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Bravo, M., & Cominetti, R. (2018). Sharp convergence rates for averaged nonexpansive maps. Isr. J. Math., 227(1), 163–188.
Abstract: We establish sharp estimates for the convergence rate of the Kranosel'skiA-Mann fixed point iteration in general normed spaces, and we use them to show that the optimal constant of asymptotic regularity is exactly . To this end we consider a nested family of optimal transport problems that provide a recursive bound for the distance between the iterates. We show that these bounds are tight by building a nonexpansive map T: [0, 1](N) -> [0, 1](N) that attains them with equality, settling a conjecture by Baillon and Bruck. The recursive bounds are in turn reinterpreted as absorption probabilities for an underlying Markov chain which is used to establish the tightness of the constant 1/root pi.
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Bravo, M., Cominetti, R., & Pavez-Signe, M. (2019). Rates of convergence for inexact Krasnosel'skii-Mann iterations in Banach spaces. Math. Program., 175(1-2), 241–262.
Abstract: We study the convergence of an inexact version of the classical Krasnosel'skii-Mann iteration for computing fixed points of nonexpansive maps. Our main result establishes a new metric bound for the fixed-point residuals, from which we derive their rate of convergence as well as the convergence of the iterates towards a fixed point. The results are applied to three variants of the basic iteration: infeasible iterations with approximate projections, the Ishikawa iteration, and diagonal Krasnosels'kii-Mann schemes. The results are also extended to continuous time in order to study the asymptotics of nonautonomous evolution equations governed by nonexpansive operators.
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Colini-Baldeschi, R., Cominetti, R., Mertikopoulos, P., & Scarsini, M. (2020). When Is Selfish Routing Bad? The Price of Anarchy in Light and Heavy Traffic. Oper. Res., 68(2), 411–434.
Abstract: This paper examines the behavior of the price of anarchy as a function of the traffic inflow in nonatomic congestion games with multiple origin/destination (O/D) pairs. Empirical studies in real-world networks show that the price of anarchy is close to 1 in both light and heavy traffic, thus raising the following question: can these observations be justified theoretically? We first show that this is not always the case: the price of anarchy may remain a positive distance away from 1 for all values of the traffic inflow, even in simple three-link networks with a single O/D pair and smooth, convex costs. On the other hand, for a large class of cost functions (including all polynomials) and inflow patterns, the price of anarchy does converge to 1 in both heavy and light traffic, irrespective of the network topology and the number of O/D pairs in the network. We also examine the rate of convergence of the price of anarchy, and we show that it follows a power law whose degree can be computed explicitly when the network's cost functions are polynomials.
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Colini-Baldeschi, R., Cominetti, R., & Scarsini, M. (2019). Price of Anarchy for Highly Congested Routing Games in Parallel Networks. Theor. Comput. Syst., 63(1), 90–113.
Abstract: We consider nonatomic routing games with one source and one destination connected by multiple parallel edges. We examine the asymptotic behavior of the price of anarchy as the inflow increases. In accordance with some empirical observations, we prove that under suitable conditions on the costs the price of anarchy is asymptotic to one. We show with some counterexamples that this is not always the case, and that these counterexamples already occur in simple networks with only 2 parallel links.
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Cominetti, R., Correa, J., & Olver, N. (2022). Long-Term Behavior of Dynamic Equilibria in Fluid Networks. Oper. Res., 70(1), 516–526.
Abstract: A fluid queuing network constitutes one of the simplest models in which to study flow dynamics over a network. In this model we have a single source-sink pair, and each link has a per-time-unit capacity and a transit time. A dynamic equilibrium (or equilibrium flow over time) is a flow pattern over time such that no flow particle has incentives to unilaterally change its path. Although the model has been around for almost 50 years, only recently results regarding existence and characterization of equilibria have been obtained. In par-ticular, the long-term behavior remains poorly understood. Our main result in this paper is to show that, under a natural (and obviously necessary) condition on the queuing capacity, a dynamic equilibrium reaches a steady state (after which queue lengths remain constant) in finite time. Previously, it was not even known that queue lengths would remain bounded. The proof is based on the analysis of a rather nonobvious potential function that turns out to be monotone along the evolution of the equilibrium. Furthermore, we show that the steady state is characterized as an optimal solution of a certain linear program. When this program has a unique solution, which occurs generically, the long-term behavior is completely predictable. On the contrary, if the linear program has multiple solutions, the steady state is more difficult to identify as it depends on the whole temporal evolution of the equilibrium.
Keywords: flows over time; dynamic equilibria; steady state
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Cominetti, R., Dose, V., & Scarsini, M. (2022). The price of anarchy in routing games as a function of the demand. Math. Program., Early Access.
Abstract: 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.
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Cominetti, R., Quattropani, M., & Scarsini, M. (2022). The Buck-Passing Game. Math. Oper. Res., Early Access.
Abstract: We consider two classes of games in which players are the vertices of a directed graph. Initially, nature chooses one player according to some fixed distribution and gives the player a buck. This player passes the buck to one of the player's out-neighbors in the graph. The procedure is repeated indefinitely. In one class of games, each player wants to minimize the asymptotic expected frequency of times that the player receives the buck. In the other class of games, the player wants to maximize it. The PageRank game is a particular case of these maximizing games. We consider deterministic and stochastic versions of the game, depending on how players select the neighbor to which to pass the buck. In both cases, we prove the existence of pure equilibria that do not depend on the initial distribution; this is achieved by showing the existence of a generalized ordinal potential. If the graph on which the game is played admits a Hamiltonian cycle, then this is the outcome of prior-five Nash equilibrium in the minimizing game. For the minimizing game, we then use the price of anarchy and stability to measure fairness of these equilibria.
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Cominetti, R., Roshchina, V., & Williamson, A. (2019). A counterexample to De Pierro's conjecture on the convergence of under-relaxed cyclic projections. Optimization, 68(1), 3–12.
Abstract: The convex feasibility problem consists in finding a point in the intersection of a finite family of closed convex sets. When the intersection is empty, a best compromise is to search for a point that minimizes the sum of the squared distances to the sets. In 2001, de Pierro conjectured that the limit cycles generated by the -under-relaxed cyclic projection method converge when towards a least squares solution. While the conjecture has been confirmed under fairly general conditions, we show that it is false in general by constructing a system of three compact convex sets in for which the -under-relaxed cycles do not converge.
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Cominetti, R., Scarsini, M., Schroder, M., & Stier-Moses, N. (2022). Approximation and Convergence of Large Atomic Congestion Games. Math. Oper. Res., Early Access.
Abstract: 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 games in which each player's weight is small. We prove that when the number of players goes to infinity and their weights to zero, the random flows in all (mixed) Nash equilibria for the finite games converge in distribution to the set of Wardrop equilibria of the corresponding nonatomic limit game. In the second setting, we consider an increasing number of players with a unit weight that participate in the game with a decreasingly small probability. In this case, the Nash equilibrium flows converge in total variation toward Poisson random variables whose expected values are War drop equilibria of a different nonatomic game with suitably defined costs. The latter can be viewed as symmetric equilibria in a Poisson game in the sense of Myerson, establishing a plausible connection between the Wardrop model for routing games and the stochastic fluctuations observed in real traffic. In both settings, we provide explicit approximation bounds, and we study the convergence of the price of anarchy. Beyond the case of congestion games, we prove a general result on the convergence of large games with random players toward Poisson games.
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Contreras, J. P., & Cominetti, R. (2022). Optimal error bounds for non-expansive fixed-point iterations in normed spaces. Math. Program., Early Access.
Abstract: This paper investigates optimal error bounds and convergence rates for general Mann iterations for computing fixed-points of non-expansive maps. We look for iterations that achieve the smallest fixed-point residual after n steps, by minimizing a worst-case bound parallel to x(n) – Tx(n)parallel to <= R-n derived from a nested family of optimal transport problems. We prove that this bound is tight so that minimizing R-n yields optimal iterations. Inspired from numerical results we identify iterations that attain the rate R-n = O(1/n), which we also show to be the best possible. In particular, we prove that the classical Halpern iteration achieves this optimal rate for several alternative stepsizes, and we determine analytically the optimal stepsizes that attain the smallest worst-case residuals at every step n, with a tight bound R-n approximate to 4/n+4. We also determine the optimal Halpern stepsizes for affine non-expansive maps, for which we get exactly R-n = 1/n+1. Finally, we show that the best rate for the classical Krasnosel'skii-Mann iteration is Si (11 Omega(1/root n), and present numerical evidence suggesting that even extended variants cannot reach a faster rate.
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Dumett, M. A., & Cominetti, R. (2018). On The Stability Of An Adaptive Learning Dynamics In Traffic Games. J. Dyn. Games, 5(4), 265–282.
Abstract: This paper investigates the dynamic stability of an adaptive learning procedure in a traffic game. Using the Routh-Hurwitz criterion we study the stability of the rest points of the corresponding mean field dynamics. In the special case with two routes and two players we provide a full description of the number and nature of these rest points as well as the global asymptotic behavior of the dynamics. Depending on the parameters of the model, we find that there are either one, two or three equilibria and we show that in all cases the mean field trajectories converge towards a rest point for almost all initial conditions.
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Rios, I., Larroucau, T., Parra, G., & Cominetti, R. (2021). Improving the Chilean College Admissions System. Oper. Res., 69(4), 1186–1205.
Abstract: In this paper we present the design and implementation of a new system to solve the Chilean college admissions problem. We develop an algorithm that obtains all applicant/program pairs that can be part of a stable allocation when preferences are not strict and when all students tied in the last seat of a program (if any) must be allocated. We use this algorithm to identify which mechanism was used in the past to perform the allocation, and we propose a new method to incorporate the affirmative action that is part of the system to correct the inefficiencies that arise from having double-assigned students. By unifying the regular admission with the affirmative action, we have improved the allocation of approximately 2.5% of students assigned every year since 2016. From a theoretical standpoint, we show that some desired properties, such as strategy-proofness and monotonicity, cannot be guaranteed under flexible quotas.
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