Genericity of Critical Types

Transcription

1 Genercty of Crtcal Types Y-Chun Chen Alfredo D Tllo Eduardo Fangold Syang Xong September 2008 Abstract Ely and Pesk 2008 offers an nsghtful characterzaton of crtcal types: a type s crtcal f and only f t has common-p belef on some closed, proper subset of the unversal type space. Thus, all types ever consdered n applcatons are crtcal. On the other hand, Ely and Pesk 2008 show that the crtcal types form a meager subset of the unversal type space under the product topology. We propose two ways to resolve ths puzzle by provng two genercty results for crtcal types. Frst, under the strategc topology due to Dekel, Fudenberg, and Morrs, 2006, Theoretcal Economcs , the set of crtcal types s open and dense n the unversal type space. Second, geometrcally, the set of crtcal types s prevalent.e. the complement of a fntely shy set. 1

2 1 Introducton Ely and Pesk 2008 offer a very clean characterzaton of crtcal types: a type s crtcal f and only f t has common-p belef on some proper closed subset of the unversal type space. Consequently, all type spaces ever consdered n applcatons conssts entrely of crtcal types. However, they also offer a puzzlng result that crtcal types are non-generc,.e., they form a meager set n the unversal type space under the product topology. We propose two ways to resolve ths puzzle by provng two genercty results for crtcal types. Frst, under the stronger strategc topology due to Dekel, Fudenberg, and Morrs, 2006, Theoretcal Economcs , crtcal types s open and dense n the unversal type space. Second, f we dentfy the set of types wth Ω U Ω under the standard Mertens-Zamr homeomorphsm, the set of regular types.e., the complement of crtcal types form a proper face n Ω U Ω and hence t s fntely shy. 2 Prelmnares Let d denote the product metrc on the unversal type space. For any set W U Ω, denote the ε-open ball contanng W under the product topology by W ε,.e., W ε := {t U Ω : d t, t < ε for some t W }. Defnton 1 Common belef convergence A sequence of types u n n=1 converges to a type u n common-p belef f for any p > 0 and any closed proper subset W U Ω wth u C p W and any ε 0, p, there exsts a postve nteger N such that u n C p ε W ε for any n N. 3 The Frst Genercty Result In ths secton, we prove crtcal types are generc n a topologcal sense. 2

3 Theorem 1 Under the strategc topology, crtcal types contan an open and dense subset of the unversal type space. By Theorem 4 of Dekel, Fudenberg, and Morrs 2006, fnte types are dense n the unversal type space under the strategc topology. Snce fnte types are crtcal by Theorem 3 of Ely and Pesk 2008, crtcal types are dense under the strategc topology. Therefore, Theorem 1 s a drect consequence of the followng Proposton. Proposton 1 The strategc closure of regular types contans no fnte types. To prove Proposton 1, we need the followng Proposton, whose proof s relegated to Appendx 1. Proposton 2 u n n=1 converges to u under the strategc topology only f u n n=1 converges to u n common-p belef. Proof of Proposton 1. Suppose nstead that a sequence of regular types u n n=1 converges to a fnte type u under the strategc topology. Snce u s a fnte type, u C p W for a fnte set W U Ω. Moreover, snce u n n=1 converges to u under the strategc topology, u n n=1 also converges to u n common-p belef by Proposton 2. Snce W s fnte, for suffcently small ε > 0, the product closure of W ε, denoted by W ε, s stll a proper closed subset of U Ω, and moreover, p ε > 0. Snce u n n=1 convergng to u n common-p belef, by Defnton 1, there exsts a postve nteger N such that u n C p ε W ε. Hence, u n s a crtcal type for all n N by Ely and Pesk 2008, whch contradcts to the assumpton that u n s regular for all n. Remark. Our proof s stll vald f we consder the soluton concept of IIR nstead of ICR. 4 The Second Genercty Result In the secton, we prove that the set of crtcal types, vewed as a subset of Ω U Ω under the standard Mertens-Zamr homeomorphsm, s generc n a geometrc sense,.e., t 3

4 s prevalent. Throughout ths secton, for any two types u, u U Ω and α 0, 1, defne αu + 1 α u π 1 απ u + 1 α π u ] where π s the Mertens-Zamr homeomorphsm between U Ω and Ω U Ω. Hence, αu + 1 α u U Ω. The followng proposton shows that crtcal types form a proper face n the unversal types space. Proposton 3 For any α 0, 1, u regular. = αu + 1 α u s regular ff both u and u are Proof. We prove frst the only f part. Suppose u s crtcal. By Theorem 3 of Ely and Pesk 2008, u C p W for some closed proper subset W U Ω. By Lemma 6 of Ely and Pesk 2008, C p W = W B p Cp W U Ω. Hence, u W {u } B p Cp W {u } U Ω where p = mn {p, α}. By Lemma 6 of Ely and Pesk 2008, u C p W {u }. Snce W {u } s a proper closed subset of U Ω, u s crtcal by Theorem 3 of Ely and Pesk We then turn to show the f part. Suppose that u s crtcal. Then, By Theorem 3 of Ely and Pesk 2008, u C p W for some closed proper subset W U Ω. By Lemma 6 of Ely and Pesk 2008, C p W = W B p Cp W U Ω. Snce u = αu + 1 α u, ether u B p Cp W U Ω or u B p Cp W U Ω. Hence, by Lemma 6 of Ely and Pesk 2008, ether u or u s crtcal. Theorem 2 The set of crtcal types s prevalent. Proof. Ths s a drect consequence of our Proposton 3 and Hefetz and Neeman 2006 s Lemma 1. 4

5 A Appendx A.1 Proof of Proposton 2 Proof. Suppose that u n n=1 does not converge to u n common-p belef. Then, there exsts a W U Ω wth u C p such that u n W and some ε > 0 such that there s a subsequence, say tself, / C p ε W ε for all n. We wll show that u n n=1 does not converge to u under strategc topology. Snce W ε s open, U Ω \W ε s closed and obvously also nonempty. Snce W and U Ω \W ε are two dsjont nonempty closed subset of U Ω, by Lemma 3, there exst γ > 0 and a game G = A =1,2, g =1,2 such that propertes 1 3 n Lemma 3 are satsfed. Defne G n the same way as defned n the proof of Lemma 2 of Ely and Pesk Defne the followng set of herarches V = { u : A 0 A 1 u R u G, 0 } ; V γ = { u : A 0 A 1 u R u G, γ = }. By the property 3 of Lemma 3, W V V γ W ε. Let Z = A 0 A 1 {1}. By the proof of Lemma 2 of Ely and Pesk 2008, 1. If u C p V, then Z R u G, 0 ; 2. If u / C p 2γ1 p V γ, then Z R u G, γ =. Note that γ can be chosen to be small so that 2γ 1 p < ε wthout losng property 3 of Lemma 3. Snce u n / C p ε W ε for all n and V γ W ε, u n / C p 2γ1 p V γ. Therefore, u n n=1 does not converge to u under strategc topology. A.2 Proof of Lemma 3 Lemma 1 Suppose that v u. There are open neghborhoods V v and U u, game G = A j, g j, acton a A, ε > 0 and m such that a R m u G, 0 for all u U and a / R m v G, ε for all v V. 5

6 Proof. Snce v u, by Lemma 17 of Ely and Pesk 2008, there s a game G, an acton a, ε > 0, and some postve nteger m such that a R m u G, 0 and a / R m v G, ε. By Lemma 16 of Ely and Pesk 2008, there s some open neghborhoods V v and U u such that a R m u G, ε/3 for all u U and a / R m v G, 2ε/3 for all v V. Then, by Lemma 4 of?, there s a game G = A j, g j and an acton a A such that a R m u G, 0 for all u U and a / R m v G, ε/3 for all v V. Lemma 2 Suppose that v u. There are ε > 0, open neghborhoods V v and U u, a game G = A j, g j such that A = A 0 A 1 and 1. For any a A, any a 1 A 1 and any a 0, a 0 A 0, any ω, g a, a 0, a 1, ω = g a, a 0, a 1, ω. 2. There are correspondence A 0 : U Ω A 0, A 1 : U Ω A 1 such that for all u, R u G, 0 = A 0 u A 1 u. 3. There s an acton a 0 A 0 such that { } a 0 A 1 u R u G, 0 for all u U ; { } a 0 A 1 v R v G, 0 = for all v V. Proof. Ths follows from the proof of Lemma 19 of?. Lemma 3 Fx player. Let W and W be nonempty closed subsets of U Ω such that W W =, there are ε > 0 and a game G = A j, g j such that A = A 0 A 1 and 1. For any a A, any a 1 A 1 and any a 0, a 0 A 0, any ω, g a, a 0, a 1, ω = g a, a 0, a 1, ω. 2. There are correspondence A 0 : U Ω A 0, A 1 : U Ω A 1 such that for all u, R u G, 0 = A 0 u A 1 u. 6

7 3. There s a nonempty subset A 0 A 0 such that A 0 A 1 u R u G, 0 for all u W ; A 0 A 1 v R v G, ε = for all v W Proof. Frst, fx u W. Snce W s an closed set and W W =, by Lemma 2, for every v W, there are ε > 0, open neghborhoods V v v and U u u, a game G v = A v j, gv j such that A v = A v 0 A v 1 and an acton a v 0 propertes 1 3 n Lemma 2 are satsfed. Snce V v v W s an open cover of W, there s a fnte subcover V v1,...v vk of W. Let U 1,..., U K be the correspondng open neghborhoods of u, G v1,..., G v K be the correspondng games for u and v k, k = 1,..., K, respectvely, and a v1 0,..., a vk 0 be the correspondng actons n property 3 of Lemma 2. Let G u be the product game of G v1,..., G v K. Defne { } A u 0 = a v1 0,..., a vk 0. Let U u K k=1 U k. Then, propertes 1 and 2 are satsfed for G u and A 0 A 1 u R u G u, 0 for all u U u ; A u 0 A 1 v R v G, 0 = for all v W. Snce U u s an open cover of W, there s a fnte subcover U u W u1,..., U W. Let G be the product game of G u1 u,..., G K. Let A 0 { } a 0 A 0 : a 0k A uk 0, a 0 A 0 u for some u. u K of We can verfy that A 0 has the desred propertes. A.3 An Alternatve Proof of Proposton 2 We prove the contra-postve statement of Proposton 2: f t n n=1 does not converge to t n common-p belef, then t n n=1 does not converge to t under the strategc topology. Suppose t n n=1 does not converge to t n common-p belef,.e., there exsts a productclosed E T wth t C p E and some ε > 0, but there s a subsequence t nk k=1 such that t nk / C p E ε for all k. For notatonal ease, we use t k to denote t nk. 7

8 We wll show that t k k=1 does not converge to t under strategc topology, whch proves that t n n=1 does not converge to t under the strategc topology. To prove Proposton 2, we need the followng proposton 1, whose proof s relegated to Appendx A.4. Proposton 4 For any two dsjont non-empty closed sets U, V T, there exst γ > 0 and a game  Ĝ =, ĝ =1,2 such that f an acton â  s 0 ratonalzable for =1,2 some t U, then â s not γ ratonalzable for t V. Snce E ε s open, ts complement denoted by E ε C s closed. Then E and E ε C are two dsjont non-empty closed sets. By Lemma??, there exst γ > 0 and a game Ĝ = Â, ĝ =1,2 such that f an acton â  s 0 ratonalzable for some =1,2 t E, then â s not γ ratonalzable for t E ε C. Defne E γ := Defne a new game G = a = â, z, â, z ], we have Λ = t E R t, Ĝ, 0 ; {t : R t, Ĝ, γ Λ }. A =1,2, g =1,2 such that A 1, f z = z = 1; g a p, θ = ĝ â, â, θ + 1 p, f z = 1 and z = 0; 0, f z = 0. =  {0, 1} and for 1, f z = z = 1 and â Λ; g a p, θ = ĝ â, â, θ + 1 p, f z = 1 and â / Λ or z = 0; 0, f z = 0. Just lke the proof n Ely and Pesk 2008, we can show that 1 Proposton 4 s a stronger verson of DFM s lemma 4, whch says that for any two dstnct types t and s, there exst γ > 0 and a game  Ĝ =, ĝ =1,2 such that f an acton â  s 0 ratonalzable =1,2 for some t, then â s not γ ratonalzable for s. 8

9 ] 1. If t C p E, then  {1} R t, G, 0 ; 2. If t / C p E γ, then ]  {1} R t, G, γ =. Note that E γ E ε, because for any â  beng 0 ratonalzable for some t E s not γ ratonalzable for t E ε C. Recall that t C p E and t k / C p E ε for all k. Hence, t k / C p E γ for all k. Thus, ]  {1} R t, G, 0 ; ]  {1} R t k, G, γ = for all k. Therefore, t k k=1 does not converge to t under strategc topology. A.4 The proof of Proposton 4 Proposton 4 s an mmedate consequence of the followng two lemmas. For the ease of exposton, we relegate the proofs of these lemmas to Appendx A.4 and A.4. For any E T, defne T k,e as T k,e := {t T : π k t = π k t for some t E}. Lemma 4 For any two dsjont closed sets U, V T, there exsts some postve nteger k such that U k and V k are two dsjont closed sets n T k. Lemma 5 For any postve nteger k and two dsjont non-empty closed sets W, Z T k, there exst γ > 0 and a game  Ĝ =, ĝ =1,2 such that f an acton â  s =1,2 0 ratonalzable for some t wth π k t W, then â s not γ ratonalzable for any t wth π k t Z. 9

10 The proof of Lemma 4 are close for any k. Frst, clearly, f U, V T are closed, the two sets U k, V k T k Second, for any E T, defne T k,e as T k,e := {t T : π k t = π k t for some t E}. We prove the followng clam. Clam 1 For any close E T, k=1 T k,e = E. Proof. Frst, we show that k=1 T k,{t} = {t}. Apparently, k=1 T k,{t} {t}. Also, k=1 T k,{t} {t}. Suppose not,.e. there s t k=1 T k,{t}, but t t. Snce t t, π k t π k t for some k, whch contradcts to t T k,{t}. Therefore, k=1 T k,{t} {t}. Second, by coherence, T 1,E T 2,E T 3,E... T k,e... E. Thus, k=1 T k,e E. Thrd, we show that k=1 T k,e E. Pck any t k=1 T k,e. Note, T k,{t} E s closed, compact and non-empty 2. Thus, k=1 Tk,{t} E ] k=1 T k,{t} = {t}. Therefore, k=1 Tk,{t} E ] = {t} and {t} = k=1 Tk,{t} E ] E. Second, we prove the followng clam. Clam 2 For any two dsjont closed sets U, V T, there exsts some postve nteger k such that T k,u and T k,v are dsjont. Proof. suppose otherwse,.e., T k,u Tk,V for all k. Hence, Then, ] T k,u Tk,V = k Z + k Z + T k,u k Z + T k,v T k,u Tk,V ]. k Z + = U V, 1 where the last equalty follows from Clam 1 and the fact that U and V are closed. 1 contradcts the the assumpton that U and V are dsjont. Thrd, note that T k,u ] k = U k and T k,v ] k = V k. Therefore, Lemma 4 s true. 2 Consder the product toplogy on T. The functon γ : T π k T such that γt = π k t s contnuous. Hence, T k,{t} s closed. Also, t s compact because T s compact. 10

11 The proof of Lemma 5 Followng DFM, we use quadratc scorng rule to elct type s k-th order belef. In ths scorng rule game  Ĝ =, ĝ =1,2, the acton set s  = B 1... B k T 1... T k, where T k set of fnte dscrete grds n T k. Note that B k B k 1. =1,2 s the space of k-th order belef and B k s a For any type t, let b 1 t,..., b k t ] denote an arbtrary 0-ratonalzable strategy for t, where b 1 t,..., b k t correspond to the 1st,..., the k-th order belefs respectvely. Let t k denote t s true k-th belef. We say b k t s t s k-th reportng belef. We defne t s k-th nduced belef, I k t as follows. I k t E] = t {t : b k 1 t E} ] for E T k 1. Wth the quadratc scorng rule, the players 0-ratonalzable strategy s to truthfully report ther belefs subject measurement errors due to fnte grds,.e., they chooses b k t to be the grd whch s mostly closed to I k t. To defne Ĝ, we only need to defne the grds of Â. We wll show that the grds can be approxmately chosen so that Lemma 5 s true. We establsh ths by the followng three steps. Step 1 For any k and δ > 0, we can choose the grds so that ρ k t k, b k t ] < δ and ρ k t k, I k t ] < δ. Step 2 There exsts a game Ĝ =  =1,2, ĝ =1,2 such that f an acton â  s 0 ratonalzable for some t wth π k t W, then â s not 0 ratonalzable for any t wth π k t Z. Step 3 There exsts γ > 0 such that n Ĝ, f an acton â  s 0 ratonalzable for some t wth π k t W, then â s not 0 ratonalzable for any t wth π k t Z. Gven Step 1 and 2, the proof of Step 3: The Ĝ n Step 2 s a quadratc scorng rule on the frst k order belefs, where k s the one n Step 1. Hence, the ICR actons n ths game s determned only by the frst k order belefs,.e., R s, Ĝ, ε = R s, k Ĝ, ε and h k s â, Ĝ = h s â, Ĝ for any s T, 11

12 â Â and ε 0. By Ely-Pesk Lemma 16, h k s â, Ĝ s a contnuous functon n s. Hence h s â, Ĝ s a contnuous functon n s too. Consder L = {s T : π k s W} R s, Ĝ, 0 Â, whch s fnte. For any â L, h s â, Ĝ > 0 for any s wth π k s Z by Step 2. Snce Z s closed hence compact and h s â, Ĝ s contnuous, there exsts γ ba > 0 such that h s â, Ĝ > γ ba for any s wth π k s Z. Let γ = mn {γ ba : â L}. Snce L s fnte and γ ba > 0 for any â L and L s fnte, we have γ > 0. Further h s â, Ĝ > γ for any s wth π k s Z and any â L. Therefore, Step 3 holds. Gven Step 1, the proof of Step 2: Snce W and Z are closed, there s 3δ > 0 such that ρ k t k, t k] > 4δ for any two types t, t wth π k t W and π k t Z. We can choose the scorng rule wth the grd A = B 1... B k 1 such that ρ k s k, I k s ] < δ for any s T. Hence, any two types t, t wth π k t W and π k t Z, ρ k I k t, I k t ] ρ k t k, I k t ] ρ k t k, I k t ] ρ k t k, t k] ρ k I k t, t k] ρ k t k, I k t ] > 3δ δ δ > δ. Let B k {0, 1 q 1,...,, q q 1} Bk 1. Choose q such that 1 < δ. q 4 B k 1 Snce ρ k I k t, I k t ] > δ, there s some E T k 1, I k t E] > I k t I k t E] + δ. Hence, I k t E B k 1] > I k t E B k 1] + δ. E] δ] + δ Then, there s some b E B k 1] such that I k t b] > I k t δ b] + E B k 1 Ik t b] + δ. Snce 1 < δ, t and B k 1 q 4 B k 1 t must report dfferent grds. Therefore, any 0-ratonalzable grd for t can not be 0-ratonalzable for t. 12

13 The proof of Step 1: We prove ths by nducton. Frst, consder k = 1. Let B 1 {0, 1 n 1,...,, 1} {0, 1 n 1,...,, 1}. Choose n such n n n n that 1 < δ. Hence, for any two adjacent grds b, n b B 1, ρ 1 b, b < δ. Because of the quadratc scorng rule, a type t report b 1 t as the most closed grd to t 1. Clearly, ρ 1 t 1, b 1 t < δ. We show that ρ 2 t 2, I 2 t ] < δ. For any E T k 1, I 2 t E] = I 2 t E B 1] = t {t : b 1 t E B 1 } ] t {t : t E B 1] δ } ] ] t {t : t E] δ } < t E] δ] + δ, where the frst nequalty follows from ρ 1 t 1, b 1 t < δ, hence {t : b 1 t E B 1 } E B 1 ] δ. Therefore, ρ 2 t 2, I 2 t ] < δ. Suppose ρ k 1 t k 1, b k 1 t ] < δ for a general k. We show that ρ k t k, I k t ] < δ. For any E T k 1, I k t E] = I k t E B k 1] = t {t : b k 1 t E B k 1 } ] t {t : t E B k 1] ] δ } ] t {t : t E] δ } < t E] δ] + δ, where the frst nequalty follows from ρ k 1 t k 1, b k 1 t ] < δ, hence {t : b k 1 t E B k 1 } E B k 1] δ. Therefore, ρ k t k, I k t ] < δ. Suppose ρ k t k, I k t ] < δ for a general k. We show ρ k t k, b k t ] < δ. Let B k {0, 1 h 1,...,, h h 1} Bk 1. Choose h such that 1 < δ. Hence, for any two h B k 1 13

14 adjacent grds b, b B k, ρ k b, b < δ. We show ρ k t k, b k t ] < δ. For any E T k 1, b k t E] = b k t E B k 1] t {t : b k 1 t E B k 1 } ] + δ E B k 1 B k 1 t {t : t E B k 1] ] δ } + δ t E] δ] + δ, where the frst nequalty follows because b k t s chosen as the closed grd to t k. Therefore, ρ k t k, b k t ] < δ. References Dekel, E., D. Fudenberg, and S. Morrs 2006: Topologes on Types, Theoretcal Economcs, 1, Ely, J. C., and M. Pesk 2008: Crtcal Types, mmeo., Northwestern Unversty. 14