Saturation and economic theory

SATURATION AND ECONOMIC THEORY ZHANG YONGCHAO (M.S., University of Science and Technology of China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2011 Contents Acknowledgement vii Summary xi Introduction 1.1 Basic Ideas of Keisler and Sun (2009) . . . . . . . . . . . . . . . . . . . . 1.2 Exact Law of Large Numbers (ELLN) . . . . . . . . . . . . . . . . . . . . 1.3 Purification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Private Information Games . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Lebesgue Extensions and Correspondences . . . . . . . . . . . . . . . . . . 14 1.6 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Preliminaries 19 2.1 Saturated Probability Space . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Lebesgue Extension 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii iv Contents 2.3 2.2.1 A General Technique . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.2 Variations of Lebesgue Extension . . . . . . . . . . . . . . . . . . . 24 A Useful Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Rich Fubini Extension with Lebesgue Extension Agent Spaces 29 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Basics on ELLN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4 Discussions: Saturation and Rich Fubini Extensions . . . . . . . . . . . . 36 3.5 Proof of Theorem 3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Purification 47 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.3 Proof of Theorem 4.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 The Private Information Games 57 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.3.1 Khan et al. (1999) Revisited . . . . . . . . . . . . . . . . . . . . . . 66 5.3.2 Necessity of Saturation . . . . . . . . . . . . . . . . . . . . . . . . 71 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.4 Lebesgue Extension and Correspondences 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 77 Contents 6.2 6.3 6.4 v Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.2.1 Claims 6.1.1 to 6.1.3 Reconsidered . . . . . . . . . . . . . . . . . . 81 6.2.2 General Properties on the Debreu Correspondence . . . . . . . . . 84 Lebesgue Extensions and Correspondences . . . . . . . . . . . . . . . . . . 88 6.3.1 Distribution on (I, I1 , λ1 ) . . . . . . . . . . . . . . . . . . . . . . . 89 6.3.2 Applications 93 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Concluding Remarks 107 Acknowledgement Many people have played important roles in the past five years of my life. I am particularly indebted to my supervisors, Prof. Sun Yeneng, Luo Xiao from NUS and M. Ali Khan (unofficial) from the Johns Hopkins University. To me, they are teachers, advisors, co-authors and great friends. Without their guidance, encouragement and help, what I have achieved would not be possible. Firstly, I would like to tell the story of Prof. Sun. I came across Prof. Sun at Hefei, China in 2006. After a ten-minute meeting with him, I made my decision, probably the most important one in my career: to study Economics from nothing. In 2007 and 2008, I worked happily with him on the paper Sun and Zhang (2009) (reported in Chapter 3). The completion of this paper gave me huge confidence to carry out research. Later, in the summer of 2009, when we wandered around the old town of Bielefield, we talked about the significance of saturated probability spaces in economic theory. Due to this talk, my research interest began to shift to the saturated probability space, which is the main subject of this thesis. From then, I found an entrance to some highway of my research. Prof. Sun supervised me step by step to complete, polish, and publish vii viii Acknowledgement the paper Wang and Zhang (2010) (reported in Chapter 4). Moreover, he introduced his co-author Prof. M. Ali Khan to me which resulted in the papers Khan and Zhang (2010a,b) (reported in Chapters and 6), and consistently, he provided golden advice and suggestion to improve the results therein. Besides, Prof. Sun never pushes me and allows me get things done at my own pace. He is always nice to me and glad to help when I turn to him. Special thanks to Prof. Sun for hiring me as his RA this semester. In 2008, I sat in a class of Prof. Luo and got interested in Game Theory. Since then, we had frequent discussions on many interesting topics and problems in game theory, be it in the classrooms, on the way back to Kent Vale after classes, in his office or in lunches, etc. His knowledge in Game theory broadens my eyesight, deepens my understanding and shapes my views for this field. The story with Ali is a little different. He asked me three questions about the construction reported in Chapter when visiting Singapore in 2009. Several months later, after getting some partial solutions, we began our frequent informative and insightful e-correspondence and face-to-face discussion during his short stay in Singapore and Beijing in 2010. This wonderful interaction resulted in two papers as mentioned above, Khan and Zhang (2010a,b) (reported in Chapter and 6). I still enjoy the whole amazing procedure to bring up these two papers from his initial simple questions. Besides, I also appreciate that Ali have shared his life experience and wisdom about research, work, family and life. In addition, I am also grateful to Prof. Sun, Luo and Khan for helping me to get my first job in Shanghai. Acknowledgement ix I also would like to take this opportunity to acknowledge Prof. Luo Xiao and Chen Yi-Chun for organizing the Theory Seminar of Economics Department. It is a regular brain storm. I have benefited and learnt a lot from Yi-Chun, Miao Bin, Wang Ben, Sun Xiang, He Wei, Qian Neng, Zhang Zilong and Qu Chen etc. Many thanks also go to Yu Haomiao, Zhang Zhixiang, Kali P. Rath and Nicholas C. Yannelis for their encouragement and help these years. In addition, some friends have contributed to my research in an indirect way, including but not limited to, Li Lu, Nian Rui, Wu Lei, Xu Ying, Yu Juan and Zhang Wenbin. Finally, my deepest gratitude and thanks are due to my parents and brothers for their unconditional love and whole hearted support. My thankfulness also goes to my wife, Zhang Ting. Her love, understanding, support and encouragement are a great source of strength for me. Zhang Yongchao April 2011 6.4 Discussions 105 probability space. Depending on the complexity of the function, even a single extension may suffice, as is the case with Claim above. This latter point attains especial salience in the context of the theory of large games. To be sure, in keeping with the necessity results in Keisler and Sun (2009), there exist games without Nash equilibria on an space extended a finite number of times, but such games may be entirely without interest in terms of the particular economic or game-theoretic phenomena being modelled. To repeat the point, as far as the specific model is at issue, a simple countably-generated extension of the space of players may suffice. We focussed mainly above on the distributions for the particular correspondence Ψ. Analogously, similar results can be established for the integral of the other particular correspondence Π involved in Claim and detailed in Equation (6.1). Define π dλ : π is an I-measurable selection of Π . Π dλ = I I We can show that I Π dλ is convex, in contrast, I Π dη is not convex; here Propo- sition 6.2.4 is a special case of this claim. With the sequence of countably generated Lebesgue extensions (I, In , λn ) as well as hn (see Section 5) at hands, we can define new correspondences Πn to be Πn = Π ◦ hn−1 , and establish that while I I Πn dλn−1 is not convex, Πn dλn is convex; and so on ad infinitum. Finally, an additional example based on a weakly compact action set in an infinite-dimensional separable Banach space is considered in Khan et al. (1997), and we leave it to the reader to use our resolution of Claim as detailed here and carry through the arguments presented in Section 6.3.2 to deal with that example. This being said, we should like to conclude this paper with two open questions. The first concerns the correspondence in Claim 6.1.1 about which we have been totally silent in the context of the necessity theory. This was inevitable in light of the fact that the two operations of distribution and integration over the entire domain space, as considered in 106 Chapter 6. Lebesgue Extension and Correspondences this paper, are not really at issue in Claim 6.1.1. What is important there is the existence of a selection for which the integral is null for each interval [0, t], t ∈ [0, 1]. Whereas such a selection can surely be found for a Loeb counting space, it is not at all clear as to the shape of this requirement for a general saturated space. In particular, there is no necessity theory here to appeal to and to try and circumvent. One suspects that the issue hinges on the homogeneity property, one that is discussed in Fajardo and Keisler (2002, Section 2D), and in the context of large non-anonymous games, in Khan and Sun (1996), but a clear formulation and possible resolution would have to await future work. Such work would presumably have to build on the reformulated theory of large games presented in Khan et al. (2005) that is constructed around the precise correspondence considered in Claim 6.1.1. Chapter Concluding Remarks The chapter concludes its comprehensive overview of the results that have been obtained by suggesting several open problems. The ELLN approach In main result, Theorem 4.2.1 in Chapter 4, compared with Podczeck (2009) and Loeb and Sun (2009), we adopt the ELLN approach. As mentioned before, one advantage of this ELLN approach is that one can simultaneously obtain many required purification mappings. In particular, these purification mappings can be indexed by a full subset in an atomless probability space. In the context of private information games, starting from one mixed strategy equilibrium of the game (the existence of mixed strategy equilibrium is well-known), then by an appeal of our purification theorem via ELLN, one can obtain many pure strategy equilibria for such a game. Here, it is of interest to investigate the problem related to the cardinality of the pure strategy equilibria in such games. One relevant line of literature to this cardinality problem in private information games might be the study of ex post Nash equilibrium of a large game with idiosyncratic uncertainty, which is already considered in Theorem of Khan and Sun (2002, p. 1792). Further results on ex post Nash equilibrium in large games are established in Khan et al. (2005) 107 108 Chapter 7. Concluding Remarks and Sun (2007a). Independence Assumption in Private Information Games It is worthwhile to mention that for game theoretical models as in Milgrom and Weber (1985) and Radner and Rosenthal (1982), and the benchmark model in Chapter as well, the (conditionally) independence condition for the information structure (Assumption 5.2.1) plays an important role. In particular, it follows from this independence condition that each player’s expected payoff depends on the others’ strategies only through the induced distributions on their action spaces, then one can apply Theorem 4.2.1 to obtain the strong purifications for any mixed strategy profile. However, one may criticize that this independence is too ideal an assumption for games with private information, correlation among private information structure conditional on some common known or publicly announced state may be common from a viewpoint of applications. In Yannelis and Rustichini (1991), a model of Bayesian game without such an independence condition was introduced where the state space is a probability space and each player’s information structure is a general measurable partition of this state space.1 A positive result on the existence of pure strategy equilibria was established in Yannelis and Rustichini (1991, Theorem 5.2) by modeling the state space as an atomless probability space and by modeling the set of strategies of each player via a Rℓ -valued correspondence which is convex-valued and integrably bounded. However, by modeling the set of strategies for each player via a more general Banach-valued correspondence, one can only obtain an approximate pure strategy Bayesian equilibrium. It is intuitively clear that by formalizing the state space as a saturated probability space, and appealing to the results in Podczeck (2008) and Sun and Yannelis (2008), one can obtain a similar positive result as in the general setting of Yannelis and Rustichini (1991). We hope to See Yannelis and Rustichini (1991, Section 6) for the comparison between this model and those in Milgrom and Weber (1985) and Radner and Rosenthal (1982), see also Yannelis (2009) for a Bayesian game theoretic model with a continuum of players. 109 take the details up in a subsequent paper. Applications in Cooperative Game Theory The theory of the value, as shaped by Aumann and Shapley (1974), and relying on a construction of Kannai (1966), hinges crucially on the set of agents being modeled by probability spaces isomorphic to the the Lebesgue unit interval, which is called Standardness Assumption (see Aumann and Shapley 1974, p.13, (2.1)). And, some problems may occur with such a standard assumption; see Aumann and Shapley (1974, Chapter 13). In one influential paper Aumann (1975), where the set of agents by Lebesgue unit interval or some probability isomorphic, Aumann showed the equivalence between the value allocation and the competitive allocation in a model of large economy. In Aumann (1975), he implicitly used the idea of nonstandard analysis in the sense that each player in the large economy occupies ǫ weight in the economy, where ǫ represents a number which is strictly smaller than any positive number. In a follow up paper, Brown and Loeb (1977) re-establish the equivalence between value and competitive allocations, while the set of agents is modeled by a hyperfinite set, or a hyperfinite Loeb counting space. 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SATURATION AND ECONOMIC THEORY ZHANG YONGCHAO NATIONAL UNIVERSITY OF SINGAPORE 2011 SATURATION AND ECONOMIC THEORY ZHANG YONGCHAO 2011 [...]... in Economic Theory can be viewed as illustrations of this general technique For example, Podczeck (2009) 4 See also Khan and Sun (2002) for a systematical discussion about the results mentioned above on the theory of large games 1.1 Basic Ideas of Keisler and Sun (2009) 5 and Loeb and Sun (2009) generalize the results in Loeb and Sun (2006) about the purification theorem; Podczeck (2008) and Sun and. .. thesis, we report on fruitful applications of the theory of saturated probability spaces developed by Hoover and Keisler (1984) into economic theory, as poineered by Keisler and Sun (2009) 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 are presented in Sun (2006) to... probability space, and the existence of pure strategy equilibria in games with many players Followed Keisler and Sun (2009), saturated probability spaces have found applications in many branches in Economic Theory For recent advancement, see Khan (2010), Khan et al (2005), Khan and Zhang (2010a,b), Loeb and Sun (2009), Noguchi (2009), Podczeck (2008, 2009), Sun and Yannelis (2008), Wang and Zhang (2010)... this, and for some recent applications in Economic Theory of ELLN, see Duffie et al (2005, 2007), Duffie and Sun (2007), Lagos and Rocheteau (2007), McLean and Postlewaite (2002, 5 See Sun (2006, Section 3) for details 1.2 Exact Law of Large Numbers (ELLN) 2005), Sun and Yannelis (2008), Weill (2007) However, unlike the case of a continuum of agents in a deterministic model, it is well known that an economic. .. players as formulated in Milgrom and Weber (1985) and Radner and Rosenthal (1982) From the methodology point of view, this modeling is parallel to the modeling of the agent spaces in the theory of large games as mentioned in Section 1.1 See also Khan and Sun (2002, Section 4) for the relationship between the private information games and large non-anonymous games.11 In game theory, for strategic-form games,... correspondences or multifunctions or random sets In the literature, one usually studies correspondences under some measure-theoretic or topological structure The study of such correspondences and their selections has widely applied in Economic Theory, see Aumann (1965), Debreu (1959, 1967), Hart and Kohlberg (1974), Hildenbrand (1974) etc 1.5 Lebesgue Extensions and Correspondences Let Ψ be a correspondence... process And this repetition can be continued ad infinitum to obtain {(I, In , λn )}n∈N , thereby giving insight into how rich a saturated probability space really is It cannot be attained in a countably infinite number of extensions 1.6 Organization of the Thesis The main results in Chapters 3, 4, 5 and 6 are based on the papers Sun and Zhang (2009), Wang and Zhang (2010), Khan and Zhang (2010b) and Khan and. .. in Keisler and Sun (2009) We draw the implications of our results for the theory of large (non-anonymous) games and private information games Chapter 1 Introduction In Economic Theory, atomless probability spaces are widely used to describe or model institutions having a large number of competing participants in political and economic life, for examples, markets, exchanges, corporations (from the shareholders’... Sun (2006, 2009) and Podczeck (2009), the payoff functions should satisfy a more restrictive condition, the Carath´odory condition e (see Section 4.2) Our proof is built heavily on the ELLN as reviewed above This ELLN approach is different from the techniques used in Loeb and Sun (2006, 2009) and Podczeck (2009) In particular, Loeb and Sun (2006) make use of the nonstandard analysis Loeb and Sun (2009)... applications of the theory of saturated probability spaces to Economic Theory In particular, we focus on the modeling of agent spaces in rich Fubini extensions as presented in Sun (2006), on general purification theorem as established in Dvorestzky et al (1950, 1951b) and on the modeling of diffused information for private information games as in Milgrom and Weber (1985), Radner and Rosenthal (1982) . developed by Hoover and Keisler (1984) into economic theory, as poineered by Keisler and Sun (2009). Many economic models include random shocks imposed on a large number (con- tinuum) of economic agents. on the theory of large games. 1.1 Basic Ideas of Keisler and Sun (2009) 5 and Loeb and Sun (2009) generalize the results in Loeb and Sun (2006) about the pu- rification theorem; Podczeck (2008) and. earlier references on this, and for some recent applications in Economic Theory of ELLN, see Duffie et al. (2005, 2007), Duffie and Sun (2007), Lagos and Rocheteau (2007), McLean and Postlewaite (2002, 5 See Sun