FlipDyn with Control: Resource Takeover Games with Dynamics

We present the FlipDyn, a dynamic game in which two opponents (a defender and an adversary) choose strategies to optimally takeover a resource that involves a dynamical system. At any time instant, each player can take over the resource and thereby control the dynamical system after incurring a state-dependent and a control-dependent costs. The resulting model becomes a hybrid dynamical system where the discrete state (FlipDyn state) determines which player is in control of the resource. Our objective is to compute the Nash equilibria of this dynamic zero-sum game. Our contributions are four-fold. First, for any non-negative costs, we present analytical expressions for the saddle-point value of the FlipDyn game, along with the corresponding Nash equilibrium (NE) takeover strategies. Second, for continuous state, linear dynamical systems with quadratic costs, we establish sufficient conditions under which the game admits a NE in the space of linear state-feedback policies. Third, for scalar dynamical systems with quadratic costs, we derive the NE takeover strategies and saddle-point values independent of the continuous state of the dynamical system. Fourth and finally, for higher dimensional linear dynamical systems with quadratic costs, we derive approximate NE takeover strategies and control policies which enable the computation of bounds on the value functions of the game in each takeover state. We illustrate our findings through a numerical study involving the control of a linear dynamical system in the presence of an adversary.Comment: 17 Pages, 2 figures. Under review at IEEE TA

The Core-Periphery Model with Forward-Looking Expectations

The 'core-periphery model' is vitiated by its assumption of static expectations. That is, migration (inter-regional or intersectoral) is the key to agglomeration, but migrants base their decision on current wage differences alone--even though migration predictably alters wages and workers are (implicitly) infinitely lived. The assumption was necessary for analytic tractability. The model can have multiple stable equilibria, so allowing forward-looking expectations would have forced consideration of the very difficult perhaps even intractable issues of global stability in non-linear dynamic systems. This paper's main contribution is to present a set of solution techniques partly analytic and partly numerical that allow us to consider forward-looking expectations. These techniques reveal a startling result. If quadratic migration costs are sufficiently high, allowing forward-looking behaviour has no impact on the main results, so static expectations are truly an assumption of convenience. If migration costs are lower, however, forward-looking behaviour creates history-vs-expectations considerations. In this case self-fulfilling prophecy.

Quadratic Multi-Dimensional Signaling Games and Affine Equilibria

This paper studies the decentralized quadratic cheap talk and signaling game problems when an encoder and a decoder, viewed as two decision makers, have misaligned objective functions. The main contributions of this study are the extension of Crawford and Sobel's cheap talk formulation to multi-dimensional sources and to noisy channel setups. We consider both (simultaneous) Nash equilibria and (sequential) Stackelberg equilibria. We show that for arbitrary scalar sources, in the presence of misalignment, the quantized nature of all equilibrium policies holds for Nash equilibria in the sense that all Nash equilibria are equivalent to those achieved by quantized encoder policies. On the other hand, all Stackelberg equilibria policies are fully informative. For multi-dimensional setups, unlike the scalar case, Nash equilibrium policies may be of non-quantized nature, and even linear. In the noisy setup, a Gaussian source is to be transmitted over an additive Gaussian channel. The goals of the encoder and the decoder are misaligned by a bias term and encoder's cost also includes a penalty term on signal power. Conditions for the existence of affine Nash equilibria as well as general informative equilibria are presented. For the noisy setup, the only Stackelberg equilibrium is the linear equilibrium when the variables are scalar. Our findings provide further conditions on when affine policies may be optimal in decentralized multi-criteria control problems and lead to conditions for the presence of active information transmission in strategic environments.Comment: 15 pages, 4 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 Price Competition with Price Adjustment Costs and Product Differentiation

We study a discrete time dynamic game of price competition with spatially differentiated products and price adjustment costs. We characterise the Markov perfect and the open-loop equilibrium of our game. We find that in the steady state Markov perfect equilibrium, given the presence of adjustment costs, equilibrium prices are always higher than prices at the repeated static Nash solution, even though, adjustment costs are not paid in steady state. This is due to intertemporal strategic complementarity in the strategies of the firms and from the fact that the cost of adjusting prices adds credibility to high price equilibrium strategies. On the other hand, the stationary open-loop equilibrium coincides always with the static solution. Furthermore, in contrast to continuous time games, we show that the stationary Markov perfect equilibrium converges to the static Nash equilibrium when adjustment costs tend to zero. Moreover, we obtain the same convergence result when adjustment costs tend to infinity.Price adjustment costs, Difference game, Markov perfect equilibrium, Open-loop equilibrium