349 research outputs found
Stochastic Game Theory: Adjustment to Equilibrium Under Noisy Directional Learning
This paper presents a dynamic model in which agents adjust their decisions in the direction of higher payoffs, subject to random error. This process produces a probability distribution of players' decisions whose evolution over time is determined by the Fokker-Planck equation. The dynamic process is stable for all potential games, a class of payoff structures that includes several widely studied games. In equilibrium, the distributions that determine expected payoffs correspond to the distributions that arise from the logit function applied to those expected payoffs. This "logit equilibrium" forms a stochastic generalization of the Nash equilibrium and provides a possible explanation of anomalous laboratory data.bounded rationality, noisy directional learning, Fokker- Planck equation, potential games, logit equilibrium, stochastic potential.
Riemannian game dynamics
We study a class of evolutionary game dynamics defined by balancing a gain
determined by the game's payoffs against a cost of motion that captures the
difficulty with which the population moves between states. Costs of motion are
represented by a Riemannian metric, i.e., a state-dependent inner product on
the set of population states. The replicator dynamics and the (Euclidean)
projection dynamics are the archetypal examples of the class we study. Like
these representative dynamics, all Riemannian game dynamics satisfy certain
basic desiderata, including positive correlation and global convergence in
potential games. Moreover, when the underlying Riemannian metric satisfies a
Hessian integrability condition, the resulting dynamics preserve many further
properties of the replicator and projection dynamics. We examine the close
connections between Hessian game dynamics and reinforcement learning in normal
form games, extending and elucidating a well-known link between the replicator
dynamics and exponential reinforcement learning.Comment: 47 pages, 12 figures; added figures and further simplified the
derivation of the dynamic
The Master Equation for Large Population Equilibriums
We use a simple N-player stochastic game with idiosyncratic and common noises
to introduce the concept of Master Equation originally proposed by Lions in his
lectures at the Coll\`ege de France. Controlling the limit N tends to the
infinity of the explicit solution of the N-player game, we highlight the
stochastic nature of the limit distributions of the states of the players due
to the fact that the random environment does not average out in the limit, and
we recast the Mean Field Game (MFG) paradigm in a set of coupled Stochastic
Partial Differential Equations (SPDEs). The first one is a forward stochastic
Kolmogorov equation giving the evolution of the conditional distributions of
the states of the players given the common noise. The second is a form of
stochastic Hamilton Jacobi Bellman (HJB) equation providing the solution of the
optimization problem when the flow of conditional distributions is given. Being
highly coupled, the system reads as an infinite dimensional Forward Backward
Stochastic Differential Equation (FBSDE). Uniqueness of a solution and its
Markov property lead to the representation of the solution of the backward
equation (i.e. the value function of the stochastic HJB equation) as a
deterministic function of the solution of the forward Kolmogorov equation,
function which is usually called the decoupling field of the FBSDE. The
(infinite dimensional) PDE satisfied by this decoupling field is identified
with the \textit{master equation}. We also show that this equation can be
derived for other large populations equilibriums like those given by the
optimal control of McKean-Vlasov stochastic differential equations. The paper
is written more in the style of a review than a technical paper, and we spend
more time and energy motivating and explaining the probabilistic interpretation
of the Master Equation, than identifying the most general set of assumptions
under which our claims are true
Compressed Sensing over -balls: Minimax Mean Square Error
We consider the compressed sensing problem, where the object x_0 \in \bR^N
is to be recovered from incomplete measurements ; here the
sensing matrix is an random matrix with iid Gaussian entries
and . A popular method of sparsity-promoting reconstruction is
-penalized least-squares reconstruction (aka LASSO, Basis Pursuit).
It is currently popular to consider the strict sparsity model, where the
object is nonzero in only a small fraction of entries. In this paper, we
instead consider the much more broadly applicable -sparsity model,
where is sparse in the sense of having norm bounded by for some fixed .
We study an asymptotic regime in which and both tend to infinity with
limiting ratio , both in the noisy () and
noiseless () cases. Under weak assumptions on , we are able to
precisely evaluate the worst-case asymptotic minimax mean-squared
reconstruction error (AMSE) for penalized least-squares: min over
penalization parameters, max over -sparse objects . We exhibit the
asymptotically least-favorable object (hardest sparse signal to recover) and
the maximin penalization.
Our explicit formulas unexpectedly involve quantities appearing classically
in statistical decision theory. Occurring in the present setting, they reflect
a deeper connection between penalized minimization and scalar soft
thresholding. This connection, which follows from earlier work of the authors
and collaborators on the AMP iterative thresholding algorithm, is carefully
explained.
Our approach also gives precise results under weak- ball coefficient
constraints, as we show here.Comment: 41 pages, 11 pdf figure
From Black-Scholes to Online Learning: Dynamic Hedging under Adversarial Environments
We consider a non-stochastic online learning approach to price financial
options by modeling the market dynamic as a repeated game between the nature
(adversary) and the investor. We demonstrate that such framework yields
analogous structure as the Black-Scholes model, the widely popular option
pricing model in stochastic finance, for both European and American options
with convex payoffs. In the case of non-convex options, we construct
approximate pricing algorithms, and demonstrate that their efficiency can be
analyzed through the introduction of an artificial probability measure, in
parallel to the so-called risk-neutral measure in the finance literature, even
though our framework is completely adversarial. Continuous-time convergence
results and extensions to incorporate price jumps are also presented
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