168 research outputs found
Stochastic Stackelberg games
In this paper, we consider a discrete-time stochastic Stackelberg game where
there is a defender (also called leader) who has to defend a target and an
attacker (also called follower). Both attacker and defender have conditionally
independent private types, conditioned on action and previous state, that
evolve as controlled Markov processes. The objective is to compute the
stochastic Stackelberg equilibrium of the game where defender commits to a
strategy. The attacker's strategy is the best response to the defender strategy
and defender's strategy is optimum given the attacker plays the best response.
In general, computing such equilibrium involves solving a fixed-point equation
for the whole game. In this paper, we present an algorithm that computes such
strategies by solving smaller fixed-point equations for each time . This
reduces the computational complexity of the problem from double exponential in
time to linear in time. Based on this algorithm, we compute stochastic
Stackelberg equilibrium of a security example.Comment: 31 pages, 6 figure
Signaling equilibria for dynamic LQG games with asymmetric information
We consider a finite horizon dynamic game with two players who observe their
types privately and take actions, which are publicly observed. Players' types
evolve as independent, controlled linear Gaussian processes and players incur
quadratic instantaneous costs. This forms a dynamic linear quadratic Gaussian
(LQG) game with asymmetric information. We show that under certain conditions,
players' strategies that are linear in their private types, together with
Gaussian beliefs form a perfect Bayesian equilibrium (PBE) of the game.
Furthermore, it is shown that this is a signaling equilibrium due to the fact
that future beliefs on players' types are affected by the equilibrium
strategies. We provide a backward-forward algorithm to find the PBE. Each step
of the backward algorithm reduces to solving an algebraic matrix equation for
every possible realization of the state estimate covariance matrix. The forward
algorithm consists of Kalman filter recursions, where state estimate covariance
matrices depend on equilibrium strategies
Dynamic Decision Problems with Cooperative and Strategic Agents and Asymmetric Information.
There exist many real world situations involving multiple decision makers with asymmetric information, such as communication systems, social networks, economic markets and many others. Through this dissertation, we attempt to enhance the conceptual understanding of such systems and provide analytical tools to characterize the optimum or equilibrium behavior.
Specifically, we study four discrete time, decentralized decision problems in stochastic dynamical systems with cooperative and strategic agents. The first problem we consider is a relay channel where nodes' queue lengths, modeled as conditionally independent Markov chains, are nodes' private information, whereas nodes' actions are publicly observed. This results in non-classical information pattern. Energy-delay tradeoff is studied for this channel through stochastic control techniques for cooperative agents. Extending this model for strategic users, in the second problem we study a general model with strategic players having conditionally independent, Markovian types and publicly observed actions. This results in a dynamic game with asymmetric information. We present a forward/backward sequential decomposition algorithm to find a class of perfect Bayesian equilibria of the game. Using this methodology, in the third problem, we study a general two player dynamic LQG game with asymmetric information, where players' types evolve as independent, controlled linear Gaussian processes and players incur quadratic instantaneous costs. We show that under certain conditions, players' strategies that are linear in their private types, together with Gaussian beliefs, form a perfect Bayesian equilibrium (PBE) of the game. Finally, we consider two sub problems in decentralized Bayesian learning in dynamic games. In the first part, we consider an ergodic version of a sequential buyers game where strategic users sequentially make a decision to buy or not buy a product. In this problem, we design incentives to align players' individual objectives with the team objective. In the second part, we present a framework to study learning dynamics and especially informational cascades for decentralized dynamic games. We first generalize our methodology to find PBE to the case when players do not perfectly observe their types; rather they make independent, noisy observations. Based on this, we characterize informational cascades for a specific learning model.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133294/1/dvasal_1.pd
Master equation of discrete-time Stackelberg mean field games with multiple leaders
In this paper, we consider a discrete-time Stackelberg mean field game with a
finite number of leaders, a finite number of major followers and an infinite
number of minor followers. The leaders and the followers each observe types
privately that evolve as conditionally independent controlled Markov processes.
The leaders are of "Stackelberg" kind which means they commit to a dynamic
policy. We consider two types of followers: major and minor, each with a
private type. All the followers best respond to the policies of the Stackelberg
leaders and each other. Knowing that the followers would play a mean field game
(with major players) based on their policy, each (Stackelberg) leader chooses a
policy that maximizes her reward. We refer to the resulting outcome as a
Stackelberg mean field equilibrium with multiple leaders (SMFE-ML). In this
paper, we provide a master equation of this game that allows one to compute all
SMFE-ML. We further extend this notion to the case when there are infinite
number of leaders.Comment: 40 pages. arXiv admin note: substantial text overlap with
arXiv:2201.05959, arXiv:2005.0199
Master Equation for Discrete-Time Stackelberg Mean Field Games with single leader
In this paper, we consider a discrete-time Stackelberg mean field game with a
leader and an infinite number of followers. The leader and the followers each
observe types privately that evolve as conditionally independent controlled
Markov processes. The leader commits to a dynamic policy and the followers best
respond to that policy and each other. Knowing that the followers would play a
mean field game based on her policy, the leader chooses a policy that maximizes
her reward. We refer to the resulting outcome as a Stackelberg mean field
equilibrium (SMFE). In this paper, we provide a master equation of this game
that allows one to compute all SMFE. Based on our framework, we consider two
numerical examples. First, we consider an epidemic model where the followers
get infected based on the mean field population. The leader chooses subsidies
for a vaccine to maximize social welfare and minimize vaccination costs. In the
second example, we consider a technology adoption game where the followers
decide to adopt a technology or a product and the leader decides the cost of
one product that maximizes his returns, which are proportional to the people
adopting that technologyComment: 25 pages. arXiv admin note: text overlap with arXiv:2005.0199
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