We study a non-cooperative two-sided facility location game in which facilities and clients behave strategically. This is in contrast to many other facility location games in which clients simply visit their closest facility. Facility agents select a location on a graph to open a facility to attract as much purchasing power as possible, while client agents choose which facilities to patronize by strategically distributing their purchasing power in order to minimize their total waiting time. Here, the waiting time of a facility depends on its received total purchasing power. We show that our client stage is an atomic splittable congestion game, which implies existence, uniqueness and efficient computation of a client equilibrium. Therefore, facility agents can efficiently predict client behavior and make strategic decisions accordingly. Despite that, we prove that subgame perfect equilibria do not exist in all instances of this game and that their existence is NP-hard to decide. On the positive side, we provide a simple and efficient algorithm to compute 3-approximate subgame perfect equilibria.

The strategic selection of resources by selfish agents is a classical research direction, with Resource Selection Games and Congestion Games as prominent examples. In these games, agents select available resources and their utility then depends on the number of agents using the same resources. This implies that there is no distinction between the agents, i.e., they are anonymous. We depart from this very general setting by proposing Resource Selection Games with heterogeneous agents that strive for a joint resource usage with similar agents. So, instead of the number of other users of a given resource, our model considers agents with different types and the decisive feature is the fraction of same-type agents among the users. More precisely, similarly to Schelling Games, there is a tolerance threshold tau in [0,1] which specifies the agents' desired minimum fraction of same-type agents on a resource. Agents strive to select resources where at least a tau-fraction of those resources' users have the same type as themselves. For tau=1, our model generalizes hedonic diversity games with single-peaked utilities with a peak at 1. For our general model, we consider the existence and quality of equilibria and the complexity of maximizing the social welfare. Additionally, we consider a bounded rationality model, where agents can only estimate the utility of a resource, since they only know the fraction of same-type agents on a given resource, but not the exact numbers. Thus, they cannot know the impact a strategy change would have on a target resource. Interestingly, we show that this type of bounded rationality yields favorable game-theoretic properties and specific equilibria closely approximate equilibria of the full knowledge setting.

We consider non-cooperative facility location games where both facilities and clients act strategically and heavily influence each other. This contrasts established game-theoretic facility location models with non-strategic clients that simply select the closest opened facility. In our model, every facility location has a set of attracted clients and each client has a set of shopping locations and a weight that corresponds to its spending capacity. Facility agents selfishly select a location for opening their facility to maximize the attracted total spending capacity, whereas clients strategically decide how to distribute their spending capacity among the opened facilities in their shopping range. We focus on a natural client behavior similar to classical load balancing: our selfish clients aim for a distribution that minimizes their maximum waiting time for getting serviced, where a facility’s waiting time corresponds to its total attracted client weight. We show that subgame perfect equilibria exist and we give almost tight constant bounds on the Price of Anarchy and the Price of Stability, which even hold for a broader class of games with arbitrary client behavior. Since facilities and clients influence each other, it is crucial for the facilities to anticipate the selfish clients’ behavior when selecting their location. For this, we provide an efficient algorithm that also implies an efficient check for equilibrium. Finally, we show that computing a socially optimal facility placement is NP-hard and that this result holds for all feasible client weight distributions.

We characterise the set of dominant strategy incentive compatible (DSIC), strongly budget balanced (SBB), and ex-post individually rational (IR) mechanisms for the multi-unit bilateral trade setting. In such a setting there is a single buyer and a single seller who holds a finite number k of identical items. The mechanism has to decide how many units of the item are transferred from the seller to the buyer and how much money is transferred from the buyer to the seller. We consider two classes of valuation functions for the buyer and seller: Valuations that are increasing in the number of units in possession, and the more specific class of valuations that are increasing and submodular.Furthermore, we present some approximation results about the performance of certain such mechanisms, in terms of social welfare: For increasing submodular valuation functions, we show the existence of a deterministic 2-approximation mechanism and a randomised e/(1 − e) approximation mechanism, matching the best known bounds for the single-item setting.

We study a two-sided network investment game consisting of two sets of players, called providers and users. The game is set in two stages. In the first stage, providers aim to maximize their profit by investing in bandwidth of cloud computing services. The investments of the providers yield a set of usable services for the users. In the second stage, each user wants to process a task and therefore selects a bundle of services so as to minimize the total processing time. We assume the total processing time to be separable over the chosen services and the processing time of each service to depend on the utilization of the service and the installed bandwidth. We provide insights on how competition between providers affects the total costs of the users and show that every game on a series-parallel graph can be reduced to an equivalent single edge game when analyzing the set of subgame perfect Nash equilibria.

We consider a scheduling problem on m identical processors sharing an arbitrarily divisible resource. In addition to assigning jobs to processors, the scheduler must distribute the resource among the processors (e.g., for three processors in shares of 20%, 15%, and 65%) and adjust this distribution over time. Each job j comes with a size pj ∈ R and a resource requirement rj > 0. Jobs do not benefit when receiving a share larger than rj of the resource. But providing them with a fraction of the resource requirement causes a linear decrease in the processing efficiency. We seek a (non-preemptive) job and resource assignment minimizing the makespan. Our main result is an efficient approximation algorithm which achieves an approximation ratio of 2 + 1/(m-2). It can be improved to an (asymptotic) ratio of 1 + 1/(m-1) if all jobs have unit size. Our algorithms also imply new results for a well-known bin packing problem with splittable items and a restricted number of allowed item parts per bin.

Max-min fairness (MMF) is a widely known approach to a fair allocation of bandwidth to each of the users in a network. This allocation can be computed by uniformly raising the bandwidths of all users without violating capacity constraints. We consider an extension of these allocations by raising the bandwidth with arbitrary and not necessarily uniform time-depending velocities (allocation rates). These allocations are used in a game-theoretic context for routing choices, which we formalize in progressive filling games (PFGs). We present a variety of results for equilibria in PFGs. We show that these games possess pure Nash and strong equilibria. While computation in general is NP-hard, there are polynomial-time algorithms for prominent classes of Max-Min-Fair Games (MMFG), including the case when all users have the same source-destination pair. We characterize prices of anarchy and stability for pure Nash and strong equilibria in PFGs and MMFGs when players have different or the same source-destination pairs. In addition, we show that when a designer can adjust allocation rates, it is possible to design games with optimal strong equilibria. Some initial results on polynomial-time algorithms in this direction are also derived.

We study the existence of approximate pure Nash equilibria in weighted congestion games and develop techniques to obtain approximate potential functions that prove the existence of alpha-approximate pure Nash equilibria and the convergence of alpha-improvement steps. Specifically, we show how to obtain upper bounds for approximation factor alpha for a given class of cost functions. For example for concave cost functions the factor is at most 3/2, for quadratic cost functions it is at most 4/3, and for polynomial cost functions of maximal degree d it is at at most d + 1. For games with two players we obtain tight bounds which are as small as for example 1.054 in the case of quadratic cost functions.

Max-min fairness (MMF) is a widely known approach to a fair allocation of bandwidth to each of the users in a network. This allocation can be computed by uniformly raising the bandwidths of all users without violating capacity constraints. We consider an extension of these allocations by raising the bandwidth with arbitrary and not necessarily uniform time-depending velocities (allocation rates). These allocations are used in a game-theoretic context for routing choices, which we formalize in progressive filling games (PFGs). We present a variety of results for equilibria in PFGs. We show that these games possess pure Nash and strong equilibria. While computation in general is NP-hard, there are polynomial-time algorithms for prominent classes of Max-Min-Fair Games (MMFG), including the case when all users have the same source-destination pair. We characterize prices of anarchy and stability for pure Nash and strong equilibria in PFGs and MMFGs when players have different or the same source-destination pairs. In addition, we show that when a designer can adjust allocation rates, it is possible to design games with optimal strong equilibria. Some initial results on polynomial-time algorithms in this direction are also derived.

We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by potential function arguments. Unfortunately, this proof of existence is inefficient and computing pure Nash equilibria in such games is a PLS-hard problem even when all players have unit weights. The situation gets worse when superlinear (e.g., quadratic) latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and pure Nash equilibria may not even exist. Given these obstacles, we consider approximate pure Nash equilibria as alternative solution concepts. Do such equilibria exist? And if so, can we compute them efficiently?

Congestion games constitute an important class of games in which computing an exact or even approximate pure Nash equilibrium is in general PLS-complete. We present a surprisingly simple polynomial-time algorithm that computes O(1)-approximate Nash equilibria in these games. In particular, for congestion games with linear latency functions, our algorithm computes (2 +ε)-approximate pure Nash equilibria in time polynomial in the number of players, the number of resources and 1/ε. It also applies to games with polynomial latency functions with constant maximum degree d: there, the approximation guarantee is do(d). The algorithm essentially identifies a polynomially long sequence of best-response moves that lead to an approximate equilibrium; the existence of such short sequences is interesting in itself. These are the first positive algorithmic results for approximate equilibria in non-symmetric congestion games. We strengthen them further by proving that, for congestion games that deviate from our mild assumptions, computing ρ-approximate equilibria is PLS-complete for any polynomial-time computable ρ.

We study the existence and computational complexity of coalitional stability concepts based on social networks. Our concepts represent a natural and rich combinatorial generalization of a recent notion termed partition equilibrium [5]. We assume that players in a strategic game are embedded in a social (or, communication) network, and there are coordination constraints defining the set of coalitions that can jointly deviate in the game. A main feature of our approach is that players act in a fashion to ignore potentially profitable (group) deviations if the change in their strategy may cause a decrease of utility to their neighbors in the network. We explore the properties of such considerate equilibria in application to the celebrated class of resource selection games (RSGs). Our main result proves existence of a super-strong considerate equilibrium in all symmetric RSGs with strictly increasing delays, for any social network among the players and feasible coalitions represented by the set of cliques. The existence proof is constructive and yields an efficient algorithm. In fact, the computed considerate equilibrium is a Nash equilibrium for a standard RSG, thus showing that there exists a state that is stable against selfish and considerate behavior simultaneously. Furthermore, we provide results on convergence of considerate dynamics.

Studying Nash dynamics is an important approach for analyzing the outcome of games with repeated selfish behavior of self-interested agents. Sink equilibria has been introduced by Goemans, Mirrokni, and Vetta for studying social cost on Nash dynamics over pure strategies in games. However, they do not address the complexity of sink equilibria in these games. Recently, Fabrikant and Papadimitriou initiated the study of the complexity of Nash dynamics in two classes of games. In order to completely understand the complexity of Nash dynamics in a variety of games, we study the following three questions for various games: (i) given a state in game, can we verify if this state is in a sink equilibrium or not? (ii) given an instance of a game, can we verify if there exists any sink equilibrium other than pure Nash equilibria? and (iii) given an instance of a game, can we verify if there exists a pure Nash equilibrium (i.e, a sink equilibrium with one state)?

We study the speed of convergence of decentralized dynamics to approximately optimal solutions in potential games. We consider α-Nash dynamics in which a player makes a move if the improvement in his payoff is more than an α factor of his own payoff. Despite the known polynomial convergence of α-Nash dynamics to approximate Nash equilibria in symmetric congestion games [7], it has been shown that the convergence time to approximate Nash equilibria in asymmetric congestion games is exponential [25]. In contrast to this negative result, and as the main result of this paper, we show that for asymmetric congestion games with linear and polynomial delay functions, the convergence time of α-Nash dynamics to an approximate optimal solution is polynomial in the number of players, with approximation ratio that is arbitrarily close to the price of anarchy of the game. In particular, we show this polynomial convergence under the minimal liveness assumption that each player gets at least one chance to move in every T steps. We also prove that the same polynomial convergence result does not hold for (exact) best-response dynamics, showing the α-Nash dynamics is required. We extend these results for congestion games to other potential games including weighted congestion games with linear delay functions, cut games (also called party affiliation games) and market sharing games.

The complexity of computing pure Nash equilibria in congestion games was recently shown to be PLS-complete. In this paper, we therefore study the complexity of computing approximate equilibria in congestion games. An alpha-approximate equilibrium, for α > 1, is a state of the game in which none of the players can make an α-greedy step, i.e., an unilateral strategy change that decreases the player's cost by a factor of at least α. Our main result shows that finding an α-approximate equilibrium of a given congestion game is sc PLS-complete, for any polynomial-time computable α > 1. Our analysis is based on a gap introducing PLS-reduction from FLIP, i.e., the problem of finding a local optimum of a function encoded by an arbitrary circuit. As this reduction is tight it additionally implies that computing an α-approximate equilibrium reachable from a given initial state by a sequence of α-greedy steps is PSPACE-complete. Our results are in sharp contrast to a recent result showing that every local search problem in PLS admits a fully polynomial time approximation scheme.