arXiv subjects

Ian Hogeboom-Burr

Publications and source records attributed to Ian Hogeboom-Burr.

4 recordsLinked to original sources

arXiv2022

Continuity Properties of Value Functions in Information Structures for Zero-Sum and General Games and Stochastic Teams

We study continuity properties of stochastic game problems with respect to various topologies on information structures, defined as probability measures characterizing a game. We will establish continuity properties of the value function under total variation, setwise, and weak convergence of information structures. Our analysis reveals that the value function for a bounded game is continuous under total variation convergence of information structures in both zero-sum games and team problems. Continuity may fail to hold under setwise or weak convergence of information structures, however, the value function exhibits upper semicontinuity properties under weak and setwise convergence of information structures for team problems, and upper or lower semicontinuity properties hold for zero-sum games when such convergence is through a Blackwell-garbled sequence of information structures. If the individual channels are independent, fixed, and satisfy a total variation continuity condition, then the value functions are continuous under weak convergence of priors. We finally show that value functions for players may not be continuous even under total variation convergence of information structures in general non-zero-sum games.

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arXiv2022

Zero-Sum Games involving Teams against Teams: Existence of Equilibria, and Comparison and Regularity in Information

Many emerging problems involve teams of agents taking part in a game. Such problems require a stochastic analysis with regard to the correlation structures among the agents belonging to a given team. In the context of Standard Borel spaces, this paper makes the following contributions for two teams of finitely many agents taking part in a zero-sum game: (i) An existence result will be presented for saddle-point equilibria in zero-sum games involving teams against teams when common randomness is assumed to be available in each team with an analysis on conditions for compactness of strategic team measures to be presented. (ii) Blackwell's ordering of information structures is generalized to $n$-player teams with standard Borel spaces, where correlated garbling of information structures is introduced as a key attribute; (iii) building on this result Blackwell's ordering of information structures is established for team-against-team zero-sum game problems. (iv) Finally, continuity of the equilibrium value of team-against-team zero-sum game problems in the space of information structures under total variation is established.

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arXiv2022

Sequential Stochastic Control (Single or Multi-Agent) Problems Nearly Admit Change of Measures with Independent Measurements

Change of measures has been an effective method in stochastic control and analysis; in continuous-time control this follows Girsanov's theorem applied to both fully observed and partially observed models, in decentralized stochastic control (or stochastic dynamic team theory) this is known as Witsenhausen's static reduction, and in discrete-time classical stochastic control Borkar has considered this method for partially observed Markov Decision processes (POMDPs) generalizing Fleming and Pardoux's approach in continuous-time. This method allows for equivalent optimal stochastic control or filtering in a new probability space where the measurements form an independent exogenous process in both discrete-time and continuous-time and the Radon-Nikodym derivative (between the true measure and the reference measure formed via the independent measurement process) is pushed to the cost or dynamics. However, for this to be applicable, an absolute continuity condition is necessary. This raises the following question: can we perturb any discrete-time sequential stochastic control problem by adding some arbitrarily small additive (e.g. Gaussian or otherwise) noise to the measurements to make the system measurements absolutely continuous, so that a change-of-measure (or static reduction) can be applicable with arbitrarily small error in the optimal cost? That is, are all sequential stochastic (single-agent or decentralized multi-agent) problems $ε$-away from being static reducible as far as optimal cost is concerned, for any $ε> 0$? We show that this is possible when the cost function is bounded and continuous in controllers' actions and the action spaces are convex. We also note that the solution and the cost obtained for the perturbed system is realizable (under a randomized policy) for the original model.

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arXiv2021

Comparison of Information Structures for Zero-Sum Games and a Partial Converse to Blackwell Ordering in Standard Borel Spaces

In statistical decision theory involving a single decision-maker, an information structure is said to be better than another one if for any cost function involving a hidden state variable and an action variable which is restricted to be conditionally independent from the state given some measurement, the solution value under the former is not worse than that under the latter. For finite spaces, a theorem due to Blackwell leads to a complete characterization on when one information structure is better than another. For stochastic games, in general, such an ordering is not possible since additional information can lead to equilibria perturbations with positive or negative values to a player. However, for zero-sum games in a finite probability space, Pęski introduced a complete characterization of ordering of information structures. In this paper, we obtain an infinite dimensional (standard Borel) generalization of Pęski's result. A corollary is that more information cannot hurt a decision maker taking part in a zero-sum game. We establish two supporting results which are essential and explicit though modest improvements on prior literature: (i) a partial converse to Blackwell's ordering in the standard Borel setup and (ii) an existence result for equilibria in zero-sum games with incomplete information.

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