Hamilton Jacobi Isaacs equations for differential games with asymmetric information on probabilistic initial condition

We investigate Hamilton Jacobi Isaacs equations associated to a two-players zero-sum differential game with incomplete information. The first player has complete information on the initial state of the game while the second player has only information of a – possibly uncountable – probabilistic nature: he knows a probability measure on the initial state. Such differential games with finite type incomplete information can be viewed as a generalization of the famous Aumann–Maschler theory for repeated games. The main goal and novelty of the present work consists in obtaining and investigating a Hamilton Jacobi Isaacs Equation satisfied by the upper and the lower values of the game. Since we obtain a uniqueness result for such Hamilton Jacobi equation, as a byproduct, this gives an alternative proof of the existence of a value of the differential game (which has been already obtained in the literature by different technics). Since the Hamilton Jacobi equation is naturally stated in the space of probability measures, we use the Wasserstein distance and some tools of optimal transport theory.     

Patent-investment games under asymmetric information

This paper analyzes preemptive patenting in a two-stage real options game where an incumbent firm competes with a potential entrant firm for the patent of a substitute product in a product market with profit flow uncertainty. The incumbent suffers loss of monopoly in the product market if the entrant acquires the patent of a substitute product and later commercializes the product. Our patent-investment game model assumes that the entrant has complete information on the incumbent’s commercialization cost while the incumbent only knows the distribution of the entrant’s cost. We investigate the impact of information asymmetry on the preemption strategies adopted by the two competing firms on patenting the substitute product by comparing the optimal preemption strategies and the real option value functions of the two competing firms under complete information and information asymmetry. Our analysis reveals that the informationally disadvantaged incumbent always suffers from loss in its real option value of investment since it tends to act more aggressively in competing for the patent. On the other hand, the real option value of investment of the informationally advantaged entrant may be undermined or enhanced. The incumbent’s aggressive response under information asymmetry may lead to reversal of winner in the patent race. We also examine how information asymmetry may affect the occurrence of sleeping patent and the corresponding expected duration between the two stages of patenting and product commercialization.     

Linear differential games with delayed and noisy information

This paper deals with a linear-quadratic-Gaussian zero-sum game in which one player has delayed and noisy information and the other has perfect information. Assuming that the player with perfect information can deduce his opponent's state estimate, the optimal closed-loop control laws are derived. Then, it is shown that the separation theorem is satisfied for the player with imperfect information and his optimal state estimate is given by a delay-differential equation.     

An agreeable collusive equilibrium in differential games with asymmetric players

We study a class of collusive equilibria in differential games with asymmetric players discounting the future at different rates. For such equilibria, at each moment, weights of players can depend on the state of the system. To fix them, we propose using a bargaining procedure according to which players can bargain again at every future moment. By choosing as threat point the feedback noncooperative outcome, the corresponding solution, if it exists, is agreeable. An exhaustible resource game illustrates the results.     

The mixed expansion of differential games with incomplete information

Kuhn's theorem [1953] about behavioral strategies is proved for the mixed expansion of a differential game.     

Positive value of information in games

We exhibit a general class of interactive decision situations in which all the agents benefit from more information. This class includes as a special case the classical comparison of statistical experiments à la Blackwell. AMS 2000 Subject Classification:Primary 91A35.The work of Bruno Bassan and Marco Scarsini was partially supported by MIUR-COFIN. The authors express their thanks to Sylvain Sorin for enlightening comments.     

残缺模糊合作对策的加权Shapley值

针对具有模糊联盟且支付值残缺的合作对策问题,给出了E-残缺模糊对策的定义.基于残缺联盟值基数集,提出了一个同时满足对称性和线性性的w-加权Shapley值公式.通过构造模糊联盟间的边际贡献,探讨了w-加权Shapley值公式的等价表示形式,指出w-加权Shapley值与完整合作对策Shapley值的兼容性.在模糊联盟框架里,探讨了w-加权Shapley值所满足的联盟单调性、零正则性等优良性质.最后通过算例验证了该公式的有效性.     

Repeated games with asymmetric information and random price fluctuations at finance markets

This paper is concerned with multistage bidding models introduced in De Meyer and Moussa Saley (Int J Game Theory 31:285–319, 2002) to analyze the evolution of the price system at finance markets with asymmetric information. The repeated games are considered modelling the biddings with the admissible bids k/m, unlike the above mentioned paper, where arbitrary bids are allowed. It is shown that the sequence of values of n-step games is bounded from above and converges to the value of the game with infinite number of steps. The optimal strategies of infinite game generate a symmetric random walk of transaction prices over admissible bids with absorbing extreme points. The value of infinite game is equal to the expected duration of this random walk multiplied by the constant one-step gain of informed Player 1. This study was supported by the grant 04-06-80430 of Russian Foundation of Basic Research which is gratefully acknowledged. I am thankful to anonymous referees and to William Thomson for instructive and helpful remarks and comments.     

Existence of value in differential games with fixed time duration

A differential game of prescribed duration with general-type phase constraints is investigated. The existence of a value in the Varaiya-Lin sense and an optimal strategy for one of the players is obtained under assumptions ensuring that the sets of all admissible trajectories for the two players are compact in the Banach space of all continuous functions. These results are next widened on more general games, examined earlier by Varaiya.The author wishes to express his thanks to an anonymous reviewer for his many valuable comments.     

Existence of value in differential games with terminal cost function

A two-person, zero-sum differential game with general type phase constraints and terminal (not fixed) cost function is investigated. Player II (possessing complete information) can choose any strategy in the Varaiya-Lin sense, while his opponent (having incomplete information) can select any lower II-strategy introduced by Friedman (Ref. 1). The existence of a value and an optimal player II's strategy is obtained under assumptions ensuring that the sets of all admissible trajectories for the two players are compact in the Banach space of all continuous functions. The present paper largely extends the results of Ref. 2.     

New results in the theory of differential games with information time lag

The results contained herein provide a rigorous formulation of a broad class of differential games with information time lag and present a theoretical analysis for treating such games. This analysis extends the so-called Hamilton-Jacobi theory of optimal control and the main equation analysis developed by Isaacs to treat differential games with information time lag. Necessary and sufficient conditions satisfied by thepotential value function are developed to indicate the strategy-synthesis procedure for differential games with information time lag.     

A BSDE approach to stochastic differential games with incomplete information

We consider a two-player zero-sum stochastic differential game in which one of the players has a private information on the game. Both players observe each other, so that the non-informed player can try to guess his missing information. Our aim is to quantify the amount of information the informed player has to reveal in order to play optimally: to do so, we show that the value function of this zero-sum game can be rewritten as a minimization problem over some martingale measures with a payoff given by the solution of a backward stochastic differential equation.     

Equilibrium existence and topology in some repeated games with incomplete information

This article proves the existence of an equilibrium in any infinitely repeated, un-discounted two-person game of incomplete information on one side where the uninformed player must base his behavior strategy on state-dependent information generated stochastically by the moves of the players and the informed player is capable of sending nonrevealing signals.

This extends our earlier result stating that an equilibrium exists if additionally the information is standard. The proof depends on applying new topological properties of set-valued mappings. Given a set-valued mapping on a compact convex set , we give further conditions which imply that every point belongs to the convex hull of a finite subset of the domain of satisfying .

     

Results in the theory of linear differential games with an information time lag

The solution to a class of differential games in which one player has imperfect information in the form of a time lag on the availability of the state vector is presented. Open-loop and closed-loop strategies are obtained, and the results are shown to reduce to the well-known perfect-information results as the time lag vanishes. The effect of lag on the existence of the solutions to the game and other important qualitative aspects of the game are discussed.The author would like to acknowledge his gratitude to Professor Celso de Renna e Souza, Department of Electrical Engineering, University of Notre Dame, South Bend, Indiana, who advised and encouraged him during the initial research efforts which led to the work presented herein. This paper is based in part on a portion of the dissertation which the author submitted in partial fulfillment of the requirement for the Ph.D. degree at the University of Notre Dame.     

A note on the value of zero-sum sequential repeated games with incomplete information

We consider repeated two-person zero-sum games with lack of information on both sides. If the one shot game is played sequentially, it is proved that the sequencev _n is monotonic,v _n being the value of then shot game. Moreover the speed of convergence is bounded byK/n, and this is the best bound.