Assignment Games with Conflicts: Robust Price of Anarchy and Convergence Results via Semi-Smoothness

We study assignment games in which jobs select machines, and in which certain pairs of jobs may conflict, which is to say they may incur an additional cost when they are both assigned to the same machine, beyond that associated with the increase in load. Questions regarding such interactions apply beyond allocating jobs to machines: when people in a social network choose to align themselves with a group or party, they typically do so based upon not only the inherent quality of that group, but also who amongst their friends (or enemies) chooses that group as well. We show how semi-smoothness, a recently introduced generalization of smoothness, is necessary to find tight bounds on the robust price of anarchy, and thus on the quality of correlated and Nash equilibria, for several natural job-assignment games with interacting jobs. For most cases, our bounds on the robust price of anarchy are either exactly 2 or approach 2. We also prove new convergence results implied by semi-smoothness for our games. Finally we consider coalitional deviations, and prove results about the existence and quality of strong equilibrium.     

Nash Equilibria and the Price of Anarchy for Flows over Time

We study Nash equilibria in the context of flows over time. Many results on static routing games have been obtained over the last ten years. In flows over time (also called dynamic flows), flow travels through a network over time and, as a consequence, flow values on edges are time-dependent. This more realistic setting has not been tackled from the viewpoint of algorithmic game theory yet; but there is a rich literature on game theoretic aspects of flows over time in the traffic community.     

The Price of Anarchy in Network Creation Games Is (Mostly) Constant

We study the price of anarchy and the structure of equilibria in network creation games. A network creation game is played by n players {1,2,…,n}, each identified with a vertex of a graph (network), where the strategy of player i, i=1,…,n, is to build some edges adjacent to i. The cost of building an edge is α>0, a fixed parameter of the game. The goal of every player is to minimize its creation cost plus its usage cost. The creation cost of player i is α times the number of built edges. In the SumGame variant, the usage cost of player i is the sum of distances from i to every node of the resulting graph. In the MaxGame variant, the usage cost is the eccentricity of i in the resulting graph of the game. In this paper we improve previously known bounds on the price of anarchy of the game (of both variants) for various ranges of α, and give new insights into the structure of equilibria for various values of α. The two main results of the paper show that for α>273?n all equilibria in SumGame are trees and thus the price of anarchy is constant, and that for α>129 all equilibria in MaxGame are trees and the price of anarchy is constant. For SumGame this answers (almost completely) one of the fundamental open problems in the field—is price of anarchy of the network creation game constant for all values of α?—in an affirmative way, up to a tiny range of α.     

Selfish Load Balancing and Atomic Congestion Games

We revisit a classical load balancing problem in the modern context of decentralized systems and self-interested clients. In particular, there is a set of clients, each of whom must choose a server from a permissible set. Each client has a unit-length job and selfishly wants to minimize its own latency (job completion time). A server's latency is inversely proportional to its speed, but it grows linearly with (or, more generally, as the pth power of) the number of clients matched to it. This interaction is naturally modeled as an atomic congestion game, which we call selfish load balancing. We analyze the Nash equilibria of this game and prove nearly tight bounds on the price of anarchy (worst-case ratio between a Nash solution and the social optimum). In particular, for linear latency functions, we show that if the server speeds are relatively bounded and the number of clients is large compared with the number of servers, then every Nash assignment approaches social optimum. Without any assumptions on the number of clients, servers, and server speeds, the price of anarchy is at most 2.5. If all servers have the same speed, then the price of anarchy further improves to We also exhibit a lower bound of 2.01. Our proof techniques can also be adapted for the coordinated load balancing problem under L₂ norm, where it slightly improves the best previously known upper bound on the competitive ratio of a simple greedy scheme.     

Design-Rule-Aware Congestion Model with Explicit Modeling of Vias and Local Pin Access Paths

As technology advances, there is a considerable gap between the congestion model used in global routing and the routing resource consumption in detailed routing. The new factors contributing to congest...     

Differential Games with Fixed Time

In this paper, we consider differential games, the termination time of which is fixed. These games constitute a special class suitable for more complete investigation.     

Handling Communication Restrictions and Team Formation in Congestion Games

There are many domains in which a multi-agent system needs to maximize a “system utility” function which rates the performance of the entire system, while subject to communication restrictions among the agents. Such communication restrictions make it difficult for agents that take actions to optimize their own “private” utilities to also help optimize the system utility. In this article we show how previously introduced utilities that promote coordination among agents can be modified to be effective in domains with communication restrictions. The modified utilities provide performance improvements of up to 75 over previously used utilities in congestion games (i.e., games where the system utility depends solely on the number of agents choosing a particular action). In addition, we show that in the presence of severe communication restrictions, team formation for the purpose of information sharing among agents leads to an additional 25 improvement in system utility. Finally, we show that agents’ private utilities and team sizes can be manipulated to form the best compromise between how “aligned” an agent’s utility is with the system utility and how easily an agent can learn that utility.     

Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms

We reconsider the well-studied Selfish Routing game with affine latency functions. The Price of Anarchy for this class of games takes maximum value 4/3; this maximum is attained already for a simple network of two parallel links, known as Pigou’s network. We improve upon the value 4/3 by means of Coordination Mechanisms. We increase the latency functions of the edges in the network, i.e., if ? _e (x) is the latency function of an edge e, we replace it by $\hat{\ell}_{e}(x)$ with $\ell_{e}(x) \le \hat{\ell}_{e}(x)$ for all x. Then an adversary fixes a demand rate as input. The engineered Price of Anarchy of the mechanism is defined as the worst-case ratio of the Nash social cost in the modified network over the optimal social cost in the original network. Formally, if $\hat{C}_{N} (r)$ denotes the cost of the worst Nash flow in the modified network for rate r and C _opt (r) denotes the cost of the optimal flow in the original network for the same rate then $$\mathit{ePoA} = \max_{r \ge 0} \frac{\hat{C}_N(r)}{C_{\mathit{opt}}(r)}. $$ We first exhibit a simple coordination mechanism that achieves for any network of parallel links an engineered Price of Anarchy strictly less than 4/3. For the case of two parallel links our basic mechanism gives 5/4=1.25. Then, for the case of two parallel links, we describe an optimal mechanism; its engineered Price of Anarchy lies between 1.191 and 1.192.     

The Impact of Social Ignorance on Weighted Congestion Games

We consider weighted linear congestion games, and investigate how social ignorance, namely lack of information about the presence of some players, affects the inefficiency of pure Nash equilibria (PNE) and the convergence rate of the ε-Nash dynamics. To this end, we adopt the model of graphical linear congestion games with weighted players, where the individual cost and the strategy selection of each player only depends on his neighboring players in the social graph. We show that such games admit a potential function, and thus a PNE. Next, we investigate the Price of Anarchy (PoA) and the Price of Stability (PoS) of graphical linear congestion games with respect to the players’ total actual cost. Our main result is that the impact of social ignorance on the PoA and on the PoS is naturally quantified by the independence number α(G) of the social graph G. In particular, we show that the PoA grows roughly as α(G)(α(G)+2), which is essentially tight as long as α(G) does not exceed half the number of players, and that the PoS lies between α(G) and 2α(G). Moreover, we show that the ε-Nash dynamics reaches an α(G)(α(G)+2)-approximate configuration in polynomial time that does not directly depend on the social graph. For unweighted graphical linear games with symmetric strategies, we show that the ε-Nash dynamics reaches an ε-approximate PNE in polynomial time that exceeds the corresponding time for symmetric linear games by a factor at most as large as the number of players.     

Self-Fulfilling Signal of an Endogenous State in Network Congestion Games

We consider the problem of coordination via signaling in network congestion games to improve social welfare deteriorated by incomplete information about traffic flow. Traditional studies on signaling, which focus on exogenous factors of congestion and ignore congestion externalities, fail to discuss the oscillations of traffic flow. To address this gap, we formulate a problem of designing a coordination signal on endogenous information about traffic flow and introduce a self-fulfilling characteristic of a signal that guarantees an outcome flow consistent with the signal itself without causing the unwanted oscillation. An instance of the self-fulfilling signal is shown in the case of a Gaussian signal distribution. In addition, we show simple numerical examples. The results reveal how a self-fulfilling signal suppresses the oscillation and simultaneously improves social welfare through improved network efficiency.     

Convergence Analysis for Symmetric Arbitration Games FOA and DOA

Up to now, results on the existence of Nash equilibrium have required either continuity of payoff functions or compactness of strategy sets. However, in the arbitration games FOA and DOA, neither condition is satisfied. This paper first gives a new existence result for a general game. The result is then applied to the symmetric arbitration games FOA and DOA. The conclusions of this paper generalize the main result of Zeng, Nakamura and Ibaraki (1996), that DOA leads to a convergence of offers but FOA does not.     

Optimal Control in Differential Games with Nonfixed Termination Time

The paper considers the problem of finding an optimal control region for a pursuer. This region guarantees that the pursuer wins the game starting from some point that belongs to a prescribed set.     

On Time Extension in Differential Games with Impulse Controls

In the development of ideas of B. N. Pshenichnyi, the paper considers a linear differential game of approach with impulse controls. A research technique is proposed, which is based on time extension and oriented to the case where the classical Pontryagin condition does not hold. Sufficient conditions for the finiteness of the guaranteed approach time are obtained. An illustrative example is given.     

Fault-Tolerant Embedding of Pairwise Independent Hamiltonian Paths on a Faulty Hypercube with Edge Faults

A Hamiltonian path in G is a path which contains every vertex of G exactly once. Two Hamiltonian paths P ₁=〈u ₁,u ₂,…,u _n 〉 and P ₂=〈v ₁,v ₂,…,v _n 〉 of G are said to be independent if u ₁=v ₁, u _n =v _n , and u _i ≠v _i for all 1<i<n; and both are full-independent if u _i ≠v _i for all 1≤i≤n. Moreover, P ₁ and P ₂ are independent starting at u ₁, if u ₁=v ₁ and u _i ≠v _i for all 1<i≤n. A set of Hamiltonian paths {P ₁,P ₂,…,P _k } of G are pairwise independent (respectively, pairwise full-independent, pairwise independent starting at u ₁) if any two different Hamiltonian paths in the set are independent (respectively, full-independent, independent starting at u ₁). A bipartite graph G is Hamiltonian-laceable if there exists a Hamiltonian path between any two vertices from different partite sets. It is well known that an n-dimensional hypercube Q _n is bipartite with two partite sets of equal size. Let F be the set of faulty edges of Q _n . In this paper, we show the following results:

Complete Convergence of Competitive Neural Networks with Different Time Scales

This paper studies the complete convergence of a class of neural networks with different time scales under the assumption that the activation functions are unsaturated piecewise linear functions. Under this assumption, there are multiple equilibrium points in the neural network. Traditional methods cannot be used in this neural network. Complete convergence is proved by constructing an energy-like function. Simulations are employed to illustrate the theory.     

On the Convergence of Multicast Games in Directed Networks

We investigate the convergence of the price of anarchy after a limited number of moves in the classical multicast communication game when the underlying communication network is directed. Namely, a subset of nodes of the network are interested in receiving the transmission from a given source node and can share the cost of the used links according to fixed cost sharing methods. At each step, a single receiver is allowed to modify its communication strategy, that is to select a communication path from the source, and assuming a selfish or rational behavior, it will make a best response move, that is it will select a solution yielding the minimum possible payment or shared cost. We determine lower and upper bounds on the price of anarchy, that is the highest possible ratio among the overall cost of the links used by the receivers and the minimum possible cost realizing the required communications, after a limited number of moves under the fundamental Shapley cost sharing method. In particular, assuming that the initial set of connecting paths can be arbitrary, we show an $O(r\sqrt{r})We investigate the convergence of the price of anarchy after a limited number of moves in the classical multicast communication game when the underlying communication network is directed. Namely, a subset of nodes of the network are interested in receiving the transmission from a given source node and can share the cost of the used links according to fixed cost sharing methods. At each step, a single receiver is allowed to modify its communication strategy, that is to select a communication path from the source, and assuming a selfish or rational behavior, it will make a best response move, that is it will select a solution yielding the minimum possible payment or shared cost. We determine lower and upper bounds on the price of anarchy, that is the highest possible ratio among the overall cost of the links used by the receivers and the minimum possible cost realizing the required communications, after a limited number of moves under the fundamental Shapley cost sharing method. In particular, assuming that the initial set of connecting paths can be arbitrary, we show an O(r?r)O(r\sqrt{r}) upper bound on the price of anarchy after 2 rounds, during each of which all the receivers move exactly once, and a matching lower bound, that we also extend to W(r^k?{r})\Omega(r\sqrt[k]{r}) for any number k≥2 rounds, where r is the number of receivers. Similarly, exactly matching upper and lower bounds equal to r are determined for any number of rounds when starting from the empty state in which no path has been selected. Analogous results are obtained also with respect to other three natural cost sharing methods considered in the literature, that is the egalitarian, path-proportional and egalitarian-path proportional ones. Most results are also extended to the undirected case in which the communication links are bidirectional.     

A k-Median Algorithm with Running Time Independent of Data Size

We give a sampling-based algorithm for the k-Median problem, with running time O(k $(\frac{{k^2 }}{ \in } \log k)^2 $ log $(\frac{k}{ \in } \log k)$ ), where k is the desired number of clusters and ∈ is a confidence parameter. This is the first k-Median algorithm with fully polynomial running time that is independent of n, the size of the data set. It gives a solution that is, with high probability, an O(1)-approximation, if each cluster in some optimal solution has Ω $(\frac{{n \in }}{k})$ points. We also give weakly-polynomial-time algorithms for this problem and a relaxed version of k-Median in which a small fraction of outliers can be excluded. We give near-matching lower bounds showing that this assumption about cluster size is necessary. We also present a related algorithm for finding a clustering that excludes a small number of outliers.