We consider a game theoretic variant of an interval scheduling problem. Every job is associated with a length, a weight, and a color. Each player controls all the jobs of a specific color, and needs to decide on a processing interval for each of its jobs. Jobs of the same color can be processed simultaneously by the machine. A job is covered if the machine is configured to its color during its whole processing interval. The goal of the machine is to maximize the sum of weights of all covered jobs, and the goal of each player is to maximize the sum of weights of covered jobs from its color. The study of this game is motivated by several applications like antenna scheduling for wireless networks. We first show that given a strategy profile of the players, the machine scheduling problem can be solved in polynomial time. We then study the game from the players’ point of view. We analyze the existence of Nash equilibria, its computation, and inefficiency. We distinguish between instances of the classical interval scheduling problem, in which every player controls a single job, and instances in which color sets may include multiple jobs.

We consider a game-theoretic variant of an interval scheduling problem. Every job is associated with a length, a weight, and a color. Each player controls all the jobs of a specific color, and needs to decide on a processing interval for each of its jobs. Jobs of the same color can be processed simultaneously by the machine. A job is covered if the machine is configured to its color during its whole processing interval. The goal of the machine is to maximize the sum of weights of all covered jobs, and the goal of each player is to place its jobs such that the sum of weights of covered jobs from its color is maximized. The study of this game is motivated by several applications like antenna scheduling for wireless networks. We first show that given a strategy profile of the players, the machine scheduling problem can be solved in polynomial time. We then study the game from the players' point of view. We analyze the existence of Nash equilibria, its computation, and inefficiency. We distinguish between instances of the classical interval scheduling problem, in which every player controls a single job, and instances in which color sets may include multiple jobs.

In classical job-scheduling games, each job behaves as a selfish player, choosing a machine to minimize its own completion time. To reduce the equilibria inefficiency, coordination mechanisms [8] are employed, allowing each machine to follow its own scheduling policy. In this paper we study the effects of incorporating rank-based utilities within coordination mechanisms across environments with either identical or unrelated machines. With rank-based utilities, players aim to perform well relative to their competitors, rather than solely minimizing their completion time. We first demonstrate that even in basic setups, such as two identical machines with unit-length jobs, a pure Nash equilibrium (NE) assignment may not exist. This observation motivates our inquiry into the complexity of determining whether a given game instance admits a NE. We prove that this problem is NP-complete, even in highly restricted cases. In contrast, we identify specific classes of games where a NE is guaranteed to exist, or where the decision problem can be resolved in polynomial time. Additionally, we examine how competition impacts the efficiency of Nash equilibria, or sink equilibria if a NE does not exist. We derive tight bounds on the price of anarchy, and show that competition may either enhance or degrade overall performance.

Job-scheduling games have traditionally assumed fixed processing times. However, in many realistic environments, ranging from cyber-security response to high-frequency trading, a task's duration depends on its starting time. We study job-scheduling games with time-dependent processing times, where job lengths are linear functions of their start times, exhibiting either positive deterioration (increasing length) or negative deterioration (decreasing length). We analyze these games under various coordination mechanisms and priority policies. By introducing the concept of delay-averse agents, we provide a unifying framework to characterize equilibrium existence. For delay-averse jobs, we show that stability is maintained and pure Nash equilibria (NE) can be computed efficiently. In contrast, for non-delay-averse jobs, we demonstrate that a NE may not exist, and prove that deciding its existence is NP-complete, even on identical machines - a fundamental departure from classical coordination mechanisms. Regarding equilibrium inefficiency, we show that the Price of Anarchy (PoA) can be significantly higher than in environments with fixed processing times. To mitigate this, we propose and analyze three coordination mechanisms: SBPT (Shortest Basic Processing Time), which reduces the PoA in games with positive deterioration to a constant, and SDR (Smallest Deterioration Rate) and LBDR (Largest Basic-Deterioration Ratio) for negative deterioration, which achieve tight constant PoA bounds of 2 and max{e/e-1, 2-1/m}, respectively. Our results bridge the gap between centralized time-dependent scheduling and decentralized game-theoretic analysis.

We consider a scheduling environment in which jobs are associated with machine-dependent due-dates. This natural setting arises in systems where clients' tolerance depends on the service provider. The objective is to maximize throughput, defined as the number of non-tardy jobs. The problem exhibits significant differences from previously studied scheduling models. We analyze its computational complexity both in general and for the special case of unit-length jobs. In the unit-length setting, we provide an optimal algorithm that also extends to cases with machine-dependent release times and machine-dependent weights (i.e., rewards depending on the machine that completes the job). For jobs with different lengths, we show that even the unweighted problem without release times, with only two different lengths, specifically, for all j, pj is an element of {1, 2}, is APX-hard. To isolate the role of machine-dependent due-dates in this hardness result, we present an optimal algorithm for the case where all pj is an element of 1, 2 and due-dates are not machine-dependent. This algorithm further extends to instances with a constant number of integer processing times. 2012 ACM Subject Classification Theory of computation -> Scheduling algorithms

We study the price of combining fairness and stability (PoFS) in resource buying games, in which players share the activation cost of the resources they are using. The PoFS is the ratio between the cost of a min–max fair NE profile, and the cost of a cheapest NE profile. We distinguish between games played on resources with fixed costs and load-dependent costs, and between fair cost-sharing and arbitrary cost-sharing mechanisms. We provide tight bound for the PoFS in various game classes. While in general, striving for fairness may lead to a significant increase in the social cost, we identify classes for which PoFS =1 or is bounded by a small constant. We show that computing a min–max fair stable profile may be NP-hard even for simple classes, for which calculating a social optimum profile and a cheapest NE can be done efficiently. On the other hand, for other classes we provide optimal algorithms for calculating the min–max fair profile among the stable ones.

Many combinatorial optimization problems arise in the context of resource allocation. In this paper, we study the problem of allocating shareable resources of different types to jobs, where each job consists of multiple tasks, and each task has a demand of a given duration to a single resource type. All resources are available over a common time interval. Several copies from each resource may be allocated. In a valid solution, at any given time, each job may be processed by at most one resource, and each resource may process at most one job. The objective is to complete all jobs while minimizing the total cost of the allocated resources.We focus on the power of preemptions in this model, analyzing how much the total cost can be reduced when jobs are allowed to be preempted — that is, when the processing of tasks can be split into multiple intervals.We present both theoretical and experimental results, distinguishing between environments where jobs may be preempted but all intervals of a task must be processed on the same resource copy (weak preemptions), and environments where jobs may split the processing of a task among different resource copies (strong preemptions). Without preemptions, the problem is clearly NP-hard, as it generalizes the classical Bin Packing problem. We provide an optimal polynomial-time algorithm for the strong-preemption model, as well as a polynomial-time algorithm for the non-preemption model under a restricted class of task durations. Our empirical evaluation investigates the performance of several greedy heuristics, showing that even simple methods can achieve near-optimal results.

In job-scheduling games, each job is a selfish player that selects a machine to minimize its own completion time. Coordination mechanisms are employed to reduce the inefficiency of equilibria that result from such decentralized decision-making. This paper contributes to the extensive body of research on coordination mechanisms by investigating their application to unrelated parallel machines, where each machine may use its own scheduling policy to determine the processing order of assigned jobs. Since pure Nash equilibria (NE) are not guaranteed to exist in this setting, we identify and characterize several classes of instances-motivated by real-world applications-in which a NE is guaranteed to exist. For each such class, we design an algorithm to compute a NE, prove the convergence of best-response dynamics, and analyze the inefficiency of equilibria with respect to the makespan. In addition, we study two fundamental problems: (1) computing a NE schedule with low makespan, and (2) selecting, given a matrix of processing times, machine-specific scheduling policies that guarantee the existence of a NE with low makespan. For both problems, we establish computational hardness results.

Let G=(U ∪ V, E) be a bipartite graph, where U represents jobs and V represents machines. We study a new variant of the bipartite matching problem in which each job in U can be matched to at most one machine in V, and the number of jobs that can be assigned to a machine depends on the specific jobs matched to it. These pair-dependent bounds reflect systems where different jobs have varying tolerance for congestion, determined by the specific machine they are assigned to. We define a bipartite PD-matching as a set of edges M ⊆ E that satisfies these job-to-machine tolerance constraints. This variant of matching extends well-known matching problems, however, despite its relevance to real-world systems, it has not been studied before. We study bipartite PD-matchings with the objective of maximizing the matching size. As we show, the problem exhibits significant differences from previously studied matching problems. We analyze its computational complexity both in the general case and for specific restricted instances, presenting hardness results alongside optimal and approximation algorithms.

We consider a natural, yet challenging variant of the parallel machine scheduling problem in which each machine imposes a preferential order over the jobs and schedules the jobs accordingly once assigned to it. We study the problem of minimizing the total completion time, distinguishing between identical and unrelated machines, machine-dependent and identical priority lists, or a constant number of different priority classes. Additionally, we consider the setting in which the priority list on a machine must satisfy longest processing time first. We resolve the computational complexity of the problem and provide a clear distinction between problems that are polynomial time solvable and APX-hard.

Entrepreneurship has long been a driving force for innovation and economic growth. While it has been extensively studied by economists, it has not received much attention in the AGT community. We define and study an entrepreneurship facility-activation game, played by a single entrepreneur and n users. The entrepreneur may activate and close facilities, and each user should select one active facility. This setting combines a weighted singleton congestion game played by the users, with a revenue maximization game played by the entrepreneur, who dynamically determines the set of active facilities in response to the users’ assignment. We analyze the resulting game from multiple perspectives. From the entrepreneur’s perspective, maximizing her profit, we provide an asymptotically tight (√(n)) -approximation algorithm with and without stability restrictions. For the total welfare problem of minimizing the total users cost and facilities activation cost, we provide tight linear bounds for the PoA and PoS. Additionally, we analyze the computational complexity of both the social optimum and the cheapest stable solution. We distinguish between games with weighted and unweighted users, with and without symmetric strategies, and between arbitrary and uniform facility activation costs. Our results highlight the challenges of revenue maximization for entrepreneurs and the high impact of entrepreneurship on the total welfare and the equilibrium efficiency.

Studies in behavioural science show that individuals are often concerned primarily about their relative welfare, rather than their absolute well-being. In this paper we define and study a variant of congestion game that reflects this phenomenon. In a cost-sharing game with rank-based utilities (CSRB-game, for short), the players are partitioned into competition sets , and the goal of every player is to minimize its cost relative to its competitors . Specifically, the primary goal of a player is to minimize the rank of its cost among its competitors, while minimizing the cost itself is a secondary objective. We show that CSRB-games are significantly different from classical cost-sharing games, and that competition may lead to a poor outcome. In particular, singleton CSRB-games need not have a pure Nash equilibrium, and even when a NE exists, natural dynamics may not converge to a NE, and the price of stability is linear in the number of players. We then analyze several natural restricted classes of singleton CSRB-games, for which we present positive results. We provide tight characterization of classes for which a NE exists and can be computed efficiently, and bound the equilibrium inefficiency, based on the competition structure, the number of players and resources, the uniformity of resources’ costs, and the strategy space of competing players.

An instance of a weighted Stackelberg load balancing game is given by a set of identical machines, a set of variable-length jobs and a parameter 0 ≤ α ≤ 1. A centralized authority, denoted the leader, selects a subset of the jobs whose total length is at most an α-fraction of the total length and determines their assignment on the machines. After the controlled jobs are assigned, the remaining jobs join the schedule. They act selfishly, each determining its own assignment.Our work combines theoretical and experimental results for this setting. We suggest various heuristics for the leader and analyze their performance.

We consider cost-sharing games in which resources’ costs are fairly shared by their users. In this type of games, the total players’ cost in a Nash Equilibrium profile may be significantly higher than the social optimum. We compare and analyze several methods to lead the players to a good Nash Equilibrium by temporal addition of dummy players. The dummy players create artificial load on some resources, that encourage other players to change their strategies. This best-response (BR) dynamic continues until reaching a Nash equilibrium profile. We show that it is NP-hard to calculate an optimal strategy for the dummy players. We then focus on symmetric singleton games, where each player needs exactly one resource. We suggest several heuristics for the problem, based on the resources costs and the initial loads on the resources. The heuristics are simulated and their performance is evaluated, distinguishing between the following measures: The social cost of the final profile, the number of dummies used, the length of the BR-sequence till convergence, and the number of times the dummy players move. Our main conclusion is that the use of dummy players may significantly improve the equilibrium inefficiency.

In many scheduling environments, some jobs have higher priority than others.Such scenarios are theoretically modelled by associating jobs with weights, or by having precedence constraints that limit jobs' processing order.In this paper we define and consider a new model, motivated by real-life behaviour, in which the priority among jobs is defined by a dominance hierarchy.Specifically, the jobs are arranged in hierarchy levels, and high ranking jobs are ready to accept only outcomes in which the service they receive is better than the service of subordinate jobs.We first define the model and the set of feasible schedules formally.We then consider two classical problems: minimizing the maximal tardiness and minimizing the number of tardy jobs.We provide optimal algorithms or hardness proofs for these problems, distinguishing between a global objective function and a multi-criteria objective.

We consider cost-sharing games in which resources’ costs are fairly shared by their users. The total players’ cost in a Nash Equilibrium profile may be significantly higher than the social optimum. We compare and analyze several methods to lead the players to a good Nash Equilibrium by temporal addition of dummy players. The dummy players create artificial load on some resources, that encourage other players to change their strategies.We show that it is NP-hard to calculate an optimal strategy for the dummy players. We then focus on symmetric singleton games for which we suggest several heuristics for the problem. We analyze their performance distinguishing between several classes of instances and several performance measures.

We study the inefficiency of equilibria of resource buying games, i.e., congestion games with arbitrary cost-sharing. Under arbitrary cost-sharing, players do not only declare the resources they will use, they also declare and submit a payment per resource. If the total payments on a resource cover its cost, the resource is activated, otherwise it remains unavailable to the players. Equilibrium existence and inefficiency under arbitrary cost-sharing is very well understood in certain models, such as network design games, where the joint cost of every resource (edge) is constant. In the case of congestion-dependent costs the understanding is not yet complete. For increasing per player cost functions, it is known that the optimal solution can be cast as a Nash equilibrium with the appropriate selection of payments and, hence, the price of stability is 1. In this work we initially focus on the price of anarchy for linear congestion games and prove that (in the direct generalization of the arbitrary cost-sharing model to congestion-dependent costs) it grows to infinity as the number of players grows large. However, we also show that with a natural modification to the cost-sharing model, the price of anarchy becomes 17/3. Turning our attention to strong Nash equilibria, we show that the worst-case inefficiency of the best and worst stable outcomes remains the same as for Nash equilibria, with the strong price of stability staying at 1 and the strong price of anarchy staying at 17/3. These results imply arbitrary cost-sharing is comparable to fair cost-sharing as it has a better best-case scenario and a (slightly) worse worst-case scenario. We also study models with restricted strategy sets (uniform matroid congestion games) and properties of best response dynamics with arbitrary cost-sharing.

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 unitand arbitrary-cost machines, and instances with unitand 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). (C) 2020 Elsevier B.V. All rights reserved.

Job scheduling on parallel machines is a well-studied singleton congestion game. We consider a variant of this game in which the jobs are partitioned into competition sets, and the goal of every player is to minimize the completion time of his job relative to his competitors. Specifically, the primary goal of a player is to minimize the rank of its completion time among his competitors, while minimizing the completion time itself is a secondary objective. This fits environments with strong competition among the participants, in which the relative performance of the players determine their welfare. We define and study the corresponding race scheduling game (RSG). We show that RSGs are significantly different from classical job-scheduling games, and that competition may lead to a poor outcome. In particular, an RSG need not have a pure Nash equilibrium, and best-response dynamics may not converge to a NE even if one exists. We identify several natural classes of games, on identical and on related machines, for which a NE exists and can be computed efficiently, and we present tight bounds on the equilibrium inefficiencies. For some classes we prove convergence of BRD, while for others, even with very limited competition, BRD may loop. Among classes for which a NE is not guaranteed to exist, we distinguish between classes for which, it is tractable or NP-hard to decide if a given instance has a NE. Striving for stability, we also study the Nashification cost of RSGs, either by adding dummy jobs, or by compensating jobs for having high rank. Our analysis provides insights and initial results for several other congestion and cost-sharing games that have a natural ‘race’ variant.