EQUILIBRIA OF LARGE GAMES AND BAYESIAN GAMES WITH PRIVATE AND PUBLIC INFORMATION FU HAIFENG (B.S., Fudan Univ. and M.A., East China Normal Univ.) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF STATISTICS AND APPLIED PROBABILITY NATIONAL UNIVERSITY OF SINGAPORE 2008 Acknowledgements At the outset, I would like to express my heartfelt gratitude to my advisor Professor Sun Yeneng for his great guidance and assistance during my doctoral research endeavor. I thank him for leading me into this wonderful area of game theory and providing me with the opportunity to work with him and other talented researchers in this area. Without his help, this thesis could not have been completed. I am indebted to my co-advisor Professor Bai Zhidong who is very knowledgable, kind and helpful. Whenever I have a question of which I think he may know the answer, I always go to his office without hesitation, knock on his door and he is always there for me. I thank my co-authors, Professor Nicholas C. Yannelis, Dr Zhang Zhixiang, Ms Xu Ying and Ms Zhang Luyi for their help and collaboration. Prof. Yannelis has ii Acknowledgements helped me in many ways and provided me with helpful feedbacks. So does Dr Zhang. Xu Ying and Luyi are my junior classmates and they have brought a lot of fun to my life in NUS. I am very grateful to Professor M. Ali Khan for his very helpful comments and suggestions on several of my research papers on which this thesis is based. His very warm encouragement also inspired me greatly. I thank my classmates, Zhao Yudong, Xu Yuhong, Yang Jialiang, Wu lei and Zhang Yongchao for their help and support at all the times, and I also thank Wanting, Jingyuan, Rongli and Hao Ying for sharing with me a quite and harmonious studying environment in our small PhD students’ room. I thank Jolene, Ziyi and other friends in NUSBS for their company and friendship, which makes my life in this doctoral research period more colorful. I am grateful to my landlord Madam Huang whose help excused me from cleaning my room and doing my laundry. Last, but not least, I would like to dedicate this thesis to my parents and my sister for their life-long love, support and understanding. iii Contents Acknowledgements Summary ii viii Introduction 1.1 Some backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Motivations and contributions . . . . . . . . . . . . . . . . . . . . . 1.3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . Pure-strategy equilibria in games with private and public information 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Contents v 2.2 Games with private and public information . . . . . . . . . . . . . . 2.3 Distribution of correspondences via vector measures . . . . . . . . . 13 2.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5.1 Proof of Theorem . . . . . . . . . . . . . . . . . . . . . . . 16 2.5.2 Proof of Proposition . . . . . . . . . . . . . . . . . . . . . 19 Mixed-strategy equilibria and strong purification 25 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Games with private and public information . . . . . . . . . . . . . . 29 3.3 The existence of mixed-strategy equilibria . . . . . . . . . . . . . . 31 3.4 Strong purification and pure-strategy equilibria . . . . . . . . . . . 32 3.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5.1 Proof of Theorem . . . . . . . . . . . . . . . . . . . . . . . 37 3.5.2 Proof of Theorem . . . . . . . . . . . . . . . . . . . . . . . 39 3.5.3 Proof of Corollary 43 . . . . . . . . . . . . . . . . . . . . . . Characterizing pure-strategy equilibria in large games 44 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3 The results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.4 A counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Contents vi 4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.6 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.6.1 Proof of Theorem . . . . . . . . . . . . . . . . . . . . . . . 53 4.6.2 Proof of Theorem . . . . . . . . . . . . . . . . . . . . . . . 55 4.6.3 Proof of Theorem . . . . . . . . . . . . . . . . . . . . . . . 57 From large games to Bayesian games: connection and generalization 59 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2 Large Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2.1 Game model . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2.2 Pure-strategy equilibrium . . . . . . . . . . . . . . . . . . . 66 Bayesian Games with countable players . . . . . . . . . . . . . . . . 69 5.3.1 Game model . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.3.2 Connecting Bayesian games with large games . . . . . . . . 72 5.3.3 Pure-strategy equilibria for Bayesian games . . . . . . . . . 75 5.4 Bayesian games with private and public information . . . . . . . . . 76 5.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.6 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.6.1 Proof of Theorem . . . . . . . . . . . . . . . . . . . . . . . 81 5.6.2 Proof of Theorem . . . . . . . . . . . . . . . . . . . . . . . 84 5.3 Contents vii 5.6.3 Proof of Theorem 11 . . . . . . . . . . . . . . . . . . . . . . 86 5.6.4 Proof of Theorem 12 . . . . . . . . . . . . . . . . . . . . . . 88 Summary This thesis studies the equilibria in large games and Bayesian games and it consists of four parts. In the first part, we generalize the traditional Bayesian games by introducing a new game form, the so-called games with private and public information. This new game model allows the players’ strategies to depend on their strategyrelevant 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 directly prove the existence of pure strategy equilibrium for such a game viii Summary by developing a distribution theory of correspondences via vector measures. In the second part, we further explore this new game model by showing the existence of mixed-strategy equilibria under general conditions. Moreover, under the additional assumptions of finiteness of action spaces and diffuseness and conditional independence of private information, a strong purification result is obtained for the mixed strategies in such games. As a corollary, the existence of purestrategy equilibria follows. This corollary generalizes the main result in our first part. In the third part, we consider a generalized large game model where the agent space is divided into countable subgroups and each players payoff depends on her own action and the action distribution in each of the subgroups. Focusing on the interaction between Nash equilibria and the best response correspondence of the players, we characterize the pure-strategy equilibrium distributions in large games endowed with countable actions, countable homogeneous groups of players, or atomless Loeb agent spaces by showing that a given distribution is an equilibrium distribution if and only if for any (Borel) subset of actions the proportion of players in each group playing this subset of actions is no larger than the proportion of players in that group having a best response in this subset. Furthermore, we also present a counterexample showing that this characterization result does not hold for a more general setting. In the fourth part, we firstly present a unified proof for the existence of pure ix Summary strategy equilibria in the three settings of large games mentioned above by showing the existence of their common characterizing counterpart. Then we show that each Bayesian game with countable players can induce a large game and the Bayesian game has a pure strategy equilibria if and only if the induced large game has one. This result enables us to apply the existence results in large games to Bayesian games and obtain existence of pure strategy equilibria in four different settings of Bayesian games. Finally, we also establish a connection between the generalized Bayesian games with private and public information and large games. Based on this connection and the existence results in large games, we obtain more generalized existence results of pure strategy equilibria in both Bayesian games and the generalized Bayesian games with private and public information. These results cover and improve the main results in Parts and 2. x 5.6 Proofs 82 is measurable. To see this, let ∆ = {u ∈ C(Θ) : u|D = f } and note that Wi−1 (BΘi (f, D, )) = {h ∈ C(Θ) : h|Θi ∈ BΘi (f, D, )} = {h ∈ C(Θ) : sup{d(f (x), h(x))|x ∈ D} < } = {h ∈ C(Θ) : sup{d(u(x), h(x))|x ∈ D} < } u∈∆ = BΘ (u, D, ). u∈∆ Since BΘ (u, D, ) is open by the definition of the topology on C(Θ), Wi−1 (BΘi (f, D, )) is also open and hence measurable. Thus our claim is verified. For all i ∈ I and any µ ∈ i∈I M (Ai ), define Γµi : C(Θi ) Ai by letting Γµi (u) = arg maxa∈Ai u(a, µ) for all u ∈ C(Θi ). Thus we have Biµ (t) = Γµi (Vi (t)) for all t ∈ Ti . By the Berge’s Maximum Theorem, Γµi is upper semicountinuous.22 Thus, (Γµi )−1 (F ) is measurable for all closed set F ∈ A.23 It is also straightforward to verify that Vi−1 [(Γµi )−1 (F )] = (Biµ )−1 (F ) for any closed set F ∈ A. Since Vi is measurable, λi Vi−1 is a Borel probability measure on C(Θi ). Define Φ : i∈I M (Ai ) i∈I M (Ai ) as M (Ai ) : ηi (E) ≤ λi [(Biµ )−1 (E)] for each i ∈ I and any E ∈ B(A)}. Φ(µ) = {η ∈ i∈I It is easy to see that Φ is nonempty,24 closed-valued and convex-valued. 22 Note that the map fµ : A × U → R defined by fµ (a, u) = u(a, µ) is continuous (see Theorem 46.10 in Munkres (2000)). 23 See, eg, Lemma 16.4 in Aliprantis and Border (1999). By the Measurable Maximum Theorem, Biµ admits a measurable selection gi . Thus η = (λi gi−1 )i∈I is a trivial element of Φ(µ). 24 5.6 Proofs 83 Now we want to show that Φ is upper semicontinuous or, equivalently, has a closed graph. i∈I Toward this end, we choose a sequence {(µm , η m )}m∈N from M (Ai ) with η m ∈ Φ(µm ) for each m and converging to (µ0 , η ). We need to show that η ∈ Φ(µ0 ). m Fix any i ∈ I. Let F be a closed subset of Ai and let Λm := (Γµi )−1 (F ) and Λ0 := (Γµi )−1 (F ). Since Γµi is upper semicontinuous and F is closed, Λ0 is also closed. Since Θi is compact, C(Θi ) is metrizable and we let dˆ be one of the compatible metrics on C(Θi ). For all k = 1, 2, . . ., let Gk = {u ∈ C(Θi ) : ˆ Λ0 )} < }. d(u, k Fix any k. We claim that Λm ⊂ Gk for large enough m. To see this, let um ∈ Λm , which, by the definition of Λm , implies that there is an am ∈ F such that um (am , µm ) = maxa∈Ai um (a, µm ). Since µm → µ0 and um is uniformly continuous on Ai × i∈I M (Ai )25 , when m is large enough we have |um (am , µ0 ) − maxa∈Ai um (a, µ0 )| < k1 . Thus it is straightforward to find a continuous real function um ∈ C(Θi ) such that um (am , µ0 ) = maxa∈Ai um (a, µ0 ) = maxa∈Ai um (a, µ0 ) ˆ m , u ) < .26 Thus u ∈ Λ0 and um ∈ Gk . and d(u m m k Hence, the above result and our hypothesis imply that ηim (F ) ≤ λi Vi−1 (Λm ) ≤ λi Vi−1 (Gk ) for large enough m. Since ηim (F ) → ηi0 (F ), we have that ηi0 (F ) ≤ λi Vi−1 (Gk ). Since Gk ↓ Λ0 , we have ηi0 (F ) ≤ λi Vi−1 (Λ0 ) = λi Vi−1 [(Γµi )−1 (F )] = λi [(Biµ )−1 (F )]. 25 26 Continuous real function on compact metric space is also uniformly continuous. Just let um be a little bit bigger than um around the area of am . 5.6 Proofs 84 Now we want to show the above result holds for all Borel set E ∈ A. To see this, recall that every probability measure on a Polish space is regular.27 Therefore, we have ηi0 (E) = ηi0 (E ∩ Ai ) = sup{ηi0 (F ) : F is closed and F ⊆ E ∩ Ai } ≤ sup{λi [(Biµ )−1 (F )] : F is closed and F ⊆ E ∩ Ai } ≤ λi [(Biµ )−1 (E ∩ Ai )] = λi [(Biµ )−1 (E)]. Since the above arguments hold for all i ∈ I, we conclude that η ∈ Φ(µ0 ). Therefore Φ also has a closed graph, hence by the Ky Fan fixed point theorem in Fan (1952), there is a fixed point µ∗ ∈ Φ(µ∗ ). 5.6.2 Proof of Theorem We have seen from Lemma 10 that Γ induces a semi-large game U Γ . Hence we only need to show the equivalence of the pure strategy equilibrium for Γ and U Γ . Sufficiency (⇐): Suppose that f : X → A is a pure strategy equilibrium of the game U Γ . Thus, for almost all x ∈ X, U Γ (x)(f (x), (λi fi−1 )i∈I ) ≥ U Γ (x)(a, (λi fi−1 )i∈I ) for all a ∈ K Γ (x) (5.7) where fi is the restriction of f to Xi . Now define gi : Ti → Ai by letting gi (ti ) = fi (ti , i) for all ti ∈ Ti , and let g = (gi )i∈I . Obviously, g is a pure strategy profile of Γ and we also have that 27 See Theorem 10.7 in Aliprantis and Border (1999). 5.6 Proofs 85 λi fi−1 = τi gi−1 . By Equations (5.5) and (5.7), we have that for any i ∈ I and τi -almost all ti ∈ Ti , Gi (ti , gi (ti ), (τj gj−1 )j∈I ) ≥ Gi (ti , a, (τj gj−1 )j∈I ) for all a ∈ Ai . Thus, by Equation (5.3), we have that for all i ∈ I, Ei (g) ≥ Ei (gi , g−i ) for all gi ∈ Meas(Ti , Ai ). Hence g is a pure strategy equilibria of Γ. Necessity (⇒): Now let g = (gi )i∈I be a pure strategy equilibrium for the Bayesian game Γ. Define fi : Xi → Ai by letting fi (x) = gi (t) for all x = (t, i) ∈ Xi and define f : X → A by f (x) = fi (x) if x ∈ Xi . Suppose now f is not a pure strategy equilibrium for the game U Γ . Then there must exists an index i ∈ I and a subset Ci ∈ Xi such that λ(Ci ) > and for all x ∈ Ci U Γ (x)[f (x), (λi fi−1 )i∈I ] < U Γ (x)[a, (λi fi−1 )i∈I ] for some a ∈ K Γ (x). Now let B µ be the best-response correspondence for the game U Γ where µ := (λi fi−1 )i∈I and let φ be a measurable selection of B µ . Define gi : Ti → Ai by letting gi (ti ) = fi (ti , i) if (ti , i) ∈ Xi − Ci and gi (ti ) = φ(ti , i) if (ti , i) ∈ Ci . Let Di = {ti : (ti , i) ∈ Ci }. Thus, by the fact that Gi does not depend on τi gi−1 (see Eq. 5.4) and Eq. (5.3), we have Gi (ti , gi (ti ), (τj gj−1 )j∈I ) ≥ Gi (ti , gi (ti ), (τj gj−1 )j∈I ) for all ti ∈ Ti , 5.6 Proofs 86 and Gi (ti , gi (ti ), (τj gj−1 )j∈I ) > Gi (ti , gi (ti ), (τj gj−1 )j∈I ) for all ti ∈ Di . Thus, by Equation (5.3), we also have Ei (gi , g−i ) > Ei (g) which contradicts with the claim that g is a pure strategy equilibrium for the Bayesian game Γ. Therefore, our hypothesis is false and f is indeed a pure strategy equilibrium for the game U Γ. 5.6.3 Proof of Theorem 11 By Theorem 10, cases (i) and (ii) of Theorem 11 follow directly form cases (i) and (iii) of Theorem 8, and cases (iii) and (iv) of Theorem 11 can be derived from case (ii) of Theorem by adopting the following transformations. Consider case (iii) first. Suppose that the condition (iii) holds, i.e., νi = τi × δi and let g = (gj )j∈I be a pure strategy profile. Then by Equation (5.1), the expected payoff for player i can be expressed as τj gj−1 ui (gi (ti ), a−i , si )dνi d Ei (g) = A−i Ti ×Si = j=i τj gj−1 ui (gi (ti ), a−i , si )dδi dτi d A−i Ti Si = j=i τj gj−1 dτi ui (gi (ti ), a−i , si )dδi d Ti = Ti A−i Si Gi (gi (ti ), (τj gj−1 )j∈I )dτi . j=i (5.8) 5.6 Proofs 87 where Gi is a function from Ai × Gi (ai , (µj )j∈I ) = for any (µj )j∈I ∈ j∈I j∈I M (Aj ) to R, defined by ui (gi (ti ), a−i , si )dδi d A−i Si µj , j=i M (Aj ). Obviously, Gi is continuous from Ai × j∈I M (Aj ) to R. Notice that Gi does not depend directly on ti now. Therefore, following the same procedure in Section 5.3.2, we can induce a generic large game U Γ of Γ with the property that all players in group i play the same payoff function Gi . Thus, by Theorem 10 and case (ii) of Theorem 8, our result in case (iii) holds. Now suppose the condition (iv) holds, that is, Si = (siq )q∈Q , where Q is a countable index set. For simplicity, let αq := δi (siq ) for all q ∈ Q, and given siq ∈ Si , let the conditional probability of νi on Ti be denoted by τiq . Let g = (gj )j∈I be a pure strategy profile. Then, the expected payoff for player i can be expressed as Ei (g) = τj gj−1 ui (gi (ti ), a−i , si )dνi d A−i Ti ×Si = j=i αq A−i q∈Q = Ti αq A−i Ti q∈Q = Ti τj gj−1 dτiq j=i αq dτiq = dτi τj gj−1 j=i ui (gi (ti ), a−i , siq )d Ti q∈Q ui (gi (ti ), a−i , siq )dτiq d τj gj−1 dτi ui (gi (ti ), a−i , siq )d A−i Gi (gi (ti ), (τj gj−1 )j∈I )dτi , j=i (5.9) 5.6 Proofs 88 where Gi is a function from Ai × Gi (ai , (µj )j∈I ) = q∈Q for any (µj )j∈I ∈ j∈I value in [0, 1] and continuous from Ai × j∈I M (Aj ) to R, defined by αq dτiq dτi ui (gi (ti ), a−i , siq )d A−i M (Aj ). Since τi = q∈Q αq dτiq (ti ) dτi j∈I µj , j=i q∈Q αq τiq , αq dτiq dτi is well defined with = for τi -almost all ti . Obviously, Gi is also M (Aj ) to R. Also notice that Gi does not depend directly on ti now. Therefore, following the similar procedure in Section 5.3.2, we can induce a generic large game U Γ of Γ with the property that all players in group i play the same payoff function Gi . Thus, by Theorem 10 and case (ii) of Theorem 8, our result in case (iv) holds. 5.6.4 Proof of Theorem 12 Fix i ∈ I. For simplicity, we denote η0 ({t0k , s0q }) by βkq . Notice that the conditional probability τikq is uniquely defined only when β kq > 0. When β kq = 0, we can redefine τikq to be τi without changing anything. Since τi = k∈K q∈Q β kq τikq , τikq is atomless and absolutely continuous with respect to τi whenever β kq > 0. Therefore, we can assume that for each k ∈ K, q ∈ Q, τikq is atomless and absolutely kq continuous with respect to τi . Let φkq i be the Radon-Nikodym derivative of τi with respect to τi , which is integrable on (Ti , Ti , τi ). Also, let (M (Ai ))|Q| be the product space of |Q| copies of M (Ai ) with the product topology, which is also a compact metrizable space. 5.6 Proofs 89 Let h = (hi )i∈I be a pure strategy profile. Denote hi (t0k , ti ) by hki (ti ) for all k ∈ K. Thus, for each k, hki is a measurable mapping from Ti to Ai . With the assumption of conditional independence in D1 , we can rewrite player i’s payoff in Equation (5.6) as Ei (h) = βkq j∈I (Tj ×Sj ) k∈K q∈Q = βkq Ti ×Si ×T−i k∈K q∈Q = βkq Ti ×Si ×T−i k∈K q∈Q ui ((hkj (tj ))j∈I , s0q , si )dρkq i ui ((hkj (tj ))j∈I , s0q , si )d νikq × Πj=i τjkq βkq = Ti ×Si ×A−i k∈K q∈Q = ui (hki (ti ), a−i , s0q , si )d νikq × Πj=i τjkq (hkj )−1 βkq A−i k∈K q∈Q ui ((hkj (tj ))j∈I , s0q , si )dη kq Ti ×Si ui (hki (ti ), a−i , s0q , si )dνikq d τjkq (hkj )−1 . (5.10) j=i Define the regular conditional expectation vikq (a, ti ) = E{ui (a, s0q , si )|ti } as we did in Section 5.3.2. Then, we have Ti ×Si ui (hki (ti ), a−i , s0q , si )dνikq = Ti vikq (hki (ti ), a−i , ti )dτikq . (5.11) Substituting (5.11) into (5.10), we get Ei (h) = βkq A−i k∈K q∈Q = k∈K Ti q∈Q βkq φkq i (ti ) Vik (hk ), = k∈K Ti vikq (hki (ti ), a−i , ti )dτikq d τjkq (hkj )−1 j=i A−i vikq (hki (ti ), a−i , ti )d τjkq (hkj )−1 dτi j=i (5.12) 5.6 Proofs 90 where hk = (hki )i∈I and Vik (hk ) = Ti q∈Q = Ti βkq φkq i (ti ) vikq (hki (ti ), a−i , ti )d A−i τjkq (hkj )−1 dτi j=i wik (ti , hki (ti ), [τjkq (hkj )−1 ]q∈Q,j∈I )dτi , where wik is a function from Ti × Ai × βkq φkq i (ti ) wik (ai , ti , γ) = q∈Q for any given γ = (γi )i∈I ∈ j∈I j∈I A−i (5.13) (M (Aj ))|Q| to R, defined by vikq (ai , a−i , ti )d( γjq )(a−i ), (5.14) j=i (M (Aj ))|Q| , where γi = (γiq )q∈Q . It is obvious that for each fixed ∈ Ai and γ ∈ j∈I (M (Aj ))|Q| , wik (ai , ti , γ) is Ti measurable. For any ti ∈ Ti , the fact that wik (ai , ti , γ) is continuous on Ai × j∈I (M (Aj ))|Q| follows from the Stone-Weierstrass theorem. Now compare Equations (5.13) and (5.14) with Equations (5.3) and (5.4) in Section 5.3.2. We see they have the same properties. Thus, following the same procedure in Section 5.3.2, we see that for each k ∈ K, the term Vik can induce a large game U k whose payoff is give by U k = wik . By the same reasoning as in Theorem 10, we have that there exists a hk∗ = (hk∗ i )i∈I such that for all i ∈ I, k Vik (hk∗ ) ≥ Vik (hki , hk∗ −i ), for all hi ∈ Meas(Ti , Ai ) (5.15) iff there is a pure strategy equilibrium for the game U k . Now suppose for each k ∈ K, hk∗ = (hk∗ i )i∈I satisfies Equation (5.15). Then the pure strategy profile h∗ = (h∗i )i∈I for game Υ defined by letting h∗i (t0k , ti ) = hk∗ i (ti ) for all k ∈ K and ti ∈ Ti is obviously a pure strategy equilibrium for game Υ. 5.6 Proofs Therefore, parts (i) and (ii) in our theorem follow from parts (i) and (iii) in Theorem and parts (iii) and (iv) in our theorem follow from part (ii) in Theorem by doing some similar transformation as illustrated in the Proof of Theorem 11. 91 Bibliography Aliprantis, C., Border, K., 1994. 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Springer-Verlag, pp. 23–48. 97 [...]... strategy-relevant private information is diffuse and their private information is conditionally independent given the public and payoff-relevant common information 1 This chapter is based on the joint publication of Fu, Sun, Yannlis and Zhang in 2007 8 2.2 Games with private and public information The proof of the existence of pure strategy equilibrium in our setting is far from trivial and requires the use of some... equilibrium distributions in large games Chapter 5 is for connecting large games and Bayesian games It has long been noted that there is a close relationship between large games and Bayesian games (see eg, Mas-Colell (1984), Khan and Sun (1995, 1999)) But no formal connection was established between the two types of games They are still regarded as two separate types of games without any direct links In... Depending on whether or not the game is played simultaneously by all the players, games are classified into static games and dynamic games The games discussed in this paper, i.e., Bayesian games and large games, all belong to noncooperative static games Bayesian games, also called games of incomplete information, are games in which at least one player is uncertain about another player’s payoff function... Pure-strategy equilibria in games with private and public information1 2.1 Introduction We introduce a generalized Bayesian game model 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 payoffrelevant private information and some payoff-relevant common information The purpose of this... (1967-68, 1973)’s framework, Milgrom and Weber (1981, 1 The information space is said to be diffuse if it is an atomless probability space 2 1.1 Some backgrounds 1985) and Radner and Rosenthal (1982) gave a comprehensive theory of Bayesian games and proved the existence of pure strategy equilibria in Bayesian games with a finite number of players and a finite number of actions Khan and Sun (1995) presented a generalized... existence proof Section 4 contains some concluding remarks All the proofs are given in the appendix 2.2 Games with private and public information Consider a game Γ with private and public information formulated as follows The game has finitely many players i = 1, , l Each player i is endowed with a finite action set Ai , a measurable space (Ti , Ti ) representing her strategy-relevant 9 2.2 Games with private. .. studied in Radner and Rosenthal (1982) Milgrom and Weber (1985) considers games with payoff-relevant common information and private information that influences players’ strategies and payoffs.3 Our model introduces the new concept of public information that influences all players’ strategies, in addition to payoff-relevant and strategy-relevant private information and payoff-relevant common information It is... Mixed-strategy equilibria and strong purification1 3.1 Introduction For a game with incomplete information, three types of information, namely strategyrelevant private information, payoff-relevant private information and payoff-relevant common information, are considered in previous literatures (see Radner and Rosenthal (1982), Milgrom and Weber (1985), Yannelis and Rustichini (1991) and Khan and Sun (1999))... of pure strategy equilibria, which allows players to have countably many (finite or countably infinite) actions Khan and Sun (1999) models the set of players as a Loeb space and shows the existence of pure strategy equilibria in Bayesian games with uncountable actions In contrast, a large game is a game where the set of players is endowed with an atomless measure Thus the number of the players in a large. .. player is endowed with finite actions Khan and Sun (1995) generalized the result of Schmeidler (1973) to allow a countable set of pure strategies The usage of hyperfinite Loeb spaces in modeling large games was systematically studied in Khan and Sun (1996, 1999) By modeling the set of players as a Loeb space, Khan and Sun (1999) shows the existence of Nash equilibria in large games without any countability . EQUILIBRIA OF LARGE GAMES AND BAYESIAN GAMES WITH PRIVATE AND PUBLIC INFORMATION FU HAIFENG (B.S., Fudan Univ. and M.A., East China Normal Univ.) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF. games with private and public information and large games. Based on this connection and the existence results in large games, we obtain more general- ized existence results of pure strategy equilibria. 3 1985) and Radner and Rosenthal (1982) gave a comprehensive theory of Bayesian games and proved the existence of pure strategy equilibria in Bayesian games with a finite number of players and a