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

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

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

Routine Left Ventricular Pacing for Patients Undergoing Transcatheter Aortic Valve Replacement

ABSTRACTBackground: Rapid ventricular pacing is often required during transcatheter aortic valve replacement (TAVR) procedures. Pacing via the retrograde left ventricular guidewire (LV-GW) is an al..

Proof of concept study on coronary microvascular function in low flow low gradient aortic stenosis

ObjectivesWe hypothesised that low flow low gradient aortic stenosis (LFLGAS) is associated with more severe coronary microvascular dysfunction (CMD) compared with normal-flow high-gradient aortic stenosis (NFHGAS) and that CMD is related to reduced cardiac performance. MethodsInvasive CMD assessment was performed in 41 consecutive patients with isolated severe aortic stenosis with unobstructed coronary arteries undergoing transcatheter aortic valve implantation (TAVI). The index of microcirculatory resistance (IMR), resistive reserve ratio (RRR) and coronary flow reserve (CFR) were measured in the left anterior descending artery before and after TAVI. Speckle tracking echocardiography was performed to assess cardiac function at baseline and repeated at 6 months. ResultsIMR was significantly higher in patients with LFLGAS compared with patients with NFHGAS (24.1 (14.6 to 39.1) vs 12.8 (8.6 to 19.2), p=0.002), while RRR was significantly lower (1.4 (1.1 to 2.1) vs 2.6 (1.5 to 3.3), p=0.020). No significant differences were observed in CFR between the two groups. High IMR was associated with low stroke volume index, low cardiac output and reduced peak atrial longitudinal strain (PALS). TAVI determined no significant variation in microvascular function (IMR: 16.0 (10.4 to 26.1) vs 16.6 (10.2 to 25.6), p=0.403) and in PALS (15.9 (9.9 to 26.5) vs 20.1 (12.3 to 26.7), p=0.222). Conversely, left ventricular (LV) global longitudinal strain increased after TAVI (-13.2 (8.4 to 16.6) vs -15.1 (9.4 to 17.8), p=0.047). In LFLGAS, LV systolic function recovered after TAVI in patients with preserved microvascular function but not in patients with CMD. ConclusionsCMD is more severe in patients with LFLGAS compared with NFHGAS and is associated with low-flow state, left atrial dysfunction and reduced cardiac performance