Designing Networks with Good Equilibria under Uncertainty

We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of k terminal vertices or players need to establish connectivity with the root. The social optimum is the Minimum Steiner Tree. We are interested in situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying metric but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model). In the adversarial model, the designer's goal is to choose a single, universal protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is: to what extent can prior knowledge of the underlying metric help in the design? We first demonstrate that there exist graphs (outerplanar) where knowledge of the underlying metric can dramatically improve the performance of good network design. Then, in our main technical result, we show that there exist graph metrics, for which knowing the underlying metric does not help and any universal protocol has PoA of $\Omega(\log k)$, which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey Theory in high dimensional hypercubes. Then we switch to the stochastic model, where each player is independently activated. We show that there exists a randomized ordered protocol that achieves constant PoA. By using standard derandomization techniques, we produce a deterministic ordered protocol with constant PoA.Comment: This version has additional results about stochastic inpu

Cost-Sharing Methods for Scheduling Games under Uncertainty

We study the performance of cost-sharing protocols in a selfish scheduling setting with load-dependent cost functions. Previous work on selfish scheduling protocols has focused on two extreme models: omnipotent protocols that are aware of every machine and every job that is active at any given time, and oblivious protocols that are aware of nothing beyond the machine they control. The main focus of this paper is on a well-motivated middle-ground model of resource-aware protocols, which are aware of the set of machines that the system comprises, but unaware of what jobs are active at any given time. Apart from considering budget-balanced protocols, to which previous work was restricted, we augment the design space by also studying the extent to which overcharging can lead to improved performance. We first show that, in the omnipotent model, overcharging enables us to enforce the optimal outcome as the unique equilibrium, which largely improves over the Θ(log n)-approximation of social welfare that can be obtained by budget-balanced protocols, even in their best equilibrium. We then transition to the resource-aware model and provide price of anarchy (PoA) upper and lower bounds for different classes of cost functions. For concave cost functions, we provide a protocol with PoA of 1+ε for arbitrarily small ε0. When the cost functions can be both convex and concave we construct an overcharging protocol that yields PoA ≤ 2; a spectacular improvement over the bounds obtained for budget-balanced protocols, even in the omnipotent model. We complement our positive results with impossibility results for general increasing cost functions. We show that any resource-aware budget-balanced cost-sharing protocol has PoA of Θ(n) in this setting and, even if we use overcharging, no resource-aware protocol can achieve a PoA of o(√n)

Designing Networks with Good Equilibria under Uncertainty

We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of $k$ terminal vertices or players needs to establish connectivity with the root. The social optimum is the minimum Steiner tree. We study situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying graph metric, but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model). In the adversarial model, the goal of the designer is to choose a single, universal cost-sharing protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is, to what extent can prior knowledge of the underlying graph metric help in the design? We first demonstrate that there exist classes of graphs where knowledge of the underlying graph metric can dramatically improve the performance of good network cost-sharing design. For outerplanar graph metrics, we provide a universal cost-sharing protocol with constant PoA, in contrast to protocols that, by ignoring the graph metric, cannot achieve PoA better than $\Omega(\log k)$. Then, in our main technical result, we show that there exist graph metrics for which knowing the underlying graph metric does not help and any universal protocol has PoA of $\Omega(\log k)$, which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey theory in high-dimensional hypercubes. Then we switch to the stochastic model, where the players are activated according to some probability distribution that is known to the designer. We show that there exists a randomized ordered protocol that achieves constant PoA. If, further, each player is activated independently with some probability, by using standard derandomization techniques, we produce a deterministic ordered protocol that achieves constant PoA. We remark that the first result holds also for the black-box model, where the probabilities are not known to the designer, but she is allowed to draw independent (polynomially many) samples. Read More: https://epubs.siam.org/doi/10.1137/16M109669

Coordinating selfish players in scheduling games

We investigate coordination mechanisms that schedule n jobs on m unrelated machines. The objective is to minimize the makespan. It was raised as an open question whether it is possible to design a coordination mechanism that has constant price of anarchy using preemption. We give a negative answer. Next we introduce multi-job players that control a set of jobs, with the aim of minimizing the sum of the completion times of theirs jobs. In this setting, previous mechanisms designed for players with single jobs are inadequate, e.g., having large price of anarchy, or not guaranteeing pure Nash equilibria. To meet this challenge, we design three mechanisms that induce pure Nash equilibria while guaranteeing relatively small price of anarchy. Then we consider multi-job players where each player\u27s objective is to minimize the weighted sum of completion time of her jobs, while the social cost is the sum of players\u27 costs. We first prove that if machines order jobs according to Smith-rule, then the coordination ratio is at most 4, moreover this is best possible among non-preemptive policies. Then we design a preemptive policy, em externality that has coordination ratio 2.618, and complement this result by proving that this ratio is best possible even if we allow for randomization or full information. An interesting consequence of our results is that an $varepsilon-$local optima of $R|,|sum w_iC_i$ for the jump neighborhood can be found in polynomial time and is within a factor of 2.618 of the optimal solution.Wir betrachten Koordinationsmechanismen um n Jobs auf m Maschinen mit individuellen Bearbeitungszeiten zu verteilen. Ziel dabei ist es den Makespan zu minimieren. Es war eine offene Frage, ob es möglich ist einen preämptiven Koordinationsmechanismus zu entwickeln, der einen konstanten Price of Anarchy hat. Wir beantworten diese Frage im negativen Sinne. Als nächstes führen wir Multi-Job-Spieler ein, die eine Menge von Jobs kontrollieren können, mit dem Ziel die Summe der Fertigstellungszeiten ihrer Jobs zu minimieren. In diesem Szenario sind bekannte Mechanismen, die für Ein-Job-Spieler entworfen worden sind, nicht gut genug, und haben beispielsweise einen hohen Price of Anarchy oder können kein reines Nash Gleichgewicht garantieren. Wir entwickeln drei Mechanismen die jeweils ein reines Nash Gleichgewicht besitzen, und einen relativ kleinen Price of Anarchy haben. Zusätzlich betrachten wir Multi-Job-Spieler, mit dem Ziel jeweils die gewichtete Summe der Fertigstellungszeiten ihrer Jobs zu minimieren, während die Gesamtkosten die Summe der Kosten der Spieler sind. Wir zeigen zuerst, dass das Koordinationsverhältnis höchstens $4$ ist, wenn die Maschinen die Jobs nach der Smith-Regel sortieren, was bei nicht-preämptiven Verfahren optimal ist. Danach entwickeln wir ein preämptives Verfahren, Externality, welches ein Koordinationsverhältnis von 2.618 hat, und ergänzen dieses Ergebniss indem wir beweisen, dass dieses Verhältnis optimal ist, auch für den Fall, dass wir Randomisierung oder volle Information erlauben. Eine interessante Folge unserer Ergebnisse ist, dass ein $varepsilon$-lokales Optimum von $R|,|sum w_iC_i$ für die Jump-Neighborhood in Polynomialzeit gefunden werden kann, und innerhalb eines Faktors von 2.618 von der optimalen Lösung ist