International Game Theory Review

This paper deals with inertia functions in control theory introduced in Aubin, Bernardo and Saint-Pierre (2004, 2005) and their adaptation to dynamical games. The inertia function associates with any initial state-control pair the smallest of the worst norms over time of the velocities of the controls regulating viable evolutions. For tychastic systems (parameterized systems where the parameters are tyches, disturbances, perturbations, etc.), the palicinesia of a tyche measure the worst norm over time of the velocities of the tyches. The palicinesia function is the largest palicinesia threshold c such that all evolutions with palicinesia smaller than or equal to c are viable. For dynamical games where one parameter is the control and the other one is a tyche (games against nature or robust control), we define the guaranteed inertia function associated with any initial state-control-tyche triple the best of the worst of the norms of the velocities of the controls and of the tyches and study their properties. Viability Characterizations and Hamilton-Jacobi equations of which these inertia and palicinesia functions are solutions are provided.

We investigate an impulse control differential game arising in a problem of option pricing in mathematical finance. In a previous paper, it was shown that its Value function in ℝ³ could be described as a pair of functions affine in one of the variables, joined on a 2D manifold. Depending on the regions of the state space, this manifold is either a dispersal one, an equivocal one or a 2D focal manifold. A pair of PDE's were derived for the focal part. Here we show that irrespective of the nature of this manifold, it has to satisfy this same set of PDE's.

In this paper we present some numerical methods for the solution of two-persons zero-sum deterministic differential games. The methods are based on the dynamic programming approach. We first solve the Isaacs equation associated to the game to get an approximate value function and then we use it to reconstruct approximate optimal feedback controls and optimal trajectories. The approximation schemes also have an interesting control interpretation since the time-discrete scheme stems from a dynamic programming principle for the associated discrete time dynamical system. The general framework for convergence results to the value function is the theory of viscosity solutions. Numerical experiments are presented solving some classical pursuit-evasion games.

In this paper we further explore the properties and computational methods of intergenerational equilibrium solutions for a class of multi-generation stochastic games that are inspired by the intergenerational equity issues typically found in cost-benefit analysis for climate change policies. We first extend previous results by showing that discounting over time and over generations yields to an intergenerational equilibrium which is equivalent to the use of a lower discount rate. We then propose a model that encompasses the different formulations proposed before to represent intergenerational equilibria in the presence of selfishness. We propose a numerical method to compute these equilibria and we test it on an adaptation of the DICE94 model proposed by Nordhaus to do cost-benefit analysis of climate policies.

The paper explores the properties of multistage games with perfect information, in which (unlike the classic Kuhnian definition) the conditional coalition partition at any vertex is determined by a chance move and remains unchanged until the random process repeats at the next vertex. A new value for such a game is proposed in terms of a PMS-vector. It is computed by backward induction using conditional partition and transition in the vertices. An illustrative example is provided.

Cooperative stochastic differential games constitute a highly complex form of decision making under uncertainty. In particular, interactions between strategic behaviors, dynamic evolution, stochastic elements and solution agreement have to be considered simultaneously. This complexity leads to great difficulties in the derivation of dynamically stable solutions. Despite urgent calls for cooperation in the global economy, the lack of formal analyses has precluded rigorous analysis of this problem. In this paper, mechanisms for the derivation of dynamically stable solutions to cooperative stochastic differential games are presented. Games with transferable payoffs and those with nontransferable payoffs are considered. Numerical illustrations are also provided.