THEORY OF CORRESPONDENCES AND GAMES HE, WEI (B.S., Peking University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2014 ii iii Declaration I hereby declare that the thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. He, Wei July 28, 2014 iv Acknowledgement I would like to express my sincere appreciation to my supervisor Prof. Sun Yeneng. For me, he is not only the greatest supervisor who always prepares to listen to my naive ideas, answers my questions, encourages me to explore various new areas, but also a perfect friend who would like to share his thinking about the way of life. Without his continuous guidance, encouragement and help, what I have achieved would not be possible. I would like to take this opportunity to thank Prof. Chen Yi-Chun, Prof. Luo Xiao, Prof. Satoru Takahashi and Prof. Nicholas Yannelis for their encouragement and help during these years. Discussions with them significantly broadens my horizon, deepens my understanding and shapes my views in the field of Game Theory. I have also benefited and learnt a lot from my research family: Prof. Yu Haomiao, Prof. Zhang Yongchao, Prof. Sun Xiang, Mr. Qiao Lei and Ms. Zeng Yishu. Special thanks must be given to Prof. Nicholas Yannelis and Sun Xiang, who gave valuable advice on my research projects. I am also grateful to my postgraduate friends in NUS, including but not limited to, Jia Xiaowei, Li Shangru, Sun Yifei, Wang Haitao, Zhang Rong, Zhou Feng, for their help, friendship and the good time we have together. Finally, my deepest appreciation and thanks are due to my family for their unconditional love and whole hearted support. In particular, I am greatly indebted to my wife Zhao Yang for her constant love, understanding, support and encouragement, which are great source of strength throughout the years of my PhD study. He, Wei July 22, 2014 v vi ACKNOWLEDGEMENT Contents Acknowledgement v Contents ix Summary xi Introduction 1.1 Modeling Infinitely Many Agents . . . . . . . . . . . . . . . . . . . . . . 1.2 Conditional Distributions/Expectations of Correspondences . . . . . . . . 1.3 Games with Incomplete Information . . . . . . . . . . . . . . . . . . . . . 1.4 Discounted Stochastic Games . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Infinitely Many Agents 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Characterizations of the Agent Space . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Setwise Coarseness . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.1 Distributional Equilibria . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.2 Standard Representation . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 vii viii CONTENTS 2.3.3 Hyperfinite Agent Space . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.4 Saturated Agent Space . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.5 Many More Players than Strategies . . . . . . . . . . . . . . . . . 22 2.4 Necessity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5.1 Proofs of Results in Section 2.2 . . . . . . . . . . . . . . . . . . . 23 2.5.2 Proof of Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5.3 Proofs of Theorem and Proposition . . . . . . . . . . . . . . . 31 Theory of Correspondences 39 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3 Regular Conditional Distributions of Correspondences . . . . . . . . . . . 43 3.3.1 Distributions of Correspondences . . . . . . . . . . . . . . . . . . 43 3.3.2 Converse Results for Distributions of Correspondences . . . . . . 46 3.3.3 Regular Conditional Distributions of Correspondences . . . . . . . 51 Conditional Expectations of Banach Valued Correspondences . . . . . . . 57 3.4.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4.2 Regularity Properties . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.4.3 Converse Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Conditional Expectations of Correspondences in Rn . . . . . . . . . . . . 74 3.5.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.5.2 Regularity Properties . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.5.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.4 3.5 Games with Incomplete Information 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 83 CONTENTS 4.2 4.3 ix Games with Incomplete Information and General Action Spaces . . . . . 85 4.2.1 Relative Di↵useness of Information . . . . . . . . . . . . . . . . . 87 4.2.2 Existence of Pure Strategy Equilibria . . . . . . . . . . . . . . . . 88 4.2.3 Undistinguishable Purification . . . . . . . . . . . . . . . . . . . . 91 4.2.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 97 Bayesian Games with Inter-player Information and Finite Actions . . . . 98 4.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.3.2 Existence of Pure Strategy Equilibria . . . . . . . . . . . . . . . . 99 4.3.3 Purification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.3.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Stochastic Games 111 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.2 Discounted Stochastic Games . . . . . . . . . . . . . . . . . . . . . . . . 113 5.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.3.1 Stochastic Games with Coarser Transition Kernels . . . . . . . . . 115 5.3.2 Stochastic Games with Decomposable Coarser Transition Kernels 117 5.3.3 Decomposable Coarser Transition Kernels on the Atomless Part . 119 5.3.4 Minimality of the Condition . . . . . . . . . . . . . . . . . . . . . 121 5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Bibliography 131 x CONTENTS 124 Chapter 5. Stochastic Games Section 5.2. Duffie et al. (1994) obtained ergodic properties of such correlated equilibria under stronger conditions. They essentially assumed that players can observe the outcome of a public randomization device before making decisions at each stage.13 Thus, the new state space can be regarded as S = S ⇥ L endowed with the product -algebra S = S ⌦ B and product measure = ⌦ ⌘, where L is the unit interval endowed with the Borel -algebra B and Lebesgue measure ⌘. Denote G = S ⌦ {;, L}. Given s0 , s01 S and x X, the new transition kernel q (s01 |s0 , x) = q(s1 |s, x), where s (resp. s1 ) is the projection of s0 (resp. s01 ) on S and q is the original transition kernel with the state space S. Thus, q (·|s0 , x) is measurable with respect to G for any s0 S and x X. It is obvious that G is setwise coarser than S . Then the condition of coarser transition kernel is satisfied for the extended state space (S , S , ), and the existence of a stationary Markov perfect equilibrium follows from Theorem 15. The drawback of this approach is that the “sunspot” is irrelevant to the fundamental parameters of the game. Our result shows that it can indeed enter the stage payo↵ u, the correspondence of feasible actions A and the transition probability Q. Decomposable constant transition kernels on the atomless part Nowak (2003) considered stochastic games with transition probabilities as combinations of finitely many measures on the atomless part. In particular, the structure of the transition probability in Nowak (2003) is as follows. 1. S2 is a countable subset of S and S1 = S \ S2 , each point in S2 is S-measurable. 2. There are atomless nonnegative measures µj concentrated on S1 , nonnegative measures k concentrated on S2 , and measurable functions qj , bk : S ⇥ X ! [0, 1],  j  J and  k  K, where J and K are positive integers. The transition probability Q(·|s, x) = (·|s, x) + Q0 (·|s, x) for each s S and x X, where P P (·|s, x) = 1kK bk (s, x) k (·) and Q0 (·|s, x) = 1jJ qj (s, x)µj (·). 3. For any j and k, qj (s, ·) and bk (s, ·) are continuous on X for any s S. We shall show that any stochastic game with the above structure satisfies the condition of decomposable coarser transition kernel on the atomless part. 13 For detailed discussions on such a public randomization device, or “sunspot”, see Duffie et al. (1994) and their references. 5.4. Discussion 125 Without loss of generality, assume that µj and k are all probability measures. Let P P (E) = J+K 1jJ µj (E) + 1kK k (E) for any E S. Then µj is absolutely continuous with respect to and assume that ⇢j is the Radon-Nikodym derivative for  j  J. Given any s S and x X, let P > > q (s, x)⇢j (s0 ), if s0 S1 ; > < 1jJ j (s0 |s,x) q(s0 |s, x) = , if s0 S2 and (s0 ) > 0; (s0 ) > > > :0, if s0 S2 and (s0 ) = 0. Then Q(·|s, x) is absolutely continuous with respect to and q(·|s, x) is the transition kernel. It is obvious that the condition of a decomposable coarser transition kernel on the atomless part is satisfied with G = {;, S1 }. Then a stationary Markov perfect equilibrium exists by Proposition 12. Noisy stochastic games Duggan (2012) proved the existence of stationary Markov perfect equilibria in stochastic games with noise – a component of the state that is nonatomically distributed and not directly a↵ected by the previous period’s state and actions. The exogenously given product structure of the state space as considered by Duggan (2012) is defined as follows: 1. The set of states can be decomposed as S = H ⇥ R and S = H ⌦ R, where H and R are complete, separable metric spaces, and H and R are the respective Borel -algebras. Qh (·|s, a) denotes the marginal of Q(·|s, a) on h H. 2. There is a fixed probability measure  on (H, H) such that for all s and a, Qh (·|s, a) is absolutely continuous with respect to  and ↵(·|s, a) is the Radon-Nikodym derivative. 3. For all s, the mapping a ! Qh (·|s, a) is norm continuous; that is, for all s, all a and each sequence {am } of action profiles converging to a, the sequence {Qh (·|s, am )} converges to Qh (·|s, a) in total variation. 4. Conditional on next period’s h0 , the distribution of r0 in next period is independent of the current state and actions. In particular, Qr : H ⇥ R ! [0, 1] 126 Chapter 5. Stochastic Games is a transition probability such that for all s, all a, and all Z S, we have R R Q(Z|s, a) = H R 1Z (h0 , r0 )Qr (dr0 |h0 )Qh (dh0 |s, a). 5. For -almost all h, Qr (·|h) (abbreviated as ⌫h ) is absolutely continuous with respect to an atomless probability measure ⌫ on (R, R), and (·|h) is the Radon-Nikodym derivative. In the following we show that the condition of a coarser transition kernel is satisfied in noisy stochastic games. Proposition 15. Every noisy stochastic game has a coarser transition kernel. R R Proof. Let (Z) = H R 1Z (h, r) (r|h) d⌫(r) d(h) for all Z S. Let G = H ⌦ {;, R}. It is clear that ↵(·|s, a) is G-measurable, we need to show that G is setwise coarser than S under . Fix any Borel D ✓ S with (D) > 0. Then there is a measurable mapping from (D, S D ) to (L, B) such that can generate the -algebra S D , where L is the unit interval endowed with the Borel -algebra B. Let g(h, r) = h for each (h, r) D, Dh = {r : (h, r) D} and HD = {h H : ⌫h (Dh ) > 0}. Denote gh (·) = g(h, ·) and h (·) = ⌫h (h, ·) for each h HD . Define a mapping ([0,l]) h f : HD ⇥ L ! [0, 1] as follow: f (h, l) = . Similarly, denote fh (·) = f (h, ·) for ⌫h (Dh ) each h HD . For -almost all h HD , the atomlessness of ⌫h implies ⌫h h ({l}) = for all l L. Thus the distribution function fh (·) is continuous on L for -almost all h HD . Let (s) = f (g(s), (s)) for each s D, and D0 = ([0, 12 ]), which is a subset of D. For h HD , let lh be max{l L : fh (l)  12 } if fh is continuous, and otherwise. It is clear that when fh is continuous, fh (lh ) = 1/2. For any E H, let D1 = (E ⇥ R) \ D, and E1 = E \ HD . If (D1 ) = 0, then = Z 1 ⌫h fh [0, ] d(h) h HD Z f (h, lh )⌫h (Dh ) d(h) h ([0, lh ]) d(h) = (D0 \ D1 ) = (D0 ) = ⌫h HD = Z Z HD ⌫h (Dh ) d(h) = (D) > 0. HD 5.5. Concluding Remarks 127 If (D1 ) > 0, then Z Z Z 1 (D1 \ D0 ) = 1D\D0 (h, r) d⌫h (r) d(h) = ⌫h fh ( , 1] d(h) h E1 R E1 Z Z 1 1 = ⌫h fh [0, 1] \ [0, ] d(h) = ⌫h (Dh ) d(h) = (D1 ) > 0. h 2 E1 E1 Hence, D is not a G-atom. Therefore, G is setwise coarser than S and the condition of coarser transition kernel is satisfied. By Proposition 15, the existence of stationary Markov perfect equilibria in noisy stochastic games follows from Theorem 15 directly. Theorem 17 (Duggan (2012)). Every noisy stochastic game possesses a stationary Markov perfect equilibrium. 5.5 Concluding Remarks We consider stationary Markov perfect equilibria in discounted stochastic games with a general state space. So far, only several special classes of stochastic games have been shown to possess equilibria, while the existence of such equilibria under some general condition has remained an open problem. In the literature, the standard approach for the existence arguments is to work with the convex hull of the collection of all selections from the equilibrium payo↵ correspondence. We adopt this approach and provide a very simple proof of some existence results under the general condition of a (decomposable) coarser transition kernel. The minimality of our condition is illustrated. As shown in Section 5.4, our results strictly generalize various previous existence results. 128 Chapter 5. Stochastic Games 5.5. Concluding Remarks 129 130 Chapter 5. Stochastic Games Bibliography D. Acemoglu and A. Wolitzky, The Economics of Labor Coercion, Econometrica 79 (2011), 555–600. G. A. Akerlof and R. E. Kranton, Economics and Identity, Quarterly Journal of Economics 115 (2000), 715–753. C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhikers Guide, Berlin, Springer, 2006. R. M. Anderson, An Elementary Core Equivalence Theorem, Econometrica 46 (1978), 1483–1487. R. M. Anderson, The Second Welfare Theorem with Nonconvex Preferences, Econometrica 56 (1988), 361–382. G. M. Angeletos, C. Hellwig and A. Pavan, Dynamic Global Games of Regime Change: Learning, Multiplicity and the Timing of Attacks, Econometrica 75 (2007), 711–756. A. Araujo and L. I. de Castro, Pure Strategy Equilibria of Single and Double Auctions with Interdependent Values, Games and Economic Behavior 65 (2009), 25–48. S. Athey, Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information, Econometrica 69 (2001), 861–889. R. J. Aumann, Markets with a Continuum of Traders, Econometrica 32 (1964), 39–50. R. J. Aumann, Integrals of Set-Valued functions, Journal of Mathematical Analysis and Applications 12 (1965), 1–12. J. P. Aubin and H. Frankowska, Set Valued Analysis, Birkhaiuser, Boston, 1990. 131 132 Bibliography E. M. Azevedo, E. G. Weyl and A. White, Walrasian Equilibrium in Large, Quasilinear Markets, Theoretical Economics (2013), 281–290. E. J. Balder, Generalized Equilibrium Results for Games with Incomplete Information, Mathematics of Operations Research 13 (1988), 265–276. P. Barelli and J. Duggan, Purification of Bayes Nash Equilibrium with Correlated Types and Interdependent Payo↵s, working paper, University of Rochester, 2013. P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968. V. I. Bogachev, Measure Theory, volume 2, Springer-Verlag Berlin Heidelberg, 2007. W. A. Brock and S. N. Durlauf, Discrete Choice with Social Interactions, Review of Economic Studies 68 (2001), 235–260. D. J. Brown and P. Loeb, The Values of Nonstandard Exchange Economies, Israel Journal of Mathematics 25 (1976), 71–86. D. J. Brown and A. Robinson, Nonstandard Exchange Economies, Econometrica 43 (1975), 41–55. C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, No. 580, Springer-Verlag, Berlin/New York, 1977. C. Castaing (Charles), P. R. de Fitte and M. Valadier, Young Measures on Topological Spaces: with Applications in Control Theory and Probability Theory, volume 571 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 2004. G. Debreu and H. Scarf, A Limit Theorem on the Core of an Economy, International Economic Review (1963), 235–246. J. Diestel, W. M. Ruess and W. Schachermayer, Weak Compactness in L1 (µ, X), Proceedings of the American Mathematical Society 118 (1993), 447–453. J. Diestel and J. J. Uhl, Vector Measures, Mathematical Surveys, vol. 15, American Mathematical Society, Providence, RI, 1977. D. Duffie, Dynamic Asset Pricing Theory, 3rd edition, Princeton University Press, Princeton, 2001. Bibliography 133 D. Duffie, N. Gˆarleanu and L. H. Pedersen, Over-The-Counter Markets, Econometrica 73 (2005), 1815–1847. D. Duffie, J. Geanakoplos, A. Mas-Colell and A. McLennan, Stationary Markov Equilibria, Econometrica 62 (1994), 745–781. D. Duffie and B. Strulovici, Capital Nobility and Asset Pricing, Econometrica 80 (2012), 2469–2509. J. Duggan, Noisy Stochastic Games, Econometrica 80 (2012), 2017–2045. R. Durrett, Probability: Theory and Examples, Cambridge, New York, 2010. A. Dvoretsky, A. Wald and J. Wolfowitz, Elimination of Randomization in Certain Statistical Decision Procedures and Zero-Sum Two-Person Games, The Annals of Mathematical Statistics 22 (1951), 1–21. E. B. Dynkin and I. V. Evstigneev, Regular Conditional Expectations of Correspondences, Theory of Probability and its Applications 21 (1977), 325–338. J. Eeckhout and P. Kircher, Sorting Versus Screening: Search Frictions and Competing Mechanisms, Journal of Economic Theory 145 (2010), 1354–1385. S. Fajardo and H. J. Keisler, Model Theory of Stochastic Processes, Lecture Notes in Logic, vol. 14, Assoc. Symbolic Logic, Urbana, IL, 2002. D. H. Fremlin, Measure Algebras, Handbook of Boolean Algebras, volume 3, Elsevier, Amsterdam, 1989. H. Fu, Mixed-Strategy Equilibria and Strong Purification for Games with Private and Public Information, Economic Theory 37 (2008), 521–532. H. Fu, Y. N. Sun, N. C. Yannelis and Z. Zhang, Pure Strategy Equilibria in Games with Private and Public Information, Journal of Mathematical Economics 43 (2007), 523–531. D. Fudenberg and J. Tirole, Game Theory, The MIT Press, 1991. E. J. Green, Continuum and Finite-Player Noncooperative Models of Competition, Econometrica 52 (1984), 975–993. 134 Bibliography R. Guesnerie and P. Jara-Moroni, Expectational Coordination in Simple Economic Contexts: Concepts and Analysis with Emphasis on Strategic Substitutabilities, Economic Theory 47 (2011), 205–246. P. J. Hammond, Straightforward Individual Incentive Compatibility in Large Economies, Review of Economic Studies 46 (1979), 263–282. C. Hara, Existence of Equilibria in Economies with Bads, Econometrica 73 (2005), 647–658. J. C. Harsanyi, Games with Incomplete Information Played by ’Bayesian’ Players, Management Science 14 (1967–68) 159–182, 320–334, and 486–502, Parts I–III. S. Hart, W. Hildenbrand and E. Kohlberg, On Equilibrium Allocations as Distributions on the Commodity Space, Journal of Mathematical Economics (1974), 159–166. W. He, X. Sun and Y. N. Sun, Modeling Infinitely Many Agents, working paper, National University of Singapore, 2013. W. He and X. Sun, On the Di↵useness of Incomplete Information Game, Journal of Mathematical Economics, forthcoming in 2014. W. He and Y. N. Sun, The Necessity of Nowhere equivalence, working paper, National University of Singapore, 2013a. W. He and Y. N. Sun, Conditional Expectations of Banach Valued Correspondences, working paper, National University of Singapore, 2013b. W. He and Y. N. Sun, Stationary Markov Perfect Equilibria in Discounted Stochastic Games, working paper, National University of Singapore, 2013c. W. He and Y. N. Sun, Conditional Expectations of Correspondences, working paper, National University of Singapore, 2013d (updated in 2014). W. Hildenbrand, Core and Equilibria of a Large Economy, Princeton University Press, Princeton, NJ, 1974. D. N. Hoover and H. J. Keisler, Adapted Probability Distribution, Transactions of the American Mathematical Society 286 (1984), 159–201. S. Kakutani, Construction of a Non-Separable Extension of the Lebesque Measure Space, Proc. Acad. Tokyo 20 (1944), 115–119. Bibliography 135 Y. Kannai, Continuity Properties of the Core of a Market, Econometrica 38 (1970), 791–815. H. J. Keisler, An Infinitesimal Approach to Stochastic Analysis, Memoirs of the American Mathematical Society 48 (297) (1984). H. J. Keisler and Y. N. Sun, Why Saturated Probability Spaces are Necessary, Advances in Mathematics 221 (2009), 1584–1607. M. A. Khan, K. P. Rath and Y. N. Sun, On the Existence of Pure Strategy Equilibria in Games with a Continuum of Players, Journal of Economic Theory 76 (1997), 13–46. M. A. Khan, K. P. Rath and Y. N. Sun, On a Private Information Game without Pure Strategy Equilibria, Journal of Mathematical Economics 31 (1999), 341–359. M. A. Khan, K. P. Rath and Y. N. Sun, The Dvoretzky-Wald-Wolfowitz Theorem and Purification in Atomless Finite-Action Games, International Journal of Game Theory 34 (2006), 91–104. M. A. Khan, K. P. Rath, Y. N. Sun and H. Yu, Large Games with a Bio-Social Typology, Journal of Economic Theory 148 (2013), 1122–1149. M. A. Khan and N. Sagara, Maharam-Types and Lyapunov’s Theorem for Vector Measures on Banach Spaces, Illinois Journal of Mathematics 57 (2013), 145–169. M. A. Khan and N. Sagara, Weak Sequential Convergence in L1 (µ, X) and an Exact Version of Fatou’s Lemma, Journal of Mathematical Analysis and Applications 412 (2014a), 554–563. M. A. Khan and N. Sagara, The Bang-Bang, Purification and Convexity Principles in Infinite Dimensions: Additional Characterizations of the Saturation Property, SetValued and Variational Analysis, forthcoming, (2014b). M. A. Khan and Y. N. Sun, Pure Strategies in Games with Private Information, Journal of Mathematical Economics 24 (1995), 633–653. M. A. Khan and Y. N. Sun, Non-Cooperative Games on Hyperfinite Loeb Spaces, Journal of Mathematical Economics 31 (1999), 455–492. 136 Bibliography M. A. Khan and Y. N. Sun, Non-Cooperative Games with Many Players, in Handbook of Game Theory, volume (R. J. Aumann and S. Hart eds.), Chapter 46, 1761–1808, North-Holland, Amsterdam, 2002. M. A. Khan and Y. Zhang, Set-Valued Functions, Lebesgue Extensions and Saturated Probability Spaces, Advances in Mathematics 229 (2012), 1080–1103. M. A. Khan and Y. Zhang, On the Existence of Pure-Strategy Equilibria in Games with Private Information: A Complete Characterization, Journal of Mathematical Economics 50 (2014), 197–202 G. Knowles, Lyapunov Vector Measures, SIAM Journal on Control and Optimization 13 (1975), 294–303. Y. Levy, A Discounted Stochastic Game with No Stationary Nash Equilibrium: Two Examples, Econometrica 81 (2013), 1973–2007. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer-Verlag, Berlin/New York, 1977. P. A. Loeb, Conversion from Nonstandard to Standard Measure Spaces and Applications in Probability Theory, Transactions of the American Mathematical Society 211 (1975), 113–122. P. Loeb and Y. N. Sun, Purification of Measure-Valued Maps, Illinois Journal of Mathematics, 50 (2006), 747–762. P. Loeb and Y. N. Sun, Purification and Saturation, Proceedings of the American Mathematical Society 137 (2009), 2719–2724. G. J. Mailath, A. Postlewaite and L. Samuelson, Pricing and Investments in Matching Markets, Theoretical Economics (2013), 535–590. A. Mas-Colell, On a Theorem of Schmeidler, Journal of Mathematical Economics 13 (1984), 201–206. R. McLean and A. Postlewaite, Informational Size and Incentive Compatibility, Econometrica 70 (2002), 2421–2454. R. McLean and A. Postlewaite, Informational Size and Efficient Auctions, Review of Economic Studies 71 (2004), 809–827. Bibliography 137 P. R. Milgrom and R. J. Weber, Distributional Strategies for Games with Incomplete Information, Mathematics of Operations Research 10 (1985), 619–632. J. W. Milnor and L. S. Shapley, Values of Large Games II: Oceanic Games, The Rand Corporation, RM 2649, February 28, (1961); published in Mathematics of Operations Research (1978), 290–307. J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, 3rd edn., Princeton University Press, 1953. M. Noguchi, Existence of Nash Equilibria in Large Games, Journal of Mathematical Economics 45 (2009), 168–184. M. Noguchi and W. R. Zame, Competitive Markets with Externalities, Theoretical Economics (2006), 143–166. A. S. Nowak, On a New Class of Nonzero-Sum Discounted Stochastic Games Having Stationary Nash Equilibrium Points, International Journal of Game Theory 32 (2003), 121–132. A. S. Nowak and T. E. S. Raghavan, Existence of Stationary Correlated Equilibria with Symmetric Information for Discounted Stochastic Games, Mathematics of Operations Research 17 (1992), 519–527. M. Peters, Non-Contractible Heterogeneity in Directed Search, Econometrica 78 (2010), 1173–1200. K. Podczeck, On the Convexity and Compactness of the Integral of a Banach Space Valued Correspondence, Journal of Mathematical Economics 44 (2008), 836–852. K. Podczeck, On Purification of Measure-Valued Maps, Economic Theory 38 (2009), 399–418. R. Radner and R. W. Rosenthal, Private Information and Pure-Strategy Equilibria, Mathematics of Operations Research (1982), 401–409. K. Rath, Y. N. Sun and S. Yamashige, The Nonexistence of Symmetric Equilibria in Anonymous Games with Compact Action Spaces, Journal of Mathematical Economics 24 (1995), 331–346. 138 Bibliography M. T. Rauh, Nonstandard Foundations of Equilibrium Search Models, Journal of Economic Theory 132 (2007), 518–529. P. J. Reny, On the Existence of Monotone Pure Strategy Equilibria in Bayesian Games, Econometrica 79 (2011), 499–553. H. L. Royden, Real Analysis, 3rd edn., Macmillan, New York, NY, 1988. H. L. Royden and P. M. Fitzpatrick, Real Analysis, 4th edn., Prentice Hall, Boston, 2010. A. Rustichini and N. C. Yannelis, What is Perfect Competition, in Equilibrium Theory in Infinite Dimensional Spaces (M. A. Khan and N. C. Yannelis eds.), Springer-Verlag, Berlin/New York, 1991, 249–265. R. Serrano, R. Vohra and O. Volij, On the Failure of Core Convergence in Economies with Asymmetric Information, Econometrica 69 (2001), 1685–1696. L. Shapley, Stochastic Games, Proceedings of the National Academy of Sciences 39 (1953), 1095–1100. X. Sun and Y. Zhang, Pure-Strategy Nash equilibria in Nonatomic Games with InfiniteDimensional Action Spaces, Economic Theory, forthcoming. Y. N. Sun, On the Theory of Vector Valued Loeb Measures and Integration, Journal of Functional Analysis 104 (1992), 327–362. Y. N. Sun, Distributional Properties of Correspondences on Loeb Spaces, Journal of Functional Analysis 139 (1996), 68–93. Y. N. Sun, Integration of Correspondences on Loeb Spaces, Transactions of the American Mathematical Society 349 (1997), 129–153. Y. N. Sun and N. C. Yannelis, Saturation and the Integration of Banach Valued Correspondences, Journal of Mathematical Economics, 44 (2008), 861–865. R. Tourky and N. C. Yannelis, Markets with Many More Agents than Commodities: Aumann’s “Hidden” Assumption, Journal of Economic Theory 101 (2001), 189–221. J. L. Walsh, A Closed Set of Normal Orthogonal Functins, American Journal of Mathematics 45 (1923), 5–24. Bibliography 139 J. Wang and Y. Zhang, Purification, Saturation and the Exact Law of Large Numbers, Economic Theory 50 (2012), 527–545. S. Xiong and C. Z. Zheng, Core Equivalence Theorem with Production, Journal of Economic Theory 137 (2007), 246–270. N. C. Yannelis, Debreu’s Social Equilibrium Theorem with Asymmetric Information and a Continuum of Agents, Economic Theory 38 (2009), 419–432. H. Yu, Rationalizability in Large games, Economic Theory 55 (2014), 457–479. [...]... Distributions/Expectations of Correspondences The theory of correspondences, which has important applications in a variety of areas (including optimization, control theory and mathematical economics), has been studied extensively in recent years However, basic regularity properties on the distributions of correspondences/ integrals of Banach valued correspondences such as convexity, closeness, compactness and preservation of. .. above, and it can be used to handle the failure of the Lebesgue unit interval More importantly, the optimality of the setwise coarseness condition will be illustrated by showing its necessity in deriving certain results in general equilibrium theory and game theory The theory of correspondences has important applications in a variety of areas However, basic regularity properties on the distributions of correspondences/ integrals... class of rich measure spaces, the so-called Loeb measure spaces constructed from the method of nonstandard analysis Keisler and Sun (2009) then showed that the abstract property of saturation on a probability space is not only 6 See Mas-Colell (1984), Hart, Hildenbrand and Kohlberg (1974), Khan and Sun (1999), Keisler and Sun (2009) and Rustichini and Yannelis (1991) 7 See Sun (1996, 1997) and Keisler and. .. the distributions of correspondences; (2) these regularity properties can be extended to regular conditional distributions of correspondences; (3) the su ciency and necessity of the setwise coarseness condition can be also demonstrated for the regularity properties of the Bochner/Gel0 fand integrals and conditional expectations of Banach valued correspondences Furthermore, if the range of the correspondence... In terms of optimality for the setwise coarseness condition, the second question we consider involves games with many agents Motivated by the consideration of social identities as in Akerlof and Kranton (2000) and Brock and Durlauf (2001), Khan et al (2013) introduced a general class of large games in which agents have names and determinate social-types and/ or biological traits For such large games, ... from general equilibrium theory and game theory: (1) determinateness property in large economies, (2) existence of equilibria in large games with traits, and (3) determinateness property in large games. 17 16 For further discussions on the relevance of this condition in general equilibrium theory, see Tourky and Yannelis (2001) 17 The proofs are given in Section 2.5 2.5 Proofs 23 The following theorem... validity of the determinateness property for large games via the setwise coarseness condition Proposition 4 G is setwise coarser than F if and only if D(G1 ) = D(G2 ) for any two G-measurable large games G1 and G2 with the same distribution, where D(Gi ) is the set of distributions of F-measurable Nash equilibria in the game Gi for i = 1, 2 2.5 2.5.1 Proofs Proofs of Results in Section 2.2 Proof of Proposition... example, Sun (1996, 1997), Sun and Yannelis (2008), Podczeck (2008), Keisler and Sun (2009) and Khan and Zhang (2012) 4 Chapter 1 Introduction su cient but also necessary for any of these regularity properties for distributions of correspondences to hold Furthermore, Sun and Yannelis (2008) found that all the existing results for Bochner/Gel0 fand integrals of Banach valued correspondences in Loeb spaces... setwise coarser than F Large games with traits Motivated by the consideration of social identities as in Akerlof and Kranton (2000) and Brock and Durlauf (2001), Khan et al (2013) provided a treatment of large games in which individual players have names as well as traits, and a player’s dependence on society is formulated as a joint probability measure on the space of actions and traits.9 The agent space... payo↵ correspondences in stochastic games and a general result on the conditional expectations of correspondences The proof is remarkably simple and our theorems cover previous existence results for stochastic games considered in Nowak and Raghavan (1992), Du e et al (1994), Nowak (2003) and Duggan (2012), while no product structure is imposed on the state space We also illustrate the minimality of our . Necessity 22 2.5 Proofs 23 2.5.1 Proofs of Results in Section 2.2 23 2.5.2 Proof of Theorem 1 26 2.5.3 Proofs of Theorem 2 and Proposition 4 31 3 Theory of Correspondences 39 3.1 Introduction. THEORY OF CORRESPONDENCES AND GAMES HE, WEI (B.S., Peking University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATI O N AL UNIVERSITY OF SINGAPORE 2014 ii iii Declaration Iherebydeclarethatthethesisismyoriginalworkandithasbeenwrittenbyme in. Distributions of Correspondences 43 3.3.1 Distributions of Correspondences 43 3.3.2 Converse Results for Distributions of Correspondences 46 3.3.3 Regular Conditional Distribu t i on s of Correspondences