International Game Theory Review

Motivated by repeated play of non-cooperative games, we study equation solving undertaken in parallel by several non-communicating agents, each dealing with his own block of variables. The process is akin to Newton's method in using derivative information. It does, however, proceed without matrix inversion and dispenses with the need to exchange information between various blocks.

This paper deals with the approximation of Nash equilibria in m-player games. We present conditions under which an approximating sequence of games admits near-equilibria that approximate near-equilibria in the limit game. We apply the results to two classes of games: (i) a duopoly game approximated by a sequence of matrix games, and (ii) a stochastic game played under the S-adapted information structure approximated by games played over a sampled event tree. Numerical illustrations show the usefulness of this approximation theory.

A two-player nonzero-sum differential game is considered. Given a pair of threat payoff functions, we characterise a set of pairs of acceptable feedback controls. Any such pair induces a history-dependent Nash δ-equilibrium as follows: the players agree to use the acceptable controls unless one of them deviates. If this happens, a feedback control punishment is implemented. The problem of finding a pair of "acceptable" controls is significantly simpler than the problem of finding a feedback control Nash equilibrium. Moreover, the former may have a solution in case the latter does not. In addition, if there is a feedback control Nash equilibrium, then our technique gives a subgame perfect Nash δ-equilibrium that might improve the payoff function for at least one player.

Doubly Invariant Equilibrium is introduced as an alternative concept to Nash equilibrium in dynamic games doing away with the notion of a payoff function. A subset of the state space enjoys the invariance property if the state can be kept in it by one player, regardless of the action of the opponent. A doubly invariant equilibrium obtains when each player can make his own subset invariant. Relationships to Nash equilibrium and viability theory are discussed and a necessary and sufficient condition for the existence of a doubly invariant equilibrium is given for the class of linear discrete-time games with polyhedral constraints on the state and strategy spaces.

This paper investigates some differential game applications to option pricing mechanisms and related problems. Two players, an investor and "Nature", play a zero-sum game. The usual uncertainty modelling (log-normality for instance) in systems describing the price evolution of stocks is replaced by "Nature", a player who counteracts the investor as much as possible. A relationship between a restricted version of the Black-Scholes and the Hamilton-Jacobi-Bellman partial differential equations is given. This paper, is a first step to possibly solve various option pricing problems (with constraints and/or transactions costs for instance) by means of the available numerical software for optimal control problems. In the second part of the paper, another model, now with three players, is considered. The third player is the bank interested in maximising its own profits by choosing the right formula for transaction costs. Thus a three-person nonzero-sum game, with a special kind of Stackelberg information structure, results. Some simple examples hint in the direction that the bank will be a clear winner.

A linear pursuit-evasion game with first-order acceleration dynamics and bounded controls is considered. In this game, the pursuer has to estimate the state variables of the game, including the lateral acceleration of the evader, based on the noise-corrupted measurements of the relative position vector. The estimation process inherently involves some delay, rendering the information structure of the pursuer imperfect. If the pursuer implements the optimal strategy of the perfect information game, an evader with perfect information can take advantage of the estimation delay. However, the performance degradation is minimised if the pursuer compensates for its own estimation delay by implementing the optimal strategy derived from the solution of the imperfect (delayed) information game. In this paper the analytical solution of the delayed information game, allowing to predict the value of the game, is presented. The theoretical results are tested in a noise-corrupted scenario by Monte Carlo simulations, using a Kalman filter type estimator. The simulation results confirm the substantial improvement achieved by the new pursuer strategy.