Interdisciplinary Studies in Intelligent Systems Department of Electrical Engineering University of Notre Dame

Title: Stochastic Di↵erential Games and Intricacy of Information Structures Author: Tamer Başar (University of Illinois at Urbana-Champaign, [email protected]) Coauthor(s): Abstract: In spite of decades long past research activity on stochastic di↵erential games, there still remain some outstanding fundamental questions on existence, uniqueness, and characterization of non-cooperative equilibria when players have access to noisy state information. This talk will identify these questions, along with the underlying challenges, and address a number of them within specific contexts by focusing on two-player zero-sum stochastic di↵erential games (ZSSDGs). One specific class the talk will dwell on is games where the state information is acquired by the players through an intermittently failing noisy channel, where the failure is governed by a time-independent Bernoulli process. When the failure rate is zero, this corresponds to the standard ZSSDG with identical noisy measurements for the players (in which case derivation of the ”complete” saddle-point solution is still quite subtle, as will be discussed in the talk). For the other extreme case when the failure rate is one, we have the open-loop ZSSDG. For the intermediate case, the saddle-point solution involves, under some conditions, a Kalman filter (or an extended Kalman filter) with intermittent (or missing) measurements; in this case, a restricted certainty equivalence holds. A variant of this is the more challenging class of problems where the failure of the transmission of the common noisy measurement of the state to the players is governed by two independent Bernoulli processes with possibly di↵erent rates. This class of ZSSDGs, which will be discussed briefly, involves two scenarios: (i) the players are not aware of the failure of links corresponding to each other, and (ii) this information is available (that is players share explicitly or implicitly the failure information) but with one step delay. Extensions to (i) multi-player ZSSDGs (with teams playing against teams, where agents in each team do not have identical information), and (ii) nonzero-sum stochastic di↵erential games (with asymmetric information among the players) constitute yet two other classes of challenging problems, which will be touched upon toward the end of the talk. In spite of decades long past research activity on stochastic di↵erential games, there still remain some outstanding fundamental questions on existence, uniqueness, and characterization of non-cooperative equilibria when players have access to noisy state information. This talk will identify these questions, along with the underlying challenges, and address a number of them within specific contexts by focusing on two-player zero-sum stochastic di↵erential games (ZSSDGs). One specific class the talk will dwell on is games where the state information is acquired by the players through an intermittently failing noisy channel, where the failure is governed by a time-independent Bernoulli process. When the failure rate is zero, this corresponds to the standard ZSSDG with identical noisy measurements for the players (in which case derivation of the ”complete” saddle-point solution is still quite subtle, as will be discussed in the talk). For the other extreme case when the failure rate is one, we have the open-loop ZSSDG. For the intermediate case, the saddle-point solution involves, under some conditions, a Kalman filter (or an extended Kalman filter) with intermittent (or missing) measurements; in this case, a restricted certainty equivalence holds. A variant of this is the more challenging class of problems where the failure of the transmission of the common noisy measurement of the state to the players is governed by two independent Bernoulli processes with possibly di↵erent rates. This class of ZSSDGs, which will be discussed briefly, involves two scenarios: (i) the players are not aware of the failure of links corresponding to each other, and (ii) this information is available (that is players share explicitly or implicitly the failure information) but with one step delay. Extensions to (i) multi-player ZSSDGs (with teams playing against teams, where agents in each team do not have identical information), and (ii) nonzero-sum stochastic di↵erential games (with asymmetric information among the players) constitute yet two other classes of challenging problems, which will be touched upon toward the end of the talk. Title: Formulating and Solving Portfolio Decision and Asset Pricing Problems Author: Thomas Cosimano (University of Notre Dame, [email protected])