A lattice-theoretical fixpoint theorem and its applications

Tarski, Alfred

doi:10.2140/pjm.1955.5.285

Cited by 2,290 publications

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“…By Tarski's fixed point theorem (Tarski, 1955), the set of fixed points of ϕ is a complete lattice with respect to ≤ . It follows that the set of fixed points has a least and a greatest element.…”

Section: Proof (⇒)mentioning

confidence: 99%

Decentralized Clearing in Financial Networks

Csóka

Herings

2016

SSRN Journal

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We consider a situation in which agents have mutual claims on each other, summarized in a liability matrix. Agents' assets might be insufficient to satisfy their liabilities leading to defaults. We assume the primitives to be denoted in some unit of account. In case of default, bankruptcy rules are used to specify the way agents are going to be rationed. We present a convenient representation of bankruptcy rules. A clearing payment matrix is a payment matrix consistent with the prevailing bankruptcy rules that satisfies limited liability and priority of creditors. Both clearing payment matrices and the corresponding values of equity are not uniquely determined. We provide bounds on the possible levels equity can take. We analyze decentralized clearing processes and show the convergence of any such process in finitely many steps to the least clearing payment matrix. When the unit of account is sufficiently small, all decentralized clearing processes lead essentially to the same value of equity as a centralized clearing procedure. As a policy implication, it is not necessary to collect and process all the sensitive data of all the agents simultaneously and run a centralized clearing procedure. Abstract We consider a situation in which agents have mutual claims on each other, summarized in a liability matrix. Agents' assets might be insufficient to satisfy their liabilities leading to defaults. We assume the primitives to be denoted in some unit of account. In case of default, bankruptcy rules are used to specify the way agents are going to be rationed. We present a convenient representation of bankruptcy rules.A clearing payment matrix is a payment matrix consistent with the prevailing bankruptcy rules that satisfies limited liability and priority of creditors. Both clearing payment matrices and the corresponding values of equity are not uniquely determined. We provide bounds on the possible levels equity can take. We analyze decentralized clearing processes and show the convergence of any such process in finitely many steps to the least clearing payment matrix. When the unit of account is sufficiently small, all decentralized clearing processes lead essentially to the same value of equity as a centralized clearing procedure. As a policy implication, it is not necessary to collect and process all the sensitive data of all the agents simultaneously and run a centralized clearing procedure.

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Section: Proof (⇒)mentioning

confidence: 99%

Decentralized Clearing in Financial Networks

Csóka

Herings

2016

SSRN Journal

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“…As A S (h)(k; k;ĥ(K)) is isotone in h on C, each 2 C; the …xed point correspondence C A S (ĥ)(k; k) is a nonempty complete lattice follows by Tarski's theorem ( [55], Theorem 1). By Veinott's …xed point comparative statics result in the appendix (e.g., Veinott [58], Theorem 14, Chapter 4), C A S (ĥ)(k; k) is strong set order isotone jointly in the parameterĥ(k; k) with all isotone selections forming a complete lattice.…”

Section: Two Step Monotone Map Methodsmentioning

confidence: 99%

Which recursive equilibrium?

Mirman

Morand²,

Reffett

2010

Proceedings of the Behavioral and Quantitative Game Theory: Conference on Future Directions

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For simple versions of the Cass-Diamond growth model where the fundamental welfare theorems of economics fail, we prove the set of minimal state space recursive equilibrium (RE) is large, with multiple RE existing in multiple subclasses of functions. This situation remains the true even when uniqueness results are known, for both OLG and in…nity-lived agent versions of the model. This implies the set of Generalized Markov equilibrium (GME) is also large. We also provide explicit iterative procedures from computing particular RE in each subclass, with some procedures globally stable from a topological persective for particular RE when their domains are restricted. All iterative methods are order stable relative to perturbations of deep parameters. Finally, we construct an simple economy where existing correspondence-based methods for computing GME fails, and propose a new method that computes the set of GME as minimal state spaceWe thank Subir Chakrabarti, Ramu Gopolan, Takashi Kamihigashi, Rajnesh Mehra, Adrian Peralta-Alva, Ed Prescott, Manuel Santos, Yiannis Vailakis, × ukasz Woźny, and especially, Robert Becker for helpful discussions, as well as seminar participants at IU-PUI, ASU, and EWGET-2012 at Exeter for their helpful comments. Manjira Datta and Kevin Re¤ett thank the Center d'Economie de la Sorbonne (CES) and the Paris School of Economics for graciously arranging their visits during the Spring terms of both 2011 and 2012.y Manjira Datta, Department of Economics, Arizona State University; Leonard J. Mirman, Department of Economics, University of Virginia; Olivier F. Morand, Department of Economics, University of Connecticut; Kevin Re¤ett, Department of Economics, Arizona State University. We are in the process of rewriting this paper, so do not recirculate, as errors, types, etc. may remain; rather write the corresponding author at Kevin.Re¤ett@asu.edu for the most recent version. 1 RE using …xed points of the expanded set of state variables. Our results point to the complicated nature of the RE approximation problem even in simple macro models.

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“…The semantics of the fixed point logic is similar to that of the modal mu-calculus [24], but formulae are now satisfied by configurations. The characterisation of greatest fixed point formula max X .φ follows from the well-known Knaster-Tarski fixed point theorem [38].…”

Section: Corollary 318 In a Bounded Wmdp S ▹ R θ If And Only Ifmentioning

confidence: 99%

Compositional reasoning for weighted Markov decision processes

Deng

Hennessy

2013

Science of Computer Programming

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h i g h l i g h t s• We develop compositional methods for reasoning about weighted Markov decision processes (MDPs).• A coinductive simulation-based behavioural preorder is proposed for weighted MDPs.• The preorder is preserved by structural operators for constructing weighted MDPs from components.• For finitary convergent processes, the preorder can be characterised by a probabilistic logic and a novel form of testing. a r t i c l e i n f o b s t r a c tWeighted Markov decision processes (MDPs) have long been used to model quantitative aspects of systems in the presence of uncertainty. However, much of the literature on such MDPs takes a monolithic approach, by modelling a system as a particular MDP; properties of the system are then inferred by analysis of that particular MDP. In contrast in this paper we develop compositional methods for reasoning about weighted MDPs, as a possible basis for compositional reasoning about their quantitative behaviour. In particular we approach these systems from a process algebraic point of view. For these we define a coinductive simulation-based behavioural preorder which is compositional in the sense that it is preserved by structural operators for constructing weighted MDPs from components.For finitary convergent processes, which are finite-state and finitely branching systems without divergence, we provide two characterisations of the behavioural preorder. The first uses a novel quantitative probabilistic logic, while the second is in terms of a novel form of testing, in which benefits are accrued during the execution of tests.

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A lattice-theoretical fixpoint theorem and its applications

Cited by 2,290 publications

References 2 publications

Decentralized Clearing in Financial Networks

Decentralized Clearing in Financial Networks

Which recursive equilibrium?

Compositional reasoning for weighted Markov decision processes

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