Stackelberg strategies in linear-quadratic stochastic differential games

This paper obtains the Stackelberg solution to a class of two-player stochastic differential games described by linear state dynamics and quadratic objective functionals. The information structure of the problem is such that the players make independent noisy measurements of the initial state and are permitted to utilize only this information in constructing their controls. Furthermore, by the very nature of the Stackelberg solution concept, one of the players is assumed to know, in advance, the strategy of the other player (the leader). For this class of problems, we first establish existence and uniqueness of the Stackelberg solution and then relate the derivation of the leader's Stackelberg solution to the optimal solution of a nonstandard stochastic control problem. This stochastic control problem is solved in a more general context, and its solution is utilized in constructing the Stackelberg strategy of the leader. For the special case Gaussian statistics, it is shown that this optimal strategy is affine in observation of the leader. The paper also discusses numerical aspects of the Stackelberg solution under general statistics and develops algorithms which converge to the unique Stackelberg solution

On the saddle-point solution of a class of stochastic differential games

This paper deals with the saddle-point solution of a class of stochastic differential games described by linear state dynamics and quadratic objective functionals. The information structure of the problem is such that both players have access to a common noisy linear measurement of the state and they are permitted to utilize only this information in constructing their controls. The saddle-point solution of such differential game problems has been discussed earlier in Ref. 1, but the conclusions arrived there are incorrect, as is explicitly shown in this paper. We extensively discuss the role of information structure on the saddle-point solution of such stochastic games (specifically within the context of an illustrative discrete-time example) and then obtain the saddle-point solution of the problem originally formulated by employing an indirect approach

Lyapunov stochastic stability and control of robust dynamic coalitional games with transferable utilities

This paper considers a dynamic game with transferable utilities (TU), where the characteristic function is a continuous-time bounded mean ergodic process. A central planner interacts continuously over time with the players by choosing the instantaneous allocations subject to budget constraints. Before the game starts, the central planner knows the nature of the process (bounded mean ergodic), the bounded set from which the coalitions' values are sampled, and the long run average coalitions' values. On the other hand, he has no knowledge of the underlying probability function generating the coalitions' values. Our goal is to find allocation rules that use a measure of the extra reward that a coalition has received up to the current time by re-distributing the budget among the players. The objective is two-fold: i) guaranteeing convergence of the average allocations to the core (or a specific point in the core) of the average game, ii) driving the coalitions' excesses to an a priori given cone. The resulting allocation rules are robust as they guarantee the aforementioned convergence properties despite the uncertain and time-varying nature of the coaltions' values. We highlight three main contributions. First, we design an allocation rule based on full observation of the extra reward so that the average allocation approaches a specific point in the core of the average game, while the coalitions' excesses converge to an a priori given direction. Second, we design a new allocation rule based on partial observation on the extra reward so that the average allocation converges to the core of the average game, while the coalitions' excesses converge to an a priori given cone. And third, we establish connections to approachability theory and attainability theory

Opinion dynamics in coalitional games with transferable utilities

© 2014 IEEE. This paper studies opinion dynamics in a large number of homogeneous coalitional games with transferable utilities (TU), where the characteristic function is a continuous-time stochastic process. For each game, which we can see as a 'small world', the players share opinions on how to allocate revenues based on the mean-field interactions with the other small worlds. As a result of such mean-field interactions among small worlds, in each game, a central planner allocates revenues based on the extra reward that a coalition has received up to the current time and the extra reward that the same coalition has received in the other games. The paper also studies the convergence and stability of opinions on allocations via stochastic stability theory

Large networks of dynamic agents: Consensus under adversarial disturbances

This paper studies interactions among homogeneous social groups within the framework of large population games. Each group is represented by a network and the behavior described by a two-player repeated game. The contribution is three-fold. Beyond the idea of providing a novel two-level model with repeated games at a lower level and population games at a higher level, we also establish a mean field equilibrium and study state feedback best-response strategies as well as worst-case adversarial disturbances in that context. © 2012 IEEE

A note on a gauge-gravity relation and functional determinants

We present a refinement of a recently found gauge-gravity relation between one-loop effective actions: on the gauge side, for a massive charged scalar in 2d dimensions in a constant maximally symmetric electromagnetic field; on the gravity side, for a massive spinor in d-dimensional (Euclidean) anti-de Sitter space. The inclusion of the dimensionally regularized volume of AdS leads to complete mapping within dimensional regularization. In even-dimensional AdS, we get a small correction to the original proposal; whereas in odd-dimensional AdS, the mapping is totally new and subtle, with the `holographic trace anomaly' playing a crucial role.Comment: 6 pages, io

Chiral Modulations in Curved Space I: Formalism

The goal of this paper is to present a formalism that allows to handle four-fermion effective theories at finite temperature and density in curved space. The formalism is based on the use of the effective action and zeta function regularization, supports the inclusion of inhomogeneous and anisotropic phases. One of the key points of the method is the use of a non-perturbative ansatz for the heat-kernel that returns the effective action in partially resummed form, providing a way to go beyond the approximations based on the Ginzburg-Landau expansion for the partition function. The effective action for the case of ultra-static Riemannian spacetimes with compact spatial section is discussed in general and a series representation, valid when the chemical potential satisfies a certain constraint, is derived. To see the formalism at work, we consider the case of static Einstein spaces at zero chemical potential. Although in this case we expect inhomogeneous phases to occur only as meta-stable states, the problem is complex enough and allows to illustrate how to implement numerical studies of inhomogeneous phases in curved space. Finally, we extend the formalism to include arbitrary chemical potentials and obtain the analytical continuation of the effective action in curved space.Comment: 22 pages, 3 figures; version to appear in JHE