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Register a new userSolving Two-State Markov Games with Incomplete Information on One Side *
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We study the optimal use of information in Markov games with incomplete information on one side and two states. We provide a finite-stage algorithm for calculating the limit value as the gap between stages goes to 0, and an optimal strategy for the informed player in the limiting game in continuous time. This limiting strategy induces an-optimal strategy for the informed player, provided the gap between stages is small. Our results demonstrate when the informed player should use his information and how.
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This work considers two-player zero-sum semi-Markov games with incomplete information on one side and perfect observation. At the beginning, the system selects a game type according to a given probability distribution and informs to Player 1 only. After each stage, the actions chosen are observed by both players before proceeding to the next stage. Firstly, we show the existence of the value function under the expected discount criterion and the optimality equation. Secondly, the existence and iterative algorithm of the optimal policy for Player 1 are introduced through the optimality equation of value function. Moreove, About the optimal policy for the uninformed Player 2, we define the auxiliary dual games and construct a new optimality equation for the value function in the dual games, which implies the existence of the optimal policy for Player 2 in the dual game. Finally, the existence and iterative algorithm of the optimal policy for Player 2 in the original game is given by the results of the dual game.
We study a two-player, zero-sum, stochastic game with incomplete information on one side in which the players are allowed to play more and more frequently. The informed player observes the realization of a Markov chain on which the payoffs depend, while the non-informed player only observes his opponents actions. We show the existence of a limit value as the time span between two consecutive stages vanishes; this value is characterized through an auxiliary optimization problem and as the solution of an Hamilton-Jacobi equation.
This paper deals with control of partially observable discrete-time stochastic systems. It introduces and studies the class of Markov Decision Processes with Incomplete information and with semi-uniform Feller transition probabilities. The important feature of this class of models is that the classic reduction of such a model with incomplete observation to the completely observable Markov Decision Process with belief states preserves semi-uniform Feller continuity of transition probabilities. Under mild assumptions on cost functions, optimal policies exist, optimality equations hold, and value iterations converge to optimal values for this class of models. In particular, for Partially Observable Markov Decision Processes the results of this paper imply new and generalize several known sufficient conditions on transition and observation probabilities for the existence of optimal policies, validity of optimality equations, and convergence of value iterations.
We study stochastic differential games of jump diffusions, where the players have access to inside information. Our approach is based on anticipative stochastic calculus, white noise, Hida-Malliavin calculus, forward integrals and the Donsker delta functional. We obtain a characterization of Nash equilibria of such games in terms of the corresponding Hamiltonians. This is used to study applications to insider games in finance, specifically optimal insider consumption and optimal insider portfolio under model uncertainty.
We consider stochastic programs conditional on some covariate information, where the only knowledge of the possible relationship between the uncertain parameters and the covariates is reduced to a finite data sample of their joint distribution. By exploiting the close link between the notion of trimmings of a probability measure and the partial mass transportation problem, we construct a data-driven Distributionally Robust Optimization (DRO) framework to hedge the decision against the intrinsic error in the process of inferring conditional information from limited joint data. We show that our approach is computationally as tractable as the standard (without side information) Wasserstein-metric-based DRO and enjoys performance guarantees. Furthermore, our DRO framework can be conveniently used to address data-driven decision-making problems under contaminated samples and naturally produces distributionally robu
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