We present a game-theoretic model that captures many of the intricacies of interdomain routing in today’s Internet. In this model, the strategic agents are source nodes located on a network, who aim to send traffic to a unique destination node. The interaction between the agents is dynamic and complex – asynchronous, sequential, and based on partial information. Best-reply dynamics in this model capture crucial aspects of the only interdomain routing protocol de facto, namely the Border Gateway Protocol (BGP). We study complexity and incentive-related issues in this model. Our main results are showing that in realistic and well-studied settings, BGP is incentive-compatible. I.e., not only does myopic behaviour of all players converge to a “stable ” routing outcome, but no player has motivation to unilaterally deviate from the protocol. Moreover, we show that even coalitions of players of any size cannot improve their routing outcomes by collaborating. Unlike the vast majority of works in mechanism design, our results do not require any monetary transfers (to or by the agents).

Computational complexity is the subfield of computer science that rigorously studies the intrinsic difficulty of computational problems. This survey explains how complexity theory defines “hard problems”; applies these concepts to several equilibrium computation problems; and discusses implications for computation, games, and behavior. We assume

Why might markets tend toward and remain near equilibrium prices? In an effort to shed light on this question from an algorithmic perspective, this paper formalizes the setting of Ongoing Markets, by contrast with the classic market scenario, which we term One-Time Markets. The Ongoing Market allows trade at non-equilibrium prices, and, as its name suggests, continues over time. As such, it appears to be a more plausible model of actual markets. For both market settings, this paper defines and analyzes variants of a simple tatonnement algorithm that differs from previous algorithms that have been subject to asymptotic analysis in three significant respects: the price update for a good depends only on the price, demand, and supply for that good, and on no other information; the price update for each good occurs distributively and asynchronously; the algorithms work (and the analyses hold) from an arbitrary starting point. Our algorithm introduces a new and natural update rule. We show that this update rule leads to fast convergence toward equilibrium prices in a broad class of markets that satisfy the weak gross substitutes property. These are the first analyses for computationally and informationally distributed algorithms that demonstrate polynomial convergence. Our analysis identifies three parameters characterizing the markets, which govern the rate of convergence of our protocols. These parameters are, broadly speaking: 1. A bound on the fractional rate of change of demand for each good with respect to fractional changes in its price. 2. A bound on the fractional rate of change of demand for each good with respect to fractional changes in wealth. 3. The closeness of the market to a Fisher market (a market with buyers starting with money alone). We give two types of protocols. The first type assumes global knowledge of only (an upper bound on) the first parameter. For

We give a brief and biased survey of the past, present, and future of research on the interface of theoretical computer science and game theory. 1

Abstract. In many real-world settings (e.g., interdomain routing in the Internet) strategic agents are instructed to follow best-reply dynamics in asynchronous environments. In such settings players learn of each other’s actions via update messages that can be delayed or even lost. In particular, several players might update their actions simultaneously, or make choices based on outdated information. In this paper we analyze the convergence of best- (and better-)reply dynamics in asynchronous environments. We provide sufficient conditions, and necessary conditions for convergence in such settings, and also study the convergence-rate of these natural dynamics. 1

Why might markets tend toward and remain near equilibrium prices? In an effort to shed light on this question from an algorithmic perspective, this paper defines and analyzes two simple tatonnement algorithms that differ from previous algorithms that have been subject to asymptotic analysis in three significant respects: the price update for a good depends only on the price, demand, and supply for that good, and on no other information; the price update for each good occurs distributively and asynchronously; the algorithms work (and the analyses hold) from an arbitrary starting point. Our algorithm introduces a new and natural update rule. We show that this update rule leads to fast convergence toward equilibrium prices in a broad class of markets that satisfy the weak gross substitutes property. These are the first analyses for computationally and informationally distributed algorithms that demonstrate polynomial convergence. Our analysis identifies three parameters characterizing the markets, which govern the rate of convergence of our protocols. These parameters are, broadly speaking: 1. A bound on the fractional rate of change of demand for each good with respect to fractional changes in its price. 2. A bound on the fractional rate of change of demand for each good with respect to fractional changes in wealth. 3. The relative demand for money at equilibrium prices. We give two protocols. The first assumes global knowledge of only the first parameter. For this protocol, we also provide a matching lower bound in terms of these parameters. Our second protocol assumes no global knowledge whatsoever.

We study a general sub-class of concave games, which we call socially concave games. We show that if each player follows any no-external regret minimization procedure then the dynamics converges in the sense that both the average action vector converges to a Nash equilibrium and that the utility of each player converges to her utility in that Nash equilibrium. We show that many natural games are socially concave games. Specifically, we show that linear Cournot competition and linear resource allocation games are socially-concave games, and therefore our convergence result applies to them. In addition, we show that a simple best response dynamic might diverge for linear resource allocation games, and is known to diverge for a linear Cournot competition. For the TCP congestion games we show that “near” the equilibrium these games are socially-concave, and using our general methodology we show convergence of specific regret minimization dynamics.

Computer systems increasingly involve the interaction of multiple self-interested agents. The designers of these systems have objectives they wish to optimize, but by allowing selfish agents to interact in the system, they lose the ability to directly control behavior. What is lost by this lack of centralized control? What are the likely outcomes of selfish behavior? In this work, we consider learning dynamics as a tool for better classifying and understanding outcomes of selfish behavior in games. In particular, when such learning algorithms exist and are efficient, we propose “regret-minimization” as a criterion for self-interested behavior and study the system-wide effects in broad classes of games when players achieve this criterion. In addition, we present a general transformation from offline approximation algorithms for linear optimization problems to online algorithms that achieve low regret.

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