Reversals of Signal-Posterior Monotonicity for Any Bounded Prior

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1 Reversls of Signl-Posterior Monotonicity for Any Bounded Prior Christopher P. Chmbers Pul J. Hely Abstrct Pul Milgrom (The Bell Journl of Economics, 12(2): ) showed tht if the strict monotone likelihood rtio property (MLRP) does not hold for conditionl distribution then there exists some non-degenerte prior nd pir of signls where the higher-signl posterior does not stochsticlly dominte the lower-signl posterior. We show tht for ny non-degenerte prior with bounded support there exists conditionl distribution (stisfying severl nturl properties) nd pir of signls such tht the lower signl s posterior stochsticlly domintes tht of the higher signl. Thus, for every bounded prior, higher signls my represent strictly worse news. Keywords: Signl extrction, Byes s rule, MLRP, stochstic dominnce, updting JEL: C11, C60, D81, D84 The uthors thnk Jim Peck, Simon Grnt, nd two nonymous referees for their vluble comments. Deprtment of Economics, University of Cliforni, Sn Diego, MC Gilmn Drive, L Joll, CA Phone: (858) Emil: cpchmbers@ucsd.edu. Corresponding uthor. Deprtment of Economics, The Ohio Stte University, 1945 North High Street, Columbus, OH Phone: (614) Fx: (614) Emil: hely.52@osu.edu. 1

2 The clssic good news, bd news result of Milgrom (1981) shows tht the strict monotone likelihood rtio property (MLRP) is both necessry nd sufficient for higher signls of noisy rndom vrible to be good news, in the sense of first-order stochstic dominnce. More formlly, suppose Z is noisy signl of X, where X is distributed ccording to F, nd conditionl on X = x, Z is distributed ccording to G x, with density g x. The fmily of ll such conditionl distributions is denoted {G x }. This fmily stisfies the strict MLRP if, for ll z > z, the likelihood rtio g x (z )/g x (z ) is incresing in x. 1 Denote the unconditionl distribution of Z by G, nd the distribution of X conditionl on Z = z by F z. Using this nottion, Milgrom s result tells us tht F z first-order stochsticlly domintes F z for ll z > z independently of F if nd only if the fmily {G x } stisfies the strict MLRP. The result is compelling, s we tend to think higher vlues of noisy signl should be good news bout the underlying prmeter. Milgrom s result tells us exctly when this is the cse. According to Milgrom s result, filure of the MLRP on {G x } does not preclude the possibility tht, for some F, F z first-order stochsticlly domintes F z for ll z > z. In fct, it merely demonstrtes tht there exists some F, nd pir z > z for which F z not first-order stochsticlly dominte F z. This is not the sme s sying tht F z first-order stochsticlly domintes F z, so this does not imply tht higher signl necessrily leds to bd news. It my depend on which prior is chosen. Here, we sk if it is possible tht filure of the MLRP cn led to n extreme filure of Milgrom s result, in the sense tht higher signl reliztion cn led to bd news, regrdless of the prior. When the prior hs known, bounded support, we show tht in fct it cn, nd we do so with signl structure tht seems resonbly close to stisfying the MLRP. Specificlly, we choose Z = X + ε, where ε is independent of X, unimodl, nd symmetric. 2 does Thus, higher vlues of x led to higher signl distributions for Z, in the sense of stochstic dominnce. Given some finite support [, b] for the prior, we show tht there exists ε nd pir z > z such tht for ny non-degenerte F whose support lies in [, b], F z first-order stochsticlly domintes F z. Thus, the higher signl reliztion is bd news, no mtter wht the prior. The intuition of our proof is simple. For ny prior distribution with bounded support, consider symmetric nd unimodl signl distribution with men equl to the prmeter reliztion nd whose support is significntly lrger thn the support of the prior (though still bounded). Thus, the signl equls the underlying prmeter reliztion plus highvrince, men-zero noise vrible. If this error distribution hs sufficiently ft tils, 1 Definition 1 below gives slightly more precise definition. 2 Unimodlity is equivlent to requiring tht the density function be qusiconcve. 2

3 g x (z) x 40 x 20 x x+20 x+40 Z Figure 1: An exmple of the conditionl density used in the proof. then ny extremely lrge positive observtion z is likely due to very lrge error term, indicting reltively smll prmeter vlue. For less extreme observtion z tht flls in the support of the prior it becomes more likely tht the observtion is indictive of lrge prmeter vlue. By crefully constructing the noise distribution one cn gurntee tht the posterior fter observing z stochsticlly domintes the posterior fter observing z. In fct, the construction of the noise distribution needs only to depend on the support of the prior. For the cse where the prior hs support on [ 10, 10], the constructed conditionl distribution (for ny x) is shown in Figure. With this conditionl distribution, the posterior fter observing z = 10 stochsticlly domintes the posterior fter observing z = 30 for ny prior with support on [ 10, 10]. Our proof relies hevily on the support of the prior being bounded. Our theorem does not hold if the prior is the (improper) uniform distribution over the entire rel line. With this prior nd signl tht is men-preserving spred, the signl reliztion simply shifts the loction of the posterior distribution. Higher signls shift the entire posterior to the right, nd so signl-posterior monotonicity is restored. Whether our result holds for integrble unbounded priors remins n open question. 3 More crefully, given re rel-vlued rndom vrible X with cumultive distribution F nd, for ech reliztion x of X, conditionl rndom vrible Z x with distribution G x. 4 Here, X represents some economiclly relevnt prmeter, nd Z x rndom signl of 3 Our current proof uses conditionl distributions with bounded support; if the prior were unbounded then conditionl distribution with bounded support would not generte the necessry reversl. Limiting rguments re problemtic becuse the spce of probbility mesures over the rel line is not compct in the wek* topology, so even if sequence of bounded priors converges to n unbounded prior, the required conditionl distributions nd signls need not converge. 4 In the interest of simplicity, we refrin from defining the underlying probbility spce on which these 3

4 tht prmeter. Ech G x is ssumed to hve well-defined density function g x, nd typicl reliztions re denoted by z. 5 fmily of conditionl densities is {g x }. The fmily of conditionl distributions is {G x }, nd the A rndom vrible X (nd its distribution F ) is sid to be bounded if there exists some, b R for which the probbility tht X lies in [, b] is equl to one. The support of X is the smllest such intervl. X is degenerte if there is some R such tht F () = 1 nd F (b) = 0 for ll b < ; it is non-degenerte otherwise. If the conditionl distributions re such tht Z x x is identicl (in distribution) for every x, nd if E[Z x] = x for ech x, then we sy tht the signl forms n independent dditive signl of X. This implies tht the (unconditionl) signl cn be modeled s rndom vrible Z = X + ε for some men-zero rndom vrible ε tht is independent of X. The distribution F is referred to s the prior ; upon observing ny signl reliztion z Byesin observer s posterior belief is given by the conditionl distribution F z, formed ccording to Byes s Lw in the usul wy. For completeness, we stte Milgrom s sufficiency result here. Definition 1 (MLRP). A fmily of density functions {g x } hs the strict monotone likelihood rtio property (MLRP) if x > x nd z > z imply g x (z ) g x (z ) > g x (z ) g x (z ). Thus, for ny z > z, g x (z )/g x (z ) is strictly incresing in x. Theorem (Milgrom 1981). If fmily of conditionl density functions {g x } does not hve the strict MLRP then there exists some non-degenerte prior distribution F nd two signls z > z such tht the posterior F z does not first-order stochsticlly dominte F z. Inspection of Milgrom s proof leds to slightly stronger version of this result. Corollry (Milgrom 1981). If fmily of conditionl density functions {g x } does not hve the strict MLRP then there exists some non-degenerte prior distribution F (which puts mss on only two points) nd two signls z > z such tht the posterior F z stochsticlly domintes F z. The following theorem is our min result. strictly first-order It shows how signl monotonicity cn be reversed for ny non-degenerte, bounded prior if the modeler cnnot commit to prticulr noise (or conditionl) distribution. Theorem. Fix ny < b. There exists fmily of conditionl density functions {g x } nd two signl reliztions z > z such tht for ll X whose support is [, b], F z strictly firstorder stochsticlly domintes F z. Furthermore, {g x } forms n independent dditive signl, nd ech g x is unimodl nd symmetric. rndom vribles re defined. 5 When x is outside the support of the prior, let g x be ny rbitrry distribution. 4

5 Proof. Let [, b] R (with < b) be the support of X, set d = b, nd for ech x [, b] let g x be given by g x (z) = 1 for z [x 2d, x d] [x + d, x + 2d] 4d+d 2 1 (1 + d + (z x)) for z (x d, x] 4d+d 2 1 (1 + d (z x)) for z (x, x + d). 4d+d 2 Note tht g x hs men of x, is symmetric, nd unimodl for ech x; n exmple of this distribution is shown in Figure. Now consider z = b nd z = b + d, which re two fesible reliztions of Z such tht z > z. Fix ny w [, b] nd note tht the posterior distribution on X given z is equl to F z (w) = w b (x + 1)dF (x). (1) (x + 1)dF (x) Moreover, note tht the posterior of X conditionl on z is distributed the sme s the prior, so tht F z F. Seprtely integrting the numertor nd denomintor of (1) by prts nd rerrnging, we obtin F z (w) = (w + 1)F (w) w (d + 1) b = F (w) w 1 b F (x)dx F (x)dx [ F (w) + F (x)]dx [ 1 + F (x)]dx = F (w) + w [F (w) F (x)]dx 1 + b [1 F (x)]dx. (2) Clerly, if F (w) = 0 then this expression evlutes to 0 t w nd hence F z (w) F z (w), consistent with F z F z (w) since F z (w) = 1. stochsticlly dominting F z. If F (w) = 1 then obviously F z (w) Finlly, consider the cse where F (w) (0, 1). For these vlues of w the following is true of the numertor of (2): F (w) + w ( [F (w) F (x)]dx = F (w) 1 + ( F (w) 1 + ( F (w) 1 + w w b [ 1 F (x) ] F (w) [1 F (x)]dx ) [1 F (x)]dx ) dx ) 5

6 If w > then the first inequlity is strict since F (w) < 1. If w = then the second inequlity is strict becuse b >. 6 Dividing by the term in prentheses, we thus estblish tht F z (w) = F (w) + w [F (w) F (x)]dx 1 + < b [1 F (x)]dx F (w). Reclling eqution (2) nd the fct tht F z F, the bove inequlity implies F z (w) < F z (w). Therefore, F z strictly first-order stochstic domintes F z, even though z < z. The following key points re importnt. As stted, requiring the signl to be n independent dditive signl of X nd to stisfy dditionl properties results in much stronger theorem thn if no such conditions were required. If the signl distribution were not required to stisfy ny conditions, setting Z = X would estblish our result trivilly. Our result is not implied by, nor does it imply, Milgrom s result. Nor do simple modifictions of either result imply the other. To be cler, the difference lies in the quntifiction. Milgrom s result shows tht for ny conditionl distribution filing MLRP there exists prior distribution generting reversl of signl monotonicity. Our result hs the quntifiers reversed: for ny prior distribution there exists (well-behved) conditionl distribution generting reversl of signl monotonicity. Actully, this conditionl distribution cn be chosen s function of the support only. This distinction in quntifiction is criticl. The corollry of Milgrom s result given bove genertes reversl of signl monotonicity using prticulr prior distribution with two-point support. Focusing on priors with two-point supports necessrily strengthens the contrpositive of Milgrom s originl theorem becuse the FOSD reltion restricted to the fmily of distributions which hve the sme two-point supports is complete. Thus, we emphsize the point lluded to in the previous bullet: Our theorem holds for ny prior distribution which is non-degenerte nd hs bounded support not just those whose support hs only two points. Although the conditionl distribution used in our proof obviously must fil the strict MLRP (see below for verifiction of this fct), we rgue tht it is nturl in most other respects. In prticulr, symmetry round x nd qusiconcvity of the density imply 6 Recll tht this cse ssumes F (w) > 0, so w = implies point mss t. Since b > it cnnot be tht F (w) = 1. 6

7 tht signls re unbised nd signls closer to x re more likely thn signls frther from x. By Milgrom s result, it must be tht, for ny bounded X, the fmily of conditionl distributions used in the proof violtes the strict MLRP. We now verify this fct directly, for completeness. Let the support of X be [, b] with < b, set d = b, nd consider x =, x = b, z = b, nd z = b + d. The strict MLRP requires tht g x (z )g x (z ) > g x (z )g x (z ), or, substituting in the bove vlues of x, x, z, nd z, g (b)g b (b + d) > g (b + d)g b (b). This expression evlutes to ( ) 2 ( ) ( ) d >, 4d + d 2 4d + d 2 4d + d 2 but the right-hnd side is strictly lrger, so the strict MLRP is violted. Finlly, we illustrte our theorem with simple ppliction borrowed from Milgrom (1981). We imgine n economy with one risky nd one riskless sset. The risky sset s returns hve density f. We hve collection of identicl gents, ech of whom possesses the sme differentible utility u. Ech consumer is endowed with one unit of the risky sset nd one unit of the riskless sset. By normlizing the price of the riskless sset to one, the price of the risky sset (in equilibrium) is given by p = E[Xu (1 + X)]. E[u (1 + X)] Note in prticulr tht by defining the density function h(x) = f(x) u (1+x), we get tht E[u (1+X)] p = E[X], where E is the expecttion for p, tken with respect to h. If, insted, gents observe informtion in the form of noisy signl Z before trding, then the equilibrium price of the risky sset is given by p(z) = E[Xu (1 + X) z]. E[u (1 + X) z] According to Milgrom, this is the sme s p(z) = E[X z]. Milgrom s theorem implies tht if the noisy signl stisfies the strict MLRP, then p(z) is monotoniclly incresing in z. The point of our theorem is to show tht there cn be 7

8 very well-behved signl structures (specificlly, where Z = X + ε nd ε is independent of X, unimodl, nd symmetric) under which monotonicity cn be reversed for some signls: There re two signls z > z where p(z ) < p(z ). References Milgrom, P. R. (1981): Goods News nd Bd News: Representtion Theorems nd Applictions, The Bell Journl of Economics, 12,