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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.

In highly congested networks, is selfishness the problem? Empirical studies in road networks reveal a fairly surprising property of congestion: when there is too much (or too little) traffic in the network, there is no difference between the best and the fairest traffic allocations (i.e., between the traffic assignment that optimizes the commuters' average travel time versus the one that no commuter would have any incentive to deviate from). In “When is Selfish Routing Bad? The Price of Anarchy in Light and Heavy Traffic”, the authors give a theoretical justification to this empirical observation: for a large class of traffic inflow patterns and cost functions (including all polynomials), the gap between social optimality and equilibrium—the network’s price of anarchy—converges to 1 in both heavy and light traffic, irrespective of the network topology and the number of origin/destination pairs in the network. 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.

Order execution is an important operational level of activity encountered in portfolio investment and risk management. We study a sequential Stackelberg order execution game which arises naturally from the practice of algorithm trading in financial markets. The game consists of two risk-neutral traders, one leader and one follower, who compete to maximize their expected payoffs respectively by trading a single risky asset whose price dynamics follows a linear-price market impact model over a finite horizon. This new Stackelberg game departs from the Nash games which have been the main focus in the algorithm trading literature. We derive a closed-form solution for the unique open-loop Stackelberg equilibrium by exploiting the special structures of the model. This analytic solution enables us to develop new and complementary managerial insights by looking at both players’ equilibrium behavior in terms of trading speeds and positions, expected price dynamics, price of anarchy, first mover’s advantage, and trading horizon effect.

We study a Stackelberg game with multiple leaders and a continuum of followers that are coupled via congestion effects. The followers’ problem constitutes a nonatomic congestion game, where a population of infinitesimal players is given and each player chooses a resource. Each resource has a linear cost function which depends on the congestion of this resource. The leaders of the Stackelberg game each control a resource and determine a price per unit as well as a service capacity for the resource influencing the slope of the linear congestion cost function. As our main result, we establish existence of pure-strategy Nash–Stackelberg equilibria for this multi-leader Stackelberg game. The existence result requires a completely new proof approach compared to previous approaches, since the leaders’ objective functions are discontinuous in our game. As a consequence, best responses of leaders do not always exist, and thus standard fixed-point arguments á la Kakutani (Duke Math J 8(3):457–458, 1941) are not directly applicable. We show that the game is C -secure (a concept introduced by Reny (Econometrica 67(5):1029–1056, 1999) and refined by McLennan et al. (Econometrica 79(5):1643–1664, 2011), which leads to the existence of an equilibrium. We furthermore show that the equilibrium is essentially unique, and analyze its efficiency compared to a social optimum. We prove that the worst-case quality is unbounded. For identical leaders, we derive a closed-form expression for the efficiency of the equilibrium.

Consider a population of customers, each of which needs to decide independently when to arrive to a facility that provides a service during a fixed period of time, say a day. This is a common scenario in many service systems such as a bank, lunch at a cafeteria, music concert, flight check-in and many others. High demand for service at a specific time leads to congestion that comes at a cost, for examlple, for waiting, earliness or tardiness. Queueing theory provides tools for the analysis of the waiting times and associated costs. If customers have the option of deciding when to join the queue, they will face a decision dilemma of when to arrive. The level of congestion one suffers from depends on others, behavior and not only that of the individual under consideration. This fact leads customers to make strategic decisions regarding their time of arrival. In addition, multiple decision makers that affect each other’s expected congestion call for noncooperative game-theoretical analysis of this strategic interaction. This common daily scenario has prompted a research stream pioneered by the ?/M/1 model of Glazer and Hassin (Eur J Oper Res 13(2):146–150, 1983) that first characterized an arrival process to a queue as a Nash equilibrium solution of a game. This survey provides an overview of the main results and developments in the literature on queueing systems with strategic timing of arrivals. Another issue is that of social optimality, namely the strategy profile used by customers that optimizes their aggregate utility. In particular, we review results concerning the price of anarchy (PoA), which is the ratio between the socially optimal and the equilibrium utilities.

We consider the classical machine scheduling, where n jobs need to be scheduled on m machines, and where job j scheduled on machine i contributes pij∈R to the load of machine i, with the goal of minimizing the makespan, i.e., the maximum load of any machine in the schedule. We study the inefficiency of schedules that are obtained when jobs arrive sequentially one by one, and the jobs choose the machine on which they will be scheduled, aiming at being scheduled on a machine with a small load. We measure the inefficiency of a schedule as the ratio of the makespan obtained in the worst-case equilibrium schedule, and of the optimum makespan. This ratio is known as the sequential price of anarchy (SPoA). We also introduce two alternative inefficiency measures, which allow for a favorable choice of the order in which the jobs make their decisions. As our first result, we disprove the conjecture of Hassin and Yovel (Oper Res Lett 43(5):530–533, 2015) claiming that the sequential price of anarchy for m=2 machines is at most 3. We show that the sequential price of anarchy grows at least linearly with the number n of players, assuming arbitrary tie-breaking rules. That is, we show SPoA∈Ω(n). At the end of the paper, we show that if an authority can change the order of the jobs adaptively to the decisions made by the jobs so far (but cannot influence the decisions of the jobs), then there exists an adaptive ordering in which the jobs end up in an optimum schedule.

Solving the road congestion problem is one of the most pressing issues in modern cities since it causes time wasting, pollution, higher industrial costs and huge road maintenance costs. Advances in ITS technologies and the advent of autonomous vehicles are changing mobility dramatically. They enable the implementation of a coordination mechanism, called coordinated traffic assignment, among the sat-nav devices aiming at assigning paths to drivers to eliminate congestion and to reduce the total travel time in traffic networks. Among possible congestion avoidance methods, coordinated traffic assignment is a valuable choice since it does not involve huge investments to expand the road network. Traffic assignments are traditionally devoted to two main perspectives on which the well-known Wardropian principles are inspired: the user equilibrium and the system optimum. User equilibrium is a user-driven traffic assignment in which each user chooses the most convenient path selfishly. It guarantees that fairness among users is respected since, when the equilibrium is reached, all users sharing the same origin and destination will experience the same travel time. The main drawback in a user equilibrium is that the system total travel time is not minimized and, hence, the so-called Price of Anarchy is paid. On the other hand, the system optimum is an efficient system-wide traffic assignment in which drivers are routed on the network in such a way the total travel time is minimized, but users might experience travel times that are higher than the other users travelling from the same origin to the same destination, affecting the compliance. Thus, drawbacks in implementing one of the two assignments can be overcome by hybridizing the two approaches, aiming at bridging users’ fairness to system-wide efficiency. In the last decades, a significant number of attempts have been done to bridge fairness among users and system efficiency in traffic assignments. The survey reviews the state-of-the-art of these trade-off approaches.

We propose a model of discrete time dynamic congestion games with atomic players and a single source-destination pair. The latencies of edges are composed of free-flow transit times and possible queuing time due to capacity constraints. We give a precise description of the dynamics induced by the individual strategies of players and of the corresponding costs, either when the traffic is controlled by a planner, or when players act selfishly. In parallel networks, optimal and equilibrium behavior eventually coincide, but the selfish behavior of the initial players has consequences that cannot be undone and are paid by all future generations. In more general topologies, our main contributions are threefold. First, we illustrate a new dynamic version of Braess paradox: the presence of initial queues in the network may decrease the long-run costs in equilibrium. This paradox can arise in networks for which no Braess paradox was previously known. Second, we show that equilibria are not unique and can induce very different long-run costs. In particular, we give a sequence of networks such that the price of stability is equal to 1, and the price of anarchy is equal to n − 1, where n is the number of vertices. Third, we propose an extension to model seasonalities by assuming that departure flows fluctuate periodically over time. We introduce a measure that captures the queues induced by periodicity of inflows. For optimal and equilibrium flows in parallel networks this measure is the increase in cost compared to uniform departures. The electronic companion is available at https://doi.org/10.1287/opre.2017.1683 .

Resource selection games provide a model for a diverse collection of applications where a set of resources is matched to a set of demands. Examples include routing in traffic and in telecommunication networks, service of requests on multiple parallel queues, and acquisition of services or goods with demand-dependent prices. In reality, demands are often submitted by selfish entities (players) and congestion on the resources results in negative externalities for their users. We consider a policy maker that can set a priori rules to minimize the inefficiency induced by selfish players. For example, these rules may assume the form of scheduling policies or pricing decisions. We explore the space of such rules abstracted as cost-sharing methods. We prescribe desirable properties that the cost-sharing method should possess and prove that, in this natural design space, the cost-sharing method induced by the Shapley value minimizes the worst-case inefficiency of equilibria.

This paper studies the inefficiency of multiplicative approximate Nash Equilibrium for scheduling games. There is a set of machines and a set of jobs. Each job could choose one machine and be processed by the chosen one. A schedule is a θ -NE if no player has the incentive to deviate so that it decreases its cost by a factor larger than 1 + θ . The θ -NE is a generation of Nash Equilibrium and its inefficiency can be measured by the θ -PoA, which is also a generalization of the Price of Anarchy. For the game with the social cost of minimizing the makespan, the exact θ -PoA for any number of machines and any θ ≥ 0 is obtained. For the game with the social cost of maximizing the minimum machine load, we present upper and lower bounds on the θ -PoA. Tight bounds are provided for cases where the number of machines is between 2 and 7 and for any θ ≥ 0 .