Virtual Robust Implementation and Strategic Revealed Preference

Transcription

1 and Strategic Revealed Preference Workshop of Mathematical Economics Celebrating the 60th birthday of Aloisio Araujo IMPA Rio de Janeiro December 2006

2 Denitions "implementation": requires ALL equilibria deliver the right outcome, a.k.a. full implementation "robust": same mechanism works independent of agents' beliefs and higher order beliefs about the environment "virtual": enough if correct outcome arises with probability 1 " 1

3 Single Good, Private Values I I agents agent i has valuation i 2 [0; 1] Second Price Sealed Bid Auction { object goes to highest bidder { winner pays second highest bid 2

4 Single Good, Private Values I I agents agent i has valuation i 2 [0; 1] Second Price Sealed Bid Auction { object goes to highest bidder { winner pays second highest bid Truth-Telling is a dominant strategy, but there are inecient equilibria 3

5 Single Good, Private Values II Modied Second Price Sealed Bid Auction With probability 1 ", { allocate object to highest bidder { winner pays second highest bid For each i, with probability "b i I { i gets object { pays 1 2 b i 4

6 Single Good, Private Values II Modied Second Price Sealed Bid Auction With probability 1 ", { allocate object to highest bidder { winner pays second highest bid For each i, with probability "b i I { i gets object { pays 1 2 b i Truth-Telling is a strictly dominant strategy, so Robust Virtual Implementation 5

7 Single Good, Interdependent Values I I bidders Bidder i has type i 2 [0; 1] Bidder i's valuation is v i = i + X j6=i j Modied Second Price Sealed Bid Auction { object goes to highest bidder { winner pays max b j + X j6=i j6=i b j 6

8 Single Good, Interdependent Values I I bidders Bidder i has type i 2 [0; 1] Bidder i's valuation is v i = i + X j6=i j Modied Second Price Sealed Bid Auction { object goes to highest bidder { winner pays max b j + X j6=i j6=i b j If 1, truth-telling is a "ex post" equilibrium, but there are inecient equilibria 7

9 Single Good, Interdependent Values II Modied Second Price Sealed Bid Auction With probability 1 ", { allocate object to highest bidder i { winner pays max b j + X b j j6=i j6=i For each i with probability "b i I, { i gets object { pays 1 2 b i+ X j6=i b j 8

10 Single Good, Interdependent Values II Modied Second Price Sealed Bid Auction With probability 1 ", { allocate object to highest bidder i { winner pays max b j + X b j j6=i j6=i For each i with probability "b i I, { i gets object { pays 1 2 b i+ X j6=i b j Truth telling is a strict ex post equilibrium 9

11 Two cases Single Good, Interdependent Values III if < 1 I 1, Robust Virtual Implementation (in the modied direct mechanism) if 1 I 1, inecient multiple equilibria in this AND ALL OTHER mechanisms 10

12 This Paper Necessary and Sucient Conditions for in General Environment: 1. Ex Post Incentive Compatibility in example, 1 2. "Robust Measurability" or Not Too Much Interdependence in example, < 1 I 1 11

13 Abreu-Matsushima (1992) Incomplete Information: Setup Standard "Bayesian" incomplete information setting, i.e., common knowledge of common prior on type space More demanding in two ways: { Iterated deletion of strictly dominated strategies { Compact mechanisms Less demanding in one way: { "Virtual" implementation 12

14 Abreu-Matsushima (1992) Incomplete Information: Results Necessary conditions for virtual implementation Bayesian incentive compatibility Abreu-Matsushima measurability: types are iteratively distinguishable { reduces to "value distinction" in private values case "Slicing" mechanism shows suciency of Bayesian incentive compatibility and Abreu-Matsushima measurability 13

15 Adding Robustness I With robustness, full implementation equivalent to belief free version of iterated deletion of strictly dominated strategies 14

16 Adding Robustness II Generalizing Abreu-Matsushima, necessary conditions become: 1. Ex post incentive compatibility (instead of Bayesian IC) Bergemann-Morris "Robust Mechanism Design" 15

17 Adding Robustness II Generalizing Abreu-Matsushima, necessary conditions become: 1. Ex post incentive compatibility (instead of Bayesian IC) Bergemann-Morris "Robust Mechanism Design" 2. A belief free version of iterative distinguishability (instead of AM measurability) 16

18 Strategic Revealed Preference I Forget implementation question... Two types are "distinguishable" if there exists a mechanism (game) such that those types choose dierent messages (action) in every equilibrium on every type space We characterize distinguishability for general environments { in single good environment every pair of types are indistinguishable if 1 I 1 every pair of distinct types are distinguishable if < 1 I 1 17

19 Strategic Revealed Preference II Gul and Pesendorfer (2005): "The Canonical Type Space for Interdependent Types." What is meant by inderdependent utility representations of preferences? 1. Neuroeconomics procedural interpretation 2. Traditional "mindless" revealed preference interpretation Our exercise diers from Gul-Pesendorfer explicitly strategic 2. static mechanism ) no counterfactual information used 18

20 3. GP require that distinguishable types MIGHT behave dierently; we require that they MUST behave dierently 19

21 Back to Virtual Implementation Denition of "Robust Measurability": indistinguishable types the same social choice function treats Ex post incentive compatibility and robust measurability are necessary for virtual robust implementation Adaption of Abreu Matsushima slicing mechanism will show them to be sucient 20

22 Exact Implementation I Following Maskin methods, necessary and sucient conditions for exact robust implementation - using ANY mechanism: (Bergemann-Morris "Robust Implementation: Spaces") The Role of Large Type 1. ex post incentive compatibility 2. "robust monotonicity": not too much interdependence 21

23 Exact Implementation II in general, robust monotonicity ; robust measurability robust measurability ; robust monotonicity gap from { virtual vs. exact { compact vs. unbounded mechanism restricting to compact mechanism, strict ex post incentive compatibility and robust measurability necessary for full implementation 22

24 Exact Implementation III in large class of economically interesting "monotonic aggregator" environments: (Bergemann-Morris "Robust Implementation: The Role of Direct Mechanisms") 1. robust monotonicity = robust measurability 2. natural generalization of < 1 I 1 condition 3. when robust virtual implemention is possible, it arises in modied direct mechanism 23

25 Outline 1. Environment and Solution Concepts 2. Strategic Revealed Preference: A Characterization Result 3. 24

26 PART 1: ENVIRONMENT AND SOLUTION CONCEPTS 25

27 Environment I agents Lottery outcome space Y = (X), X nite Finite types i vnm utilities: u i : Y! R Economic Assumption (can be relaxed) { y is uniform lottery over X { for each i, there exists b i 2 Y s.t. u i (y; ) > u i (b i ; ) for all 26

28 Mechanism A mechanism M is a collection ((M i ) I i=1 ; g) each M i is a nite message set outcome function g : M! Y 27

29 Rationalizable Messages S M;0 i ( i ) = M i ; S M;k+1 ( 8i i ) = 9 i 2 ( i M i ) s.t.: >< (1) m i i ( i ; m i ) > X 0 ) m i 2 S M;k i ( i ) >: (2) m i 2 arg max i ( i ; m i ) u i (g (m 0 m 0 i ; m i) ; ( i ; i )) i i ;m i 9 >= >; S M i ( i ) = \ k1 S M;k i ( i ). S M i ( i ) are rationalizable actions of type i in mechanism M 28

30 Type Space T = T i ; b i ; b I i i=1 1. T i countable types of agent i 2. b i : T i! (T i ) 3. b i : T i! i Epistemic Foundations I Incomplete information game (T ; M) { i's strategy: i : T i! (M i ) { strategy prole is an equilibrium if i (m i jt i ) > 0 implies m i is in 0 1 X arg max b i [t i ] (t i Y j (m j jt j ) A u i g (m 0 i; m i ) ; b (t) m 0 i t i ;m i j6=i 29

31 Epistemic Foundations II PROPOSITION. m i 2 S M i ( i ) if and only if there exist 1. a type space T, 2. an equilibrium of (T ; M) and 3. a type t i 2 T i, such that (a) i (m i jt i ) > 0 and (b) b i (t i ) = i. Brandenburger and Dekel (1987), Battigalli (1996), Bergemann and Morris (2001), Battigalli and Siniscalchi (2003). 30

32 PART 2: STRATEGIC REVEALED PREFERENCE 31

33 Dening Distinguishability I DEFINITION. R i ; i is preference relation of agent i with payo type i and conjecture i 2 ( i ) about types of others: X yr i ; i y 0, i 2 i i ( i ) u i (y; ( i ; i )) X i 2 i i ( i ) u i (y 0 ; ( i ; i )) DEFINITION. Type set i is distinguishable given type set prole for all R i there exists i 2 i such that R i ; i 6= R i for all i 2 ( i if i). 32

34 Dening Distinguishability II 0 i = 2 i k+1 i = i 2 2 i 9 i 2 k i s.t. i is indistinguishable given i i = \ k1 k i DEFINITION. Type i is indistinguishable from 0 i if i ; 0 i 2 i. 33

35 Single Good Example I R i ; i = R 0 i ; 0 i if and only if i + E i X j6=i j 1 0 A = 0 i + E 0 X j6=i j 1 A Suppose i < 0 i; there exist i ; 0 i 2 ( if and only if i) such that R i ; i = R 0 i ; 0 i i + X j6=i max j 0 i + X j6=i min j 34

36 Single Good Example II i is distinguishable given i if and only if max i 0 min i X max j j6=i 1 min A j Thus k i = n i max i min i [ (I 1)] ko If 1 I 1, all i; 0 i are indistinguishable If < 1 I 1, i 6= 0 i ) i and 0 i are distinguishable 35

37 Fixed Point Distinguishability Consider a collection of sets = ( i ) I i=1, each i 2 i. DEFINITION. The collection is mutually indistinguishable if, for each i and i 2 i, there exists i 2 i such that i is indistinguishable given i. LEMMA. Type i is indistinguishable from 0 i if and only if there does not exist mutually indistinguishable such that i ; 0 i i for some i 2 i. 36

38 Strategic Revealed Preference Characterization PROPOSITION 1: If i and 0 i are indistinguishable, then S M i ( i ) \ S M i 0 i 6=? in any mechanism M. PROPOSITION 2: There exists a mechanism M such that if i and 0 i are distinguishable, then S M i ( i ) \ S M i 0 i =?. 37

39 Proof of Proposition 1 PROPOSITION 1: If i and 0 i are indistinguishable, then S M i ( i ) \ S M i 0 i 6=? in any mechanism M. Suppose is mutually indistinguishable Fix any nite mechanism 38

40 Proof of Proposition 1 By induction on k, for each k, i and i 2 i, there exists m k i ( i) such that m k i ( i) 2 S k i ( i) for each i 2 i 1. True by denition for k = Suppose true for k 1 x any i and i 2 i since is mutually indistinguishable, 9 i 2 i, R i and, for each i 2 i, i i 2 ( i) such that R i ; i = R i i m k i ( i) be any element of the argmax under R i of g m i ; m k 1 i ( i) by construction, m k i ( i) 2 S M;k i ( i ) for all i 2 i. 39

41 Strategic Revealed Preference PROPOSITION 2: There exists a mechanism M such that if i and 0 i are distinguishable, then S M i ( i ) \ S M i 0 i =?. PROOF: By construction of "maximally revealing mechanism". 40

42 Proof of Proposition 2 KEY LEMMA: Set i is distinguishable given ey : i! Y such that X (ey ( i ) y) = 0 i if and only if there exists i 2 i and, for each i 2 i and i 2 ( i), ey ( i ) P i ; i y. 41

43 Proof of Proposition 2 LEMMA (Morris 1994, Samet 1998): Let V 1 ; :::; V L be closed, convex, subsets of the N-dimensional simplex N. These sets have an empty intersection if and only if there exist z 1 ; :::; z L 2 R N such that LX z l = 0 l=1 and v:z l > 0 for each l = 1; :::; L and v 2 V l. Key lemma follows from this lemma, letting i = f1; :::; Lg and V l be the set of possible utility weights of type i = l with any i 2 ( i). 42

44 Proof of Proposition 2 TEST SET LEMMA. There exists a nite set Y Y such that 1. for each i, i and i 2 ( i ), B i ( i ; i ) 6= Y 2. for each i, i and i, if i is distinguishable given i, then for each i 2 i and i 2 ( i), there exists 0 i 2 i such that B i ( i ; i ) \ B i 0 i; i =? where B i ( i ; i ) = fy 2 Y jyr i ; i y 0 for all y 0 2 Y g B i ( i ; i) = [ i 2( i) B i ( i ; i ) 43

45 Maximally Revealing Mechanism: In Words Each player makes K simultaneous announcements: 1. an element of test set Y 2. a prole of rst round announcements of other players he thinks possible, plus an element of Y 3. a prole of second round announcements of other players he thinks possible, plus an element of Y All chosen outcomes selected with positive probability, with much higher weight on "earlier" announcements 44

46 Maximally Revealing Mechanism: In Algebra Mechanism M K;" = M K i I i=1 ; gk;" parameterized by 1. " > 0 2. integer K 45

47 Maximally Revealing Mechanism: In Algebra Message Sets: i's message set is M K i where { Mi 0 = m 0 i { M k+1 i = M k i Y typical element m k i = m 0 i ; r 1 i ; y1 i ; ::::; rk i ; yk i Allocation Rule: g K;" (m) = y + 1 "K (1 ") I KX IX " k 1 k=1 i=1 I ri k ; m k 1 i y k i y 46

48 Conclusion of Proof of Proposition 2 1. Let k i m k i = k i m k 1 i ; r k i ; y k i 8> < = i >: i 2 k 1 i m k 1 i k 1 i ri k 6=? yi k 2 B i i ; k 1 i r k i 9 >= >; 2. There exists " > 0 such that o n i 2 i m k i 2 Si Mk;" ( i ) k i m k i for all " " and m k i 2 M k i. 47

49 3. There exists " > 0 and K such that o n i 2 i m K i 2 Si MK;" ( i ) 2 i for all " " and m K i 2 M K i. 48

50 PART 3: VIRTUAL ROBUST IMPLEMENTATION 49

51 Robust Virtual Implementation Denitions DEFINITION: A social choice function f :! Y: Write ky y 0, i.e., y 0 k for the Euclidean distance between a pair of lotteries y and s X ky y 0 k = (y (x) y 0 (x)) 2. x2x DEFINITION: Social choice function f is robustly "-implementable if there exists a mechanism M such that m 2 S M () ) kg (m) f ()k ". DEFINITION: Social choice function f is robustly virtually implementable if, for every " > 0, f is robustly "-implementable. 50

52 Robust Virtual Implementation Result DEFINITION: Social choice function f satises ex post incentive compatibility if, for all i, i, i and 0 i: u i (f ( i ; i ) ; ( i ; i )) u i f 0 i; i ; (i ; i ). DEFINITION: Social choice function f satises robust measurability if f ( i ; i ) 6= f 0 i; i ) i is distinguishable from 0 i. THEOREM. Social choice function f is robustly virtually implementable if and only if f satises ex post incentive compatibility and robust measurability. 51

53 Canonical Mechanism: In Words Fix small > 0 and large integer L. Announce message from maximally revealing mechanism plus L copies of set of indistinguishable types With prob., follow allocation rule of maximally revealing mechanism For each l = 1; :::L, with prob. 1 2 L, apply f to lth announcments of players For each i = 1; :::; I, with prob. 2 I, give player i a small ne if he was one of the "rst" players to report a type inconsistent with his message in maximally revealing mechanism 52

54 Canonical Mechanism: In Algebra Write M = (M i )Ii=1 ; g for the maximally revealing mechanism Let P i be partition of i's types into indistinguishable sets; let i ( i ) be the element of P i containing i Let f () = f ( ()) where f : P! Y Let M i = M i PL i, with typical element m i = m 0 i ; m1 i ; :::; ml i 53

55 Let Canonical Mechanism: In Algebra g (m) = 1 2 L LX l=1 f m l + g m I IX r i (m) i=1 where bi, if 9k s.t. m r i (m) = k i 6= i y, otherwise m 0 i and m j = m 0 for j = 1; ::; k 1 54

56 Monotonic Aggregator Environment There exist h i :! R and v i : Y R! R such that 1. h i is continuous and strictly increasing in i 2. v i is continuous and, for any y; y 0, there exists at most one x with v i (y; x) = v i (y 0 ; x) 3. u i (y; ) = v i (y; h i ()) 55

57 Monotonic Aggregator Environment Results 1. Robust Monotonicity = Robust Measurability = "Contraction Property" 2. If h i () = i + X j6=i ij j, contraction property holds if and only if largest eigenvalue of matrix 0 0 j 12 j j 13 j j 1I j j 21 j 0 j 23 j j 2I j j 31 j j 32 j 0 j 3I j j I1 j j I2 j j I3 j 0 1 C A is greater than one... 56