Selfish Routing Game Research Articles

Selfish Routing Game-Based Multi-Resource Allocation and Fair Scheduling of Indivisible Jobs in Edge Environments

Distributed and heterogeneous edge computing environments require efficient allocation and scheduling of multiple users’ applications. This paper presents a game-theoretic solution to model the competition between time-sensitive internet of things (IoT) applications with indivisible loads to be allocated and scheduled to edge servers. We model the allocation problem as a selfish routing game such that no job is unsatisfied with its allocation. Also, the allocated jobs are scheduled using a weighted time-sharing approach, which assigns resources to jobs proportional to their demands and deadlines. We proved the existence of pure Nash equilibrium in the introduced game and presented a greedy and polynomial time algorithm to obtain the solution called SAFSA. Using an extended version of the CloudSim simulator, we assess our approach in different situations by employing a real-world dataset of the Google cluster. Since the proposed algorithm considers jobs’ deadlines beside resource requests in allocation and scheduling decisions, it can simultaneously minimize average response time and maximize the number of jobs completed within their deadlines.The simulation results confirm the ability of SAFSA to efficiently execute the jobs within their deadlines, especially when there are many IoT devices and edge servers in the system, which is the main characteristic of IoT environments.

Uncertainty in Multicommodity Routing Networks: When Does It Help?

We study the equilibrium behavior in a multi-commodity selfish routing game with many types of uncertain users where each user over- or under-estimates their congestion costs by a multiplicative factor. Surprisingly, we find that uncertainties in different directions have qualitatively distinct impacts on equilibria. Namely, contrary to the usual notion that uncertainty increases inefficiencies, network congestion actually decreases when users over-estimate their costs. On the other hand, under-estimation of costs leads to increased congestion. We apply these results to urban transportation networks, where drivers have different estimates about the cost of congestion. In light of the dynamic pricing policies aimed at tackling congestion, our results indicate that users' perception of these prices can significantly impact the policy's efficacy, and "caution in the face of uncertainty" leads to favorable network conditions.

Improving Selfish Routing for Risk-Averse Players

We investigate how and to which extent one can exploit risk-aversion and modify the perceived cost of the players in selfish routing so that the Price of Anarchy (PoA) wrt. the total latency is improved. The starting point is to introduce some small random perturbations to the edge latencies so that the expected latency does not change, but the perceived cost of the players increases, due to risk-aversion. We adopt the simple model of γ-modifiable routing games, a variant of selfish routing games with restricted tolls. We prove that computing the best γ-enforceable flow is NP-hard for parallel-link networks with affine latencies and two classes of heterogeneous risk-averse players. On the positive side, we show that for parallel-link networks with heterogeneous players and for series-parallel networks with homogeneous players, there exists a nicely structured γ-enforceable flow whose PoA improves fast as γ increases. We show that the complexity of computing such a γ-enforceable flow is determined by the complexity of computing a Nash flow for the original game. Moreover, we prove that the PoA of this flow is best possible in the worst-case, in the sense that there are instances where (i) the best γ-enforceable flow has the same PoA, and (ii) considering more flexible modifications does not lead to any improvement on the PoA.

Pruning Algorithms for A Replicator Dynamics Method in Multiple OD Selfish Routing Games

A traffic flow allocation has been studied by many researches. This problem is treated by both urban planning research and game theoretical approaches. We stand on game theory to consider the traffic flow allocation problem by treating a class of congestion games on the network. The traffic flow allocation is called, in context the congestion game, a selfish routing game. In this game, our proposal is to find an equilibria of decision making of the players. The player's decision is amounts of flows for each origin-destination path. It is known that the equilibrium searching problem as edge based modeling is able to compute easily by using Frank-Wolfe method, however, the edge based model has weak expressiveness. Thus we employ a path based modeling that can treat some complex phenomena. In this model, since we need to handle many paths in a network, it is known that the equilibrium searching problem is difficult. In this paper, we study a solving method for a multi OD selfish routing game, and a method for solving standard routing games and its high speeding method. Our algorithm employs a replicator dynamics which is one of iterative optimization techniques. In the solution based on the replicator dynamics, the calculation time is very large, since the calculation is also performed for all paths. Therefore, as a preprocessing of solving by replicator dynamics, the policy of the proposed method is to make computation time faster by deleting unused paths. This paper evaluates the algorithm by numerical experiment.

Shortest Paths from a Group Perspective—A Note on Selfish Routing Games with Cognitive Agents

This paper presents an analysis of the effects of cognitive agents employing selfish routing behavior in traffic networks with linear latency functions. Selfish routing occurs when each agent traveling on a network acts in a purely selfish manner, therefore the Braess Paradox is likely to occur. The Braess Paradox describes a situation where an additional edge with positive capacity is added to a given network, which leads to higher total system delay. By applying the concept of cognitive agents, each agent is able to make a range of non-selfish and selfish decisions. In addition, each agent has to cope with uncertainty in terms of travel time information associated with the traffic system, a factor in real-world traffic networks. This paper evaluates the influence of travel time uncertainty, and possible non-selfish decisions of the agents on overall network delay. The results indicate that both non-selfish behavior and uncertainty have an influence on overall travel delay. In addition, understanding the influence of cognitive agents on delay can help to better plan and influence traffic flows resulting in “closer to optimal” flows involving overall lower delays.

The Impact of Worst-Case Deviations in Non-Atomic Network Routing Games

We introduce a unifying model to study the impact of worst-case latency deviations in non-atomic selfish routing games. In our model, latencies are subject to (bounded) deviations which are taken into account by the players. The quality deterioration caused by such deviations is assessed by the Deviation Ratio, i.e., the worst case ratio of the cost of a Nash flow with respect to deviated latencies and the cost of a Nash flow with respect to the unaltered latencies. This notion is inspired by the Price of Risk Aversion recently studied by Nikolova and Stier-Moses (Nikolova and Stier-Moses 2015). Here we generalize their model and results. In particular, we derive tight bounds on the Deviation Ratio for multi-commodity instances with a common source and arbitrary non-negative and non-decreasing latency functions. These bounds exhibit a linear dependency on the size of the network (besides other parameters). In contrast, we show that for general multi-commodity networks an exponential dependency is inevitable. We also improve recent smoothness results to bound the Price of Risk Aversion.

OrganJet: Overcoming Geographical Disparities in Access to Deceased Donor Kidneys in the United States

There are over 90,000 patients in the United States waiting for a kidney transplant. Under the current allocation policy, the vast majority of deceased organs are allocated locally. This causes significant disparities in waiting times and access to transplant across different geographical areas. To ameliorate this inequity, we propose an operational solution that offers affordable jet services (OrganJet) to patients on the transplant waiting list, allowing them to list in multiple different, and possibly very distant, donation service areas (DSAs) of their choosing. First, using a fluid approximation, we formulate the patients’ problem of choosing a location to multiple list as a selfish routing game in which each patient tries to minimize his “congestion cost,” i.e., maximize his life expectancy. Through a combination of numerical, simulation, and analytical results, we show that multiple listing can lead to a significant improvement in geographic equity. In the special case when sufficiently many patients can multiple list, the geographic inequity disappears. Moreover, the supply of deceased donor organs increases under multiple listing, leading to more transplants and saved lives. We also consider a diffusion approximation and study the resulting multiple-listing game. The equilibrium outcome under the diffusion approximation is a second-order perturbation of that under the selfish routing formulation. In particular, the geographic equity metric, waiting times, and probabilities of receiving a transplant at various DSAs in equilibrium are second-order perturbations of those predicted by the selfish routing equilibrium. Hence, the analysis under the diffusion approximation also supports the finding that multiple listing leads to an improvement in geographic equity. In addition, restricting attention to the special case of sufficiently many patients multiple listing leads to an explicit characterization of the equilibria, which in turn yields additional structural insights. Last, we undertake a simulation study that supports aforementioned findings. This paper was accepted by Assaf Zeevi, stochastic models and simulation.

An Efficiency Improvement of the Equilibrium Solution Search on the Selfish Routing Game by Removing Redundant Paths

The selfish routing game is a mathematical model to represent the behavior of selfish players who select a path in a congested network. In the equilibrium solution search on the selfish routing gam...

Balancing Load via Small Coalitions in Selfish Ring Routing Games

This paper concerns the asymmetric atomic selfish routing game for load balancing in ring networks. In the selfish routing, each player selects a path in the ring network to route one unit traffic between its source and destination nodes, aiming at a minimum maximum link load along its own path. The selfish path selections by individuals ignore the system objective of minimizing the maximum load over all network links. This selfish ring load (SRL) game arises in a wide variety of applications in decentralized network routing, where network performance is often measured by the price of anarchy (PoA), the worst-case ratio between the maximum link loads in an equilibrium routing and an optimal routing. It has been known that the PoA of SRL with respect to classical Nash Equilibrium (NE) cannot be upper bounded by any constant, showing large loss of efficiency at some NE outcome. In an effort to improve the network performance in the SRL game, we generalize the model to so-called SRL with collusion (SRLC) which allows coordination within any coalition of up to k selfish players on the condition that every player of the coalition benefits from the coordination. We prove that, for m-player game on n-node ring, the PoA of SRLC is n - 1 when k ≤ 2, drops to 2 when k = 3 and is at least 1 + 2/m for k ≥ 4. Our study shows that on one hand, the performance of ring networks, in terms of maximum load, benefits significantly from coordination of self-interested players within small-sized coalitions; on the other hand, the equilibrium routing in SRL might not reach global optimum even if any number of players can coordinate.

Shortest Paths for Groups: Introducing a Predictive Memory for Cognitive Agents

This paper elaborates on the effect of predictive memory on cognitive agents that are acting in selfish routing games. Selfish routing describes a situation, where agents are moving in a network with defined latency functions, and act in a strictly selfish manner. Under certain situations – i.e. specific definition of the network, associated latency functions, and agents acting strictly selfish – the Braess Paradox occurs. The Braess Paradox contradicts human intuition by the fact that adding a new low-latency edge to a given network does not reduce overall latency. By incorporating cognitive agents in the game theoretical approach, agents could overcome their strictly selfish behaviour, which in turn reduces overall latency. By incorporating a kind of predictive memory, each agent can learn from a number of personal experiences and alter their present behaviour accordingly, which can reduce overall latency.

Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms

We reconsider the well-studied Selfish Routing game with affine latency functions. The Price of Anarchy for this class of games takes maximum value 4/3; this maximum is attained already for a simple network of two parallel links, known as Pigou’s network. We improve upon the value 4/3 by means of Coordination Mechanisms.We increase the latency functions of the edges in the network, i.e., if ℓ e (x) is the latency function of an edge e, we replace it by \(\hat{\ell}_{e}(x)\) with \(\ell_{e}(x) \le \hat{\ell}_{e}(x)\) for all x. Then an adversary fixes a demand rate as input. The engineered Price of Anarchy of the mechanism is defined as the worst-case ratio of the Nash social cost in the modified network over the optimal social cost in the original network. Formally, if \(\hat{C}_{N} (r)\) denotes the cost of the worst Nash flow in the modified network for rate r and C opt (r) denotes the cost of the optimal flow in the original network for the same rate then $$\mathit{ePoA} = \max_{r \ge 0} \frac{\hat{C}_N(r)}{C_{\mathit{opt}}(r)}. $$ We first exhibit a simple coordination mechanism that achieves for any network of parallel links an engineered Price of Anarchy strictly less than 4/3. For the case of two parallel links our basic mechanism gives 5/4=1.25. Then, for the case of two parallel links, we describe an optimal mechanism; its engineered Price of Anarchy lies between 1.191 and 1.192.

A Control Method of Dynamic Selfish Routing Based on a State-Dependent Tax

SUMMARY A selfish routing game is a simple model of selfish behaviors in networks. It is called that Braess’s paradox occurs in the selfish routing game if an equilibrium flow achieved by players’ selfish behaviors is not the optimal minimum latency flow. In order to make the minimum latency flow a Nash equilibrium, a marginal cost tax has been proposed. Braess graphs have also been proposed to discuss Braess’s paradox. In a large population of selfish players, conflicts between purposes of each player and the population causes social dilemmas. In game theory, to resolve the social dilemmas, a capitation tax and/or a subsidy has been introduced, and players’ dynamical behaviors have been formulated by replicator dynamics. In this paper, we formulate replicator dynamics in the Braess graphs and investigate stability of the minimum latency flow with and without the marginal cost tax. An additional latency caused by the marginal cost tax is also shown. To resolve the problem of the additional latency, we extend the capitation tax and the subsidy to a state-dependent tax and apply it to the stabilization problem of the minimum latency flow.

Nonlinear Min-Cost-Pursued Route-Swapping Dynamic System

This study proposes a nonlinear min-cost-pursued swapping dynamic (NMSD) system to model the evolution of selfish routing games on traffic network where travelers only swap from previous costly routes to the least costly ones. NMSD is a rational behavior adjustment process with stationary link flow pattern being the Wardrop user equilibrium. NMSD is able to prevent two behavioral deficiencies suffered by the existing min-cost-oriented models and keep solution invariance. NMSD relaxes the homogeneous user assumption, and the continuous-time NMSD (CNMSD) and discrete-time NMSD (DNMSD) share the same revision protocol. Moreover, CNMSD is Lyapunov-stable. Two numerical examples are conducted. The first one is designed to characterize the NMSD-conducted network traffic evolution and test the stability of day-to-day NMSD. The second one aims to explore the impacts of network scale on the stability of route-swaps conducted by pairwise and min-cost-pursed swapping behaviors.

Collusion in Atomic Splittable Routing Games

We study how collusion affects the social cost in atomic splittable routing games. Suppose that players form coalitions and each coalition behaves as if it were a single player controlling all the flows of its participants. We investigate the following question: under what conditions would the social cost of the post-collusion equilibrium be bounded by the social cost of the pre-collusion equilibrium? We show that if (i) the network is "well-designed" (satisfying a natural condition), and (ii) the delay functions are affine, then collusion is always beneficial for the social cost in the equilibrium flows. On the other hand, if either of the above conditions is unsatisfied, collusion can worsen the social cost. Our main technique is a novel flow-augmenting algorithm to build equilibrium flows. Our positive result for collusion is obtained by applying this algorithm simultaneously to two different flow value profiles of players and observing the difference in the derivatives of their social costs. Moreover, for a non-trivial subclass of selfish routing games, this algorithm finds the exact equilibrium flows in polynomial time.

Efficiency Loss of Mixed Equilibrium Behaviors with Polynomial Cost Functions

This paper investigates the efficiency loss of selfish routing simultaneously with user equilibrium (UE) player and Cournot-Nash (CN) players. The upper bound of the efficiency loss of UE-CN mixed equilibrium with polynomial cost functions is obtained by the scaling method and the nonlinear programming method, respectively. It is shown that the upper bound of efficiency loss obtained by the scaling method only depends on the polynomial degree of the link travel cost function and the upper bound of efficiency loss obtained by the nonlinear programming method depends on the number of the CN players besides the aforementioned factors. The numerical tests validate our analytical results. KEY WORDS: efficiency loss, UE-CN mixed equilibrium, variational inequality, selfish routing game

Convergence and approximation in potential games

We study the speed of convergence to approximately optimal states in two classes of potential games. We provide bounds in terms of the number of rounds, where a round consists of a sequence of movements, with each player appearing at least once in each round. We model the sequential interaction between players by a best-response walk in the state graph, where every transition in the walk corresponds to a best response of a player. Our goal is to bound the social value of the states at the end of such walks. In this paper, we focus on two classes of potential games: selfish routing games, and cut games (or party affiliation games (Fabrikant et al. 2004 [12])).Other than bounding the price of anarchy of selfish routing games (Roughgarden and Tardos, 2002 [25], Awerbuch et al. 2005 [2], Christodoulou and Koutsoupias, 2005 [9]), there are many interesting problems about game dynamics in these games. It is known that exponentially long best-response walks may exist to pure Nash equilibria (Fabrikant et al. 2004 [12]), and random best-response walks converge to solutions with good approximation guarantees after polynomially many best responses (Goemans et al. 2005 [17]). In this paper, we study the speed of convergence on deterministic best-response walks in these games and prove that starting from an arbitrary configuration, after one round of best responses of players, the resulting configuration is a Θ(n)-approximate solution. Furthermore, we show that starting from an empty configuration, the solution after any round of best responses is a constant-factor approximation. We also provide a lower bound for the multi-round case, where we show that for any constant number of rounds t, the approximation guarantee cannot be better than nϵ(t), for some ϵ(t)>0.We also study cut games, that provide an illustrative example of potential games. The convergence of potential games to locally optimum solutions has been studied in the context of local search algorithms (Johnson et al. 1988 [19], Schaffer and Yannakakis, 1991 [28]). In these games, we consider two social functions: the cut (defined as the weight of the edges in the cut), and the total happiness (defined as the weight of the edges in the cut, minus the weight of the remaining edges). For the cut social function, we prove that the expected social value after one round of a random best-response walk is at least a constant factor approximation to the optimal social value. We also exhibit exponentially long best-response walks with poor social value. For the unweighted version of this cut game, we prove Ω(n) and O(n) lower and upper bounds on the number of rounds of best responses to converge to a constant-factor cut. In addition, we suggest a way to modify the game to enforce a fast convergence in any fair best-response walk. For the total happiness social function, we show that for unweighted graphs of sufficiently large girth, starting from a random configuration, greedy behavior of players in a random order converges to an approximate solution after one round.

Pairwise cooperations in selfish ring routing for minimax linear latency

This paper studies the selfish routing game in ring networks with a load-dependent linear latency on each link. We adopt the asymmetric atomic routing model. Each player selfishly chooses a route to connect his source-destination pair, aiming at the lowest latency of his route, while the system objective is to minimize the maximum latency among all routes of players. The effectiveness of these routing games is often measured by the price of anarchy (PoA), the worst-case ratio between the maximum latencies in a Nash equilibrium (NE) and in a system optimum, where NE refers to a “stable state” among all players, from which no player has the incentive to deviate unilaterally. In classical setting, no cooperation is allowed and 16 stands as the current best upper bound on the PoA of such selfish ring routing. In this paper we show that the PoA is at most 10.16 provided cooperations within pairs of players are allowed, where any two players could change their routes simultaneously if neither would experience a longer latency and at least one would experience a shorter latency.

Tradeoffs and Average-Case Equilibria in Selfish Routing

We study Nash equilibria in a selfish routing game on m parallel links with transmission speeds. Each player seeks to communicate a message by choosing one of the links, and each player desires to minimize his experienced transmission time (latency). For evaluating the social cost of Nash equilibria, we consider the price of anarchy, which is the largest ratio of the cost of any Nash equilibrium compared to the optimum solution. Similarly, we consider the price of stability, which is the smallest ratio. The main purpose of this article is to quantify the influence of three parameters upon the prices of the game: the total traffic in the network; restrictions of the players in terms of link choice; and fluctuations in message lengths. Our main interest is to bound the sum of all player latencies, which we refer to as collective latency . For this cost, the prices of anarchy and stability are Θ ( n / t ), where n is the number of players and t the sum of message lengths (total traffic); that is, Nash equilibria approximate the optimum solution up to a constant factor if the traffic is high. If each player is restricted to choose from a subset of links, these link restrictions can cause a degradation in performance of order Θ(\sqrt{ m }). The prices of anarchy and stability increase to Θ( n \sqrt{ m }/ t ). We capture fluctuations in message lengths through a stochastic model, in which we valuate Nash equilibria in terms of their expected price of anarchy. The expected price is Θ( n /\mathbb{E}[ T ]), where \mathbb{E}[ T ] is the expected traffic. The stochastic model resembles the deterministic one, even for the efficiency loss of order Θ(\sqrt{ m }) for link restrictions. For the social cost function maximum latency, the (expected) price of anarchy is 1 + m 2 / t . In this case, Nash equilibria are almost optimal solutions for congested networks. Similar results hold when the cost function is a polynomial of the link loads.

Replicator dynamics with Pigovian subsidy and capitation tax

The selfish behaviors of individuals can cause an inefficient society in which the total amount of the individuals’ utilities is not maximized. To resolve the problem, a government sometimes tries to control the population by imposing a tax on and/or offering a subsidy to the individuals who belong to the population. The tax is roughly classified into rate taxations and capitation taxes. Using an evolutionary game theory, the authors have proposed a differential equation model with rate taxations to analyze their effects on players’ behaviors. In this paper, we propose a differential equation model with the capitation taxes, and derive stability conditions of the target state. Moreover, we also discuss an application of our model to selfish routing games.

A new model for selfish routing

In this work, we introduce and study a new, potentially rich model for selfish routing over non-cooperative networks, as an interesting hybridization of the two prevailing such models, namely the KP model [E. Koutsoupias, C.H. Papadimitriou, Worst-case equilibria, in: G. Meinel, S. Tison (Eds.), Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science, in: Lecture Notes in Computer Science, vol. 1563, Springer-Verlag, 1999, pp. 404–413] and the W model [J.G. Wardrop, Some theoretical aspects of road traffic research, Proceedings of the of the Institute of Civil Engineers 1 (Pt. II) (1952) 325–378]. In the hybrid model, each of n users is using a mixed strategy to ship its unsplittable traffic over a network consisting of m parallel links. In a Nash equilibrium, no user can unilaterally improve its Expected Individual Cost. To evaluate Nash equilibria, we introduce Quadratic Social Cost as the sum of the expectations of the latencies, incurred by the squares of the accumulated traffic. This modeling is unlike the KP model, where Social Cost [E. Koutsoupias, C.H. Papadimitriou, Worst-case equilibria, in: G. Meinel, S. Tison (Eds.), Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science, in: Lecture Notes in Computer Science, vol. 1563, Springer-Verlag, 1999, pp. 404–413] is the expectation of the maximum latency incurred by the accumulated traffic; but it is like the W model since the Quadratic Social Cost can be expressed as a weighted sum of Expected Individual Costs. We use the Quadratic Social Cost to define Quadratic Coordination Ratio. Here are our main findings: • Quadratic Social Cost can be computed in polynomial time. This is unlike the # P -completeness [D. Fotakis, S. Kontogiannis, E. Koutsoupias, M. Mavronicolas, P. Spirakis, The structure and complexity of Nash equilibria for a selfish routing game, in: P. Widmayer, F. Triguero, R. Morales, M. Hennessy, S. Eidenbenz, R. Conejo (Eds.), Proceedings of the 29th International Colloquium on Automata, Languages and Programming, in: Lecture Notes in Computer Science, vol. 2380, Springer-Verlag, 2002, pp. 123–134] of computing Social Cost for the KP model. • For the case of identical users and identical links, the fully mixed Nash equilibrium [M. Mavronicolas, P. Spirakis, The price of selfish routing, Algorithmica 48 (1) (2007) 91–126], where each user assigns positive probability to every link, maximizes Quadratic Social Cost. • As our main result, we present a comprehensive collection of tight, constant (that is, independent of m and n ), strictly less than 2 , lower and upper bounds on the Quadratic Coordination Ratio for several, interesting special cases. Some of the bounds stand in contrast to corresponding super-constant bounds on the Coordination Ratio previously shown in [A. Czumaj, B. Vöcking, Tight bounds for worst-case equilibria, ACM Transactions on Algorithms 3 (1) (2007); E. Koutsoupias, M. Mavronicolas, P. Spirakis, Approximate equilibria and ball fusion, Theory of Computing Systems 36 (6) (2003) 683–693; E. Koutsoupias, C.H. Papadimitriou, Worst-case equilibria, in: G. Meinel, S. Tison (Eds.), Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science, in: Lecture Notes in Computer Science, vol. 1563, Springer-Verlag, 1999, pp. 404–413; M. Mavronicolas, P. Spirakis, The price of selfish routing, Algorithmica 48 (1) (2007) 91–126] for the KP model.