784 research outputs found
Metastability of Logit Dynamics for Coordination Games
Logit Dynamics [Blume, Games and Economic Behavior, 1993] are randomized best
response dynamics for strategic games: at every time step a player is selected
uniformly at random and she chooses a new strategy according to a probability
distribution biased toward strategies promising higher payoffs. This process
defines an ergodic Markov chain, over the set of strategy profiles of the game,
whose unique stationary distribution is the long-term equilibrium concept for
the game. However, when the mixing time of the chain is large (e.g.,
exponential in the number of players), the stationary distribution loses its
appeal as equilibrium concept, and the transient phase of the Markov chain
becomes important. It can happen that the chain is "metastable", i.e., on a
time-scale shorter than the mixing time, it stays close to some probability
distribution over the state space, while in a time-scale multiple of the mixing
time it jumps from one distribution to another.
In this paper we give a quantitative definition of "metastable probability
distributions" for a Markov chain and we study the metastability of the logit
dynamics for some classes of coordination games. We first consider a pure
-player coordination game that highlights the distinctive features of our
metastability notion based on distributions. Then, we study coordination games
on the clique without a risk-dominant strategy (which are equivalent to the
well-known Glauber dynamics for the Curie-Weiss model) and coordination games
on a ring (both with and without risk-dominant strategy)
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 β describes the behavior of a complex system whose individual components act selfishly 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 parameter β. 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 bounded by an exponential in β and in the maximum potential difference. Instead, for games with dominant strategies the mixing time cannot grow arbitrarily with β. Finally, we refine our analysis for a subclass of potential games called graphical coordination games, often used for modeling the diffusion of new technologies. 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 β. Moreover, we consider two specific and popular network topologies, the clique and the ring. For the clique, we prove an almost matching lower bound on the mixing time of the logit dynamics that is exponential in β and in the maximum potential difference, while for the ring we prove that the time of convergence of the logit dynamics to its stationary distribution is significantly shorter
Price Competition, Fluctuations, and Welfare Guarantees
In various markets where sellers compete in price, price oscillations are
observed rather than convergence to equilibrium. Such fluctuations have been
empirically observed in the retail market for gasoline, in airline pricing and
in the online sale of consumer goods. Motivated by this, we study a model of
price competition in which an equilibrium rarely exists. We seek to analyze the
welfare, despite the nonexistence of an equilibrium, and present welfare
guarantees as a function of the market power of the sellers.
We first study best response dynamics in markets with sellers that provide a
homogeneous good, and show that except for a modest number of initial rounds,
the welfare is guaranteed to be high. We consider two variations: in the first
the sellers have full information about the valuation of the buyer. Here we
show that if there are items available across all sellers and is
the maximum number of items controlled by any given seller, the ratio of the
optimal welfare to the achieved welfare will be at most
. As the market power of the largest seller
diminishes, the welfare becomes closer to optimal. In the second variation we
consider an extended model where sellers have uncertainty about the buyer's
valuation. Here we similarly show that the welfare improves as the market power
of the largest seller decreases, yet with a worse ratio of
. The exponential gap in welfare between the two
variations quantifies the value of accurately learning the buyer valuation.
Finally, we show that extending our results to heterogeneous goods in general
is not possible. Even for the simple class of -additive valuations, there
exists a setting where the welfare approximates the optimal welfare within any
non-zero factor only for fraction of the time, where is the number
of sellers
Social Pressure in Opinion Games
Motivated by privacy and security concerns in online social networks, we study the role of social pressure in opinion games. These are games, important in economics and sociology, that model the formation of opinions in a social network. We enrich the definition of (noisy) best-response dynamics for opinion games by introducing the pressure, increasing with time, to reach an agreement. We prove that for clique social networks, the dynamics always converges to consensus (no matter the level of noise) if the social pressure is high enough. Moreover, we provide (tight) bounds on the speed of convergence; these bounds are polynomial in the number of players provided that the pressure grows sufficiently fast. We finally look beyond cliques: we characterize the graphs for which consensus is guaranteed, and make some considerations on the computational complexity of checking whether a graph satisfies such a condition
Independent Lazy Better-Response Dynamics on Network Games
International audienceWe study an independent best-response dynamics on network games in which the nodes (players) decide to revise their strategies independently with some probability. We provide several bounds on the convergence time to an equilibrium as a function of this probability, the degree of the network, and the potential of the underlying games. These dynamics are somewhat more suitable for distributed environments than the classical better- and best-response dynamics where players revise their strategies "sequentially'", i.e., no two players revise their strategies simultaneously
Distributed Adaptive Routing in Communication Networks
In this report, we present a new adaptive multi-flow routing algorithm to select end- to-end paths in packet-switched networks. This algorithm provides provable optimality guarantees in the following game theoretic sense: The network configuration converges to a configuration arbitrarily close to a pure Nash equilibrium. In this context, a Nash equilibrium is a configuration in which no flow can improve its end-to-end delay by changing its network path. This algorithm has several robustness properties making it suitable for real-life usage: it is robust to measurement errors, outdated information and clocks desynchronization. Furthermore, it is only based on local information and only takes local decisions, making it suitable for a distributed implementation. Our SDN-based proof-of-concept is built as an Openflow controller. We set up an emulation platform based on Mininet to test the behavior of our proof-of-concept implementation in several scenarios. Although real-world conditions do not conform exactly to the theoretical model, all experiments exhibit satisfying behavior, in accordance with the theoretical predictions
Distributed Adaptive Routing in Communication Networks
In this report, we present a new adaptive multi-flow routing algorithm to select end- to-end paths in packet-switched networks. This algorithm provides provable optimality guarantees in the following game theoretic sense: The network configuration converges to a configuration arbitrarily close to a pure Nash equilibrium. In this context, a Nash equilibrium is a configuration in which no flow can improve its end-to-end delay by changing its network path. This algorithm has several robustness properties making it suitable for real-life usage: it is robust to measurement errors, outdated information and clocks desynchronization. Furthermore, it is only based on local information and only takes local decisions, making it suitable for a distributed implementation. Our SDN-based proof-of-concept is built as an Openflow controller. We set up an emulation platform based on Mininet to test the behavior of our proof-of-concept implementation in several scenarios. Although real-world conditions do not conform exactly to the theoretical model, all experiments exhibit satisfying behavior, in accordance with the theoretical predictions
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