ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sat, 18 May 2024 13:42:13 GMT2024-05-18T13:42:13Z50531- Individual risk and Lebesgue extension without aggregate uncertaintyhttps://scholarbank.nus.edu.sg/handle/10635/19965Title: Individual risk and Lebesgue extension without aggregate uncertainty
Authors: Sun, Y.; Zhang, Y.
Abstract: Many economic models include random shocks imposed on a large number (continuum) of economic agents with individual risk. In this context, an exact law of large numbers and its converse is presented in [Y.N. Sun, The exact law of large numbers via Fubini extension and characterization of insurable risks, J. Econ. Theory 126 (2006) 31-69] to characterize the cancellation of individual risk via aggregation. However, it is well known that the Lebesgue unit interval is not suitable for modeling a continuum of agents in the particular setting. The purpose of this paper is to show that an extension of the Lebesgue unit interval does work well as an agent space with various desirable properties associated with individual risk.©2008 Elsevier Inc. All rights reserved.
Thu, 01 Jan 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/199652009-01-01T00:00:00Z
- Saturation and the integration of Banach valued correspondenceshttps://scholarbank.nus.edu.sg/handle/10635/19972Title: Saturation and the integration of Banach valued correspondences
Authors: Sun, Y.; Yannelis, N.C.
Abstract: This note illustrates that the saturation property of a probability space can be used to routinely generalize results on the integration of Banach valued correspondences over a Loeb measure space to those over an arbitrary saturated probability space. On the other hand, the saturation property is also necessary for the validity of those results when the target space is infinite dimensional.©2007 Elsevier B.V. All rights reserved.
Tue, 01 Jan 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/199722008-01-01T00:00:00Z
- Purification of measure-valued mapshttps://scholarbank.nus.edu.sg/handle/10635/20063Title: Purification of measure-valued maps
Authors: Loeb, P.; Sun, Y.
Sun, 01 Jan 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/200632006-01-01T00:00:00Z
- Similarity of differential information with subjective prior beliefshttps://scholarbank.nus.edu.sg/handle/10635/20028Title: Similarity of differential information with subjective prior beliefs
Authors: Khan, M.A.; Zhang, Z.; Sun, Y.; Tourky, R.
Abstract: We present a complete, separable and metrizable topology on the product space of information and (subjective) beliefs. Such a topology formalizes similarity of differential information without the assumption of a common prior, but under the assumption that objectively impossible events are considered impossible by subjective beliefs. As an application to the theory of the consumer, we provide results on the continuity of expected utility and demand functions. We also provide continuity results for the value of information and the insurance premium as defined in the literature.©2008 Elsevier B.V. All rights reserved.
Tue, 01 Jan 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/200282008-01-01T00:00:00Z
- Monte Carlo simulation of macroeconomic risk with a continuum of agents: The general casehttps://scholarbank.nus.edu.sg/handle/10635/20014Title: Monte Carlo simulation of macroeconomic risk with a continuum of agents: The general case
Authors: Hammond, P.J.; Sun, Y.
Tue, 01 Jan 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/200142008-01-01T00:00:00Z
- Pure strategy equilibria in games with private and public informationhttps://scholarbank.nus.edu.sg/handle/10635/20040Title: Pure strategy equilibria in games with private and public information
Authors: Fu, H.; Sun, Y.; Yannelis, N.C.; Zhang, Z.
Abstract: We introduce a new game form which allows the players' strategies to depend on their strategy-relevant private information as well as on some publicly announced information. The players' payoffs depend on their own payoff-relevant private information and some payoff-relevant common information. Under the assumption that the players' strategy-relevant private information is diffuse and their private information is conditionally independent given the public and payoff-relevant common information, we prove the existence of a pure strategy equilibrium for such a game by developing a distribution theory of correspondences via vector measures.©2006 Elsevier B.V. All rights reserved.
Mon, 01 Jan 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/200402007-01-01T00:00:00Z
- A general Fatou Lemmahttps://scholarbank.nus.edu.sg/handle/10635/20038Title: A general Fatou Lemma
Authors: Loeb, P.A.; Sun, Y.
Abstract: A general Fatou Lemma is established for a sequence of Gelfand integrable functions from a vector Loeb space to the dual of a separable Banach space or, with a weaker assumption on the sequence, a Banach lattice. A corollary sharpens previous results in the finite-dimensional setting even for the case of scalar measures. Counterexamples are presented to show that the results obtained here are sharp in various aspects. Applications include systematic generalizations of the distribution of correspondences from the case of scalar Loeb spaces to the case of vector Loeb spaces and a proof of the existence of a pure strategy equilibrium in games with private and public information and with compact metric action spaces.©2007 Elsevier Inc. All rights reserved.
Mon, 01 Jan 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/200382007-01-01T00:00:00Z
- Ex ante efficiency implies incentive compatibilityhttps://scholarbank.nus.edu.sg/handle/10635/20043Title: Ex ante efficiency implies incentive compatibility
Authors: Sun, Y.; Yannelis, N.C.
Tue, 01 Jan 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/200432008-01-01T00:00:00Z
- Existence of independent random matchinghttps://scholarbank.nus.edu.sg/handle/10635/20044Title: Existence of independent random matching
Authors: Duffie, D.; Sun, Y.
Mon, 01 Jan 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/200442007-01-01T00:00:00Z
- Perfect competition in asymmetric information economies: compatibility of efficiency and incentiveshttps://scholarbank.nus.edu.sg/handle/10635/19962Title: Perfect competition in asymmetric information economies: compatibility of efficiency and incentives
Authors: Sun, Y.; Yannelis, N.C.
Abstract: The idea of perfect competition for an economy with asymmetric information is formalized via an idiosyncratic signal process in which the private signals of almost every individual agent can influence only a negligible group of agents, and the individual agents' relevant signals are essentially pairwise independent conditioned on the true states of nature. Thus, there is no incentive for an individual agent to manipulate her private information. The existence of incentive compatible, ex post Walrasian allocations is shown for such a perfectly competitive asymmetric information economy with or without "common values". Consequently, the conflict between incentive compatibility and Pareto efficiency is resolved exactly, and its asymptotic version is derived for a sequence of large, but finite private information economies.©2006 Elsevier Inc. All rights reserved.
Mon, 01 Jan 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/199622007-01-01T00:00:00Z
- Core, equilibria and incentives in large asymmetric information economieshttps://scholarbank.nus.edu.sg/handle/10635/19990Title: Core, equilibria and incentives in large asymmetric information economies
Authors: Sun, Y.; Yannelis, N.C.
Abstract: We consider a perfectly competitive ex ante economy with a continuum of agents and negligible asymmetric information. For such an economy we recast the basic classical results on the existence of Walrasian equilibrium, core equivalence, and the blocking size of coalitions. Moreover, we examine the incentive compatibility of the ex ante Pareto, core and Walrasian allocations.©2007 Elsevier Inc. All rights reserved.
Mon, 01 Jan 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/199902007-01-01T00:00:00Z
- Large games with a bio-social typologyhttps://scholarbank.nus.edu.sg/handle/10635/52093Title: Large games with a bio-social typology
Authors: Khan, M.A.; Rath, K.P.; Sun, Y.; Yu, H.
Abstract: We present a comprehensive theory of large games in which players have names and determinate social-types and/or biological traits, and identify through four decisive examples, essentially based on a matching-pennies type game, pathologies arising from the use of a Lebesgue interval for player's names. In a sufficiently general context of traits and actions, we address this dissonance by showing a saturated probability space as being a necessary and sufficient name-space for the existence and upper hemi-continuity of pure-strategy Nash equilibria in large games with traits. We illustrate the idealized results by corresponding asymptotic results for an increasing sequence of finite games. © 2013 Elsevier Inc.
Wed, 01 May 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/520932013-05-01T00:00:00Z
- Strategic uncertainty and the ex post Nash property in large gameshttps://scholarbank.nus.edu.sg/handle/10635/123786Title: Strategic uncertainty and the ex post Nash property in large games
Authors: Khan, Mohammedali; Rath, Kali P; Sun, Yeneng; Yu, Haomiao
Thu, 01 Jan 2015 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1237862015-01-01T00:00:00Z
- Dynamic games with (almost) perfect informationhttps://scholarbank.nus.edu.sg/handle/10635/196163Title: Dynamic games with (almost) perfect information
Authors: He, W.; Sun, Y.
Abstract: This paper aims to solve two fundamental problems on finite- or infinite-horizon dynamic games with complete information. Under some mild conditions, we prove the existence of subgame-perfect equilibria and the upper hemicontinuity of equilibrium payoffs in general dynamic games with simultaneous moves (i.e., almost perfect information), which go beyond previous works in the sense that stagewise public randomization and the continuity requirement on the state variables are not needed. For alternating move (i.e., perfect-information) dynamic games with uncertainty, we show the existence of pure-strategy subgame-perfect equilibria as well as the upper hemicontinuity of equilibrium payoffs, extending the earlier results on perfect-information deterministic dynamic games. Copyright © 2020 The Authors.
Wed, 01 Jan 2020 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1961632020-01-01T00:00:00Z
- A remark on the trace of some Riesz operatorshttps://scholarbank.nus.edu.sg/handle/10635/102750Title: A remark on the trace of some Riesz operators
Authors: Sun, Y.
Thu, 01 Dec 1994 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1027501994-12-01T00:00:00Z
- A theory of hyperfinite processes: The complete removal of individual uncertainty via exact LLNhttps://scholarbank.nus.edu.sg/handle/10635/102773Title: A theory of hyperfinite processes: The complete removal of individual uncertainty via exact LLN
Authors: Sun, Y.
Abstract: The aim of this paper is to provide a viable measure-theoretic framework for the study of random phenomena involving a large number of economic entities. The work is based on the fact that processes which are measurable with respect to hyperfinite Loeb product spaces capture the limiting behaviors of triangular arrays of random variables and thus constitute the 'right' class for general stochastic modeling. The primary concern of the paper is to characterize those hyperfinite processes satisfying the exact law of large numbers by using the basic notions of conditional expectation, orthogonality, uncorrelatedness and independence together with some unifying multiplicative properties of random variables. The general structure of the processes is also analyzed via a biorthogonal expansion of the Karhunen-Loéve type and via the representation in terms of the simpler hyperfinite Loeb counting spaces. A universality property for atomless Loeb product spaces is formulated to show the abundance of processes satisfying the law. Generalizations to a hyperfinite number of continuous (or discrete) parameter stochastic processes are considered. The various necessary and sufficient conditions for the validity of the law provide a rather complete understanding about the cancelation of individual risks or uncertainty in general settings. Some explicit asymptotic interpretations are also given.
Fri, 01 May 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1027731998-05-01T00:00:00Z
- A metric on probabilities, and products of Loeb spaceshttps://scholarbank.nus.edu.sg/handle/10635/102678Title: A metric on probabilities, and products of Loeb spaces
Authors: Keisler, H.J.; Sun, Y.
Abstract: Two functions on finitely additive probability spaces that behave well under products are introduced: discrepancy, which measures how close one space comes to extending another, and bi-discrepancy, which is a pseudo-metric on the collection of all spaces on a given set, and a metric on the collection of complete spaces. These are then applied to show that the Loeb space of the internal product of two internal finitely additive probability spaces depends only on the Loeb spaces of the two original internal spaces. Thus the notion of a Loeb product of two Loeb spaces is well defined. The Loeb operation induces an isometry from the nonstandard hull of the space of internal probability spaces on a given set to the space of Loeb spaces on that set, with the metric of bi-discrepancy.
Sun, 01 Feb 2004 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1026782004-02-01T00:00:00Z
- Distributional properties of correspondences on Loeb spaceshttps://scholarbank.nus.edu.sg/handle/10635/103148Title: Distributional properties of correspondences on Loeb spaces
Authors: Sun, Y.
Abstract: We present some regularity properties for the set of distributions induced by the measurable selections of a correspondence over a Loeb space, which include closedness, convexity, compactness, purification, and semicontinuity. We also note that all the properties reported in the main theorems are not satisfied by some correspondences on the unit Lebesgue interval. © 1996 Academic Press, Inc.
Wed, 10 Jul 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031481996-07-10T00:00:00Z
- Asymptotic arbitrage and the APT with or without measure-theoretic structureshttps://scholarbank.nus.edu.sg/handle/10635/102886Title: Asymptotic arbitrage and the APT with or without measure-theoretic structures
Authors: Khan, M.A.; Sun, Y.
Abstract: We present a version of the APT based on an asset index set of an arbitrary infinite cardinality. Under assumptions due to Ross (1976, J. Econ. Theory 13, 341-360) and Chamberlain and Rothschild (1983, Econometrica 51, 1281-1303), we show that, in the absence of gains from asymptotic arbitrage, the square of the deviations of the individual rates of return from a factor-pricing formula sum to a finite number and that this absence, while sufficient, is not necessary for the formula to hold. We relate these results to recent work and explain, in particular, how a version of the APT exhibits several inconsistencies when the index set is the Lebesgue unit interval. Journal of Economic Literature Classification Numbers: G12, C60. © 2001 Academic Press.
Mon, 01 Jan 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028862001-01-01T00:00:00Z
- Hyperfinite law of large numbershttps://scholarbank.nus.edu.sg/handle/10635/103391Title: Hyperfinite law of large numbers
Authors: Sun, Y.
Abstract: The Loeb space construction in nonstandard analysis is applied to the theory of processes to reveal basic phenomena which cannot be treated using classical methods. An asymptotic interpretation of results established here shows that for a triangular array (or a sequence) of random variables, asymptotic uncorrelatedness or asymptotic pairwise independence is necessary and sufficient for the validity of appropriate versions of the law of large numbers. Our intrinsic characterization of almost sure pairwise independence leads to the equivalence of various multiplicative properties of random variables.
Sat, 01 Jun 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1033911996-06-01T00:00:00Z
- Isomorphisms for Convergence Structureshttps://scholarbank.nus.edu.sg/handle/10635/103448Title: Isomorphisms for Convergence Structures
Authors: Sun, Y.N.
Abstract: For a nonatomic Borel probability measure μ on a Polish space X, an isomorphism from (X, μ) to the unit Lebesgue interval ([0, 1]), λ) is constructed such that weak convergence of measures and almost sure convergence of random variables are preserved. Thus the unit interval together with the Lebesgue measure has a sort of universality for some structures involving convergence, which reveals to some extent the mystery of convergence problems on relatively sophisticated spaces (for example, function spaces). Implications of such a result in ergodic theory, probability theory, and probabilistic number theory are discussed. In particular, the study of generic orbits of an ergodic measure preserving transformation on a general Polish space is equivalent to the same problem on the unit Lebesgue interval. The fact that stochastic processes can be regarded as random elements of spaces of functions allows us to claim that the convergence of a sequence of stochastic processes is equivalent to the convergence of some sequence of random variables taking values in the unit interval [0, 1]. Furthermore, the isomorphism studied in this paper preserves uniform distribution of sequences. Thus many results in the abstract theory of uniform distribution can be obtained by transferring the corresponding results in the simplest case of uniform distribution mod 1. © 1995 Academic Press. All rights reserved.
Fri, 01 Dec 1995 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034481995-12-01T00:00:00Z
- Integrals of set-valued functions with a countable rangehttps://scholarbank.nus.edu.sg/handle/10635/103433Title: Integrals of set-valued functions with a countable range
Authors: Khan, M.A.; Sun, Y.
Abstract: Approximate versions, both of Lyapunov-type results on the compactness and convexity of the integral of a correspondence, and Fatou-type results on the preservation of upper semicontinuity by integration, are well known in the context of an infinite dimensional space. We report exact versions of these two types of results for integrals of Banach space valued correspondences with a countable range. We present results on both Bochner and Gel'fand integration.
Fri, 01 Nov 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034331996-11-01T00:00:00Z
- Integration of correspondences on loeb spaceshttps://scholarbank.nus.edu.sg/handle/10635/103434Title: Integration of correspondences on loeb spaces
Authors: Sun, Y.
Abstract: We study the Bochner and Gel'fand integration of Danach space valued correspondences on a general Loeb space. Though it is well known that the Lyapunov type result on the compactness and convexity of the integral of a correspondence and the Fatou type result on the preservation of upper semicontinuity by integration are in general not valid in the setting of an infinite dimensional space, we show that exact versions of these two results hold in the case we study. We also note that our results on a hyperfinite Loeb space capture the nature of the corresponding asymptotic results for the large finite case; but the unit Lebesgue interval fails to provide such a framework. © 1997 American Mathematical Society.
Wed, 01 Jan 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034341997-01-01T00:00:00Z
- Joint measurability and the one-way Fubini property for a continuum of independent random variableshttps://scholarbank.nus.edu.sg/handle/10635/103457Title: Joint measurability and the one-way Fubini property for a continuum of independent random variables
Authors: Hammond, P.J.; Sun, Y.
Abstract: As is well known, a continuous parameter process with mutually independent random variables is not jointly measurable in the usual sense. This paper proposes an extension of the usual product measure-theoretic framework, using a natural "one-way Fubini" property. When the random variables are independent even in a very weak sense, this property guarantees joint measurability and defines a unique measure on a suitable minimal σr-algebra. However, a further extension to satisfy the usual (two-way) Fubini property, as in the case of Loeb product measures, may not be possible in general. Some applications are also given. © 2005 American Mathematical Society.
Wed, 01 Mar 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034572006-03-01T00:00:00Z
- On the theory of vector valued Loeb measures and integrationhttps://scholarbank.nus.edu.sg/handle/10635/103844Title: On the theory of vector valued Loeb measures and integration
Authors: Sun, Y.
Abstract: Ideas and techniques from nonstandard theories of measure spaces and Banach spaces are brought together to give a further study of nonstandard vector measures. An integration theory of real valued functions with respect to Banach space valued measures is developed. Geometric properties of nonstandard hulls are also investigated. Some applications are given. © 1992.
Sun, 01 Mar 1992 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1038441992-03-01T00:00:00Z
- Why saturated probability spaces are necessaryhttps://scholarbank.nus.edu.sg/handle/10635/114178Title: Why saturated probability spaces are necessary
Authors: Keisler, H.J.; Sun, Y.
Abstract: An atomless probability space (Ω, A, P) is said to have the saturation property for a probability measure μ on a product of Polish spaces X × Y if for every random element f of X whose law is margX (μ), there is a random element g of Y such that the law of (f, g) is μ. (Ω, A, P) is said to be saturated if it has the saturation property for every such μ. We show each of a number of desirable properties holds for every saturated probability space and fails for every non-saturated probability space. These include distributional properties of correspondences, such as convexity, closedness, compactness and preservation of upper semi-continuity, and the existence of pure strategy equilibria in games with many players. We also show that any probability space which has the saturation property for just one "good enough" measure, or which satisfies just one "good enough" instance of the desirable properties, must already be saturated. Our underlying themes are: (1) There are many desirable properties that hold for all saturated probability spaces but fail everywhere else; (2) Any probability space that out-performs the Lebesgue unit interval in almost any way at all is already saturated. © 2009 Elsevier Inc. All rights reserved.
Sat, 01 Aug 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1141782009-08-01T00:00:00Z
- On the existence of pure strategy equilibria in games with a continuum of playershttps://scholarbank.nus.edu.sg/handle/10635/103796Title: On the existence of pure strategy equilibria in games with a continuum of players
Authors: Khan, M.A.; Rath, K.P.; Sun, Y.
Abstract: We present results on the existence of pure strategy Nash equilibria in nonatomic games. We also show by means of counterexamples that the stringent conditions on the cardinality of action sets cannot be relaxed, and thus resolve questions which have remained open since Schmeidler's 1973 paper.Journal of Economic LiteratureClassification Number: C72. © 1997 Academic Press.
Mon, 01 Sep 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1037961997-09-01T00:00:00Z
- Martingale property of empirical processeshttps://scholarbank.nus.edu.sg/handle/10635/114167Title: Martingale property of empirical processes
Authors: Albeverio, S.; Sun, Y.; Wu, J.-L.
Abstract: It is shown that for a large collection of independent martingales, the martingale property is preserved on the empirical processes. Under the assumptions of independence and identical finite-dimensional distributions, it is proved that a large collection of stochastic processes are martingales essentially if and only if the empirical processes are also martingales. These two results have implications on the testability of the martingale property in scientific modeling. Extensions to submartingales and supermartingales are given. © 2006 American Mathematical Society.
Thu, 01 Feb 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1141672007-02-01T00:00:00Z
- The exact law of large numbers for independent random matchinghttps://scholarbank.nus.edu.sg/handle/10635/114175Title: The exact law of large numbers for independent random matching
Authors: Duffie, D.; Sun, Y.
Abstract: This paper provides a mathematical foundation for independent random matching of a large population, as widely used in the economics literature. We consider both static and dynamic systems with random mutation, partial matching arising from search, and type changes induced by matching. Under independence assumptions at each randomization step, we show that there is an almost-sure constant cross-sectional distribution of types in a large population, and moreover that the multi-period cross-sectional distribution of types is deterministic and evolves according to the transition matrices of the type process of a given agent. We also show the existence of a joint agent-probability space, and randomized mutation, partial matching and match-induced type-changing functions that satisfy appropriate independence conditions, where the agent space is an extension of the classical Lebesgue unit interval. © 2012.
Tue, 01 May 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1141752012-05-01T00:00:00Z
- Existence, incentive compatibility and efficiency of the rational expectations equilibriumhttps://scholarbank.nus.edu.sg/handle/10635/114189Title: Existence, incentive compatibility and efficiency of the rational expectations equilibrium
Authors: Sun, Y.; Wu, L.; Yannelis, N.C.
Abstract: The rational expectations equilibrium (REE), as introduced in Radner (1979) in a general equilibrium setting à la Arrow-Debreu-McKenzie, often fails to have desirable properties such as universal existence, incentive compatibility and efficiency. We resolve those problems by providing a new model which makes the REE a desirable solution concept. In particular, we consider an asymmetric information economy with a continuum of agents whose private signals are independent conditioned on the macro states of nature. For such an economy, agents are allowed to augment their private information by the available public signals. We prove the existence, incentive compatibility and efficiency for this new REE concept. © 2012 Elsevier Inc..
Sat, 01 Sep 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1141892012-09-01T00:00:00Z
- Extremal structures and symmetric equilibria with countable actionshttps://scholarbank.nus.edu.sg/handle/10635/103262Title: Extremal structures and symmetric equilibria with countable actions
Authors: Khan, M.A.; Sun, Y.
Abstract: In this paper we show that a Cournot-Nash equilibrium distribution τ of an atomless anonymous game with countable actions is a symmetric equilibrium if and only if it is an extreme point of the set of all Cournot-Nash equilibrium distributions of the game with the same marginals as τ. This characterization allows us to show, as an application of the Krein-Milman theorem, that any particular Cournot-Nash equilibrium of such a game can be reallocated such that players with the same characteristics always take the same action, which is to say that it can be symmetrized. As a consequence of the usual result on the existence of distributional equilibria, we also obtain the existence of symmetric equilibria for the games under consideration. © 1995.
Sun, 01 Jan 1995 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1032621995-01-01T00:00:00Z
- On a private information game without pure strategy equilibriahttps://scholarbank.nus.edu.sg/handle/10635/103677Title: On a private information game without pure strategy equilibria
Authors: Khan, M.A.; Rath, K.P.; Sun, Y.
Abstract: We present an example of a two-person game of private information in which there is no equilibrium in pure strategies. Our example satisfies all the hypotheses of the existence theorems present in the literature on the subject of pure strategy equilibria, except for the fact that the action set of each player is given by the interval [-1,1]. As such, it illustrates the limitations that pertain to the purification of equilibria in a standard setting.
Thu, 01 Apr 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036771999-04-01T00:00:00Z
- Nonatomic games on Loeb spaceshttps://scholarbank.nus.edu.sg/handle/10635/103609Title: Nonatomic games on Loeb spaces
Authors: Khan, M.A.; Sun, Y.
Abstract: In the setting of noncooperative game theory, strategic negligibility of individual agents, or diffuseness of information, has been modeled as a nonatomic measure space, typically the unit interval endowed with Lebesgue measure. However, recent work has shown that with uncountable action sets, for example the unit interval, there do not exist pure-strategy Nash equilibria in such nonatomic games. In this brief announcement, we show that there is a perfectly satisfactory existence theory for nonatomic games provided this nonatomicity is formulated on the basis of a particular class of measure spaces, hyperfinite Loeb spaces. We also emphasize other desirable properties of games on hyperfinite Loeb spaces, and present a synthetic treatment, embracing both large games as well as those with incomplete information.
Tue, 24 Dec 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036091996-12-24T00:00:00Z
- Non-cooperative games on hyperfinite Loeb spaceshttps://scholarbank.nus.edu.sg/handle/10635/103611Title: Non-cooperative games on hyperfinite Loeb spaces
Authors: Khan, M.A.; Sun, Y.
Abstract: We present a particular class of measure spaces, hyperfinite Loeb spaces, as a model of situations where individual players are strategically negligible, as in large non-anonymous games, or where information is diffused, as in games with imperfect information. We present results on the existence of Nash equilibria in both kinds of games. Our results cover the case when the action sets are taken to be the unit interval, results now known to be false when they are based on more familiar measure spaces such as the Lebesgue unit interval. We also emphasize three criteria for the modelling of such game-theoretic situations - asymptotic implementability, homogeneity and measurability - and argue for games on hyperfinite Loeb spaces on the basis of these criteria. In particular, we show through explicit examples that a sequence of finite games with an increasing number of players or sample points cannot always be represented by a limit game on a Lebesgue space, and even when it can be so represented, the limit of an existing approximate equilibrium may disappear in the limit game. Thus, games on hyperfinite Loeb spaces constitute the 'right' model even if one is primarily interested in capturing the asymptotic nature of large but finite game-theoretic phenomena.
Sat, 01 May 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036111999-05-01T00:00:00Z
- On nonstandard product measure spaceshttps://scholarbank.nus.edu.sg/handle/10635/103729Title: On nonstandard product measure spaces
Authors: Berger, J.; Osswald, H.; Sun, Y.; Wu, J.-L.
Abstract: The aim of this paper is to investigate systematically the relationship between the two different types of product probability spaces based on the Loeb space construction. For any two atomless Loeb spaces, it is shown that for fixed r < s in [0, 1] there exists an increasing sequence (At)r
Fri, 01 Mar 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1037292002-03-01T00:00:00Z
- Purification and saturationhttps://scholarbank.nus.edu.sg/handle/10635/114199Title: Purification and saturation
Authors: Loeb, P.; Sun, Y.
Abstract: This paper illustrates the general technique established in 1984 by Hoover and Keisler for extending certain types of results from atomless Loeb measure spaces to measure spaces that we shall call "nowhere countably generated". The Hoover-Keisler technique is applied here to further extend the authors' 2006 generalization of a theorem of Dvoretzky, Wald and Wolfowitz on the purification of measure-valued maps. The authors' 2006 result was first extended to these more general spaces by K. Podczeck in 2007; he used new results in functional analysis produced for that purpose. This paper demonstrates that, in general, such extensions follow from the Hoover-Keisler technique. Moreover, adaptations of counterexamples from earlier papers show that the extension obtained here holds only for nowhere countably generated spaces. © 2009 American Mathematical Society.
Sat, 01 Aug 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1141992009-08-01T00:00:00Z
- The almost equivalence of pairwise and mutual independence and the duality with exchangeabilityhttps://scholarbank.nus.edu.sg/handle/10635/104249Title: The almost equivalence of pairwise and mutual independence and the duality with exchangeability
Authors: Sun, Y.
Sun, 01 Nov 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042491998-11-01T00:00:00Z
- Exact arbitrage, well-diversified portfolios and asset pricing in large marketshttps://scholarbank.nus.edu.sg/handle/10635/103218Title: Exact arbitrage, well-diversified portfolios and asset pricing in large markets
Authors: Khan, M.A.; Sun, Y.
Abstract: For a market with an atomless continuum of assets, we formulate the intuitive idea of a "well-diversified" portfolio, and present a notion of "exact arbitrage", strictly weaker than the more conventional notion of "asymptotic arbitrage", and necessary and sufficient for the validity of an APT pricing formula. Our formula involves "essential" risk, one based on a specific index portfolio constructed from factors and factor loadings that are endogenously extracted to satisfy an optimality property involving a finite number of factors. We illustrate how our results can be translated to markets with a large but finite number of assets. © 2003 Elsevier Science (USA). All rights reserved.
Sun, 01 Jun 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1032182003-06-01T00:00:00Z
- Exact arbitrage and portfolio analysis in large asset marketshttps://scholarbank.nus.edu.sg/handle/10635/103217Title: Exact arbitrage and portfolio analysis in large asset markets
Authors: Ali Khan, M.; Sun, Y.
Abstract: We provide a detailed portfolio analysis for a financial market with an atomless continuum of assets. In the context of an exact arbitrage pricing theory (EAPT), we go beyond the characterization of the existence of important portfolios (normalized riskless, mean, cost, factor and mean-variance efficient portfolios) to furnish exact portfolio compositions in terms of explicit portfolio weights. Such an analysis has not been furnished before in the context of the asymptotic arbitrage pricing theory (APT). We also characterize conditions under which a mean-variance efficient portfolio is a benchmark portfolio used in the EAPT to proxy essential risk. We illustrate our results with several examples of specific financial markets.
Wed, 01 Oct 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1032172003-10-01T00:00:00Z
- Weak measurability and characterizations of riskhttps://scholarbank.nus.edu.sg/handle/10635/104467Title: Weak measurability and characterizations of risk
Authors: Ali Khan, M.; Sun, Y.
Abstract: In the context of a continuum of random variables, arising, for example, as rates of return in financial markets with a continuum of assets, or as individual responses in games with a continuum of players, an important economic issue is to show how idiosyncratic risk can be removed through some device of aggregation or diversification when such risk is explicitly introduced into the model. In this paper, we use recent work of Al-Najjar [1] as a general backdrop to provide a review of the basic issues involved when the continuum is formulated as the Lebesgue interval. We present two examples to argue that the fundamental problem of the non-measurability of sample functions, originally identified by Doob, and further elaborated by Feldman, Gilles and Judd in the economic literature, simply cannot be bypassed by reinterpretations of standard results. We also provide an equivalence result in the spirit of Al-Najjar's efforts; but argue that this elementary result does not go beyond the standard law of large numbers for a sequence of real-valued iid random variables, and as such, is incapable of yielding anything of substantive economic interest beyond this law.
Fri, 01 Jan 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1044671999-01-01T00:00:00Z
- Uncorrelatedness and orthogonality for vector-valued processeshttps://scholarbank.nus.edu.sg/handle/10635/104420Title: Uncorrelatedness and orthogonality for vector-valued processes
Authors: Loeb, P.A.; Osswald, H.; Sun, Y.; Zhang, Z.
Abstract: For a square integrable vector-valued process f on the Loeb product space, it is shown that vector orthogonality is almost equivalent to componentwise scalar orthogonality. Various characterizations of almost sure uncorrelatedness for f are presented. The process f is also related to multilinear forms on the target Hilbert space. Finally, a general structure result for f involving the biorthogonal representation for the conditional expectation of f with respect to the usual product σ-algebra is presented.
Sun, 01 Aug 2004 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1044202004-08-01T00:00:00Z
- The Dvoretzky-Wald-Wolfowitz theorem and purification in atomless finite-action gameshttps://scholarbank.nus.edu.sg/handle/10635/104283Title: The Dvoretzky-Wald-Wolfowitz theorem and purification in atomless finite-action games
Authors: Khan, M.A.; Rath, K.P.; Sun, Y.
Abstract: In 1951, Dvoretzky, Wald and Wolfowitz (henceforth DWW) showed that corresponding to any mixed strategy into a finite action space, there exists a pure-strategy with an identical integral with respect to a finite set of atomless measures. DWW used their theorem for purification: the elimination of randomness in statistical decision procedures and in zero-sum two-person games. In this short essay, we apply a consequence of their theorem to a finite-action setting of finite games with incomplete and private information, as well as to that of large games. In addition to simplified proofs and conceptual clarifications, the unification of results offered here re-emphasizes the close connection between statistical decision theory and the theory of games.
Sat, 01 Apr 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042832006-04-01T00:00:00Z
- The capital-asset-pricing model and arbitrage pricing theory: A unificationhttps://scholarbank.nus.edu.sg/handle/10635/104259Title: The capital-asset-pricing model and arbitrage pricing theory: A unification
Authors: Khan, M.A.; Sun, Y.
Abstract: We present a model of a financial market in which naive diversification, based simply on portfolio size and obtained as a consequence of the law of large numbers, is distinguished from efficient diversification, based on mean-variance analysis. This distinction yields a valuation formula involving only the essential risk embodied in an asset's return, where the overall risk can be decomposed into a systematic and an unsystematic part, as in the arbitrage pricing theory; and the systematic component further decomposed into an essential and an inessential part, as in the capital-asset-pricing model. The two theories are thus unified, and their individual asset-pricing formulas shown to be equivalent to the pervasive economic principle of no arbitrage. The factors in the model are endogenously chosen by a procedure analogous to the Karhunen-Loeve expansion of continuous time stochastic processes; it has an optimality property justifying the use of a relatively small number of them to describe the underlying correlational structures. Our idealized limit model is based on a continuum of assets indexed by a hyperfinite Loeb measure space, and it is asymptotically implementable in a setting with a large but finite number of assets. Because the difficulties in the formulation of the law of large numbers with a standard continuum of random variables are well known, the model uncovers some basic phenomena not amenable to classical methods, and whose approximate counterparts are not already, or even readily, apparent in the asymptotic setting.
Tue, 15 Apr 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042591997-04-15T00:00:00Z
- The complete removal of individual uncertainty: Multiple optimal choices and random exchange economieshttps://scholarbank.nus.edu.sg/handle/10635/104267Title: The complete removal of individual uncertainty: Multiple optimal choices and random exchange economies
Authors: Sun, Y.
Abstract: The aim of this paper is to develop some measure-theoretic methods for the study of large economic systems with individual-specific randomness and multiple optimal actions. In particular, for a suitably formulated continuum of correspondences, an exact version of the law of large numbers in distribution is characterized in terms of almost independence, which leads to several other versions of the law of large numbers in terms of integration of correspondences. Widespread correlation due to multiple optimal actions is also shown to be removable via a redistribution. These results allow the complete removal of individual risks or uncertainty in economic models where non-unique best choices are inevitable. Applications are illustrated through establishing stochastic consistency in general equilibrium models with idiosyncratic shocks in endowments and preferences. In particular, the existence of "global" solutions preserving microscopic independence structure is shown in terms of competitive equilibria for the cases of divisible and indivisible goods as well as in terms of core for a case with indivisible goods where a competitive equilibrium may not exist. An important feature of the idealized equilibrium models considered here is that standard results on measure-theoretic economies are now directly applicable to the case of random economies. Some asymptotic interpretation of the results are also discussed. It is also pointed out that the usual unit interval [0,1] can be used as an index set in our setting, provided that it is endowed together with some sample space a suitable larger measure structure.
Mon, 01 Nov 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042671999-11-01T00:00:00Z
- The essential equivalence of pairwise and mutual conditional independencehttps://scholarbank.nus.edu.sg/handle/10635/104287Title: The essential equivalence of pairwise and mutual conditional independence
Authors: Hammond, P.J.; Sun, Y.
Abstract: For a large collection of random variables, pairwise conditional independence and mutual conditional independence are shown to be essentially equivalent - i.e., equivalent to up to null sets. Unlike in the finite setting, a large collection of random variables remains essentially conditionally independent under further conditioning. The essential equivalence of pairwise and multiple versions of exchangeability also follows as a corollary. Our result relies on an iterated extension of Bledsoe and Morse's completion of the product of two measure spaces.
Sat, 01 Jul 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042872006-07-01T00:00:00Z
- The nonexistence of symmetric equilibria in anonymous games with compact action spaceshttps://scholarbank.nus.edu.sg/handle/10635/104321Title: The nonexistence of symmetric equilibria in anonymous games with compact action spaces
Authors: Rath, K.P.; Yeneng, S.; Shinji, Y.
Abstract: In an anonymous game the payoff of a player depends upon the player's own action and the action distribution of all the players. If the game is atomless and the set of actions is finite, or countably infinite and compact, then there is a symmetric equilibrium distribution. Furthermore, every equilibrium distribution can be symmetrized. This note provides three examples to the effect that the conditions in these results cannot be relaxed. The first example is a game with atoms and finite actions which has no symmetric equilibrium distribution. In the second example, the game is atomless, the action space is uncountable and not every equilibrium distribution can be symmetrized. The third example shows that a symmetric equilibrium distribution may not exist in an atomless game with the interval [ - 1, 1]as the action space. A general construction then exhibits that given any uncountable compact metric space, there is an atomless game over that space which has no symmetric equilibrium distribution. A sufficient condition for an equilibrium distribution to be symmetrized is also given. © 1995.
Sun, 01 Jan 1995 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043211995-01-01T00:00:00Z
- The exact law of large numbers via Fubini extension and characterization of insurable riskshttps://scholarbank.nus.edu.sg/handle/10635/104288Title: The exact law of large numbers via Fubini extension and characterization of insurable risks
Authors: Sun, Y.
Abstract: A Fubini extension is formally introduced as a probability space that extends the usual product probability space and retains the Fubini property. Simple measure-theoretic methods are applied to this framework to obtain various versions of the exact law of large numbers and their converses for a continuum of random variables or stochastic processes. A model for a large economy with individual risks is developed; and insurable risks are characterized by essential pairwise independence. The usual continuum product based on the Kolmogorov construction together with the Lebesgue measure as well as the usual finitely additive measure-theoretic framework is shown further to be not suitable for modeling individual risks. Measurable processes with essentially pairwise independent random variables that have any given variety of distributions exist in a rich product probability space that can also be constructed by extending the usual continuum product. © 2005 Elsevier Inc. All rights reserved.
Sun, 01 Jan 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042882006-01-01T00:00:00Z
- Pure strategies in games with private informationhttps://scholarbank.nus.edu.sg/handle/10635/103988Title: Pure strategies in games with private information
Authors: Khan, M.A.; Sun, Y.
Abstract: Pure strategy equilibria of finite player games with informational constraints have been discussed under the assumptions of finite actions, and of independence and diffuseness of information. We present a mathematical framework, based on the notion of a distribution of a correspondence, that enables us to handle the case of countably infinite actions. In this context, we extend the Radner-Rosenthal theorems on the purification of a mixed-strategy equilibrium, and present a direct proof, as well as a generalized version of Schmeidler's large games theorem, on the existence of a pure strategy equilibrium. Our mathematical results pertain to the set of distributions induced by the measurable selections of a correspondence with a countable range, and rely on the Bollobás-Varopoulos extension of the marriage lemma. © 1995.
Sun, 01 Jan 1995 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1039881995-01-01T00:00:00Z
- Purification of measure-valued mapshttps://scholarbank.nus.edu.sg/handle/10635/103990Title: Purification of measure-valued maps
Authors: Loeb, P.; Sun, Y.
Abstract: Given a measurable mapping f from a nonatomic Loeb probability space (T, Τ, P) to the space of Borel probability measures on a compact metric space A, we show the existence of a measurable mapping g from (T, Τ, P) to A itself such that f and g yield the same values for the integrals associated with a countable class of functions on T × A. A corollary generalizes the classical result of Dvoretzky-Wald-Wolfowitz on purification of measure-valued maps with respect to a finite target space; the generalization holds when the domain is a nonatomic, vector-valued Loeb measure space and the target is a complete, separable metric space. A counterexample shows that the generalized result fails even for simple cases when the restriction of Loeb measures is removed. As an application, we obtain a strong purification for every mixed strategy profile in finite-player games with compact action spaces and diffuse and conditionally independent information. ©2006 University of Illinois.
Fri, 01 Sep 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1039902006-09-01T00:00:00Z
- Some Properties of Uniformly Distributed Sequenceshttps://scholarbank.nus.edu.sg/handle/10635/104172Title: Some Properties of Uniformly Distributed Sequences
Authors: Sun, Y.N.
Abstract: We show that if a transformation T on the unit interval is surjective, piecewise continuously differentiable, and uniform distribution preserving, then every u.d. sequence mod 1 is the image of some other u.d. sequence mod 1 induced by T. As a consequence, we obtain that every ud. sequence mod 1 can be decomposed into finitely many sequences with given asymptotic densities and asymptotic distribution functions. © 1993 Academic Press. All rights reserved.
Thu, 01 Jul 1993 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1041721993-07-01T00:00:00Z