Robust Predictions in Gmes with Incomplete Informtion Dirk Bergemnn nd Stephen Morris Northwestern University Mrch, 30th, 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?
Agend: Robust Predictions x "pyo relevnt environment" = ction sets, pyo -relevnt vribles ("sttes"), pyo functions, distribution over sttes = incomplete informtion gme without higher order beliefs bout sttes nlyze 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
Agend: Robust Identi ction hving identi ed mpping from "pyo relevnt environment" to ction-stte distributions, we cn nlyze its inverse: sssume pyo relevnt environment is observed by the econometricin given knowledge of the ction-stte distribution distribution, or some moments of it, wht cn be deduced bout the pyo relevnt environment? prtil identi ction / set identi ction
Agend: 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? robust informtion policy robust txtion policy
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
Pyo Environment plyers i = 1; :::; I (pyo relevnt) sttes ctions (A i ) I i=1 utility functions (u i ) I i=1, ech u i : A! R common prior stte distribution 2 () "bsic gme", "belief-free gme" G = (A i ; u i ) I i=1 ;
Informtion Environment signls (types) (T i ) I i=1 signl distribution :! (T 1 T 2 ::: T I ) "higher order beliefs", "type spce," "signl spce" S = (T i ) I i=1 ;
Byes Nsh Equilibrium stndrd Byesin gme is described by (u; ; T ) behvior strtegy of plyer i is de ned by: i : T i! (A i ) De nition (Byes Nsh Equilibrium (BNE)) A strtegy pro le is Byes Nsh equilibrium of (u; ; T ) if X u i (( i (t i ) ; i (t i )) ; ) () [t i ; t i ] () t i ; X t i ; u i (( i ; i (t i )) ; ) () [t i ; t i ] (). for ech i, t i nd i.
Byes Nsh Equilibrium Distribution given Byesin gme (u; ; T ), BNE genertes joint probbility distribution over outcomes nd sttes A, (; ) = () X t [t] ()! IY i ( i jt i ) equilibrium distribution (; ) is speci ed without reference to informtion structure T which gives rise to (; ) i=1 De nition (Byes Nsh Equilibrium Distribution) A probbility distribution 2 (A ) is Byes Nsh equilibrium distribution (over ction nd sttes) of (u; ; T ) if there exists BNE of (u; ; T ) such tht = :
Implictions of BNE recll the originl equilibrium conditions on (u; ; T ): X u i (( i (t i ) ; i (t i )) ; ) () [t i ; t i ] () t i ; X t i ; u i (( i ; i (t i )) ; ) () [t i ; t i ] (). with the equilibrium distribution (; ) = () X t [t] ()! IY i ( i jt i ) n impliction of BNE of (u; ; T ) : for ll i 2 supp (; ) : X u i i 0 ; i ; (; ) ; i ; u i (( i ; i ) ; ) (; ) X i ; i=1
De nition (Byes Correlted Equilibrium (BCE)) Byes Correlted Equilibrium 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 i 2A i ; 2 u i (( i ; i ) ; ) u i 0 i ; i ; ((i ; i ) ; ) 0 nd consistent, i.e., for ech X (; ) = (). 2A no restrictions (or lower bound) on the privte informtion of the gents; in prticulr, zero privte informtion is possible BCE is de ned in terms of the pyo environment nd without reference to type spces BCE is de ned on smll pyo spce nd chrcterized s
Bsic Epistemic Result now given (u; ), wht is the set of equilibrium distributions cross ll possible informtion structures T Theorem (Equivlence ) A probbility distribution 2 (A ) is Byes correlted equilibrium of (u; ) if nd only if it is Byes Nsh Equilibrium distribution of (u; ; T ) for some informtion system T. BCE ) BNE uses the richness of the possible informtion structure to complete the equivlence result Aumnn (1987) estblished the bove chrcteriztion result for complete informtion gmes
Legitimte De nitions... Forges (1993): "Five Legitimte De nitions of Correlted Equilibrium in Gmes with Incomplete Informtion"; Forges (2006) gives #6 our de nition is "illegitimte" becuse it fils "join fesibility" De nition 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 in compnion pper, Correlted Equilibrium in Gmes with Incomplete Informtion we relte it to erlier de nitions nd estblish comprtive results with respect to informtion environments
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
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 u i ( i ; A; ) = 0 @ A 1 A 0 0 @ i A 1 0 A + @ A = i A 1 A Z 1 0 0 0 @ i di A A A A A gme is completely described by liner returns nd interction mtrix = ij 1 0 A @ i A 1 A
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 x 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
Byesin Nsh equilibrium Theorem The unique Byesin Nsh equilibrium (given the bivrite informtion structure) is liner equilibrium, 0 + x x + y y, with nd y = x = 2 x A 2 x + 2 ; y 2 + A A x 2 + : 2
Joint Action Stte Distribution there is n implied joint distribution of (; A; ) 0 @ i A 1 00 A N @@ A 1 0 A ; @ nd in terms of the equilibrium coe cients: 2 A A A A 2 A A A A A 2 11 AA 2 x 2 x + 2 y 2 y + 2 ( x + y ) 2 2 y 2 y + 2 ( x + y ) 2 2 ( x + y ) 2 y 2 y + 2 ( x + y ) 2 2 y 2 y + 2 ( x + y ) 2 2 ( x + y ) 2 ( x + y ) 2 ( x + y ) 2
Given Public Informtion movements long level curve re vritions in 2 x given 2 y : 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 Byes Nsh Equilibri of beuty contest with public precision of y 2 nd r=.25
Given Privte Informtion movements long level curve re vritions in 2 y given 2 1.0 0.8 45 o 0.6 1 0.4.5 0.2.2.1 2 x :.01 0.2 0.4 0.6 0.8 1.0 Byes Nsh Equilibri of beuty contest with privte precision 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 correltedequilibri.
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
Informtion Bounds the nlyst my not know how much privte informtion the gents hve, yet he my hve lower bound on how much informtion the gents hve how does the set of BCE chnge with the lower bound ssume tht ll gents observe public signl y : y = + " nd privte signl x i x i = + " i with " " i N 0 0 2 ; y 0 0 2 x
Informtion Bounds nd Correlted Equilibrium the equilibrium conditions re now ugmented from for ll to for ll ; x; y : = 1 ( E [ j; x; y ] + A E [A j; x; y ]) ; 8; x; y. we determine ; x ; y in terms of ;, e.g.: y = + A y + A 2 set of correlted equilibri is given by the inequlities: 2 2 y 0; 1 x 0;
Given Public Informtion describe the equilibrium set C () 2 [0; 1] 2 in terms of the noise of the signl pir = ( x ; y ) the interior of ech level curve describes the correlted equilibri with given mount of public correltion 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 Eq u ilibri of beu ty contest with min iml precsion of y 2 nd r=.25
Given Privte nd Public Informtion the interior intersection of the level curves genertes the corresponding equilibrium set more informtion reduces the set of possible outcomes, becuse it dds incentive constrints but does not remove ny correltion possibilities 2 1.0 0.8 0.6 0.4 x 2.5 y 2.5 0.2 x 2.1 y 2.1 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
Given Informtion nd the Equilibrium Set describe the equilibrium set C () 2 [0; 1] 2 in terms of the noise of the signl pir = ( s ; i ) Theorem (Informtion Bounds) 1 For ll < 0, C ( s ) C ( 0 s) ; 2 For ll < 0 : 3 For ll < 0 : min > min ; 2C () 2C ( 0 ) min > min : 2C () 2C ( 0 ) more privte informtion shrinks the equilibrium set nd mkes predictions shrper generl result in "Correlted Equilibrium in Gmes of Incomplete Informtion
Identi ction 1 Predictions: Wht restrictions re imposed by the structurl model (u; ) on the observble endogenous sttistics ( ; ; ; )? 2 Identi ction: Wht restrictions cn be imposed/inferred on the structurl model (u) by the observtions of the outcome vribles ( ; ; ; ; ; )? cn we identify sign nd size of interction? cn we identify the nture of the informtionl externlity nd the strtegic externlity A
Benchmrk: Complete Informtion the stte is observble by ll gents the econometricin observes: ( ; ; ; ; ; ) the informtion bout the men: = + + A nd covrince = + A permits identi ction of slope nd intercept of best response: + A ; + A
Identi ction with Complete Informtion Proposition (Sign Identi ction) 1. Nsh equilibrium identi es the sign of ; but not the sign of A. 2. Nsh equilibrium point identi es the rtios: + A nd + A : the sign of the strtegic interction, A, cnnot be identi ed, due to the perfect correltion of the gents ction in the complete informtion gme
Identi ction with Incomplete Informtion: Byes Nsh the informtion structure 2 x ; 2 y of the Byesin gme is ssumed to be known the identi ction, given the hypothesis of Byes Nsh equilibrium, uses the vrince-covrince mtrix of ctions nd sttes: 2 A; = y 2 y + 2 ( x + y ) 2 2 ( x + y ) 2 ( x + y ) the reltionship between equilibrium coe cients y = x 2 x 2 y + A lends informtion bout the sign of A 2
Identi ction with Incomplete Informtion: Byes Nsh Proposition (Sign Identi ction) If 0 < 2 x ; 2 y < 1, then: 1. BNE identi es the sign of nd of A. 2. BNE point identi es the rtios: + A nd + A : Byes Nsh equilibrium cn recover sign of strtegic interction s opposed to Nsh equilibrium
Identi ction with Incomplete Informtion: Byes Correlted we cn recover from the equilibrium conditions men nd vrince: = + + A nd = A + s well nd notice the rte t which A nd substitute di er in men nd stndrd devition, provided < 1 source of prtil identi ction
Prtil Identi ction Proposition (Prtil Identi ction) If < 1, the interction rtios re prtilly identi ed in BCE: 2 ; 1 ; + A nd 2 + A 1; : Moreover, the sign of, but not the sign of A is identi ed. discontinuity of identi ed set t = 1
Prtil Identi ction A 0< < < <1 Set identifiction of interction rtio A filure to identify the strtegic nture of the gme, strtegic complements or strtegic substitutes
Prior Informtion nd Identi ction erlier, we nlyzed how informtion bounds in terms of privte nd public informtion, represented by 2 x nd 2 y, shrpen the prediction similrly, the informtion bounds shrpen the identi ction of the interction rtios Proposition (Prior Informtion nd Identi ction) With prior informtion, the interction rtios re shrper identi ed with: 2 (r ( x ; y ) ; R ( x ; y )) ; 1 ; + A with @r ( x ; y ) @ x < 0; @R ( x ; y ) @ y > 0:
Prior Informtion nd Set Identi ction A y 0< y < y < y Set identifiction of prior informtion y y A the privte informtion set provides the lower bound on A the public informtion set provides the upper bound on A eventully, s 2 x ; 2 y! 0, the sign of the strtegic interction is identi ed
Aggregte Informtion nd Identi ction suppose we could only observe the ggregte (verge) ction, but not idiosyncrtic ction pro le would we lose (much) in identi ction power in Byes Nsh equilibrium, we would not lose ny identi ction result, in Byes Correlted equilibrium, we lose informtion, s remining restriction is wek: 2
Demnd nd Supply Identi ction demnd is given by: supply is given by: P d = d 0 + d 1 Q + d 2 d P s = s 0 + s 1 Q + s 2 s the exogenous rndom vribles re d nd s re demnd nd supply shocks ( demnd, supply shifters ) complete informtion: ech rm observes shocks ( d ; s ) nd mkes supply decisions ccordingly econometricin only observes relized ggregte vribles, P nd Q, but not individul choices
Incomplete Informtion ech rm observes vector of signls: y i = (y d ; y di ; y s ; y si ) regrding the true cost nd demnd shocks: y s = s + " s ; y si = s + " si ; y d = d + " d ; y di = d + " di before its supply decision mintin normlity nd independence of ( d ; s ; " d ; " id ; " s ; " is ) the vrince of the rndom vribles: I = 2 d ; 2 s ; 2 d ; 2 id ; 2 s ; 2 is represents the informtion structure I in the economy
Competitive Equilibrium with Incomplete Informtion in Byes Nsh equilibrium of competitive economy rm i supplies q i (y i ) on the bsis of its privte, but noisy, informtion equilibrium price is given by equilibrium condition: P d = d 0 + d 1 Z q i (y i ) di + d 2 d = d 0 + d 1 Q + d 2 d with respect to the relized demnd shock d
Identi ction with Incomplete Informtion cn we identify nd estimte the slope of supply nd demnd function in the presence of incomplete informtion? for every informtion structure I ech rm observes noisy signl of the true cost nd demnd shock nd mkes supply decision on the bsis of the noisy informtion Theorem (Point Identi ction) For every informtion structure I; the demnd nd supply functions re point identi ed if the rms hve noisy informtion bout their cost: min 2 s ; 2 is < 1: symmetry rises s relized price vries with relized demnd
Robust Identi ction now we sk cn identi ction be ccomplished for ll (or subset of) informtion structures, i.e. cn we chieve robust identi ction demnd function remins point identi ed s ggregte quntity, ggregte price nd demnd shock re observed Theorem (Set Identi ction) 1 For every informtion bound, the supply function (s 1 ; s 2 ) is set identi ed. 2 If the informtion bounds increse, then the identi ed set decreses. concern for robustness wekens the bility to identify the structurl prmeter to llow for prtil identi ction only
Discussion Byes correlted equilibrium encodes concern for robustness to informtion environment identi ction: complete vs. incomplete informtion demnd nd supply identi ction: the mrket prticipnts hve complete informtion, but the nlyst hs noisy informtion (uses instrument to recover the informtion) uction identi ction: ech bidder hs privte informtion bout his vlution, but the bidders hve the sme informtion bout ech other s the nlyst presumbly neither informtionl ssumption is vlid, suggesting role for robust identi ction robust welfre improving policy how responsive is robust policy to informtionl conditions