Robust Predictions in Gmes with Incomplete Informtion Dirk Bergemnn nd Stephen Morris Collegio Crlo Alberto, Turin 16 Mrch 2011
Introduction Gme Theoretic Predictions re very sensitive to "higher order beliefs" or (equivlently) informtion structure Higher order beliefs re rrely observed Wht predictions cn we mke nd nlysis cn we do if we do not observe higher order beliefs?
Robust Predictions Agend Fix "pyo relevnt environment" = ction sets, pyo -relevnt vribles ("sttes"), pyo functions, distribution over sttes = incomplete informtion gme without higher order beliefs bout sttes Assume pyo relevnt environment is observed by the econometricin Anlyze wht could hppen for ll possible higher order beliefs (mintining common prior ssumption nd equilibrium ssumptions) Mke set vlued predictions bout joint distribution of ctions nd sttes
Prtil Identi ction Hving identi ed mpping from "pyo relevnt environment" to ction-stte distributions, we cn nlyze its inverse: Given knowledge of the ction-stte distribution distribution, or some moments of it, wht cn be deduced bout the pyo relevnt environment?
Robust Comprtive Sttics / Policy Anlysis Wht cn sy bout the impct of chnges in prmeters of the pyo relevnt environment (for exmple, policy choices) for the set of possible outcomes?
Prtil Informtion bout Informtion Structure Perhps you don t observe ll higher order beliefs but you re sure of some spects of the informtion structure: You re sure tht bidders know their privte vlues of n object in n uction, but you hve no ide wht their beliefs nd higher order beliefs bout others privte vlues re... You re sure tht oligopolists know their own costs, but you hve no ide wht beliefs nd higher order beliefs bout demnd nd others privte vlues. Wht cn you sy then?
This Tlk 1 Generl Approch 2 Illustrtion with Continuum Plyer, Symmetric, Liner Best Response, Norml Distribution Gmes
Results Preview 1 Generl Approch Set vlued prediction is set of "Byes Correlted Equilibri" Prtil informtion monotoniclly reduces the set of "Byes Correlted Equilibri" 2 Illustrtion with Continuum Plyer, Symmetric, Liner Best Response, Norml Distribution Gmes These sets re trctble nd intuitive Cnnot distinguish strtegic substitutes nd complements
Setting plyers i = 1; :::; I (pyo relevnt) sttes
Pyo Relevnt Environment ctions (A i ) I i=1 utility functions (u i ) I i=1, ech u i : A! R stte distribution 2 () G = (A i ; u i ) I i=1 ; ("bsic gme", "pre-gme")
Informtion Environment signls (types) (T i ) I i=1 signl distribution :! (T 1 T 2 ::: T I ) S = (T i ) I i=1 ; ("higher order beliefs", "type spce," "signl spce")
Gmes with Incomplete Informtion The pir (G; S) is stndrd gme of incomplete informtion A (behviorl) strtegy for plyer i is mpping b i : T i! (A i ) DEFINITION. A strtegy pro le b is Byes Nsh Equilibrium (BNE) of (G; S) if, for ll i, t i nd i with b i ( i jt i ) > 0, 0 1 X @ Y b j ( j jt j ) A u i (( i ; i ) ; ) () (tj) i 2A i ;t i 2T i ;2 j6=i 0 1 X b j ( j jt j ) A u i i 0 ; i ; () (tj) i 2A i ;t i 2T i ;2 for ll 0 i 2 A i. @ Y j6=i
BNE Action Stte Distributions DEFINITION. An ction stte distribution 2 (A ) is BNE ction stte distribution of (G; S) if there exists BNE strtegy pro le b such tht (; ) = X! IY () (tj) b i ( i jt i ). t2t i=1
Byes Correlted Equilibrium (with Null Informtion) DEFINITION. An ction stte distribution 2 (A ) is Byes Correlted Equilibrium (BCE) of G if is obedient, i.e., for ech i, i nd 0 i, X u i (( i ; i ) ; ) (( i ; i ) ; ) 2 X 2 u i 0 i ; i ; ((i ; i ) ; ) nd consistent, i.e., for ech X (; ) = (). 2A
Result PROPOSITION 1. Action stte distribution is BNE ction stte distribution of (G; S) for some S if nd only if it is BCE of G. c.f. Aumnn 1987, Forges 1993
Augmented Informtion System We know plyers observe S but we dont know wht dditionl informtion they observe. Augmented informtion system S e = (Z i ) I i=1 ;, where : T! (Z) Augmented informtion informtion gme G; S; S e Plyer i s behviorl strtegy i : T i Z i! (A i ) DEFINITION. A strtegy pro le is Byes Nsh Equilibrium (BNE) of (G; S; S 0 ) if, for ll i, t i ; z i nd i with b i ( i jt i ; z i ) > 0, 0 1 X @ Y b j ( j jt j ; z j ) A u i (( i ; i ) ; ) () (tj) (zj i ;t i ;z i ; j6=i 0 1 X b j ( j jt j ; z j ) A u i i 0 ; i ; () (tj) (z i ;t i ;z i ; for ll 0 2 A. @ Y j6=i
BNE Action Type Stte Distributions DEFINITION. An ction type stte distribution 2 (A T ) is BNE ction type stte distribution of (G; S; S 0 ) if there exists BNE strtegy pro le such tht (; t; ) = () (tj) X z2z! IY b i ( i jt i ; z i ) (zjt; ). i=1
Byes Correlted Equilibrium DEFINITION. An ction type stte distribution 2 (A T ) is Byes Correlted Equilibrium (BCE) of (G; S) it is obedient, i.e., for ech i, t i, i nd 0 i, X i 2A i ;t i 2T i ;2 X i 2A i ;t i 2T i ;2 u i (( i ; i ) ; ) (( i ; i ) ; (t i ; t i ) ; ) u i 0 i ; i ; ((i ; i ) ; (t i ; t i ) ; ) nd consistent, i.e., X (; t; ) = () (tj). 2A If S is null informtion system, reduces to erlier de nition.
Result PROPOSITION 2. Action type stte distribution is BNE ction type stte distribution of (G; S; S 0 ) for some S 0 if nd only if it is BCE of (G; S). c.f. Forges 1993 If S is null informtion system, reduces to Proposition 1.
Legitimte De nitions Forges (1993): "Five Legitimte De nitions of Correlted Equilibrium in Gmes with Incomplete Informtion"; Forges (2006) gives #6 This de nition is "illegitimte" becuse it fils "join fesibility" DEFINITION. Action type stte distribution is join fesible for (G; S) if there exists f : T! (A) such tht for ech ; t;. (; t; ) = () (tj) f (jt) BCE fils join fesibility, Forges wekest de nition (Byesin solution) is BCE stisfying join fesibility
Trivil One Plyer Exmple I = 1 = ; 0 () = 0 = 1 2 Pyo s u 1 0 1 2 1 0 1 0 0 unique Byesin solution: ( 1 ; ) = 1 ; 0 = 1 2 BCE: ( 1 ; ) = 0 1 ; 0 = 1 2
Result PROPOSITION. (Informl Sttement). If informtion system S 0 is less informed thn S, then S 0 hs lrger set of BCE ction stte distributions.
Compre Gossner 00, Lehrer, Rosenberg nd Schmy 06, 10; in Gossner, "more" informtion led to more equilibri
Pyo Environment: Qudrtic Pyo s utility of ech gent i is given by qudrtic pyo function: determined by individul ction i 2 R, stte of the world 2 R, nd verge ction A 2 R: nd thus: A = 0 u i ( i ; A; ) = ( i ; A; ) @ Z 1 0 i di A A AA A A 1 A ( i ; A; ) T gme is completely described by interction mtrix = ij
Pyo Environment: Norml Pyo s the stte of the world is normlly distributed N ; 2 with men 2 R nd vrince 2 2 R + the distribution of the stte of the world is commonly known common prior
Interction Mtrix given the interction mtrix, complete informtion gme is potentil gme (Monderer nd Shpley (1996)): 0 = @ A A A A A digonl entries: ; A ; describe own e ects o -digonl entries: ; A ; A interction e ects fundmentls mtter, return shocks : 6= 0; strtegic complements nd strtegic substitutes: A > 0 vs. A < 0 1 A
Concve Gme concvity t the individul level (well-de ned best response): < 0 concvity t the ggregte level (existence of n interior equilibrium) + A < 0 concve pyo s imply tht the complete informtion gme hs unique Nsh nd unique correlted equilibrium (Neymn (1997))
Exmple 1: Beuty Contest continuum of gents: i 2 [0; 1] ction (= messge): 2 R stte of the world: 2 R pyo function u i = (1 r) ( i ) 2 r ( i A) 2 with r 2 (0; 1) see Morris nd Shin (2002), Angeletos nd Pvn (2007)
Exmple 2: Competitive Mrket ction ( = quntity): i 2 R cost of production c ( i ) = 1 2 ( i ) 2 stte of the world ( = demnd intercept): 2 R inverse demnd ( = price): p (A) = A A where A is verge supply: A = Z 1 0 i di see Guesnerie (1992) nd Vives (2008)
Stndrd Approch Fix n informtion system 1 every gent i observes public signl y bout : y N ; 2 y 2 every gent i observes privte signl x i bout : x i N ; 2 x
Stndrd Approch the best response of ech gent is: = 1 ( E [ jx; y ] + A E [A jx; y ]) suppose the equilibrium strtegy is given by liner function: (x; y) = 0 + x x + y y, denote the sum of the precisions: 2 = 2 + x 2 + y 2
Stndrd Approch Theorem The unique Byesin Nsh equilibrium (given the bivrite informtion structure) is liner equilibrium, 0 + x x + y y, with x = 2 x A 2 x + 2 ; nd y = y 2 + A A x 2 + : 2 There is n implied joint distribution of (; A; )
Stndrd Approch There is n implied joint distribution of (; A; ) 0 @ i A 1 00 A N @@ A 1 0 A ; @ 2 A A A A 2 A A A A A 2 11 AA
Given Public Informtion movements long level curve re vritions in x 2 given y 2 2 1.0 0.8 0.6 1 0.4.5.2 0.2.1.001.01 0.2 0.4 0.6 0.8 1.0 Correlted Equilibri of beuty contest with mini ml precsion of y 2 nd r=.25
Given Privte Informtion movements long level curve re vritions in y 2 given x 2 2 1.0 0.8 45 o 0.6 1 0.4.5 0.2.2.1.01 0.2 0.4 0.6 0.8 1.0 Correlted Equilibri of beuty contest with miniml precsion of x 2 nd r=.25
Byes Correlted Equilibri the object of nlysis: joint distribution over ctions nd sttes: (; A; ) chrcterize the set of (normlly distributed) BCE: 0 @ i A 1 00 A N @@ A 1 0 A ; @ 2 A A A A 2 A A A A A 2 2 A is the ggregte voltility (common vrition) 2 2 A is the cross-section dispersion (idiosyncrtic vrition) sttisticl representtion of equilibrium in terms of rst nd second order moments 11 AA
Symmetric Byes Correlted Equilibri with focus on symmetric equilibri: A = ; 2 A = 2 ; A A = 2 where is the correltion coe cient cross individul ctions the rst nd second moments of the correlted equilibri re: 0 1 00 1 0 i 2 2 11 @ A A N @@ A ; @ 2 2 AA 2 correlted equilibri re chrcterized by: f ; ; ; g
Equilibrium Anlysis in the complete informtion gme, the best response is: = A A best response is weighted liner combintion of fundmentl nd verge ction A reltive to the cost of ction: = ; A = in the incomplete informtion gme, nd A re uncertin: E [] ; E [A] given the correlted equilibrium distribution (; ) we cn use the conditionl expecttions: E [ j] ; E [A j]
Equilibrium Conditions in the incomplete informtion gme, the best response is: = E [ j] E [A j] A best response property hs to hold for ll 2 supp (; ) fortiori, the best response property hs to hold in expecttions over ll : E [] = E E [ j] + E [A j] A by the lw of iterted expecttion, or lw of totl expecttion: E [E [ j]] = ; E [E [A j]] = E [A] = E [] ;
Equilibrium Moments: Men the best response property implies tht for ll (; ) : E [] = E E [ j] + E [A j] A or by the lw of iterted expecttion: = A Theorem (First Moment) In ll Byes correlted equilibri, the men ction is given by: E [] = + A : result bout men ction is independent of symmetry or norml distribution
Equilibrium Moments: Vrince in ny correlted equilibrium (; ), best response demnds = E [ j] + E [A j] A ; 8 2 supp (; ) or vrying in 1 = @E [ j] @ + @E [A j] @ A ; the chnge in the conditionl expecttion @E [ j] ; @ @E [A j] @ is sttement bout the correltion between ; A;
Equilibrium Moment Restrictions the best response condition nd the condition tht ;A; forms multivrite distribution, mening tht the vrince-covrince mtrix hs to be positive de nite we need to determine: f ; ; g Theorem (Second Moment) The triple ( ; ; ) forms Byes correlted equilibrium i : 2 0; nd = A + :
Moment Restrictions: Correltion Coe cients the equilibrium set is chrcterized by inequlity 2 0 : correltion of ctions cross gents; : correltion of ctions nd fundmentl 1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 Set of correlted equilibri.
Equivlence between BCE nd BNE bivrite informtion structure which genertes voltility (common signl) nd dispersion (idiosyncrtic signl) Theorem There is BCE with ( ; ) if nd only if there is BNE with 2 x ; 2 y. public nd privte signl re su cient to generte the entire set of correlted equilibri... but given BCE does not uniquely identify the informtion environment of BNE
Known Miniml Informtion for noise terms = ( x ; y ), write C ( x ; y ) 2 [0; 1] 2 for possible vlues of ( ; ) 1 For ll < 0, C () C ( 0 ) ; 2 For ll 0 : min > min ; 2C () 2C ( 0 ) Proposition For ll 0 : min > min : 2C () 2C ( 0 )
Known Miniml Informtion 3 grph 2 1.0 0.8 0.6 0.4 x 2.5 y 2.5 0.2 x 2.1 y 2.1 4:pdf 0.2 0.4 0.6 0.8 1.0 Correlted Equilibri of beuty contest with r=.25 nd miniml precsions of x 2 nd y 2 Figure: Set of BCE with given public nd privte informtion
Identi ction Exercise 1 suppose only the ctions re observble: ( ; ; ) but the reliztion of the stte is unobservble, nd hence we do not hve ccess to covrite informtion between nd : the identi ction then uses the men: = + A nd vrince = A + :
Sign Identi ction = = + A A +
Sign Identi ction cn we identify sign of interction? recll: informtionl externlity, A strtegic externlity Theorem (Sign Identi ction) The Byes Nsh Equilibrium identi es the sign of nd A. identi ction in Byes Nsh equilibrium uses vrince-covrince given informtion structure 2 x ; 2 y Theorem (Prtil Sign Identi ction) The Byes Correlted Equilibrium identi es the sign of but it does not identify the sign of A. filure to identify the strtegic nture of the gme, strtegic complements or strtegic substitutes
Identi ction Exercise 2 Now suppose is observble But now llow to hve unknown intercept insted of 0 now we hve nd vrince = 0 + A = A + : cn identify sign of but not sign of A
Clssicl Supply nd Demnd Identi ction If the informtion structure is known, slope of supply nd demnd cn be identi ed uncertinty bout informtion structure gives bounds on slopes