A Topology for Team Policies and Existence of Optimal Team Policies in Stochastic Team Theory

Saldi, Naci

doi:10.1109/tac.2019.2911798

Cited by 20 publications

(12 citation statements)

References 32 publications

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“…The ideas from weak convergence in probability theory is used to show convergence of measure of joint probability of actions. In [16], author extended study of [15], further weaken assumptions their. They have shown the existence of optimal strategies for static teams and topology on set of policies are introduced.…”

Section: A Related Workmentioning

confidence: 99%

Role of Externally Provided Randomness in Stochastic Teams and Zero-sum Team Games

Meshram¹

2021

Preprint

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Stochastic team decision problem is extensively studied in literature and the existence of optimal solution is obtained in recent literature. The value of information in statistical problem and decision theory is classical problem. Much of earlier does not qualitatively describe role of externally provided private and common randomness in stochastic team problem and team vs team zero sum game.In this paper, we study the role of extrenally provided private or common randomness in stochastic team decision. We make observation that the randomness independent of environment does not benefit either team but randomness dependent on environment benefit teams and decreases the expected cost function. We also studied LQG team game with special information structure on private or common randomness. We extend these study to problem team vs team zero sum game. We show that if a game admits saddle point solution, then private or common randomness independent of environment does not benefit either team. We also analyze the scenario when a team with having more information than other team which is dependent on environment and game has saddle point solution, then team with more information benefits. This is also illustrated numerically for LQG team vs team zero sum game. Finally, we show for discrete team vs team zero sum game that private randomness independent of environment benefits team when there is no saddle point condition. Role of common randomness is discussed for discrete game.

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Section: A Related Workmentioning

confidence: 99%

Role of Externally Provided Randomness in Stochastic Teams and Zero-sum Team Games

Meshram¹

2021

Preprint

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“…Furthermore, [23, Proposition III.4.2.] builds on a topology construction on policies which is different from what we present here (we note that in control theory a related construction has been utilized in [8]; see also [33]); the construction in [23] is rather abstract and we would like to caution that in the absence of absolute continuity conditions on the information structure, this construction may lead to a lack of closedness on the sets of admissible policies (or strategic measures) as the counterexample [43, Theorem 2.7] reveals: in this counterexample, which would reduce to the setup studied here with y 1 = y 2 = y, a sequence of policies is constructed so that for each element of the sequence the action variables of the two decision makers are conditionally independent given their measurements, but the setwise (and hence, weak) limit of the sequence is not conditionally, or otherwise, independent; and thus the limit measure does not belong to the original information structure. Theorem 3.1 (Existence of Equilibria) For a given game, assume that Assumption 2.1 holds.…”

Section: On Existence Of Saddle-points and Equilibriamentioning

confidence: 99%

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

Hogeboom-Burr¹,

Yüksel²

2020

Preprint

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In statistical decision theory involving a single decision-maker, one says that an information structure is better than another one if for any cost function involving a hidden state variable and an action variable which is restricted to be only a function of some measurement, the solution value under the former is not worse than the value under the latter. For finite probability spaces, Blackwell's celebrated theorem on comparison of information structures leads to a complete characterization on when one information structure is better than another. For stochastic games with incomplete information, due to the presence of competition among decision makers, 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 of our analysis is that more information cannot hurt a decision maker taking part in a zero-sum game in standard Borel spaces. During our analysis, we establish two novel supporting results: (i) a partial converse to Blackwell's ordering of information structures in the standard Borel space setup and (ii) a refined existence result for equilibria in zero-sum games with incomplete information when compared with the prior literature.

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“…Relative compactness and closedness of strategic measures, and existence of optimal policies. Existence of optimal policies for static and a class of sequential dynamic teams have been studied recently in [68,149,117]. More specific setups have been studied in [142], [135], [148] and [147].…”

Section: Measurability Properties Of Sets Of Strategic Measuresmentioning

confidence: 99%

Geometry of Information Structures, Strategic Measures and associated Control Topologies

Saldi,

Yuksel

2020

Preprint

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In many areas of applied mathematics, engineering, and social and natural sciences, decentralization of information is a key aspect determining how to approach a problem. In this review article, we study information structures in a probability theoretic and geometric context. We define information structures, place various topologies on them, and study closedness, compactness and convexity properties on the strategic measures induced by information structures and decentralized control/decision policies under varying degree of relaxations with regard to access to private or common randomness. Ultimately, we present existence and tight approximation results for optimal decision/control policies. We discuss various lower bounding techniques, through relaxations and convex programs ranging from classically realizable and classically non-realizable (such as quantum and non-signaling) relaxations. For each of these, we establish closedness and convexity properties and also a hierarchy of correlation structures. As a second main theme, we review and introduce various topologies on decision/control strategies defined independent of information structures, but for which information structures determine whether the topologies entail utility in arriving at existence, compactness, convexification or approximation results. These approaches, which we term as the strategic measures approach and the control topology approach, lead to complementary results on existence, approximations and upper and lower bounds in optimal decentralized decision and control.

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A Topology for Team Policies and Existence of Optimal Team Policies in Stochastic Team Theory

Cited by 20 publications

References 32 publications

Role of Externally Provided Randomness in Stochastic Teams and Zero-sum Team Games

Role of Externally Provided Randomness in Stochastic Teams and Zero-sum Team Games

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

Geometry of Information Structures, Strategic Measures and associated Control Topologies

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