Posterior Probabilities: Dominance and Optimism∗

Posterior Probabilities: Dominance and Optimism ^∗

Sergiu Hart ^† Yosef Rinott ^‡ November 11, 2019

Abstract

The distribution of the posterior probability of the true state is dominated in the likelihood ratio order, a strengthening of first-order stochastic dominance, by the distribution of that same posterior prob- ability under the probability law of the true state. This generalizes to notions of “optimism” about posterior probabilities.

1 Introduction

Alice is a classic Bayesian decision-maker, in a setup with a state that is unknown, a prior probability distribution over the states, and signals whose distribution depends on the state. Alice is interested in a particular “good”

state γ, and so she computes, by Bayes’ rule, the posterior probability of γ for each possible signal; call this posterior probability Q γ . Before getting the

∗

First version: September 2018. The authors thank Marco Scarsini for many discussions on this topic, and Alex Frankel, Drew Fudenberg, Ilan Kremer, Motty Perry, and Benji Weiss for useful suggestions and references.

†

The Hebrew University of Jerusalem (Federmann Center for the Study of Rationality, Department of Economics, and Institute of Mathematics). E-mail : hart@huji.ac.il Web site : http://www.ma.huji.ac.il/hart

‡

The Hebrew University of Jerusalem (Federmann Center for the Study of Rationality,

and Department of Statistics). E-mail : yosef.rinott@mail.huji.ac.il Web site :

http://pluto.huji.ac.il/~rinott

signal, Alice thus believes that Q γ will be either higher or lower than the prior probability p of γ, with an expectation exactly equal to p.

Now suppose that Carroll (an outside observer, say) knows that the state is in fact γ, and he considers what will be the posterior probability Q γ com- puted by Alice (who does not know that the state is γ). How does Carroll’s belief on Q γ compare to Alice’s belief on Q γ (both beliefs are of course before the signal)? ¹ It turns out that Carroll always gives higher probability than Alice to high values of Q γ , and lower probabilities to low values; formally, we will show that Carroll’s probability distribution of Q γ stochastically dom- inates Alice’s in the so-called “likelihood ratio” order, a strengthening of first-order stochastic dominance. ² What may come as a surprise here is that this fact, whose proof is nothing but immediate, seems to be not known.

Even its immediate consequence that Carroll’s expectation of Q γ is higher than the prior p (which, as pointed out above, is equal to Alice’s expectation of Q γ ) seems to be still widely unknown despite it having appeared in the literature. ³

This dominance result is stated and proved formally in Section 2 imme- diately below. It naturally leads to notions of “optimism” about posterior probabilies, for single sets in Section 3, and for multiple sets with MLRP signals in Section 4. For a study of sequences of signals, see Hart and Rinott (2019).

2 Dominance

The standard Bayesian setup that Alice faces is as follows. The space of states of nature is Θ, and ⁴ π ∈ ∆(Θ) is the prior probability distribution

1

Lest there be any misunderstanding, we are not considering Carroll’s posterior proba- bility of γ (which equals 1), but rather Carroll’s view of Alice’s posterior probability Q of γ. For a simple setup where this is a natural question, with Alice the uninformed market and Carroll an informed agent, see the end of Section 2 below.

2

A standard reference on the various stochastic dominance orders is Shaked and Shan- thikumar (2010); see in particular Section 1.C there.

3

See the references to Corollary 2 below. None of the colleagues whom we asked knew this result, just as those whom Francetich and Kreps asked didn’t.

4

We denote by ∆(X) the set of probability distributions over the set X.

on Θ. The state γ in Θ is a state of interest, with prior p := π(γ); to avoid trivialities, assume that 0 < p < 1. Let S be the set of signals and σ(s|θ) the probability of signal s in state θ (thus P

s∈S σ(s|θ) = 1 for all θ). For simplicity, all spaces are assumed to be finite. The prior probability π and the signalling probabilities σ induce a probability P ≡ P π,σ on Θ × S.

For each signal s in S, Alice’s posterior probability of γ, which we denote by Q γ (s), is

Q γ (s) = P(γ|s) = P(γ ∩ s)

P(s) = π(γ)σ(s|γ) P

θ∈Θ π(θ)σ(s|θ) .

The posterior of γ is a random variable that takes the value Q γ (s) with probability P(s); its expectation equals the prior, i.e., E [Q γ ] = π(γ) = p.

Carroll knows that the state is in fact γ. He thus assigns probability ⁵ P ^γ (s) := P(s|γ) = σ(s|γ) to the signal s, and so he believes that Alice’s pos- terior of γ takes the value Q γ (s) with probability P ^γ (s) (instead of P(s)). We denote this random variable by (Q γ , P ^γ ), whereas Alice’s posterior random variable, which takes the value Q γ (s) with probability P(s), is denoted by (Q γ , P).

What is the relation between (Q γ , P ^γ ) and (Q γ , P) ? It turns out that the former is always higher ⁶ than the latter, in the following precise sense.

Proposition 1 (Q γ , P ^γ ) dominates (Q γ , P) in the likelihood ratio order.

The likelihood ratio stochastic order is a strengthening of first-order stochastic dominance. Formally, let X and Y be two random variables. Then X first-order stochastically dominates Y if P(X ≥ v) ≥ P(Y ≥ v) for every v; and X dominates Y in the likelihood ratio order, denoted by X ≥ lr Y, if P(X = v)/P(Y = v) is an increasing function of ⁷ v. The former is equivalent to E [ϕ(X)] ≥ E [ϕ(Y )] for every increasing function ϕ, and the latter to

5

Superscripts on π, P, and E are used throughout to denote conditioning.

6

All comparisons such as “higher” and “increasing” are meant in the weak-inequality sense (unless, of course, they are preceded by “strict”).

7

For nonnegative functions g and h, the statement that the ratio g(v)/h(v) is increasing

in v is always understood as g(v

)h(v) ≥ g(v)h(v

) for every v

> v (this takes care of the

possible 0/0 ratios).

E [ϕ(X)|X ∈ A] ≥ E [ϕ(Y )|Y ∈ A] for every increasing function ϕ and every measurable set of values A ⊆ R.

Proposition 1 says that high values of the posterior Q γ are more probable under P ^γ than under P, whereas low values of the posterior are more probable under P than under P ^γ . This is readily seen from Bayes’ formula

Q γ (s) = p · P ^γ (s)

P(s) (1)

(recall that π(γ) = p): the higher Q γ (s) is, the relatively more probability mass goes to P ^γ (s) than to P(s). Formally:

Proof of Proposition 1. By (1), Q γ (s) = q if and only if P ^γ (s)/P(s) = q/p, and so summing over all s with Q γ (s) = q yields P ^γ (Q γ = q)/P(Q γ = q) = q/p, showing that the likelihood ratio is an increasing function of ⁸ q.

Remarks. (a) The result and proof of Proposition 1 trivially extend from a single state γ ∈ Θ to a subset of states Γ ⊂ Θ (cf. Remark (a) following Proposition 5 below).

(b) If the signal is completely uninformative about γ (that is, the pos- terior of γ is independent of the signal, and thus equal to the prior; i.e., Q γ (s) = p for every ⁹ s in S), then the constant random variables (Q γ , P ^γ ) and (Q γ , P) are of course identical. In all other cases the proof above shows that the dominance is strict, in the sense that the likelihood ratio P ^γ (Q γ = q)/P(Q γ = q) is a strictly increasing function of the value q of Q γ ; we denote this by (Q γ , P ^γ ) > 1r (Q γ , P). This strict dominance implies that ¹⁰ E ^γ [ϕ(Q γ )|Q γ ∈ A] > E [ϕ(Q ^γ )|Q γ ∈ A] for every strictly increasing function ϕ and every set A ⊆ [0, 1].

Despite its simplicity, this result appears not to be known. One implica- tion of the first-order stochastic dominance that is known (see Mailath and

8

This argument shows that (R, P

) ≥

(R, P

) for any two probability measures P

and P

and R an increasing function of the likelihood ratio P

/P

(on the union of their supports).

9

This happens if and only if σ(s|γ) = P

θ∈Θ

π(θ)σ(s|θ) for all s ∈ S.

10

E

denotes expectation with respect to P

, that is, expectation conditional on the

state γ; i.e., E

[X] = E [X|γ] .

Samuelson 2006, Lemma 15.4.2; Francetich and Kreps 2014, Corollary 1) is the corresponding inequality on expectations:

Corollary 2

E ^γ [Q γ ] ≥ E [Q γ ] = π(γ).

The conditional-on-γ expectation of the posterior of γ is thus higher than the prior of γ; that is, the posterior probability of γ is a submartingale relative to P ^γ . Moreover, the inequality is strict except when the signal is completely uninformative about γ; see remark (b) above.

When does this stochastic domination matter? For a simple setup, con- sider an uncertain asset whose market price is determined by the estimated probability that the state is the “good” state γ; specifically, the price is an increasing function ϕ of the probability of γ. Suppose that a signal (such as a quarterly report or a management announcement), whose distribution depends on the state, is forthcoming. Before the signal the expectation of the price by the uninformed market (represented by Alice) is thus E [ϕ(Q γ )] , whereas for an informed trader, Carroll, who knows that the state is in fact γ, this expectation is E ^γ [ϕ(Q γ )] . Our result says that Carroll’s expectation of the price is always higher than that of the market. Moreover, this holds even if one conditions on a certain set of values of the posterior, such as the pos- terior being within certain bounds: E ^γ [ϕ(Q γ )|Q γ ∈ A] ≥ E [ϕ(Q ^γ )|Q γ ∈ A]

for every set A ⊆ [0, 1]. Thus, no matter what the information structure is, the informed trader who knows that the state is γ is ex-ante (i.e., be- fore the signal) more “optimistic” than the market about the ex-post (i.e., after the signal) price. ¹¹ He will therefore buy the asset now. By the same token, if Carroll knows that the state is not γ (i.e., he assigns probability π(θ)/(1 − π(γ)) to each state θ 6= γ), then all the above inequalities are reversed, ¹² and he therefore sells the asset. But if Carroll knows more than

11

We emphasize that the optimism is not about γ per se (which goes without saying), but rather about the distribution of the market’s posterior of γ after the signal, and thus about the market’s behavior then.

12

The expectation E[X] is a (weighted) average of the conditional-on-γ expectation

E

[X] and the conditional-on-not-γ expectation E

[X], and so E

[X] ≥ E[X] if and

only if E

[X] ≤ E[X].

just “it is not γ”—for instance, he knows the precise state θ 6= γ—then the conclusion is no longer true in general, and he may well want to buy rather than sell the asset (see the next section for a simple example).

Another relevant setup is that of reputation models with imperfect mon- itoring; see Fudenberg and Levine (1992) and Mailath and Samuelson (2006, Section 15.4). Let Alice represent the short-run player (who plays only once) and Carroll the long-run player. Suppose that Carroll plays like the “com- mitment” type ¹³ γ; then Alice’s posterior probability that Carroll is in fact of type γ becomes higher in the likelihood ratio order, thus strengthening his

“reputation.”

3 Optimism

Consider a general setup where Carroll has a prior distribution eπ ∈ ∆(Θ) over the states that may be different from Alice’s prior π (the special case where Carroll knows that the state is γ corresponds to eπ putting probability 1 on γ and 0 on every θ 6= γ), and there is a set Γ ⊂ Θ of states of nature that is of interest. To avoid trivialities, assume that p := π(Γ) satisfies 0 < p < 1.

For every signal s ∈ S, Alice’s posterior probability Q ^Γ of Γ is given by

Q Γ (s) := P(Γ|s) = P(Γ ∩ s) P(s) =

P

θ∈Γ π(θ)σ(s|θ) P

θ∈Θ π(θ)σ(s|θ) = p · P ^Γ (s) P(s) ,

where P ^Γ denotes the conditional-on-Γ probability, i.e., P ^Γ (s) = P(s|θ ∈ Γ) = P(Γ and s)/P(Γ) = P(Γ and s)/π(Γ). The posterior is a random vari- able (Q Γ , P) that takes the value Q ^Γ (s) ≡ P(Γ|s) with probability P(s); its expectation equals the prior, i.e.,

E [Q ^Γ ] = π(Γ) = p. (2)

Let e P ≡ P π,σ e denote Carroll’s probability, i.e., the probability on Θ × S

13

That is, Carroll plays so as to generate a distribution of signals to Alice that is identical

to the one that obtains from playing the fixed mixed action of type γ (as we will see in the

next section, it may well suffice to have the distribution of signals move in the direction

of the distribution corresponding to γ).

that is induced by eπ and σ. Before the signal s, Carroll believes that Alice’s posterior probability of Γ will be Q ^Γ (s) with probability e P(s) (rather than P(s)), i.e., the random variable (Q ^Γ , e P).

We will now allow the signalling to vary, and so a pair (S, σ) as above (formally, S is a finite set and σ is a Markov transition matrix from Θ to S) will be called a signalling structure on the fixed state space Θ.

We define two concepts.

Definition 3 Carroll is Γ-optimistic if (Q Γ , e P) dominates (Q ^Γ , P) in the like- lihood ratio order for every signalling structure (S, σ) on the state space Θ.

Definition 4 Carroll’s prior eπ is a Γ-strengthening of the prior π if eπ is an average of π and π ^Γ , i.e., eπ = aπ ^Γ + (1 − a)π for some a ∈ [0, 1] (where π ^Γ (θ) ≡ π(θ|Γ) is the prior probability of θ conditional on ¹⁴ Γ).

It is readily seen that a Γ-strengthening is a uniform rescaling upwards of the probabilities of all states θ ∈ Γ and a uniform rescaling downwards of the probabilities of all states θ ^′ ∈ Γ (specifically, the former are all multiplied / by a/p + 1 − a ≥ 1, and the latter by 1 − a ≤ 1). It is also equivalent to Carroll starting with the same prior π and then receiving (before the signal S) a specific signal t 0 ∈ T from another signalling structure (T, τ ) with τ (t 0 |θ) = b ≥ c = τ (t 0 |θ ^′ ) for every θ ∈ Γ and every θ ^′ ∈ Γ (thus t / 0 is a

“Γ-signal”). ¹⁵ A special case is when Carroll knows exactly Γ, i.e., eπ = π ^Γ (this corresponds to b = 1 and c = 0).

The two concepts of optimism and strengthening are equivalent.

Proposition 5 Carroll is Γ-optimistic if and only if Carroll’s prior eπ is a Γ-strengthening of the prior π.

Proof. If eπ = aπ ^Γ + (1 − a)π then e P(s) = aP(s|Γ)+(1−a)P(s) = (a/p)P(Γ∩

s) + (1 − a)P(s) for every s, and so, dividing by P(s), e P(s)

P(s) = a

p Q Γ (s) + (1 − a),

14

Thus π

is an element of ∆(Θ), with π

(θ) = π(θ)/π(Γ) for θ ∈ Γ and π

(θ) = 0 for θ / ∈ Γ.

15

Indeed, a = (b − c)p/((b − c)p + c).

which is increasing in Q Γ (s). This readily yields likelihood ratio dominance:

P(Q e ^Γ = q)/P(Q ^Γ = q) = (a/p) · q + 1 − a is increasing in q (cf. the proof of Proposition 1).

Conversely, assume that Carroll is Γ-optimistic, and take S = {0, 1} with σ(1|θ) = 1/2 + x θ and σ(0|θ) = 1/2 − x θ with |x θ | ≤ 1/2. If P

θ∈Θ π(θ)x θ = 0 and P

θ∈Γ π(θ)x θ > 0 we have P(1) = P(0) = 1/2 and P(Γ ∩ 1) > P(Γ ∩ 0), and thus P(Γ|1) > P(Γ|0). Then Γ-optimism implies that ¹⁶ e P(1) ≥ P(1), i.e., P

θ π(θ)x e θ ≥ 0.

We thus have the following: if P

θ∈Θ π(θ)x θ = 0 and P

θ∈Θ π ^Γ (θ)x θ > 0 (we have divided by π(Γ)) then P

θ∈Θ π(θ)x e θ ≥ 0 (this holds for any x = (x θ ) θ∈Θ even if some x θ do not satisfy |x θ | ≤ 1/2: just rescale x so that they all do, which does not affect any of the above three homogenenous relations).

Equivalently, there is no x such that P

θ π(θ)x θ = 0, P

θ π ^Γ (θ)x θ > 0, and P

θ π(θ)x e θ < 0 (the sums are over all θ ∈ Θ), i.e., no x such that π · x = 0, π ^Γ · x > 0, and (−eπ) · x > 0. Motzkin’s Theorem of the Alternative (see, e.g., Mangasarian 1969) therefore yields y, z, w such that

yπ + zπ ^Γ + w(−eπ) = 0 (3)

and z, w ≥ 0 with at least one of them strictly positive. Take θ / ∈ Γ (recall that π(Γ) < 1); then yπ(θ) = weπ(θ) by (3), and so w ≥ 0 yields y ≥ 0. If w = 0 then we must have z > 0, but then the sum in (3) is positive for any θ ∈ Γ (recall that π(Γ) > 0), a contradiction. Therefore w > 0, and dividing by w yields

e

π = aπ ^Γ + a ^′ π

with a = z/w ≥ 0 and a ^′ = y/w ≥ 0. Summing over all θ in Θ gives 1 = a+a ^′ , and the proof is complete.

Remarks. (a) The special case where Carroll knows Γ, and thus eπ = π ^Γ and e P = P ^Γ , yields

(Q Γ , P ^Γ ) ≥ lr (Q Γ , P), (4)

16

For binary random variables, likelihood ratio dominance and first-order dominance

are equivalent: both amount to the high value having a higher probability.

which is the straightforward extension of Proposition 1 from single states to sets of states (the direct proof is the same as for Proposition 1; see Remark (a) there).

(b) The proof of the second direction above shows that when defining optimism it suffices to require first-order stochastic dominance (instead of likelihood ratio dominance) to hold for all binary signalling structures, i.e.,

|S| = 2 (instead of all signalling structures).

(c) Applying Proposition 5 to Γ ^C = Θ\Γ, the complement of Γ, and using P(Γ ^C |s) = 1 − P(Γ|s)), yields the following: “Γ-pessimism” (which is defined by reversing the direction of the dominance relation in Γ-optimism) is equiv- alent to Γ ^C -strengthening (which, in turn, is nothing but “Γ-weakening”); in terms of the above signal t ⁰ , it means that b ≤ c.

Interestingly, if Carroll knows more than Γ—for instance, if Γ = {β, γ}

and Carroll knows that the state is γ—then some of these inequalities may well be reversed: while knowing Γ makes one optimistic (relative to Alice, or the market) about the posterior of Γ, knowing strictly more than Γ may well turn an optimist into a pessimist. For a simple example, suppose that the set of states is Θ = {α, β, γ}, the prior π is uniform on Θ (i.e., each state has probability 1/3), and Γ = {β, γ}. Let S = {0, 1} be the set of signals, with signalling probabilities σ(s|θ) as given in the table below (the last two columns then give the total probability P(s) of each signal s and the posterior Q Γ (s) ≡ P(Γ|s)):

σ(s|α) σ(s|β) σ(s|γ) P(s) P ^Γ (s) Q Γ (s)

s = 1 3/8 3/4 1/2 13/24 5/8 10/13

s = 0 5/8 1/4 1/2 11/24 3/8 6/11

Proposition 5 (see (4)) says that (Q Γ , P ^Γ ) > lr (Q Γ , P), which is easily checked

here: the posterior Q Γ is higher after signal s = 1 than after signal s = 0 (i.e.,

Q Γ (1) = 10/13 > 6/11 = Q Γ (0)), and P ^Γ gives a higher probability than P

to s = 1 (i.e., P ^Γ (1) ≡ P(1|Γ) = 5/8 > 13/24 = P(1)). However, if Carroll

were to know that the state is γ, then the dominance would be reversed, i.e.,

(Q ^Γ , P ^γ ) < ^lr (Q ^Γ , P); indeed, the high-posterior signal s = 1 would then get

a lower probability under P ^γ than under ¹⁷ P (i.e., P ^γ (1) = 1/2 < 13/24 = P(1)).

4 Monotonic Optimism

Suppose now that more than one set of states matters; for instance, the asset price above is an increasing function ϕ of the probabilities of several different sets of states Γ 1 , ..., Γ n ⊂ Θ. What does it take to be optimistic in this case?

Proposition 5 implies that even for two sets Γ 1 and Γ 2 that are distinct, which is taken to mean π(Γ 1 \Γ 2 ) > 0 or π(Γ 2 \Γ 1 ) > 0, Carroll cannot be both Γ 1 -optimistic and Γ 2 -optimistic unless eπ = π; that is, Carroll must be the same as Alice, a standard Bayesian with no additional information.

The definition of optimism requires taking into account all possible sig- nalling structures, which suggests restricting them to a natural subclass of interest. We consider the much used class of signalling structures that have the monotone likelihood ratio property (MLRP). Let thus Θ and S be (finite) subsets of, say, the real line R with the usual order. Then (S, σ) satisfies MLRP if σ(s ^′ |θ)/σ(s|θ) is an increasing function of θ for every s ^′ > s, i.e., σ(s ^′ |θ ^′ )σ(s|θ) ≥ σ(s ^′ |θ)σ(s|θ ^′ ) for every s ^′ > s and θ ^′ > θ.

A set Γ ⊆ Θ is an upper set (also called an increasing set) if θ ∈ Γ and θ < θ ^′ ∈ Θ implies θ ^′ ∈ Γ. We note that the posterior Q Γ (s) is an increasing function of s for any upper set Γ and any MLRP signalling structure; this is easily seen, for instance, from P(Γ ∩ s ^′ )P(s) − P(Γ ∩ s)P(s ^′ ) = P(Γ ∩ s ^′ )P(Γ ^C ∩ s) − P(Γ ∩ s)P(Γ ^C ∩ s ^′ ) = P P

π(θ ^′ )π(θ) [σ(s ^′ |θ ^′ )σ(s|θ) − σ(s ^′ |θ)σ(s|θ ^′ )] , where the double sum goes over all θ ^′ ∈ Γ and θ ∈ Γ ^C , all of which satisfy θ ^′ > θ because Γ is an upper set.

The monotonic counterparts of Definitions 3 and 4 are:

Definition 6 Carroll is monotonic-optimistic if (Q ^Γ , e P) ≥ ^lr (Q ^Γ , P) for ev- ery upper set Γ ⊆ Θ and every MLRP signalling structure (S, σ) on the state space Θ.

17

For α as the state of interest we have (Q

, P

) <

(Q

, P) but (Q

, P

) >

(Q

, P)

(because Q

(s) = 1 − Q

(s) for all s), which in the asset market setup says that if Carroll

knows “not-α” then he sells and if he knows more, specifically, γ, then he buys.

Definition 7 Carroll’s prior eπ is a monotonic strengthening of the prior π if eπ is obtained by a finite sequence of Γ ⁱ -strengthenings from π where all the Γ i ⊂ Θ are upper sets.

The following lemma characterizes monotonic strengthening.

Lemma 8 Carroll’s prior eπ is a monotonic strengthening of the prior π if and only if eπ ∈ conv{π ^Γ : Γ ⊆ Θ upper set} if and only if eπ ≥ ^lr π (as probability distributions on the space Θ).

Proof. A Γ-strengthening of π is nothing but a convex combination of π and π ^Γ , and so iterating this yields the first equivalence (note that π ≡ π ^Θ ).

Next, if eπ = aπ ^Γ + (1 − a)π then ¹⁸ e

π(θ) π(θ) = a

π(Γ) 1 Γ (θ) + 1 − a;

when Γ is an upper set this ratio is increasing in θ, and so eπ ≥ ^lr π. Since π 1 ≥ lr π 2 and π 2 ≥ lr π 2 imply π 1 ≥ lr π 3 (because π 1 (θ)/π 3 (θ) = π 1 (θ)/π 2 (θ) · π 2 (θ)/π 3 (θ)), iterating shows that if eπ is a monotonic strengthening of π then eπ ≥ ^lr π. Conversely, let θ 1 < θ 2 < ... < θ n be the elements of Θ. If e

π ≥ lr π then the increasing function h(θ) := eπ(θ)/π(θ) can be represented as P n

i=1 a i 1 Γ

(θ) where Γ i := {θ i , ..., θ n } (an upper set) and a i ≥ 0 for all i. Therefore eπ(θ) = P n

i=1 a i 1 Γ

(θ)π(θ) = P n

i=1 a i π(Γ i )π ^Γ

(θ), showing that eπ is a convex combination of the π ^Γ

(the coefficients add up to 1 because all these are probability distributions).

We again get an equivalence between the two concepts of optimism and strengthening.

Proposition 9 Carroll is monotonic-optimistic if and only if Carroll’s prior e

π is a monotonic strengthening of the prior π.

Proof. If eπ is a monotonic strengthening of π then eπ ≥ ^lr π (on Θ) by Lemma 8, which implies that e P ≥ ^lr P (on S) for every MLRP (S, σ) by a

18

The indicator 1

(θ) takes the value 1 if θ ∈ Γ and value 0 otherwise.

standard composition argument (e P and P are the compositions of σ with e π and π, respectively; see, e.g., Karlin 1968). ¹⁹ Therefore (R, e P) ≥ ^lr (R, P) for every increasing function R : S → R, which includes, as we have seen above, R = Q Γ with Γ an upper set.

Conversely, let θ 1 < θ 2 be two adjacent elements of Θ (i.e., there is no θ ∈ Θ between θ 1 and θ 2 ), and take the following signalling structure:

S = {0, 1, 2, 3}; for θ < θ 1 the signal is s = 0, for θ > θ 2 it is s = 3, for θ = θ 1 it is s = 1 with probability 2/3 and s = 2 with probability 1/3, and for θ = θ 2 it is s = 1 with probability 1/3 and s = 2 with probability 2/3. Let Γ = {θ ∈ Θ : θ ≥ θ 2 }; then Q Γ (s) takes four distinct values, ²⁰ 0 < α ¹ < α ² < 1 for s = 0, 1, 2, 3, respectively, and we have

e

π(θ ¹ ) + 2eπ(θ ² )

π(θ 1 ) + 2π(θ 2 ) = e P(Q ^Γ = α ² )

P(Q ^Γ = α 2 ) ≥ P(Q e ^Γ = α ¹ )

P(Q ^Γ = α 1 ) = 2eπ(θ ¹ ) + eπ(θ ² ) 2π(θ 1 ) + π(θ 2 )

where the inequality is by (Q Γ , e P) ≥ ^lr (Q Γ , P). Simplifying yields eπ(θ ² )/π(θ 2 ) ≥ e

π(θ ¹ )/π(θ ¹ ); since this holds for any adjacent θ ¹ < θ ² , we get eπ ≥ ^lr π.

Remark. The above proof shows that monotonic optimism is equivalent to P ≥ e ^lr P for every MLRP signalling structure (S, σ), and also to (R, e P) ≥ ^lr (R, P) for every increasing function R : S → R and every MLRP signalling structure (S, σ).

References

Francetich, A. and D. Kreps (2014), “Bayesian Inference Does Not Lead You Astray... on Average,” Economics Letters 125, 444–446.

Fudenberg, D. and D. Levine, “Maintaining a Reputation when Strategies are Imperfectly Observed,” Review of Economic Studies 59, 561–579.

Hart, S. and Y. Rinott (2019), “Monotonicity and Unimodality over Time of Expected Posterior Probabilities,” (forthcoming)

19

A direct proof is immediate: use the identity e P(s

)P(s)− e P(s)P(s) = P P

[ e π(θ

)π(θ)−

π(θ)π(θ e

)] · [σ(s

|θ

)σ(s|θ) − σ(s|θ

)σ(s

|θ)], where the sum is over all θ

> θ.

20

With α

= π(θ

)/(2π(θ

) + π(θ

)) and α

= 2π(θ

)/(π(θ

) + 2π(θ

)).

Karlin, S. (1968), Total Positivity, Stanford University Press.

Mailath, G. J. and L. Samuelson (2006), Repeated Games and Reputations, Oxford University Press.

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