Worst Nash Equilibrium Research Articles

A Coordination Mechanism for Scheduling Game on Parallel-Batch Machines with Deterioration Jobs

In this study, we consider a parallel-batch machines scheduling game problem with deterioration jobs. The processing time of a job is a simple linear function of its starting time. Each of the parallel-batch machines can process up to B jobs simultaneously as a batch. The processing time of a batch is the processing time of the job with the longest deteriorating rate in the batch. All jobs in the same batch start and complete at the same time. Each job as an agent and its individual cost is the completion time of the job. We present a coordination mechanism for the scheduling game problem with social cost of minimizing the makespan in this paper, namely fully batch longest deteriorating rate. For this problem, we precisely quantify the inefficiency of Nash equilibrium by the logarithm price of anarchy. It is defined to be the ratio between the logarithm of social cost of the worst Nash equilibrium and the logarithm of social cost of an optimum schedule. In addition, we discuss the existence of Nash equilibrium and present an upper bound and lower bounds on the logarithm price of anarchy of the coordination mechanism. We show that the mechanism has a logarithm price of anarchy at most 2 − 1 / 3 m a x m , B − 2 / 3 B .

Price of Pareto Optimality in hedonic games

The Price of Anarchy measures the welfare loss caused by selfish behavior: it is defined as the ratio of the social welfare in a socially optimal outcome and in a worst Nash equilibrium. Similar measures can be derived for other classes of stable outcomes. We observe that Pareto optimality can be seen as a notion of stability: an outcome is Pareto optimal if and only if it does not admit a deviation by the grand coalition that makes all players weakly better off and some players strictly better off. Motivated by this observation, we introduce the concept of Price of Pareto Optimality: this is an analogue of the Price of Anarchy, with the worst Nash equilibrium replaced with the worst Pareto optimal outcome. We then study this concept in the context of hedonic games, and provide lower and upper bounds on the Price of Pareto Optimality in three classes of hedonic games: additively separable hedonic games, fractional hedonic games, and modified fractional hedonic games.

The power of one evil secret agent

I am a job. In job-scheduling applications, my friends and I are assigned to machines that can process us. In the last decade, thanks to our strong labor union, and the rise of algorithmic game theory, we are getting more and more freedom regarding our assignment. Each of us acts to minimize his own cost, rather than to optimize a global objective.My goal is different. I am a secret agent operated by the system. I do my best to lead my fellow jobs to an outcome with a high social cost. My naive friends keep doing the best they can, each of them performs his best-response move whenever he gets the opportunity to do so. Luckily, I am a charismatic guy. I can determine the order according to which the naive jobs perform their best-response moves. In this paper, I analyze my power, formalized as the Price of a Traitor (PoT), in cost-sharing scheduling games – in which we need to cover the cost of the machines that process us.Starting from an initial Nash Equilibrium (NE) profile, I join the instance and hurt its stability. A sequence of best-response moves is performed until I vanish, leaving the naive jobs in a new NE. For an initial NE assignment, S0, the PoT measures the ratio between the social cost of a worst NE I can lead the jobs to, starting from S0, and the social cost of S0. The PoT of a game is the maximal such ratio among all game instances and initial NE assignments.My analysis distinguishes between instances with unit- and arbitrary-cost machines, and instances with unit- and arbitrary-length jobs. I give exact bounds on the PoT for each setting, in general and in symmetric games. While it turns out that in most settings my power is really impressive, my task is computationally hard (and also hard to approximate).

A new lower bound on the price of anarchy of selfish bin packing

In standard bin packing, the goal is to partition or pack items with positive sizes of at most 1 into a minimum number of subsets, called bins, each of a total size no larger than 1. We study bin packing games. In these games, given a valid partition of the items, each item has a cost associated with it, based on the partition and on its size. Specifically, the cost of an item is the ratio between its size and the total size of items packed into its bin, that is, the cost sharing is in proportion to item sizes. We study pure Nash equilibria, which exist for all such games, and prove a new lower bound on the price of anarchy, which is the asymptotic worst-case ratio between the cost of the worst Nash equilibrium and a socially optimal packing.

Price of anarchy and price of stability in multi-agent project scheduling

We consider a project scheduling environment in which the activities are partitioned among a set of agents. The owner of each activity can decide its length, which is linearly related to its cost within a minimum (crash) and a maximum (normal) length. For each day the project makespan is reduced with respect to its normal value, a reward is offered to the agents, and each agent receives a given ratio of the reward. As in classical game theory, we assume that the agents’ parameters are common knowledge. We study the Nash equilibria of the corresponding non-cooperative game as a desired state where no agent is motivated to change his/her decision. Regarding project makespan as an overall measure of efficiency, here we consider the worst and the best Nash equilibria (i.e., for which makespan is maximum and, respectively, minimum among Nash equilibria). We show that the problem of finding the worst Nash equilibrium is NP-hard (finding the best Nash equilibrium is already known to be strongly NP-hard), and propose an ILP formulation for its computation. We then investigate the values of the price of anarchy and the price of stability in a large sample of realistic size problems and get useful insights for the project owner.

Limits and limitations of no-regret learning in games

AbstractWe study the limit behavior and performance of no-regret dynamics in general game theoretic settings. We design protocols that achieve both good regret and equilibration guarantees in general games. We also establish a strong equivalence between them and coarse correlated equilibria (CCE). We examine structured game settings where stronger properties can be established for no-regret dynamics and CCE. In congestion games with non-atomic agents (each contributing a fraction of the flow), as we decrease the individual flow of agents, CCE become closely concentrated around the unique equilibrium flow of the non-atomic game. Moreover, we compare best/worst case no-regret learning behavior to best/worst case Nash equilibrium (NE) in small games. We prove analytical bounds on these inefficiency ratios for 2×2 games and unboundedness for larger games. Experimentally, we sample normal form games and compute their measures of inefficiency. We show that the ratio distribution has sharp decay, in the sense that most generated games have small ratios. They also exhibit strong anti-correlation between each other, that is games with large improvements from the best NE to the best CCE present small degradation from the worst NE to the worst CCE.

Price of Pareto Optimality in Hedonic Games

Price of Anarchy measures the welfare loss caused by selfish behavior: it is defined as the ratio of the social welfare in a socially optimal outcome and in a worst Nash equilibrium. A similar measure can be derived for other classes of stable outcomes. In this paper, we argue that Pareto optimality can be seen as a notion of stability, and introduce the concept of Price of Pareto Optimality: this is an analogue of the Price of Anarchy, where the maximum is computed over the class of Pareto optimal outcomes, i.e., outcomes that do not permit a deviation by the grand coalition that makes all players weakly better off and some players strictly better off. As a case study, we focus on hedonic games, and provide lower and upper bounds of the Price of Pareto Optimality in three classes of hedonic games: additively separable hedonic games, fractional hedonic games, and modified fractional hedonic games; for fractional hedonic games on trees our bounds are tight.

Workload Factoring: A Game-Theoretic Perspective

Contention among users utilizing a single shared resource arises in multiple contexts of computing and computer communications. We consider a setup in which users can split their work between a shared resource and a private resource. Unlike the private resource, which provides guaranteed performance, the performance of the shared resource is highly dependent on the usage pattern of other users, which in turn influences a user's decision if and to what extent to make use of the shared resource. The intrinsic relation between the utility that a user perceives from the shared resource and the usage pattern followed by other users gives rise to a noncooperative game, which we model and investigate. We show that the considered game admits a Nash equilibrium. Moreover, we show that this equilibrium is unique. We investigate the ratio between the worst Nash equilibrium and the social optimum, known as the of anarchy, and show that, while in some cases of interest the Nash equilibrium coincides with a social optimum, in other cases the price of anarchy can be arbitrarily large. We demonstrate that, somewhat counterintuitively, exercising admission control to the shared resource may deteriorate its performance. Furthermore, we demonstrate that certain (heavy) users may scare off other, potentially large, communities of users. Accordingly, we propose a resource allocation scheme that addresses this problem and opens the shared resource to a wide range of user types.

The ring design game with fair cost allocation

In this paper we study the network design game when the underlying network is a ring. In a network design game we have a set of players, each of them aims at connecting nodes in a network by installing links and equally sharing the cost of the installation with other users. The ring design game is the special case in which the potential links of the network form a ring. It is well known that in a ring design game the price of anarchy may be as large as the number of players. Our aim is to show that, despite the worst case, the ring design game always possesses good equilibria. In particular, we prove that the price of stability of the ring design game is at most 3/2, and such bound is tight. Moreover, we observe that the worst Nash equilibrium cannot cost more than 2 times the optimum if the price of stability is strictly larger than 1. We believe that our results might be useful for the analysis of more involved topologies of graphs, e.g., planar graphs.

Equilibria in Epidemic Containment Games

The spread of epidemics and malware is commonly modeled by diffusion processes on networks. Protective interventions such as vaccinations or installing anti-virus software are used to contain their spread. Typically, each node in the network has to decide its own strategy of securing itself, and its benefit depends on which other nodes are secure, making this a natural game-theoretic setting. There has been a lot of work on network security game models, but most of the focus has been either on simplified epidemic models or homogeneous network structure. We develop a new formulation for an epidemic containment game, which relies on the characterization of the SIS model in terms of the spectral radius of the network. We show in this model that pure Nash equilibria (NE) always exist, and can be found by a best response strategy. We analyze the complexity of finding NE, and derive rigorous bounds on their costs and the Price of Anarchy or PoA (the ratio of the cost of the worst NE to the optimum social cost) in general graphs as well as in random graph models. In particular, for arbitrary power-law graphs with exponent $\beta>2$, we show that the PoA is bounded by $O(T^{2(\beta-1)})$, where $T=\gamma/\alpha$ is the ratio of the recovery rate to the transmission rate in the SIS model. We prove that this bound is tight up to a constant factor for the Chung-Lu random power-law graph model. We study the characteristics of Nash equilibria empirically in different real communication and infrastructure networks, and find that our analytical results can help explain some of the empirical observations.

Auction-based distributed efficient economic operations of microgrid systems

This paper studies the economic operations of the microgrid in a distributed way such that the operational schedule of each of the units, like generators, load units, storage units, etc., in a microgrid system, is implemented by autonomous agents. We apply and generalise the progressive second price (PSP) auction mechanism which was proposed by Lazar and Semret to efficiently allocate the divisible network resources. Considering the economic operation for the microgrid systems, the generators play as sellers to supply energy and the load units play as the buyers to consume energy, while a storage unit, like battery, super capacitor, etc., may transit between buyer and seller, such that it is a buyer when it charges and becomes a seller when it discharges. Furthermore in a connected mode, each individual unit competes against not only the other individual units in the microgrid but also the exogenous main grid possessing fixed electricity price and infinite trade capacity; that is to say, the auctioneer assigns the electricity among all individual units and the main grid with respect to the submitted bid strategies of all individual units in the microgrid in an economic way. Due to these distinct characteristics, the underlying auction games are distinct from those studied in the literature. We show that under mild conditions, the efficient economic operation strategy is a Nash equilibrium (NE) for the PSP auction games, and propose a distributed algorithm under which the system can converge to an NE. We also show that the performance of worst NE can be bounded with respect to the system parameters, say the energy trading price with the main grid, and based upon that, the implemented NE is unique and efficient under some conditions.

Multi-Regional Transmission Planning as a Non-Cooperative Decision-Making

This paper discusses the transmission planning problem in a transmission network with multiple transmission planners. Each transmission planner is responsible for a region of the transmission network and maximizes its own region social welfare taking into account the transmission planning decisions of other transmission planners. The mathematical formulation of non-cooperative transmission planning problem is proposed. This problem is modeled using the multiple-leaders single-follower game in applied mathematics. The solution concept of the worst-Nash equilibrium is introduced to solve the set-up game. Different mathematical techniques are employed to formulate the worst-Nash equilibrium solution as a mixed-integer linear programming problem. A discussion on two possible applications of the derived mathematical structure is provided. The computational complexity is also discussed. The Three-Region IEEE-RTS96 example system is employed and modified to suit the purpose of analysis. The cooperative solution where the transmission planners merge into one single transmission planner is assumed as the benchmark in this study. The cooperative transmission planning is formulated in Appendix A of the paper. The results demonstrate the utility of the proposed solution in multi-regional transmission planning.

Circumventing the Price of Anarchy: Leading Dynamics to Good Behavior

Many natural games have a dramatic difference between the quality of their best and worst Nash equilibria, even in pure strategies. Yet, nearly all results to date on dynamics in games show only convergence to some equilibrium, especially within a polynomial number of steps. In this work we initiate a theory of how well-motivated multiagent dynamics can make use of global information about the game---which might be common knowledge or injected into the system by a helpful central agency---and show that in a wide range of interesting games this can allow the dynamics to quickly reach (within a polynomial number of steps) states of cost comparable to the best Nash equilibrium. We present several natural models for dynamics that can use such additional information and analyze their ability to reach low-cost states for two important and widely studied classes of potential games: network design with fair cost-sharing and party affiliation games (which include consensus and cut games). From the perspective of a...

A note on a selfish bin packing problem

In this paper, we consider a selfish bin packing problem, where each item is a selfish player and wants to minimize its cost. In our new model, if there are k items packed in the same bin, then each item pays a cost 1/k, where k ? 1. First we find a Nash Equilibrium (NE) in time O(n log n) within a social cost at most 1.69103OPT + 3, where OPT is the social cost of an optimal packing; where n is the number of items or players; then we give tight bounds for the worst NE on the social cost; finally we show that any feasible packing can be converged to a Nash Equilibrium in O(n 2) steps without increasing the social cost.

On the Convergence of Fictitious Play in Channel Selection Games

The channel selection (CS) problem in decentralized parallel multiple access channels considers the mutual interference of the different radio devices, and this interaction can be modeled by strategic-form games. Here, we show that the CS problem is a potential game (PG) and thus the fictitious play (FP) converges to a Nash equilibrium (NE) either in pure or mixed strategies. Using a 2-player 2-channel game, it is shown that convergence in mixed strategies might lead to cycles of action profiles which lead to individual spectral efficiencies (SE) which are worse than the SE at the worst NE in mixed and pure strategies. Finally, taking advantage from the fact that the CS problem is a PG and an aggregation game, we show a method to implement FP with local information and minimum feedback.

Transmission Augmentation with the Competition Benefit Modelling of Additional Transmission Capacity

Additional transmission capacity can improve the performance of the electricity market from two perspectives: firstly, efficiency benefit in terms of improving the social welfare of the electricity industry; and, secondly, competition benefit that leads to increasing competition among generating companies. This paper introduces an algorithm to model both the efficiency and competition benefits of additional transmission capacity in the process of transmission augmentation. The economic mathematical model is developed based on game theory in applied mathematics and the concept of social welfare in microeconomics. Transmission network service providers, generating companies and market management companies are placed in different stages of the developed model. The multiple Nash equilibria problem is tackled through the introduced concept of worst-Nash equilibrium. A numerical algorithm is designed to solve the developed model. The Garver’s and IEEE 14-bus example systems are carefully modified to suit the purpose of this study; the former is used for conceptual evaluation of the algorithm, and the latter for testing the developed numerical solution. The results show that the proposed approach can improve the efficiency of the electricity market and reduce the market power in the generation sector using the transmission augmentation policies.

On spectrum sharing games

Efficient spectrum-sharing mechanisms are crucial to alleviate the bandwidth limitation in wireless networks. In this paper, we consider the following question: can free spectrum be shared efficiently? We study this problem in the context of 802.11 or WiFi networks. Each access point (AP) in a WiFi network must be assigned a channel for it to service users. There are only finitely many possible channels that can be assigned. Moreover, neighboring access points must use different channels so as to avoid interference. Currently these channels are assigned by administrators who carefully consider channel conflicts and network loads. Channel conflicts among APs operated by different entities are currently resolved in an ad hoc manner (i.e., not in a coordinated way) or not resolved at all. We view the channel assignment problem as a game, where the players are the service providers and APs are acquired sequentially. We consider the price of anarchy of this game, which is the ratio between the total coverage of the APs in the worst Nash equilibrium of the game and what the total coverage of the APs would be if the channel assignment were done optimally by a central authority. We provide bounds on the price of anarchy depending on assumptions on the underlying network and the type of bargaining allowed between service providers. The key tool in the analysis is the identification of the Nash equilibria with the solutions to a maximal coloring problem in an appropriate graph. We relate the price of anarchy of these games to the approximation factor of local optimization algorithms for the maximum k-colorable subgraph problem. We also study the speed of convergence in these games.

Transmission System Augmentation Based on the Concepts of Quantity Withheld and Monopoly Rent for Reducing Market Power

This paper proposes two mathematical structures for considering the market power effect of transmission capacity in transmission augmentation assessment. These mathematical structures use the concepts of monopoly rent and quantity withheld in economics for market power modeling in the assessment process of transmission augmentation. The simultaneous-move and sequential-move games in applied mathematics are used to model the interactions of the transmission network service provider, generating companies, and the market management company in the proposed mathematical structures. The solution concept of Nash equilibria is reformulated as an optimization problem, and the multiple Nash equilibria is tackled through an introduced concept termed worst Nash equilibrium. A numerical solution is developed to solve the proposed mathematical structures. The numerical solution is an island parallel genetic algorithm nested in a standard genetic algorithm. The six-bus Garver's example system and the IEEE 14-bus test system are modified and studied. The results prove the strong mechanism of the developed structures for modeling the market power effect of transmission capacity in the assessment of transmission augmentation.

Cost of cooperation for scheduling meetings

Scheduling meetings among agents can be represented as a game - the Meetings Scheduling Game (MSG). In its simplest form, the two-person MSG is shown to have a price of anarchy (PoA) which is bounded by 0.5. The PoA bound provides a measure on the efficiency of the worst Nash Equilibrium in social (or global) terms. The approach taken by the present paper introduces the Cost of Cooperation (CoC) for games. The CoC is defined with respect to different global objective functions and provides a measure on the efficiency of a solution for each participant (personal). Applying an ?egalitarian? objective, that maximizes the minimal gain among all participating agents, on our simple example results in a CoC which is non positive for all agents. This makes the MSG a cooperation game. The concepts are defined and examples are given within the context of the MSG. Although not all games are cooperation games, a game may be revised by adding a mediator (or with a slight change of its mechanism) so that it behaves as a cooperation game. Rational participants can cooperate (by taking part in a distributed optimization protocol) and receive a payoff which will be at least as high as the worst gain expected by a game theoretic equilibrium point.

Two-person knapsack game

In this paper, we study a two-person knapsack game. Two investors, each with an individual budget, bid on a common pool of potential projects. To undertake a project, investors have their own cost estimation to be charged against their budgets. Associated with each project, there is a potential market profit that can be taken by the only bidder or shared proportionally by both bidders. The objective function of each investor is assumed to be a linear combination of the two bidders' profits. Both investors act in a selfish manner with best-response to optimize their own objective functions by choosing portfolios under the budget restriction. We show that pure Nash equilibrium exists under certain conditions. In this case, no investor can improve the objective by changing individual bid unilaterally. A pseudo polynomial-time algorithm is presented for generating a pure Nash equilibrium. We also investigate the price of anarchy (the ratio of the worst Nash equilibrium to the social optimum) associated with a simplified two-person knapsack game.