11 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)
A risk-security tradeoff in graphical coordination games
A system relying on the collective behavior of decision-makers can be
vulnerable to a variety of adversarial attacks. How well can a system operator
protect performance in the face of these risks? We frame this question in the
context of graphical coordination games, where the agents in a network choose
among two conventions and derive benefits from coordinating neighbors, and
system performance is measured in terms of the agents' welfare. In this paper,
we assess an operator's ability to mitigate two types of adversarial attacks -
1) broad attacks, where the adversary incentivizes all agents in the network
and 2) focused attacks, where the adversary can force a selected subset of the
agents to commit to a prescribed convention. As a mitigation strategy, the
system operator can implement a class of distributed algorithms that govern the
agents' decision-making process. Our main contribution characterizes the
operator's fundamental trade-off between security against worst-case broad
attacks and vulnerability from focused attacks. We show that this tradeoff
significantly improves when the operator selects a decision-making process at
random. Our work highlights the design challenges a system operator faces in
maintaining resilience of networked distributed systems.Comment: 13 pages, double column, 4 figures. Submitted for journal publicatio
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
Rejuvenation and the Spread of Epidemics in General Topologies
International audienceEpidemic models have received significant atten-tion in the past few decades to study the propagation of viruses, worms and ideas in computer and social networks. In the case of viruses, the goal is to understand how the topology of the network and the properties of the nodes that comprise the network, together, impact the spread of the epidemics. In this paper, we propose rejuvenation as a way to cope with epidemics. Then, we present a model to study the effect of rejuvenation and of the topology on the steady-state number of infected and failed nodes. We distinguish between a state in which the virus is incubating and in which symptoms might not be visible and yet they may be contagious and infecting other nodes, and a state of failure where symptoms are clear. Sampling costs might be incurred to examine nodes in search for viruses at an early stage. Using the proposed model, we show that the sampling rate admits at most one local minimum greater than zero. Then, we numerically illustrate the impact of different system parameters on the optimal sampling rate, indicating when rejuvenation is more beneficial
Local Mixing Time: Distributed Computation and Applications
The mixing time of a graph is an important metric, which is not only useful
in analyzing connectivity and expansion properties of the network, but also
serves as a key parameter in designing efficient algorithms. We introduce a new
notion of mixing of a random walk on a (undirected) graph, called local mixing.
Informally, the local mixing with respect to a given node , is the mixing of
a random walk probability distribution restricted to a large enough subset of
nodes --- say, a subset of size at least for a given parameter
--- containing . The time to mix over such a subset by a random walk
starting from a source node is called the local mixing time with respect to
. The local mixing time captures the local connectivity and expansion
properties around a given source node and is a useful parameter that determines
the running time of algorithms for partial information spreading, gossip etc.
Our first contribution is formally defining the notion of local mixing time
in an undirected graph. We then present an efficient distributed algorithm
which computes a constant factor approximation to the local mixing time with
respect to a source node in rounds, where is
the local mixing time w.r.t in an -node regular graph. This bound holds
when is significantly smaller than the conductance of the local mixing
set (i.e., the set where the walk mixes locally); this is typically the
interesting case where the local mixing time is significantly smaller than the
mixing time (with respect to ). We also present a distributed algorithm that
computes the exact local mixing time in rounds,
where and is the diameter of the graph. We
further show that local mixing time tightly characterizes the complexity of
partial information spreading.Comment: 16 page
Distributed Community Detection via Metastability of the 2-Choices Dynamics
We investigate the behavior of a simple majority dynamics on networks of
agents whose interaction topology exhibits a community structure. By leveraging
recent advancements in the analysis of dynamics, we prove that, when the states
of the nodes are randomly initialized, the system rapidly and stably converges
to a configuration in which the communities maintain internal consensus on
different states. This is the first analytical result on the behavior of
dynamics for non-consensus problems on non-complete topologies, based on the
first symmetry-breaking analysis in such setting. Our result has several
implications in different contexts in which dynamics are adopted for
computational and biological modeling purposes. In the context of Label
Propagation Algorithms, a class of widely used heuristics for community
detection, it represents the first theoretical result on the behavior of a
distributed label propagation algorithm with quasi-linear message complexity.
In the context of evolutionary biology, dynamics such as the Moran process have
been used to model the spread of mutations in genetic populations [Lieberman,
Hauert, and Nowak 2005]; our result shows that, when the probability of
adoption of a given mutation by a node of the evolutionary graph depends
super-linearly on the frequency of the mutation in the neighborhood of the node
and the underlying evolutionary graph exhibits a community structure, there is
a non-negligible probability for species differentiation to occur.Comment: Full version of paper appeared in AAAI-1
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
Metastability of Logit Dynamics for Coordination Games
Logit Dynamics [Blume, Games and Economic Behavior, 1993] is a 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. In several cases it happens that on a time-scale shorter than mixing time the chain is “quasistationary”, meaning that it stays close to some small set of the state space, while in a time-scale multiple of the mixing time it jumps from one quasi-stationaryconfiguration to another; this phenomenon is usually called “metastability”. 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. In particular, we study no-risk-dominant coordination games on the clique (which is equivalent to the well-known Glauber dynamics for the Ising model) and coordination games on a ring (both the risk-dominant and norisk-dominant case). We also describe a simple “artificial” game that highlights the distinctive features of our metastability notion based on distributions