Price of Anarchy in Bernoulli Congestion Games with Affine Costs

We consider an atomic congestion game in which each player participates in the game with an exogenous and known probability $p_{i}\in[0,1]$, independently of everybody else, or stays out and incurs no cost. We first prove that the resulting game is potential. Then, we compute the parameterized price of anarchy to characterize the impact of demand uncertainty on the efficiency of selfish behavior. It turns out that the price of anarchy as a function of the maximum participation probability $p=\max_{i} p_{i}$ is a nondecreasing function. The worst case is attained when players have the same participation probabilities $p_{i}\equiv p$. For the case of affine costs, we provide an analytic expression for the parameterized price of anarchy as a function of $p$. This function is continuous on $(0,1]$, is equal to $4/3$ for $0<p\leq 1/4$, and increases towards $5/2$ when $p\to 1$. Our work can be interpreted as providing a continuous transition between the price of anarchy of nonatomic and atomic games, which are the extremes of the price of anarchy function we characterize. We show that these bounds are tight and are attained on routing games -- as opposed to general congestion games -- with purely linear costs (i.e., with no constant terms).Comment: 29 pages, 6 figure

Monotonicity of equilibria in nonatomic congestion games

This paper studies the monotonicity of equilibrium costs and equilibrium loads in nonatomic congestion games, in response to variations of the demands. The main goal is to identify conditions under which a paradoxical non-monotone behavior can be excluded. In contrast with routing games with a single commodity, where the network topology is the sole determinant factor for monotonicity, for general congestion games with multiple commodities the structure of the strategy sets plays a crucial role. We frame our study in the general setting of congestion games, with a special focus on singleton congestion games, for which we establish the monotonicity of equilibrium loads with respect to every demand. We then provide conditions for comonotonicity of the equilibrium loads, i.e.,we investigate when they jointly increase or decrease after variations of the demands. We finally extend our study from singleton congestion games to the larger class of constrained series-parallel congestion games, whose structure is reminiscent of the concept of a series-parallel network

The Price of Anarchy in Routing Games as a Function of the Demand

Most of the literature on the price of anarchy has focused on worst-case bounds for specific classes of games, such as routing games or more general congestion games. Recently, the price of anarchy in routing games has been studied as a function of the traffic demand, providing asymptotic results in light and heavy traffic. In this paper we study the price of anarchy in nonatomic routing games in the intermediate region of the demand. We begin by establishing some smoothness properties of Wardrop equilibria and social optima for general smooth costs. In the case of affine costs we show that the equilibrium is piecewise linear, with break points at the demand levels at which the set of active paths changes. We prove that the number of such break points is finite, although it can be exponential in the size of the network. Exploiting a scaling law between the equilibrium and the social optimum, we derive a similar behavior for the optimal flows. We then prove that in any interval between break points the price of anarchy is smooth and it is either monotone, or unimodal with a minimum attained on the interior of the interval. We deduce that for affine costs the maximum of the price of anarchy can only occur at the break points. For general costs we provide counterexamples showing that the set of break points is not always finite.Comment: 22 pages, 6 figure

The price of anarchy in routing games as a function of the demand

The price of anarchy has become a standard measure of the efficiency of equilibria in games. Most of the literature in this area has focused on establishing worst-case bounds for specific classes of games, such as routing games or more general congestion games. Recently, the price of anarchy in routing games has been studied as a function of the traffic demand, providing asymptotic results in light and heavy traffic. The aim of this paper is to study the price of anarchy in nonatomic routing games in the intermediate region of the demand. To achieve this goal, we begin by establishing some smoothness properties of Wardrop equilibria and social optima for general smooth costs. In the case of affine costs we show that the equilibrium is piecewise linear, with break points at the demand levels at which the set of active paths changes. We prove that the number of such break points is finite, although it can be exponential in the size of the network. Exploiting a scaling law between the equilibrium and the social optimum, we derive a similar behavior for the optimal flows. We then prove that in any interval between break points the price of anarchy is smooth and it is either monotone (decreasing or increasing) over the full interval, or it decreases up to a certain minimum point in the interior of the interval and increases afterwards. We deduce that for affine costs the maximum of the price of anarchy can only occur at the break points. For general costs we provide counterexamples showing that the set of break points is not always finite

376 Incomplete functional revascularization is associated with adverse clinical outcomes after TAVI

Abstract Aims Whether incomplete functional revascularization has an impact on the clinical outcome of patients treated with transcatheter aortic valve implantation (TAVI) is still unknown. We aim to assess the prognostic value of residual functional Syntax score (rFSS) in a cohort of patients undergoing TAVI. Methods and results One-hundred-twenty-four patients (229 lesions) with severe aortic stenosis and coronary artery disease (CAD) underwent fractional flow reserve (FFR)-guided revascularization. The primary endpoint of the study was the composite of cardiac death, myocardial infarction and revascularization at last available follow-up after TAVI. Median Syntax score (SS) and Functional Syntax score (FSS) at baseline were 7 (range 5–12) and 0 (range 0–7) respectively. After revascularization or deferral according to FFR, residual SS (rSS) and rFSS were 5 (range 0–8) and 0 (range 0–0), respectively. At COX regression analysis, angiographic incomplete revascularization (rSS = 0) was not associated with the primary endpoint (HR: 1.26; 95% CI: 0.40; 3.95; P-value 0.698), whereas functional incomplete revascularization was associated with worse event-free survival at Follow-up after adjusting for clinical confounders (HR: 3.74, 95% CI: 1.02–13.75, P = 0.047). Conclusions Incomplete functional revascularization is associated with adverse clinical outcome after TAVI. rFSS may be regarded as a treatment goal for patients with CAD undergoing TAVI. Further studies are warranted to confirm our hypothesis. 376 Central FigureMACEs free survival analysis of patients stratified according to complete revascularization vs. incomplete revascularization assessed according to anatomy (residual SYNTAX score) (A) or physiology (residual functional SYNTAX score) (B)

Convergence of Large Atomic Congestion Games

We consider the question of whether, and in what sense, Wardrop equilibria provide a good approximation for Nash equilibria in atomic unsplittable congestion games with a large number of small players. We examine two different definitions of small players. In the first setting, we consider a sequence of games with an increasing number of players where each player's weight tends to zero. We prove that all (mixed) Nash equilibria of the finite games converge to the set of Wardrop equilibria of the corresponding nonatomic limit game. In the second setting, we consider again an increasing number of players but now each player has a unit weight and participates in the game with a probability tending to zero. In this case, the Nash equilibria converge to the set of Wardrop equilibria of a different nonatomic game with suitably defined costs. The latter can also be seen as a Poisson game in the sense of Myerson (1998), establishing a precise connection between the Wardrop model and the empirical flows observed in real traffic networks that exhibit stochastic fluctuations well described by Poisson distributions. In both settings we give explicit upper bounds on the rates of convergence, from which we also derive the convergence of the price of anarchy. Beyond the case of congestion games, we establish a general result on the convergence of large games with random players towards Poisson games.Comment: 34 pages, 3 figure

Coronary Artery Plaque Phenotype and 5-Year Clinical Outcomes in Older Patients with Non-ST Elevation Acute Coronary Syndrome.

BACKGROUND Lesions with thin-cap fibroatheroma (TCFA), small luminal area and large plaque burden (PB) have been considered at high risk of cardiovascular events. Older patients were not represented in studies which demonstrated correlation between clinical outcome and plaque characteristics. This study aims to investigate the prognostic role of high-risk plaque characteristics and long-term outcome in older patients presenting with non-ST elevation acute coronary syndrome (NSTEACS). METHODS This study recruited older patients aged 75 years with NSTEACS undergoing virtual-histology intravascular ultrasound (VH-IVUS) imaging from the Improve Clinical Outcomes in high-risk patieNts with acute coronary syndrome (ICON-1). Primary endpoint was the composite of major adverse cardiovascular events (MACE) consisting of all-cause mortality, myocardial infarction (MI), and any revascularisation. Every component of MACE and target vessel failure (TVF) including MI and any revascularisation were considered as secondary endpoints. RESULTS Eighty-six patients with 225 vessels undergoing VH-IVUS at baseline completed 5-year clinical follow-up. Patients with minimal lumen area (MLA) 4 demonstrated increased risk of MACE (hazard ratio [HR] 2.37, 95% confidence interval [CI] 1.00-5.59, p = 0.048) with a worse event-free survival (Log Rank 4.17, p = 0.041) than patients with MLA 4 . Patients with combination of TCFA, MLA 4 and PB 70% showed high risk of MI (HR 5.23, 95% CI 1.05-25.9, p = 0.043). Lesions with MLA 4 had 6-fold risk of TVF (HR 6.16, 95% CI 1.24-30.5, p = 0.026). CONCLUSIONS Small luminal area appears as the major prognostic factor in older patients with NSTEACS at long-term follow-up. Combination of TCFA, MLA 4 and PB 70% was associated with high risk of MI. CLINICAL TRIAL REGISTRATION NCT01933581

Why, When and How Should Clinicians Use Physiology in Patients with Acute Coronary Syndromes?

Current data support the use of coronary physiology in patients with acute coronary syndrome (ACS). In patients with ST-elevation MI, the extent of myocardial damage and microvascular dysfunction create a complex conundrum to assimilate when considering clinical management and risk stratification. In this setting, the index of microcirculatory resistance emerged as an accurate tool to identify patients at risk of suboptimal myocardial reperfusion after primary percutaneous coronary intervention who may benefit from novel adjunctive therapies. In the context of non-ST-elevation ACS, coronary physiology should be carefully interpreted and often integrated with intracoronary imaging, especially in cases of ambiguous culprit lesion. Conversely, the functional assessment of bystander coronary disease is favoured by the available evidence, aiming to achieve complete revascularisation. Based on everyday clinical scenarios, the authors illustrate the available evidence and provide recommendations for the functional assessment of infarct-related artery and non-culprit lesions in patients with ACS