The curse of simultaneity

Internet and Network Economics, 2010

We study Congestion Games with non-increasing cost functions (Cost Sharing Games) from a complexity perspective and resolve their computational hardness, which has been an open question. Specifically we prove that when the cost functions have the form f (x) = cr/x (Fair Cost Allocation) then it is PLS-complete to compute a Pure Nash Equilibrium even in the case where strategies of the players are paths on a directed network. For cost functions of the form f (x) = cr(x)/x, where cr(x) is a non-decreasing concave function we also prove PLS-completeness in undirected networks. Thus we extend the results of [7, 1] to the non-increasing case. For the case of Matroid Cost Sharing Games, where tractability of Pure Nash Equilibria is known by [1] we give a greedy polynomial time algorithm that computes a Pure Nash Equilibrium with social cost at most the potential of the optimal strategy profile. Hence, for this class of games we give a polynomial time version of the Potential Method introduced in [2] for bounding the Price of Stability.

Communications of the ACM, 2009

How long does it take until economic agents converge to an equilibrium? By studying the complexity of the problem of computing a mixed Nash equilibrium in a game, we provide evidence that there are games in which convergence to such an equilibrium takes prohibitively long. Traditionally, computational problems fall into two classes: those that have a polynomial-time algorithm and those that are NP-hard. However, the concept of NP-hardness cannot be applied to the rare problems where "every instance has a solution"---for example, in the case of games Nash's theorem asserts that every game has a mixed equilibrium (now known as the Nash equilibrium, in honor of that result). We show that finding a Nash equilibrium is complete for a class of problems called PPAD, containing several other known hard problems; all problems in PPAD share the same style of proof that every instance has a solution.

Journal of Economic Dynamics and Control, 1997

We present some algorithmic unsolvability and incompleteness results in game theory and discuss their significance. The main theorem presents a class of n-person games, where each player's strategy set is the real line and payoffs are continuous functions, for which there could not possibly exist an algorithm to compute either a Nash equilibrium or an E-equilibrium. Conditions sufficient to ensure solvability are also discussed.

We study coordination mechanisms for Scheduling Games (with unrelated machines). In these games, each job represents a player, who needs to choose a machine for its execution, and intends to complete earliest possible. Our goal is to design scheduling policies that always admit a pure Nash equilibrium and guarantee a small price of anarchy for the l_k-norm social cost --- the objective balances overall quality of service and fairness. We consider policies with different amount of knowledge about jobs: non-clairvoyant, strongly-local and local. The analysis relies on the smooth argument together with adequate inequalities, called smooth inequalities. With this unified framework, we are able to prove the following results. First, we study the inefficiency in l_k-norm social costs of a strongly-local policy SPT and a non-clairvoyant policy EQUI. We show that the price of anarchy of policy SPT is O(k). We also prove a lower bound of Omega(k/log k) for all deterministic, non-preemptive, ...

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) choose that group as well. We show how semi-smoothness, a recently introduced generalization of smoothness, is necessary to find tight bounds on the price of total 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 price of total 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.

International Journal of Game Theory

We introduce set packing games as an abstraction of situations in which n selfish players select disjoint subsets of a finite set of indivisible items, and analyze the quality of several equilibria for this basic class of games. Special attention is given to a subclass of set packing games, namely throughput scheduling games, where the items represent jobs, and the subsets that a player can select are those jobs that this player can schedule feasibly. We show that the quality of three types of equilibrium solutions is only moderately suboptimal. Specifically, the paper gives tight bounds on the price of anarchy for Nash equilibria, subgame perfect equilibria of games with sequential play, and k-collusion Nash equilibria. Under the assumption that players are allowed to play suboptimally and achieve an α-approximate equilibrium, our tight price of anarchy bounds are α + 1 for Nash and subgame perfect equilibria, but less than α + 1/(e − 1) for subgame perfect equilibria when games are symmetric. For k-collusion Nash equilibria, the price of anarchy equals α + (n − k)/(n − 1), where 1 ≤ k ≤ n.

Journal of Computer and System Sciences, 2011

We study the computational complexity of problems involving equilibria in strategic games and in perfect information extensive games when the number of players is large. We consider, among others, the problems of deciding the existence of a Pure Nash Equilibrium in strategic games or deciding the existence of a pure Nash or a subgame perfect Nash equilibrium with a given payoff in finite perfect information extensive games. We address the fundamental question of how can we represent a game with a large number of players? We propose three ways of representing a game with different degrees of succinctness for the components of the game. For perfect information extensive games we show that when the number of moves of each player is large and the input game is represented succinctly these problems are PSPACE-complete. In contraposition, when the game is described explicitly by means of its associated tree all these problems are decidable in polynomial time. For strategic games we show that the complexity of deciding the existence of a Pure Nash equilibrium depends on the succinctness of the game representation and then on the size of the action sets. In particular we show that it is NP-complete when the number of players is large and the number of actions for each player is constant, while the problem is Σ p 2-complete when the number of players is a constant and the size of the action sets is exponential.

Proceedings of the 23rd ACM Conference on Economics and Computation

A central goal in algorithmic game theory is to analyze the performance of decentralized multiagent systems, like communication and information networks. In the absence of a central planner who can enforce how these systems are utilized, the users can strategically interact with the system, aiming to maximize their own utility, possibly leading to very inefficient outcomes, and thus a high price of anarchy. To alleviate this issue, the system designer can use decentralized mechanisms that regulate the use of each resource (e.g., using local queuing protocols or scheduling mechanisms), but with only limited information regarding the state of the system. These information limitations have a severe impact on what such decentralized mechanisms can achieve, so most of the success stories in this literature have had to make restrictive assumptions (e.g., by either restricting the structure of the networks or the types of cost functions). In this paper, we overcome some of the obstacles that the literature has imposed on decentralized mechanisms, by designing mechanisms that are enhanced with predictions regarding the missing information. Specifically, inspired by the big success of the literature on "algorithms with predictions", we design decentralized mechanisms with predictions and evaluate their price of anarchy as a function of the prediction error, focusing on two very well-studied classes of games: scheduling games and multicast network formation games. CCS Concepts: • Theory of computation → Quality of equilibria; Network games.

Lecture Notes in Computer Science, 2005

Journal of Economic Dynamics and Control, 1988