arXiv subjects

Diodato Ferraioli

Publications and source records attributed to Diodato Ferraioli.

25 records · Page 2Linked to original sources

arXiv2015

Sequential Posted Price Mechanisms with Correlated Valuations

We study the revenue performance of sequential posted price mechanisms and some natural extensions, for a general setting where the valuations of the buyers are drawn from a correlated distribution. Sequential posted price mechanisms are conceptually simple mechanisms that work by proposing a take-it-or-leave-it offer to each buyer. We apply sequential posted price mechanisms to single-parameter multi-unit settings in which each buyer demands only one item and the mechanism can assign the service to at most k of the buyers. For standard sequential posted price mechanisms, we prove that with the valuation distribution having finite support, no sequential posted price mechanism can extract a constant fraction of the optimal expected revenue, even with unlimited supply. We extend this result to the the case of a continuous valuation distribution when various standard assumptions hold simultaneously. In fact, it turns out that the best fraction of the optimal revenue that is extractable by a sequential posted price mechanism is proportional to ratio of the highest and lowest possible valuation. We prove that for two simple generalizations of these mechanisms, a better revenue performance can be achieved: if the sequential posted price mechanism has for each buyer the option of either proposing an offer or asking the buyer for its valuation, then a Omega(1/max{1,d}) fraction of the optimal revenue can be extracted, where d denotes the degree of dependence of the valuations, ranging from complete independence (d=0) to arbitrary dependence (d=n-1). Moreover, when we generalize the sequential posted price mechanisms further, such that the mechanism has the ability to make a take-it-or-leave-it offer to the i-th buyer that depends on the valuations of all buyers except i's, we prove that a constant fraction (2-sqrt{e})/4~0.088 of the optimal revenue can be always be extracted.

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arXiv2014

Metastability of Asymptotically Well-Behaved Potential Games

One of the main criticisms to game theory concerns the assumption of full rationality. Logit dynamics is a decentralized algorithm in which a level of irrationality (a.k.a. "noise") is introduced in players' behavior. In this context, the solution concept of interest becomes the logit equilibrium, as opposed to Nash equilibria. Logit equilibria are distributions over strategy profiles that possess several nice properties, including existence and uniqueness. However, there are games in which their computation may take time exponential in the number of players. We therefore look at an approximate version of logit equilibria, called metastable distributions, introduced by Auletta et al. [SODA 2012]. These are distributions that remain stable (i.e., players do not go too far from it) for a super-polynomial number of steps (rather than forever, as for logit equilibria). The hope is that these distributions exist and can be reached quickly by logit dynamics. We identify a class of potential games, called asymptotically well-behaved, for which the behavior of the logit dynamics is not chaotic as the number of players increases so to guarantee meaningful asymptotic results. We prove that any such game admits distributions which are metastable no matter the level of noise present in the system, and the starting profile of the dynamics. These distributions can be quickly reached if the rationality level is not too big when compared to the inverse of the maximum difference in potential. Our proofs build on results which may be of independent interest, including some spectral characterizations of the transition matrix defined by logit dynamics for generic games and the relationship of several convergence measures for Markov chains.

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arXiv2014

Imperfect best-response mechanisms

Best-response mechanisms (Nisan, Schapira, Valiant, Zohar, 2011) provide a unifying framework for studying various distributed protocols in which the participants are instructed to repeatedly best respond to each others' strategies. Two fundamental features of these mechanisms are convergence and incentive compatibility. This work investigates convergence and incentive compatibility conditions of such mechanisms when players are not guaranteed to always best respond but they rather play an imperfect best-response strategy. That is, at every time step every player deviates from the prescribed best-response strategy according to some probability parameter. The results explain to what extent convergence and incentive compatibility depend on the assumption that players never make mistakes, and how robust such protocols are to "noise" or "mistakes".

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arXiv2013

Logit Dynamics with Concurrent Updates for Local-Interaction Games

Logit choice dynamics are a family of randomized best response dynamics based on the logit choice function [McFadden, 1974], used for modeling players with limited rationality and knowledge. In this paper we study the all-logit dynamics, where at each time step all players concurrently update their strategies according to the logit choice function. In the well studied one-logit dynamics [Blume, 1993] instead at each step only one randomly chosen player is allowed to update. We study properties of the all-logit dynamics in the context of local interaction games, a class of games that has been used to model complex social phenomena and physical systems. In a local interaction game, players are the vertices of a social graph whose edges are two-player potential games. Each player picks one strategy to be played for all the games she is involved in and the payoff of the player is the sum of the payoffs from each of the games. We prove that local interaction games characterize the class of games for which the all-logit dynamics is reversible. We then compare the stationary behavior of one-logit and all-logit dynamics. Specifically, we look at the expected value of a notable class of observables, that we call decomposable observables. We prove that the difference between the expected values of the observables at stationarity for the two dynamics depends only on the rationality level beta and on the distance of the social graph from a bipartite graph. In particular, if the social graph is bipartite then decomposable observables have the same expected value. Finally, we show that the mixing time of the all-logit dynamics has the same twofold behavior that has been highlighted in the case of the one-logit: for some games it exponentially depends on the rationality level beta, whereas for other games it can be upper bounded by a function independent from beta.

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arXiv2013

Decentralized Dynamics for Finite Opinion Games

Game theory studies situations in which strategic players can modify the state of a given system, due to the absence of a central authority. Solution concepts, such as Nash equilibrium, are defined to predict the outcome of such situations. In multi-player settings, it has been pointed out that to be realistic, a solution concept should be obtainable via processes that are decentralized and reasonably simple. Accordingly we look at the computation of solution concepts by means of decentralized dynamics. These are algorithms in which players move in turns to improve their own utility and the hope is that the system reaches an "equilibrium" quickly. We study these dynamics for the class of opinion games, recently introduced by Bindel et al. [Bindel et al., FOCS2011]. These are games, important in economics and sociology, that model the formation of an opinion in a social network. We study best-response dynamics and show upper and lower bounds on the convergence to Nash equilibria. We also study a noisy version of best-response dynamics, called logit dynamics, and prove a host of results about its convergence rate as the noise in the system varies. To get these results, we use a variety of techniques developed to bound the mixing time of Markov chains, including coupling, spectral characterizations and bottleneck ratio.

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arXiv2012

Convergence to Equilibrium of Logit Dynamics for Strategic Games

We present the first general bounds on the mixing time of the Markov chain associated to the logit dynamics for wide classes of strategic games. The logit dynamics with inverse noise beta describes the behavior of a complex system whose individual components act selfishly and keep responding according to some partial ("noisy") knowledge of the system, where the capacity of the agent to know the system and compute her best move is measured by the inverse of the parameter beta. In particular, we prove nearly tight bounds for potential games and games with dominant strategies. Our results show that, for potential games, the mixing time is upper and lower bounded by an exponential in the inverse of the noise and in the maximum potential difference. Instead, for games with dominant strategies, the mixing time cannot grow arbitrarily with the inverse of the noise. Finally, we refine our analysis for a subclass of potential games called graphical coordination games, a class of games that have been previously studied in Physics and, more recently, in Computer Science in the context of diffusion of new technologies. We give evidence that the mixing time of the logit dynamics for these games strongly depends on the structure of the underlying graph. We prove that the mixing time of the logit dynamics for these games can be upper bounded by a function that is exponential in the cutwidth of the underlying graph and in the inverse of noise. Moreover, we consider two specific and popular network topologies, the clique and the ring. For games played on a clique we prove an almost matching lower bound on the mixing time of the logit dynamics that is exponential in the inverse of the noise and in the maximum potential difference, while for games played on a ring we prove that the time of convergence of the logit dynamics to its stationary distribution is significantly shorter.

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arXiv2011

Mixing Time and Stationary Expected Social Welfare of Logit Dynamics

We study "logit dynamics" [Blume, Games and Economic Behavior, 1993] for strategic games. This dynamics works as follows: at every stage of the game a player is selected uniformly at random and she plays according to a "noisy" best-response where the noise level is tuned by a parameter $β$. Such a dynamics defines a family of ergodic Markov chains, indexed by $β$, over the set of strategy profiles. We believe that the stationary distribution of these Markov chains gives a meaningful description of the long-term behavior for systems whose agents are not completely rational. Our aim is twofold: On the one hand, we are interested in evaluating the performance of the game at equilibrium, i.e. the expected social welfare when the strategy profiles are random according to the stationary distribution. On the other hand, we want to estimate how long it takes, for a system starting at an arbitrary profile and running the logit dynamics, to get close to its stationary distribution; i.e., the "mixing time" of the chain. In this paper we study the stationary expected social welfare for the 3-player CK game, for 2-player coordination games, and for two simple $n$-player games. For all these games, we also give almost tight upper and lower bounds on the mixing time of logit dynamics. Our results show two different behaviors: in some games the mixing time depends exponentially on $β$, while for other games it can be upper bounded by a function independent of $β$.

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