Reversals of signal-posterior monotonicity for any bounded prior

PaulMilgrom [Milgrom, P.R., 1981. Goods news and bad news: representation theorems and applications. The Bell Journal of Economics 12, 380–391] showed that if the strict monotone likelihood ratio property (MLRP) does not hold for a conditional distribution then there exists some non-degenerate prior and pair of signals where the higher-signal posterior does not stochastically dominate the lower-signal posterior. We show that for any non-degenerate prior with bounded support there exists a conditional distribution (satisfying several natural properties) and pair of signals such that the lower signal’s posterior stochastically dominates that of the higher signal. Thus, for every bounded prior, higher signals may represent strictly ‘‘worse’’ news. © 2011 Elsevier B.V. All rights reserved. The classic ‘‘good news, bad news’’ result of Milgrom (1981) shows that the strict monotone likelihood ratio property (MLRP) is both necessary and sufficient for higher signals of a noisy random variable to be ‘‘good news’’, in the sense of first-order stochastic dominance. More formally, suppose Z is a noisy signal of X , where X is distributed according to F , and conditional on X = x, Z is distributed according to Gx, with density gx. The family of all such conditional distributions is denoted {Gx}. This family satisfies the strict MLRP if, for all z ′′ > z , the likelihood ratio gx(z ′′)/gx(z ) is increasing in x.2 Denote the unconditional distribution of Z by G, and the distribution of X conditional on Z = z by Fz . Using this notation, Milgrom’s result tells us that Fz′′ first-order stochastically dominates Fz′ for all z ′′ > z ′ independently of F if and only if the family {Gx} satisfies the strict MLRP. The result is compelling, as we tend to think higher values of a noisy signal should be ‘‘good news’’ ✩ The authors thank Jim Peck, Curtis E. Bear, Simon Grant, and two anonymous referees for their valuable comments. ∗ Corresponding author. Tel.: +1 614 247 8876; fax: +1 614 292 3906. E-mail addresses: [email protected] (C.P. Chambers), [email protected] (P.J. Healy). 1 Tel.: +1 858 534 2988. 2 Definition 1 gives a slightly more precise definition. about the underlying parameter. Milgrom’s result tells us exactly when this is the case. According toMilgrom’s result, a failure of theMLRP on {Gx} does not preclude the possibility that, for some F , Fz′′ first-order stochastically dominates Fz′ for all z ′′ > z . In fact, it merely demonstrates that there exists some F , and a pair z ′′ > z ′ for which Fz′′ does not first-order stochastically dominate Fz′ . This is not the same as saying that Fz′ first-order stochastically dominates Fz′′ , so this does not imply that a higher signal necessarily leads to ‘‘bad news’’. It may depend on which prior is chosen. Here, we ask if it is possible that a failure of the MLRP can lead to an ‘‘extreme’’ failure of Milgrom’s result, in the sense that a higher-signal realization can lead to ‘‘bad news’’, regardless of the prior. When the prior has a known, bounded support, we show that in fact it can, and we do so with a signal structure that seems reasonably ‘‘close’’ to satisfying the MLRP. Specifically, we choose Z = X+ε̃, where ε̃ is independent of X , unimodal, and symmetric.3 Thus, higher values of x lead to higher-signal distributions for Z , in the sense of stochastic dominance. Given some finite support [a, b] for the prior, we show that there exists ε̃ and a pair z ′′ > z ′ 3 Unimodality is equivalent to requiring that the density function be quasiconcave. 0165-4896/$ – see front matter© 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2011.02.003 2 C.P. Chambers, P.J. Healy / Mathematical Social Sciences ( ) – Fig. 1. An example of the conditional density used in the proof. such that for any non-degenerate F whose support lies in [a, b], Fz′ first-order stochastically dominates Fz′′ . Thus, the higher-signal realization is ‘‘bad news’’, no matter what the prior. The intuition of our proof is simple. For any prior distribution with bounded support, consider a symmetric and unimodal signal distribution with mean equal to the parameter realization and whose support is significantly larger than the support of the prior (though still bounded). Thus, the signal equals the underlying parameter realization plus a high-variance, mean-zero ‘noise’ variable. If this error distribution has sufficiently ‘fat’ tails, then any extremely large positive observation z ′′ is likely due to a very large error term, indicating a relatively small parameter value. For a less extreme observation z ′ that falls in the support of the prior it becomes more likely that the observation is indicative of a large parameter value. By carefully constructing the noise distribution one can guarantee that the posterior after observing z ′ stochastically dominates the posterior after observing z . In fact, the construction of the noise distribution needs only to depend on the support of the prior. For the case where the prior has support on [−10, 10], the constructed conditional distribution (for any x) is shown in Fig. 1. With this conditional distribution, the posterior after observing z ′ = 10 stochastically dominates the posterior after observing z ′′ = 30 for any prior with support on [−10, 10]. Our proof relies heavily on the support of the prior being bounded. Our theorem does not hold if the prior is the (improper) uniform distribution over the entire real line. With this prior and a signal that is a mean-preserving spread, the signal realization simply shifts the location of the posterior distribution. Higher signals shift the entire posterior to the right, and so signal-posterior monotonicity is restored. Whether our result holds for integrable unbounded priors remains an open question.4 More carefully, given are a real-valued random variable X with cumulative distribution F and, for each realization x of X , a conditional random variable Z |x with distribution Gx. Here, X represents some economically relevant parameter, and Z |x a randomsignal of that parameter. EachGx is assumed to have awelldefined density function gx, and typical realizations are denoted by z.6 The family of conditional distributions is {Gx}, and the family of conditional densities is {gx}. 4 Our current proof uses conditional distributions with bounded support; if the prior were unbounded then a conditional distributionwith bounded support would not generate the necessary reversal. Limiting arguments are problematic because the space of probability measures over the real line is not compact in the weak topology, so even if a sequence of bounded priors converges to an unbounded prior, the required conditional distributions and signals need not converge. 5 In the interest of simplicity, we refrain from defining the underlying probability space on which these random variables are defined. 6 When x is outside the support of the prior, let gx be any arbitrary distribution. A randomvariableX (and its distribution F ) is said to be bounded if there exists some a, b ∈ R for which the probability that X lies in [a, b] is equal to one. The support of X is the smallest such interval. X is degenerate if there is some a ∈ R such that F(a) = 1 and F(b) = 0 for all b < a; it is non-degenerate otherwise. If the conditional distributions are such that Z |x − x is identical (in distribution) for every x, and if E[Z |x] = x for each x, then we say that the signal forms an independent additive signal of X . This implies that the (unconditional) signal can bemodeled as a random variable Z = X + ε̃ for some mean-zero random variable ε̃ that is independent of X . The distribution F is referred to as the ‘prior’; upon observing any signal realization z a Bayesian observer’s posterior belief is given by the conditional distribution Fz , formed according to Bayes’ Law in the usual way. For completeness, we state Milgrom’s sufficiency result here. Definition 1 (MLRP). A family of density functions {gx} has the strict monotone likelihood ratio property (MLRP) if x > x and z ′′ > z ′ imply gx′′(z ′′)gx′(z ) > gx′′(z ′)gx′(z ). Thus, for any z ′′ > z , gx(z ′′)/gx(z ) is strictly increasing in x. Theorem (Milgrom, 1981). If a family of conditional density functions {gx} does not have the strict MLRP then there exists some non-degenerate prior distribution F and two signals z ′′ > z ′ such that the posterior Fz′′ does not first-order stochastically dominate Fz′ . Inspection of Milgrom’s proof leads to a slightly stronger version of this result. Corollary (Milgrom, 1981). If a family of conditional density functions {gx} does not have the strict MLRP then there exists some non-degenerate prior distribution F (which puts mass on only two points) and two signals z ′′ > z ′ such that the posterior Fz′ strictly first-order stochastically dominates Fz′′ . The following theorem is our main result. It shows how signal monotonicity can be reversed for any non-degenerate, bounded prior if the modeler cannot commit to a particular noise (or conditional) distribution. Theorem. Fix any a < b. There exists a family of conditional density functions {gx} and two signal realizations z ′′ > z ′ such that for all X whose support is [a, b], Fz′ strictly first-order stochastically dominates Fz′′ . Furthermore, {gx} forms an independent additive signal, and each gx is unimodal and symmetric. Proof. Let [a, b] ⊂ R (with a < b) be the support of X , set d = b − a, and for each x ∈ [a, b] let gx be given by gx(z) =  1 4d + d2 for z ∈ [x − 2d, x − d] ∪ [x + d, x + 2d] 1 4d + d2 (1 + d + (z − x)) for z ∈ (x − d, x] 1 4d + d2 (1 + d − (z − x)) for z ∈ (x, x + d). Note that gx has a mean of x, is symmetric, and unimodal for each x; an example of this distribution is shown in Fig. 1. Now consider z ′ = b and z ′′ = b + d, which are two feasible realizations of Z such that z ′′ > z . Fix any w ∈ [a, b] and note that the posterior distribution on X given z ′ is equal to Fz′(w) =  w a (x − a + 1)dF(x)  b a (x − a + 1)dF(x) . (1) Moreover, note that the posterior of X conditional on z ′′ is distributed the same as the prior, so that Fz′′ ≡ F . C.P. Chambers, P.J. Healy / Mathematical Social Sciences ( ) – 3 Separately integrating the numerator and denominator of (1) by parts and rearranging, we obtain Fz′(w) = (w − a + 1)F(w) −  w a F(x)dx (d + 1) −  b a F(x)dx = F(w) −  w a [−F(w) + F(x)]dx 1 −  b a [−1 + F(x)]dx = F(w) +  w a [F(w) − F(x)]dx 1 +  b a [1 − F(x)]dx . (2) Clearly, if F(w) = 0 then this expression evaluates to 0 at w and hence Fz′(w) ≤ Fz′′(w), consistent with Fz′ stochastically dominating Fz′′ . If F(w) = 1 then obviously Fz′(w) ≤ Fz′′(w) since Fz′′(w) = 1. Finally, consider the case where F(w) ∈ (0, 1). For these values of w the following is true of the numerator of (2):