Robust Predictions in Gmes with Incomplete Informtion Dirk Bergemnn nd Stephen Morris April 2011
Introduction in gmes of incomplete informtion, privte informtion represents informtion bout: pyo environment nd strtegic environment rst order beliefs vs. higher order beliefs gme theoretic predictions re very sensitive to speci ction of the strtegic environment, the higher order beliefs", or (equivlently) the informtion structure emil gme, coordintion gmes in gme theory revenue equivlence, surplus extrction in mechnism design
The Role of Higher Order Beliefs: An Exmple rst price seled bid uction with independent, privte vlues identicl, independent common prior over vlutions of bidders rst order beliefs of bidder i is bout: vlution of bidder i himself; vlutions of bidders j; k second order beliefs of bidder i is bout: the rst order belief of bidder j bout the vlution of bidder i in rst price seled bid uction, bid of i depends on his rst-order, second-order nd higher-order beliefs revenue equivlence - ssuming second-order belief of i puts probbility 1 on j s rst order belief being equl to common prior i.e. privte informtion of bidder i doesn t contin ny dditionl informtion bout bidder j beyond common prior; revenue equivlence fils without this restrictive ssumption wht predictions cn we o er independent of the higher order beliefs?
Pyo Environment nd Strtegic Environment x "pyo relevnt environment" = ction sets, pyo -relevnt vribles ("sttes"), pyo functions, distribution over sttes = incomplete informtion gme without higher order beliefs bout sttes there re mny strtegic environments consistent with xed pyo environment consistent: fter integrting over the higher-order beliefs, the mrginl over pyo relevnt sttes coincides with common prior over pyo relevnt sttes the possible informtion environments vry widely: from zero informtion, where every gent knows nothing beyond common prior to complete informtion, where every gents knows reliztion of pyo relevnt stte
Agend: Robust Predictions nlyze wht could hppen for ll possible higher order beliefs (mintining common prior ssumption nd equilibrium ssumptions) ech speci c informtion environment genertes speci c predictions regrding equilibrium behvior given tht the di erent strtegic environments shre the sme pyo environment, does the predicted behvior disply common fetures? cn nlyst mke predictions which re robust to the exct speci ction of the strtegic environment? mke set vlued predictions bout joint distribution of ctions nd sttes
Agend: Prior Informtion nd Prediction perhps nlyst doesn t observe ll higher order beliefs but is sure of some spects of the informtion structure: is sure tht bidders know their privte vlues of n object in n uction, but hs no ide wht their beliefs nd higher order beliefs bout others privte vlues re... is sure tht oligopolists know their own costs, but hs no ide wht beliefs nd higher order beliefs bout demnd nd others privte vlues... wht cn you sy then?
Agend: Robust Identi ction the observble outcomes of the gme re the ctions nd the pyo relevnt sttes the chosen ction revels the preference of the gent given his higher-order belief, but typiclly does not revel his higher order belief hving identi ed mpping from "pyo relevnt environment" to ction-stte distributions, we cn nlyze its inverse: ssume pyo relevnt environment is observed by the econometricin given knowledge of the ction-stte distribution, or some moments of it, wht cn be deduced bout the pyo relevnt environment? prtil identi ction / set identi ction
This Tlk 1 Generl Approch set vlued prediction is set of "Byes Correlted equilibri" epistemic result linking Byes Nsh nd Byes Correlted equilibrium prtil prior informtion monotoniclly reduces the set of "Byes Correlted equilibri" 2 Illustrtion with Continuum Plyer, Symmetric, Liner Best Response, Norml Distribution Gmes the resulting equilibrium sets re trctble nd intuitive cnnot distinguish between gmes with 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 ;
Strtegic Environment signls (types) (T i ) I i=1 signl distribution :! (T 1 T 2 ::: T I ) "higher order beliefs", "type spce," "signl spce" T = (T i ) I i=1 ; common prior () nd conditionl distribution [t i ; t i ] () llow gent i to hold privte informtion t i in terms of posterior beliefs bout the pyo relevnt stte posterior beliefs bout the beliefs t i of the other gents
Byes Nsh Equilibrium stndrd Byesin gme is described by (G; 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 (G; 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 (G; 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 type spce 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 (G; T ) if there exists BNE of (G; T ) such tht = :
Implictions of BNE recll the originl equilibrium conditions on (G; 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 ) i=1 n impliction of BNE of (G; T ) : for ll i; i 2 supp (; ) : X u i (( i ; i ) ; ) (; ) X u i 0 i ; i ; (; ) ; i ; i ;
Byes Correlted Equilibrium joint distribution over ction-stte (; ) describes choices, but is silent bout reson (informtion) for choice De nition (Byes Correlted Equilibrium (BCE)) 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 on privte informtion beyond (); zero informtion nd complete informtion of re possible
Byes Correlted Equilibrium BCE is de ned in terms of the pyo environment nd without reference to type spce: (; ) without uncertinty, = fg, Byes correlted equilibrium reduces to correlted equilibrium (Aumnn (1974)) erlier de nitions of correlted equilibrium for gmes of incomplete informtion, see Forges (1993), (2006), de ne solution concepts for (G; T ), integrte out pyo relevnt sttes, nd describe correlted equilibrium s ction type distributions (A T ) we work with bsic gme G, describe correlted equilibrium s ction stte distributions (A ) in compnion pper, Correlted Equilibrium in Gmes with Incomplete Informtion we relte de nitions nd estblish comprtive results wrt informtion environments
Bsic Epistemic Result now given (G), 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 G if nd only if it is Byes Nsh Equilibrium distribution of (G; T ) for some informtion system T. BCE ) BNE uses the richness of the possible informtion structure to complete the equivlence result: 8i; 8 i : t i = (; i j i ) ; i (t i ) = i Aumnn (1987) estblished the bove chrcteriztion result for complete informtion gmes
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 the mtrix de nes the nture of the interction: 0 1 A = @ A A A A A digonl entries: ; A ; describe own e ects o -digonl entries: ; A ; A interction e ects best response of gent i depends on ; A ; : cost of djustment: informtionl externlity: strtegic externlity: A ; strtegic complements nd strtegic substitutes, A > 0 vs. A < 0
suppose is commonly known best response of gent i : Complete Informtion Gme = + + A :A equilibrium response of gent i: () = mintined ssumptions: intercept z } { + A slope z } { + A concvity t individul level (well-de ned best response): < 0 concvity t ggregte level (existence of interior equilibrium): + A < 0 concve pyo s imply tht complete informtion gme hs unique Nsh nd correlted equilibrium (Neymn (1997))
Exmple 1: 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)
Exmple 2: Beuty Contest, Mcroeconomic Coordintion Gmes 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)
Stndrd Approch x n rbitrry 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 binry informtion structure with privte nd public component every pir 2 x ; 2 y genertes di erent informtion system, type spce
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: s (x; y) = 0 + x x + y y with nd y = x = 2 x A 2 x + 2 ; y 2 + A A x 2 + : 2 the liner coe cients x nd y stisfy the reltionship: y x = y 2 x 2 ; w/o strtegic interction + A = 0 : y A x = 2 y 2 x
Equilibrium Response nd Informtion liner coe cients x nd y respond to signls x i nd y: x i = + " i nd y = + ", nd hence to stte of the world sum of liner coe cients x nd y disply ttenution :! x + y = 1 2 + A 2 + A x 2 < ; + A yet men ction: 0 + x + y = + A ; for ll 2 x ; 2 y rst equilibrium moment is constnt cross informtion structures second equilibrium moments vry cross informtion structures.
Joint Action Stte Distribution given normlly distributed rndom vribles 2 ; 2 x ; 2 y nd liner strtegy, the joint distribution of ction nd stte ( i ; j ; ) is multivrite norml s well: 0 i ; j ; = @ 2 2 2 2 2 1 A nd in terms of equilibrium coe cients: 0 @ 2 2 1 0 A = @ 2 x 2 x + 2 y 2 y + 2 ( x + y ) 2 2 y 2 y + 2 ( x + y ) 2 2 ( x + y ) 1 A represent equilibrium in terms of correltion coe cients ;
Equilibrium nd Privte Informtion equilibrium covrinces vi correltion coe cients ; consider given precision x 2 of privte informtion 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 privte precision of x 2 nd r=.25 increse in precision of public informtion y 2 movement long level curve is upwrd
Equilibrium nd Public Informtion equilibrium covrinces vi correltion coe cients ; consider given precision y 2 of public informtion 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 By es Nsh Eq u ilib ri of b eu ty co n test w ith public precision of y 2 nd r=.2 5 increse in precision of privte informtion x 2 movement long level curve is upwrd
Multitude of Informtion Environments every type t i of gent i could contin mny pieces of informtion t i = (s; s i ; s ij ; s ijk; ::::) every gent i my observe public (common) signl s centered round the stte of the world : s N ; 2 s every gent i my observe privte signl s i centered round the stte of the world : s i N ; 2 i every gent i my observe privte signl s i;j bout the signl of gent j : s i;j N s j ; 2 i;j every gent i my observe privte signl s i;j;k bout...: s i;j;k N s j;k ; 2 i;j;k
Byes Correlted Equilibri object of nlysis: joint distribution over ctions nd sttes (; ) for exposition, focus on symmetric Byes correlted equilibri: ( i ; A; ) which re normlly distributed: 0 1 00 1 0 i 2 A A @ A A N @@ 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 ; = A A 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 chrcterized by rst nd second moments: 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; lw of totl vrince
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: Theorem (Second Moment) f ; ; g The triple ( ; ; ) forms Byes correlted equilibrium i : 2 0; nd = A + : correltion of ctions cross gents: correltion of ctions nd fundmentls:
Moment Restrictions: Correltion Coe cients the equilibrium set is completely chrcterized by inequlity: 2 0 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 in 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 s dditionl incentive constrints = 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;
Lower Bound on Public Informtion movements long level curve re vritions in x 2 given y 2 the interior of ech level curve describes the correlted equilibri for given lower bound on public informtion 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
more informtion reduces set of possible distributions, s it dds incentive constrints but does not remove correltion possibilities Lower Bounds on Privte nd Public Informtion interior of intersection of level curves is the set of Byes correlted equilibrium subject to lower bounds on privte nd public informtion 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
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
Identi ction nd Informtion: Single Agent zero interction: A = 0, normlize cost of djustment: = 1 complete informtion: the stte is observble to the gent the best response of the gent is given by: () = + u, where u is n error, observble to the gent, but unobservble to the econometricin unobservble error u, E [u] = u, u? the econometricin observes nd but not u the covrince between nd point identi es
Identi ction nd Noisy Informtion noisy informtion: the stte is unobservble, the gent only observes noisy signl s = + " with " N 0; 2 " the gent chooses upon observing s nd u, (s) = s " 2 + 2 " 2 + 2 + u the econometricin observes nd but neither s nor u the best response of the gent hs the slope: 2 " 2 " + 2 < ; the best response is ttenuted by the signl to noise rtio
Robust Identi ction the covrince between nd point identi es: " 2 b = " 2 + 2 ; how to identify without knowing the informtion of the gent, i.e. without knowing the vrince 2 " of " : without knowing 2 ", there is only set identi ction: = b " 2 + 2 " 2 ) 2 [b ; 1) noise in the predictor vrible induces bis: ttenution bis, regression dilution in the presence of interction, ttenution bis impcts identi ction of other structurl prmeters, here the strtegic externlity
Identi ction nd Informtion: Mny Agents non-zero interction: A 6= 0 informtion structure 2 x ; 2 y of Byesin gme is ssumed to be known the identi ction, given the hypothesis of BNE, uses 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 2 y x = 2 x 2 y + A lends informtion bout the sign of A
Identi ction with Incomplete Informtion: informtion structure 2 x ; 2 y of Byesin gme is ssumed to be known Proposition (Sign nd Point 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 slope of the equilibrium response: + A.
Robust Identi ction with Incomplete Informtion informtion structure of Byesin gme is ssumed to be unknown we cn use observed men, vrince, nd covrince of (; ) nd equilibrium conditions: = + + A nd = A +
Prtil Identi ction Proposition (Prtil Identi ction) 1 The sign of is, but the sign of A is not identi ed. 2 If < 1, the slope of the equilibrium response is prtilly identi ed: 2 ; 1 : + A filure to identify the strtegic nture of the gme, the sign of A describes whether the gme is one of 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 nd @r ( x ; y ) @ x < 0; @R ( x ; y ) @ y > 0:
Prior Informtion nd Set Identi ction A y 0< y < y < y Set identifiction with 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 s 2 x ; 2 y! 0, sign of strtegic interction is identi ed
liner inverse demnd is given by: Demnd nd Supply Identi ction P d = d + A Q + d d liner inverse supply is given by: P s = s + Q + s s two-dimensionl uncertinty: d nd s re demnd nd supply shocks ( demnd, supply shifters ) rm i mkes supply decision with noisy informtion bout ( d ; s ) only ggregte dt is observed: ggregte quntity nd price in Byes Nsh equilibrium identi ed ; A ; d ; s re point in Byes Correlted equilibrium nd s re only set identi ed
Discussion Byes correlted equilibrium encodes concern for robustness to strtegic informtion environment next items on the gend: robust comprtive sttics, robust policy nlysis 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