Price of Anarchy in Bernoulli Congestion Games with Affine Costs

We consider an atomic congestion game in which each player participates in the game with an exogenous and known probability $p_{i}\in[0,1]$, independently of everybody else, or stays out and incurs no cost. We first prove that the resulting game is potential. Then, we compute the parameterized price of anarchy to characterize the impact of demand uncertainty on the efficiency of selfish behavior. It turns out that the price of anarchy as a function of the maximum participation probability $p=\max_{i} p_{i}$ is a nondecreasing function. The worst case is attained when players have the same participation probabilities $p_{i}\equiv p$. For the case of affine costs, we provide an analytic expression for the parameterized price of anarchy as a function of $p$. This function is continuous on $(0,1]$, is equal to $4/3$ for $0<p\leq 1/4$, and increases towards $5/2$ when $p\to 1$. Our work can be interpreted as providing a continuous transition between the price of anarchy of nonatomic and atomic games, which are the extremes of the price of anarchy function we characterize. We show that these bounds are tight and are attained on routing games -- as opposed to general congestion games -- with purely linear costs (i.e., with no constant terms).Comment: 29 pages, 6 figure

Designing cost-sharing methods for Bayesian games

We study the design of cost-sharing protocols for two fundamental resource allocation problems, the Set Cover and the Steiner Tree Problem, under environments of incomplete information (Bayesian model). Our objective is to design protocols where the worst-case Bayesian Nash equilibria, have low cost, i.e. the Bayesian Price of Anarchy (PoA) is minimized. Although budget balance is a very natural requirement, it puts considerable restrictions on the design space, resulting in high PoA. We propose an alternative, relaxed requirement called budget balance in the equilibrium (BBiE).We show an interesting connection between algorithms for Oblivious Stochastic optimization problems and cost-sharing design with low PoA. We exploit this connection for both problems and we enforce approximate solutions of the stochastic problem, as Bayesian Nash equilibria, with the same guarantees on the PoA. More interestingly, we show how to obtain the same bounds on the PoA, by using anonymous posted prices which are desirable because they are easy to implement and, as we show, induce dominant strategies for the players

Network uncertainty in selfish routing

We study the problem of selfish routing in the presence of incomplete network information. Our model consists of a number of users who wish to route their traffic on a network of m parallel links with the objective of minimizing their latency. However, in doing so, they face the challenge of lack of precise information on the capacity of the network links. This uncertainty is modelled via a set of probability distributions over all the possibilities, one for each user. The resulting model is an amalgamation of the KP-model of [13] and the congestion games with user-specific functions of [17]. We embark on a study of Nash equilibria and the price of anarchy in this new model. In particular, we propose polynomial-time algorithms for computing some special cases of pure Nash equilibria and we show that negative results of [17], for the non-existence of pure Nash equilibria in the case of three users, do not apply to our model. Consequently, we propose an interesting open problem in this area, that of the existence of pure Nash equilibria in the general case of our model. Furthermore, we consider appropriate notions for the social cost and the price of anarchy and obtain upper bounds for the latter. With respect to fully mixed Nash equilibria, we propose a method to compute them and show that when they exist they are unique. Finally we prove that the fully mixed Nash equilibrium maximizes the social welfare. 1

The Value of Information in Selfish Routing

Path selection by selfish agents has traditionally been studied by comparing social optima and equilibria in the Wardrop model, i.e., by investigating the Price of Anarchy in selfish routing. In this work, we refine and extend the traditional selfish-routing model in order to answer questions that arise in emerging path-aware Internet architectures. The model enables us to characterize the impact of different degrees of congestion information that users possess. Furthermore, it allows us to analytically quantify the impact of selfish routing, not only on users, but also on network operators. Based on our model, we show that the cost of selfish routing depends on the network topology, the perspective (users versus network operators), and the information that users have. Surprisingly, we show analytically and empirically that less information tends to lower the Price of Anarchy, almost to the optimum. Our results hence suggest that selfish routing has modest social cost even without the dissemination of path-load information.Comment: 27th International Colloquium on Structural Information and Communication Complexity (SIROCCO 2020

Price of Anarchy for Non-atomic Congestion Games with Stochastic Demands

We generalize the notions of user equilibrium and system optimum to non-atomic congestion games with stochastic demands. We establish upper bounds on the price of anarchy for three different settings of link cost functions and demand distributions, namely, (a) affine cost functions and general distributions, (b) polynomial cost functions and general positive-valued distributions, and (c) polynomial cost functions and the normal distributions. All the upper bounds are tight in some special cases, including the case of deterministic demands.Comment: 31 page

Computer Science and Game Theory: A Brief Survey

There has been a remarkable increase in work at the interface of computer science and game theory in the past decade. In this article I survey some of the main themes of work in the area, with a focus on the work in computer science. Given the length constraints, I make no attempt at being comprehensive, especially since other surveys are also available, and a comprehensive survey book will appear shortly.Comment: To appear; Palgrave Dictionary of Economic

Bounding the Inefficiency of Altruism Through Social Contribution Games

We introduce a new class of games, called social contribution games (SCGs), where each player's individual cost is equal to the cost he induces on society because of his presence. Our results reveal that SCGs constitute useful abstractions of altruistic games when it comes to the analysis of the robust price of anarchy. We first show that SCGs are altruism-independently smooth, i.e., the robust price of anarchy of these games remains the same under arbitrary altruistic extensions. We then devise a general reduction technique that enables us to reduce the problem of establishing smoothness for an altruistic extension of a base game to a corresponding SCG. Our reduction applies whenever the base game relates to a canonical SCG by satisfying a simple social contribution boundedness property. As it turns out, several well-known games satisfy this property and are thus amenable to our reduction technique. Examples include min-sum scheduling games, congestion games, second price auctions and valid utility games. Using our technique, we derive mostly tight bounds on the robust price of anarchy of their altruistic extensions. For the majority of the mentioned game classes, the results extend to the more differentiated friendship setting. As we show, our reduction technique covers this model if the base game satisfies three additional natural properties