cycle and that it terminates after O(n^2) steps where n denotes the number of resources. For the class of games represented as cycles, we show that the best response dynamics can cycle. However, we also show that the random best response dynamics terminates after O(n^2) steps in expectation. Additionally, we conjecture that in general player-specific singleton congestion games there exists no polynomial upper bound on the expected number of steps until the random best response dynamics terminates. We support our conjecture by presenting a family of games for which simulations indicate a super-polynomial convergence time

Ultrasound for Distal Forearm Fracture:A Systematic Review and Diagnostic Meta-Analysis

STUDY OBJECTIVE:To determine the diagnostic accuracy of ultrasound for detecting distal forearm fractures. METHODS:A systematic review and diagnostic meta-analysis was performed according to the PRISMA statement. We searched MEDLINE, Web of Science and the Cochrane Library from inception to September 2015. All prospective studies of the diagnostic accuracy of ultrasound versus radiography as the reference standard were included. We excluded studies with a retrospective design and those with evidence of verification bias. We assessed the methodological quality of the included studies with the QUADAS-2 tool. We performed a meta-analysis of studies evaluating ultrasound to calculate the pooled sensitivity and specificity with 95% confidence intervals (CI95%) using a bivariate model with random effects. Subgroup and sensitivity analysis were used to examine the effect of methodological differences and other study characteristics. RESULTS:Out of 867 publications we included 16 studies with 1,204 patients and 641 fractures. The pooled test characteristics for ultrasound were: sensitivity 97% (CI95% 93-99%), specificity 95% (CI95% 89-98%), positive likelihood ratio (LR) 20.0 (8.5-47.2) and negative LR 0.03 (0.01-0.08). The corresponding pooled diagnostic odds ratio (DOR) was 667 (142-3,133). Apparent differences were shown for method of viewing, with the 6-view method showing higher specificity, positive LR, and DOR, compared to the 4-view method. CONCLUSION:The present meta-analysis showed that ultrasound has a high accuracy for the diagnosis of distal forearm fractures in children when used by proper viewing method. Based on this, ultrasound should be considered a reliable alternative, which has the advantages of being radiation free

A First Step towards Analyzing the Convergence Time in PlayerSpecific Singleton congestion games

Abstract. We initiate studying the convergence time to Nash equilibria in playerspecific singleton congestion games. We consider simple games that have natural representations as graphs as we assume that each player chooses between two resources. We are not able to present an analysis for general graphs. However, we present first results for interesting classes of graphs. For the class of games that are represented as trees, we show that every best-response schedule terminates after O(n 2) steps. We also consider games that are represented as circles. We show that deterministic best response schedules may cycle, whereas the random best response schedule, which selects the next player to play a best response uniformly at random, terminates after O(n 2) steps in expectation. These results imply that in player-specific congestion games in which each player chooses between two resources, and each resource is allocated by at most two players, the random best response schedule terminates quickly. Our analysis reveals interesting relationships between random walks on lines and the random best response schedule.

Nash Equilibria and Improvement Dynamics in Congestion Games

Communication infrastructures and markets are maintained and used by millions of entities each of them facing a private objective. The vast number of participants in conjunction with their individual goals to choose the best alternative gave rise to study such scenarios in the framework of game theory as it is rather unrealistic to assume that a centrally computed solution can be implemented. In this thesis, we follow this line of research and study congestion games as introduced by Rosenthal in 1973 and several modifications of his original approach. Congestion games model scenarios in which a finite number of players individually strives to allocate resources maximizing their utility. Here, the resources can correspond to quite different types of objects, e. g. to edges in a network or to machines processing tasks. Given a set of resources a player’s goal is to select a feasible subset of the resources, subsequently named a strategy, that minimizes the sum of the latencies of the resources in the set. Thereby, a subset is feasible if it possesses a predefined combinatorial structure, e. g., corresponds to a path or a tree in a network. Th