Stackelberg Routing Research Articles

New Game-Theoretic Approach to Decentralized Path Selection and Sleep Scheduling for Mobile Edge Computing

Network function virtualization (NFV) implements mobile edge computing (MEC) services as software appliances, and allows resources to be adaptively allocated to accommodate demand variations. Scalability and network cost (including operational cost and response latency) are key challenges. This paper presents a new game-theoretic approach to minimizing the network cost, where access points (APs) select MEC servers and routes in a decentralized manner, and unloaded routers and links are deactivated for cost saving. The key idea is that we interpret the minimization of network cost as a mixed game with a non-monotonic cost function capturing both the operational cost and response latency. We prove that the game is conditionally an ordinary potential game and converges to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -approximate equilibriums. A closed-form expression is derived for the convergence delay. Another important aspect is that we integrate Stackelberg routing into the proposed mixed game to avoid inefficient equilibriums (with high cost or latency). We prove that the mixed game can converge faster to better equilibriums under linear response latency models. Extensive simulations corroborate the new game-theoretic approach can significantly outperform existing techniques in terms of efficiency, convergence, and scalability.

On Social Optimal Routing Under Selfish Learning

We consider a repeated routing game over a finite horizon with partial control under selfish response, in which a central authority can control a fraction of the flow and seeks to improve a network-wide objective, while the remaining flow applies an online learning algorithm. This finite horizon control problem is inspired from the one-shot Stackelberg routing game. Our setting is different in that we do not assume that the selfish players play a Nash equilibrium; instead, we assume that they apply an online learning algorithm. This results in an optimal control problem under learning dynamics. We propose different methods for approximately solving this problem: A greedy algorithm and a mirror descent algorithm based on the adjoint method. In particular, we derive the adjoint system equations of the Hedge dynamics and show that they can be solved efficiently. We compare the performance of these methods (in terms of achieved cost and computational complexity) on parallel networks, and on a model of the Los Angeles highway network.

Stackelberg Routing on Parallel Networks With Horizontal Queues

This paper presents a game theoretic framework for studying Stackelberg routing games on parallel networks with horizontal queues, such as transportation networks. First, we introduce a new class of latency functions that models congestion due to the formation of physical queues. For this new class, some results from the classical congestion games literature (in which latency is assumed to be a non-decreasing function of the flow) do not hold. In particular, we find that there may exist multiple Nash equilibria that have different total costs. We provide a simple polynomial-time algorithm for computing the best Nash equilibrium, i.e., the one which achieves minimal total cost. Then we study the Stackelberg routing game: assuming a central authority has control over a fraction of the flow on the network (compliant flow), and that the remaining flow (non-compliant) responds selfishly, what is the best way to route the compliant flow in order to minimize the total cost? We propose a simple Stackelberg strategy, the Non-Compliant First (NCF) strategy, that can be computed in polynomial time. We show that it is optimal for this new class of latency on parallel networks. This work is applied to modeling and simulating congestion relief on transportation networks, in which a coordinator (traffic management agency) can choose to route a fraction of compliant drivers, while the rest of the drivers choose their routes selfishly.

Altruism in Atomic Congestion Games

This article studies the effects of altruism, a phenomenon widely observed in practice, in the model of atomic congestion games. Altruistic behavior is modeled by a linear trade-off between selfish and social objectives. Our model can be embedded in the framework of congestion games with player-specific latency functions. Stable states are the pure Nash equilibria of these games, and we examine their existence and the convergence of sequential best-response dynamics. In general, pure Nash equilibria are often absent, and existence is NP-hard to decide. Perhaps surprisingly, if all delay functions are affine, the games remain potential games, even when agents are arbitrarily altruistic. The construction underlying this result can be extended to a class of general potential games and social cost functions, and we study a number of prominent examples. These results give important insights into the robustness of multi-agent systems with heterogeneous altruistic incentives. Furthermore, they yield a general technique to prove that stabilization is robust, even with partly altruistic agents, which is of independent interest. In addition to these results for uncoordinated dynamics, we consider a scenario with a central altruistic institution that can set incentives for the agents. We provide constructive and hardness results for finding the minimum number of altruists to stabilize an optimal congestion profile and more general mechanisms to incentivize agents to adopt favorable behavior. These results are closely related to Stackelberg routing and answer open questions raised recently in the literature.

The effectiveness of stackelberg strategies and tolls for network congestion games

It is well known that in a network with arbitrary (convex) latency functions that are a function of edge traffic, the worst-case ratio, over all inputs, of the system delay caused due to selfish behavior versus the system delay of the optimal centralized solution may be unbounded even if the system consists of only two parallel links. This ratio is called the price of anarchy (PoA). In this article, we investigate ways by which one can reduce the performance degradation due to selfish behavior. We investigate two primary methods (a) Stackelberg routing strategies , where a central authority, for example, network manager, controls a fixed fraction of the flow, and can route this flow in any desired way so as to influence the flow of selfish users; and (b) network tolls , where tolls are imposed on the edges to modify the latencies of the edges, and thereby influence the induced Nash equilibrium. We obtain results demonstrating the effectiveness of both Stackelberg strategies and tolls in controlling the price of anarchy. For Stackelberg strategies, we obtain the first results for nonatomic routing in graphs more general than parallel-link graphs, and strengthen existing results for parallel-link graphs. (i) In series-parallel graphs, we show that Stackelberg routing reduces the PoA to a constant (depending on the fraction of flow controlled). (ii) For general graphs, we obtain latency-class specific bounds on the PoA with Stackelberg routing, which give a continuous trade-off between the fraction of flow controlled and the price of anarchy. (iii) In parallel-link graphs, we show that for any given class L of latency functions, Stackelberg routing reduces the PoA to at most α + (1-α)ċρ( L ), where α is the fraction of flow controlled and ρ( L ) is the PoA of class L (when α = 0). For network tolls, motivated by the known strong results for nonatomic games, we consider the more general setting of atomic splittable routing games. We show that tolls inducing an optimal flow always exist, even for general asymmetric games with heterogeneous users, and can be computed efficiently by solving a convex program . This resolves a basic open question about the effectiveness of tolls for atomic splittable games. Furthermore, we give a complete characterization of flows that can be induced via tolls.

Stackelberg Routing in Arbitrary Networks

We investigate the impact of Stackelberg routing to reduce the price of anarchy in network routing games. In this setting, an α fraction of the entire demand is first routed centrally according to a predefined Stackelberg strategy and the remaining demand is then routed selfishly by (nonatomic) players. Although several advances have been made recently in proving that Stackelberg routing can, in fact, significantly reduce the price of anarchy for certain network topologies, the central question of whether this holds true in general is still open. We answer this question negatively by constructing a family of single-commodity instances such that every Stackelberg strategy induces a price of anarchy that grows linearly with the size of the network. Moreover, we prove upper bounds on the price of anarchy of the largest-latency-first (LLF) strategy that only depend on the size of the network. Besides other implications, this rules out the possibility to construct constant-size networks to prove an unbounded price of anarchy. In light of this negative result, we consider bicriteria bounds. We develop an efficiently computable Stackelberg strategy that induces a flow whose cost is at most the cost of an optimal flow with respect to demands scaled by a factor of 1 + √1−α. Finally, we analyze the effectiveness of an easy-to-implement Stackelberg strategy, called SCALE. We prove bounds for a general class of latency functions that includes polynomial latency functions as a special case. Our analysis is based on an approach that is simple yet powerful enough to obtain (almost) tight bounds for SCALE in general networks.

Stackelberg thresholds in network routing games or the value of altruism

It is well known that the Nash equilibrium in network routing games can have strictly higher cost than the optimum cost. In Stackelberg routing games, where a fraction of flow is centrally-controlled, a natural problem is to route the centrally-controlled flow such that the overall cost of the resulting equilibrium is minimized. We consider the scenario where the network administrator wants to know the minimum amount of centrally-controlled flow such that the cost of the resulting equilibrium solution is strictly less than the cost of the Nash equilibrium. We call this threshold the Stackelberg threshold and prove that for networks of parallel links with linear latency functions, it is equal to the minimum of the Nash flows on links carrying more optimum flow than Nash flow. Our approach also provides a simpler proof of characterization of the minimum fraction that must be centrally controlled to induce the optimum solution.

The price of optimum in Stackelberg games on arbitrary single commodity networks and latency functions

Let M be a single s – t network of parallel links with load dependent latency functions shared by an infinite number of selfish users. This may yield a Nash equilibrium with unbounded Coordination Ratio [E. Koutsoupias, C. Papadimitriou, Worst-case equilibria, in: 16th Annual Symposium on Theoretical Aspects of Computer Science, STACS, vol. 1563, 1999, pp. 404–413; T. Roughgarden, É. Tardos, How bad is selfish routing? in: 41st IEEE Annual Symposium of Foundations of Computer Science, FOCS, 2000, pp. 93–102]. A Leader can decrease the coordination ratio by assigning flow α r on M , and then all Followers assign selfishly the ( 1 − α ) r remaining flow. This is a Stackelberg Scheduling Instance ( M , r , α ) , 0 ≤ α ≤ 1 . It was shown [T. Roughgarden, Stackelberg scheduling strategies, in: 33rd Annual Symposium on Theory of Computing, STOC, 2001, pp. 104–113] that it is weakly NP-hard to compute the optimal Leader’s strategy. For any such network M we efficiently compute the minimum portion β M of flow r > 0 needed by a Leader to induce M ’s optimum cost, as well as her optimal strategy. This shows that the optimal Leader’s strategy on instances ( M , r , α ≥ β M ) is in P . Unfortunately, Stackelberg routing in more general nets can be arbitrarily hard. Roughgarden presented a modification of Braess’s Paradox graph, such that no strategy controlling α r flow can induce ≤ 1 α times the optimum cost. However, we show that our main result also applies to any s – t net G . We take care of the Braess’s graph explicitly, as a convincing example. Finally, we extend this result to k commodities. A conference version of this paper has appeared in [A. Kaporis, P. Spirakis, The price of optimum in stackelberg games on arbitrary single commodity networks and latency functions, in: 18th annual ACM symposium on Parallelism in Algorithms and Architectures, SPAA, 2006, pp. 19–28]. Some preliminary results have also appeared as technical report in [A.C. Kaporis, E. Politopoulou, P.G. Spirakis, The price of optimum in stackelberg games, in: Electronic Colloquium on Computational Complexity, ECCC, (056), 2005].

Stackelberg Strategies for Selfish Routing in General Multicommodity Networks

A natural generalization of the selfish routing setting arises when some of the users obey a central coordinating authority, while the rest act selfishly. Such behavior can be modeled by dividing the users into an α fraction that are routed according to the central coordinator’s routing strategy (Stackelberg strategy), and the remaining 1−α that determine their strategy selfishly, given the routing of the coordinated users. One seeks to quantify the resulting price of anarchy, i.e., the ratio of the cost of the worst traffic equilibrium to the system optimum, as a function of α. It is well-known that for α=0 and linear latency functions the price of anarchy is at most 4/3 (J. ACM 49, 236–259, 2002). If α tends to 1, the price of anarchy should also tend to 1 for any reasonable algorithm used by the coordinator. We analyze two such algorithms for Stackelberg routing, LLF and SCALE. For general topology networks, multicommodity users, and linear latency functions, we show a price of anarchy bound for SCALE which decreases from 4/3 to 1 as α increases from 0 to 1, and depends only on α. Up to this work, such a tradeoff was known only for the case of two nodes connected with parallel links (SIAM J. Comput. 33, 332–350, 2004), while for general networks it was not clear whether such a result could be achieved, even in the single-commodity case. We show a weaker bound for LLF and also some extensions to general latency functions. The existence of a central coordinator is a rather strong requirement for a network. We show that we can do away with such a coordinator, as long as we are allowed to impose taxes (tolls) on the edges in order to steer the selfish users towards an improved system cost. As long as there is at least a fraction α of users that pay their taxes, we show the existence of taxes that lead to the simulation of SCALE by the tax-payers. The extension of the results mentioned above quantifies the improvement on the system cost as the number of tax-evaders decreases.

Stackelberg games and multiple equilibrium behaviors on networks

The classical Wardropian principle assumes that users minimize either individual travel cost or overall system cost. Unlike the pure Wardropian equilibrium, there might be in reality both competition and cooperation among users, typically when there exist oligopoly Cournot-Nash (CN) firms. In this paper, we first formulate a mixed behavior network equilibrium model as variational inequalities (VI) that simultaneously describe the routing behaviors of user equilibrium (UE), system optimum (SO) and CN players, each player is presumed to make routing decision given knowledge of the routing strategies of other players. After examining the existence and uniqueness of solutions, the diagonalization approach is applied to find a mixed behavior equilibrium solution. We then present a Stackelberg routing game on the network in which the SO player is the leader and the UE and CN players are the followers. The UE and CN players route their flows in a mixed equilibrium behavior given the SO player’s routing strategy. In contrast, the SO player, realizing how the UE and CN players react to the given strategy, routes its flows to minimize total system travel cost. The Stackelberg game of network flow routing is formulated as a mathematical program with equilibrium constraints (MPEC). Using a marginal function approach, the MPEC is transformed into an equivalent, continuously differentiable single-level optimization problem, where the lower level VI is represented by a differentiable gap function constraint. The augmented Lagrangian method is then used to solve the resulting single-level optimization problem. Some numerical examples are presented to demonstrate the proposed models and algorithms.

Stackelberg Routing in Atomic Network Games

We consider network games with atomic players, which indicates that some players control a positive amount of flow. Instead of studying Nash equilibria as previous work has done, we consider that players with considerable market power will make decisions before the others because they can predict the decisions of players without market power. This description fits the framework of Stackelberg games, where those with market power are leaders and the rest are price-taking followers. As Stackelberg equilibria are difficult to characterize, we prove bounds on the inefficiency of the solutions that arise when the leader uses a heuristic that approximate its optimal strategy.

INCENTIVE STACKELBERG STRATEGIES FOR FLOW CONFIGURATION OF PARALLEL NETWORKS

This work consider the problem of flow control using incentive strategy in Stackelberg game theory. The network model employed here is that users route their flows from a common source to a common destination node, each of them trying to optimize its individual performance objective. First, the existing Stackelberg routing strategy is briefly introduced. And then, the linear Stackelberg incentive strategies are presented for both single-follower and multi-follower systems, by which the leader (manager) force the followers (the noncooperative users) adopt the team optimal flow configuration as their reply strategies. It is shown that the incentive strategy improves the existing Stackelberg routing strategy. A numerical example is given to illustrate the results of the strategy.