The Price of Anarchy for Selfish Ring Routing is Two

We analyze the network congestion game with atomic players, asymmetric strategies, and the maximum latency among all players as social cost. This important social cost function is much less understood than the average latency. We show that the price of anarchy is at most two, when the network is a ring and the link latencies are linear. Our bound is tight. This is the first sharp bound for the maximum latency objective.Comment: Full version of WINE 2012 paper, 24 page

On Existence and Properties of Approximate Pure Nash Equilibria in Bandwidth Allocation Games

In \emph{bandwidth allocation games} (BAGs), the strategy of a player consists of various demands on different resources. The player's utility is at most the sum of these demands, provided they are fully satisfied. Every resource has a limited capacity and if it is exceeded by the total demand, it has to be split between the players. Since these games generally do not have pure Nash equilibria, we consider approximate pure Nash equilibria, in which no player can improve her utility by more than some fixed factor $\alpha$ through unilateral strategy changes. There is a threshold $\alpha_\delta$ (where $\delta$ is a parameter that limits the demand of each player on a specific resource) such that $\alpha$-approximate pure Nash equilibria always exist for $\alpha \geq \alpha_\delta$, but not for $\alpha < \alpha_\delta$. We give both upper and lower bounds on this threshold $\alpha_\delta$ and show that the corresponding decision problem is ${\sf NP}$-hard. We also show that the $\alpha$-approximate price of anarchy for BAGs is $\alpha+1$. For a restricted version of the game, where demands of players only differ slightly from each other (e.g. symmetric games), we show that approximate Nash equilibria can be reached (and thus also be computed) in polynomial time using the best-response dynamic. Finally, we show that a broader class of utility-maximization games (which includes BAGs) converges quickly towards states whose social welfare is close to the optimum

A modification of Honoré's triple-link model in the synoptic problem

In New Testament studies, the synoptic problem is concerned with the relationships between the gospels of Matthew, Mark and Luke. In an earlier paper a careful specification in probabilistic terms was set up of Honoré's triple-link model. In the present paper, a modification of Honoré's model is proposed. As previously, counts of the numbers of verbal agreements between the gospels are examined to investigate which of the possible triple-link models appears to give the best fit to the data, but now using the modified version of the model and additional sets of data

Atypical coeliomesenteric anastomosis: The presence of an anomalous fourth coeliac trunk branch.

The presence of more than three coeliac trunk branches is a commonly encountered variant. Literature occasionally describes cases of middle or left colic arteries originating from the celiac trunks or its branches; however, the presence of an anomalous arterial connection between the celiac trunk and both the superior and inferior mesenteric arteries (SMA and IMA, respectively) has yet to be reported. Routine abdominal dissection of a male Caucasian cadaver, revealed the presence of an anomalous fourth arterial branch on the 4-cm long coeliac trunk. The course of this artery was traced, and it terminated by anastomosing with the marginal artery of the mesenteric circulation. The distal termination point of this anomalous fourth coeliac branch was the marginal artery, 5 cm medial of the splenic flexure, anastomosing almost perpendicularly. The diameter of this anomalous artery was comparable with the left gastric artery at their origins. The artery coursed inferiorlaterally toward the splenic flexure, passing immediately posterior to both the pancreas and the splenic vein. The anastomosis point of this artery, near Griffith's Point, is normally considered a watershed region with dual arterial supply from both the SMA and IMA, allowing collateral circulation. This region has a relatively higher susceptibility to irreversible damage in ischemic diseases because of lower perfusion, thus, the anastomosis of atypical coeliac branches represents a rare case for consideration. Awareness of the possibility of embryological variants will minimize the risk of complications in surgical or clinical procedures, and exploration of rare variants will benefit the understanding of vascular embryology