Discussion Paper No. 894

CAREERIST EXPERTS
AND POLITICAL INCORRECTNESS
Chia-Hui Chen
Junichiro Ishida

March 2014

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Careerist Experts and Political Incorrectness
Chia-Hui Chen∗ and Junichiro Ishida†
March 7, 2014

Abstract
While political correctness is a dominant norm in many public situations, we also
observe behaviors that are apparently “politically incorrect,” often from professionals
and experts. This paper examines the flip side of political correctness as analyzed in
Morris (2001) to shed some light on the elusive notion of political incorrectness andelucidate its equilibrium and welfare properties. We show that there are circumstances inwhich unbiased experts deliberately take a politically incorrect stance out of reputational
concerns and identify key elements which give rise to this perverse reputational incentive.
The results suggest that political incorrectness cannot necessarily be viewed as a signof blunt honesty when informed experts have long-term reputational concerns. We also
examine the welfare consequences of political incorrectness and argue that this form of
information manipulation can be beneficial under some conditions.

JEL Classification Number: D82, D83
Keywords: cheap talk, career concerns, political correctness, political incorrectness, contrarianism.
∗†

Institute of Economics, Academia Sinica. email: chchen@econ.sinica.edu.tw
Institute of Social and Economic Research, Osaka University. email: jishida@iser.osaka-u.ac.jp

1

Introduction

In a seminal analysis of political correctness, Morris (2001) eloquently shows how the incentive
to appear politically correct obstructs truthful information transmission. He makes thispoint in an environment where an uninformed decision maker needs to solicit advice froman informed expert repeatedly over time. The expert in question may be biased in favor ofsome politically incorrect alternative, but his predispositions are only privately known. In
this dynamic context, there naturally arises an incentive for the agent to present himself as
unbiased in the early stages so as to remain credible in the eyes of the decision maker. This
reputational incentive is actually self-defeating, however, as it forces the expert to take apolitically correct stance regardless of the true state of nature. As such, political correctness
generally entails the loss of socially valuable information, illustrating why politically correct
opinions are often uninformative and unreliable as a source of knowledge. Thanks in no smallpart to this contribution, we now have a fairly clear understanding of (at least one form of)political correctness.
The situation contrasts sharply with its counterpart, i.e., political incorrectness, which hasreceived far less attention in the literature. Even then, the lack of attention per se is largely
inconsequential if we can apply this same line of reasoning to its “flip side” to gain a sense ofpolitical incorrectness. To elaborate on this possibility, consider the case where the true stateof nature happens to favor a politically correct alternative. In this contingency, the unbiased
expert should have no incentive to take a politically incorrect stance against his belief becausethat can only lower his reputation, not to mention his current payoff. Given this, because
any expert who makes a false recommendation is more likely to be perceived as biased, eventhe biased expert now has a reputational incentive to reveal the true information. Note that,
unlike in the case of political correctness, reputational concerns now discipline the expert tobe more truthful. According to this reasoning, political incorrectness should be regarded asa sign of blunt honesty, or “intellectual integrity,” in environments where reputation matters
because an informed expert would take a politically incorrect stance only when he firmly
believes in it.1
As convincing as it may sound, however, the validity of this conclusion is not necessarily
clear. At the very least, the conclusion seems rather too extreme to hold in general,2 sug1
This perception is perhaps exemplified most symbolically by a popular book series The Politically Incorrect Guide which presents conservative or so-called “politically incorrect” views on various topics such as
Darwinism, the Constitution, the Bible, and so on.
2
Although it is certainly not easy to quantify this claim, some people express an even harsher view against
political incorrectness in general. For instance, in a widely-read political blog Crooked Timber, John Quiggin
is quoted as saying “politically incorrect views are almost always incorrect in every way: literally, scientificallyand morally.”

1gesting that there may be a gap to be filled in the aforementioned argument. Particularly
suspicious in this regard is an implicit presumption that the reputational effect of politicalcorrectness (incorrectness) is invariably weakly positive (negative), which effectively rulesout the possibility that the expert intentionally takes a politically incorrect stance out ofreputational concerns. We argue that this presumption may trivialize the intricate natureof reputation formation in a dynamic setting because what “reputation” can mean in reality
is potentially very broad and diverse, and the expert can gain or lose the decision maker’s
trust along many different dimensions. In fact, quite contrary to the original intent of theword, political correctness is now often associated with a negative connotation where people
who express politically correct views are perceived as manipulative or even dishonest; put it
differently, taking a politically correct stance is not necessarily a sure way to improve one’s
reputation, broadly defined. Given this negative perception, we may have a situation where
the unbiased expert strategically deviates from the norm of political correctness to show that
he is, at least, not manipulative.
In this paper, we construct a dynamic model of strategic communication to see whether
and under what conditions this rough intuition would indeed survive in formal equilibrium
analysis. To this end, we extend Morris (2001), which we refer to as the “original setup” forclarity, by incorporating an additional period and an additional expert type to capture a morediverse process of reputation formation. As in the original setup, the expert can be either
good (unbiased) or bad (biased): if the expert is good, he has the same payoff function as
the decision maker; if bad, he always wants a higher action than the decision maker. On topof these two strategic types, as another key departure from the original setup, we introduce
the possibility that the expert may be inherently honest, in which case he simply reveals thetrue information in every opportunity he comes across.
The sequence of events within each period proceeds as follows. At the beginning of eachperiod, the expert observes the state of nature, which takes a value of either 0 or 1, and sends
a cheap-talk message, again 0 or 1, to the decision maker. Upon receiving the message, thedecision maker then chooses an action from some continuous interval. The state is publicly
observed after the action is taken, and the decision maker updates her belief about theexpert’s type conditional on all the available information. Without loss of generality, we letmessage 1 represent the “politically incorrect stance,” i.e., the message that induces a higher(more politically incorrect) action, and say that an expert is “politically incorrect” whenever
he announces 1.3
3Note that our notion of political correctness is defined in the ex ante sense (before the true state is publicly
observed). It is also defined in a different way from political correctness in Morris (2001), who takes a muchbroader view: he defines political correctness as an act of altering what to say in order to avoid adverse2

Under this setup, it is not overly surprising to see the bad type occasionally announce
1 in state 0 because he can always derive a current benefit from inducing a higher action.
It is a totally different story, however, if the unbiased good type ever chooses to do so inequilibrium for some strategic reasons. For the analysis, we label this particular form of political incorrectness as anti-political correctness, and say that anti-political correctness arises
in equilibrium whenever the good type announces 1 in state 0 with any positive probability.
Since the good type derives no current benefit from misreporting in any situation, the emergence of anti-political correctness means that there must be a reputational gain from falsely
announcing 1 in state 0. We emphasize this notion of anti-political correctness because it
necessarily yields a profound impact on our interpretation of political incorrectness: without anti-political correctness in equilibrium, all the reputational forces would point in one
direction, only working to discipline the expert to be more truthful as we discussed at theoutset; if it could ever be supported in equilibrium, on the other hand, the reputational effectof political incorrectness would be reversed, resulting in qualitatively different outcomes andimplications.
We obtain several results concerning the equilibrium and welfare properties of political
incorrectness. First, we derive a necessary condition for our notion of anti-political correctness
in a fairly general environment, and then establish as a corollary of this result that antipolitical correctness can never arise in any two-period variant of our model, including theoriginal setup, even with the addition of the honest type. There are roughly two reasons forthis. First, in any two-period model, there is only one opportunity for the expert to establish
his reputation. Second, since the good type and honest types are strategically equivalent in
the final period, it is unambiguously better to be perceived as good for both of the strategic
types. These two features altogether imply that there is only one route through which theexpert can gain the decision maker’s trust: the good type would like to separate from thebad type whereas the bad type would like to pool with the good type. Under this static
reputational structure, there is simply no room for supporting anti-political correctness inequilibrium, suggesting that it inherently calls for a dynamic process of reputation formation
which offers more routes to be taken by the expert.
By contrast, the two extensions of the current model allow us to depart from the dichotomous structure of the original setup and substantially enrich the process of reputation
formation. With the additional period, the process is dynamic, as the expert now has multiple opportunities to build his reputation. Moreover, in the presence of the honest type,there is a potential reputational gain from announcing 0 in state 0 because that allows theinferences.

3

expert to pool with the honest type. If the bad type’s incentive to pool with the honest typeis sufficiently strong, the good type may be able to separate from the bad type by falsely
announcing 1 in state 0. We show that there indeed exists an equilibrium which supports
this form of anti-political correctness and identify its sufficient condition. This result indicates that to the extent that these conditions hold, we cannot regard political incorrectness
naively as a sign of blunt honesty since it can easily be an attempt to signal one’s hidden
characteristics rather than the true state of the world.
Aside from this equilibrium analysis, we also briefly discuss welfare implications of antipolitical correctness with emphasis on the sorting effect it generates. Note that in evaluating
the welfare impact, Morris (2001) focuses on pooling equilibria where political correctness
attracts experts into a false message. In its worst case, only the babbling equilibrium survives, conveying no information about the expert’s preference type despite the good type’s
initial intention to separate from the bad. The situation is clearly different for the case ofanti-political correctness because it is a form of separating equilibrium which conveys some
useful information for future interactions. Early separation of types, induced by anti-politicalcorrectness, can indeed be socially beneficial when the decision maker’s expected payoff issufficiently convex with respect to her posterior belief, i.e., in environments where precise information is more valuable. We obtain some sufficient conditions for anti-political correctness
to be welfare-enhancing even though it is yet another form of information manipulation.
The paper proceeds as follows. In the remainder of this section, we briefly review the
related literature. Section 2 outlines the basic model. Section 3 analyzes the general model
to show that anti-political correctness never emerges in a two-period model, including theoriginal setup, and identify key necessary conditions for anti-political correctness. Section 4
considers a more specific environment, called the model with asymmetric states, to obtainsufficient conditions for anti-political correctness and derive welfare implications. Finally,
section 5 provides some concluding remarks.
Related Literature: The paper is broadly related to the literature which illustrates perverse
effects of career concerns, initiated by Holmstrom and Ricart i Costa (1986) and Holmstrom(1999). There are roughly two strands of this literature. One strand is concerned withthe case where agents attempt to signal their competence, such as the ability to acquire
or interpret information, through their actions or payoff-irrelevant (cheap-talk) messages.
Scharfstein and Stein (1990) show that managers ignore their private information and instead
simply mimic the investment decisions of other managers under some conditions. Prendergast(1993) provides a model of “yes men” and argues that workers have an incentive to conformto the opinion of their supervisors when firms can only use subjective performance evaluation.
4

There are now many works along this line, showing that agents sometimes fail to make fulluse of their private information when they are concerned about their reputations.4
By contrast, this paper falls into the other strand where agents attempt to signal their
payoff congruence or “unbiasedness.” Earlier examples of this strand are Sobel (1985) andBenabou and Laroque (1992), who consider a repeated cheap-talk game where the sender’spreference type is not known to the receiver. Both of these works assume that the senderis either honest or strategic and focus on the behavior of the strategic type. Morris (2001),
on which our model is based, considers a different setting in which the sender is either good
(unbiased) or bad (biased), while both types are strategic. Similarly, Ely and Valimaki (2003)
consider a situation where a long-run player interacts with a sequence of short-run players
who are uncertain about the long-run player’s preference type. As in Morris (2001), thelong-run player is either good (unbiased) or bad (biased) and attempts to signal his payoffcongruence though payoff-relevant actions.5 In terms of the type space, the current analysis
can be regarded as a hybrid of these previous settings as we consider two strategic types(good and bad) as well as a commitment type (honest).
From a broader perspective, our notion of anti-political correctness can be interpreted
as a form of contrarian behavior. There are now many works which analyze the origin andnature of conformism (or “herding”) in economic environments (Akerlof, 1980; Banerjee,
1992; Bikhchandani et al., 1992; Bernheim, 1994). To a large extent, Morris (2001) andEly and Valimaki (2005) are also models of conformism where agents conform to a certain
standard of behavior. By contrast, our model can be seen as a model of contrarianism
where (some) agents deliberately deviate from a social norm out of reputational concerns.
Although the issue of contrarianism has received relatively less attention in the literature,
there is a handful of works which examine why agents sometimes divert, or “anti-herd,”from the majority (Avery and Chevalier, 1999; Levy 2004, 2005). The key aspect of theseworks is that agents differ in the accuracy of information which allows some talented ones
to separate by moving against the herd to signal their competence.6 This contrasts sharply
with our model where agents move against a social norm in an attempt to signal their payoffcongruence. We provide a more detailed account of this interpretation in section 4.3.
4

Other examples include Prendergast and Stole (1996), Ottaviani and Sorensen (2001, 2006), Visser andSwank (2007), and Fu and Li (2014), just to name some.
5
Ely and Valimaki (2003) consider two cases, one in which the bad type is a commitment type who always
chooses a certain stage-game strategy and the other in which he is a standard strategic player, but the main
conclusion is essentially the same.
6
Effinger and Polborn (2001) consider a different setup in which agents do not know their own type. In
this setup, they derive a benefit function such that an agent is most valuable if he is the only smart one, sothat career concerns are not individualistic, and show that a form of contrarianism emerges in this case.

5

2

Model

Environment: Consider a three-period advice game with an uninformed decision makerand an expert who is perfectly informed about the current state of the world ωt ∈ {0, 1}.7
For simplicity, we assume that each state occurs equally likely with probability one half. In
each period t = 1, 2, 3, the decision maker solicits information about the current state of theworld from the expert and chooses an action at ∈ R. Let mt ∈ {0, 1} denote the expert’s
message, or his “opinion,” about the state in period t. After the action is chosen, the true
state is realized and publicly observed.
Decision maker: In each period t, the decision maker’s utility uωt (at ) depends on therealized state ωt and his action at , where uωt (at ) is differentiable, concave in at (with at least
one of u0 (at ) and u1 (at ) being strictly concave), and attains a unique maximum for each ωt .
Leta∗ (ω) := arg max uω (a) ∈ R ∪ {−∞, ∞} ,adenote the optimal action under full information when the true state is known to be ω.
Without loss of generality, we assume a∗ (1) > a∗ (0), so that the decision maker prefers a(weakly) higher action when the state is ωt = 1. The decision maker’s total utility is givenby3X

xt uωt (at ),t=1

where xt > 0 is the weight attached to the payoff in period t. The weight is meant to capture
the player’s time preference as well as the salience of the decision problem at hand.
Expert: As in the original setup, the nature of information asymmetry is two dimensional.
Aside from the true state, the expert privately observes his own type θ ∈ {G, B, H}, which
can be good (θ = G), bad (θ = B), or honest (θ = H). The expert type is drawn only
once at the beginning and is time-invariant. If the expert is good, he has a utility function
identical to the decision maker’s, so that their interests are perfectly aligned. If the expert isbad, on the other hand, he always prefers a (weakly) higher action than the decision maker,
regardless of the true state, where his instantaneous utility is always given by u1 (at ).8 The
7This is another difference from the original setup which assumes that the expert observes a possibly noisy
signal. By contrast, we assume that the expert can perfectly identify the true state. The assumption is madestrictly to simplify the analysis and does not affect our results in any qualitative way.
8
Note that the payoff functions are specified in a slightly different, and more flexible, way from the original
setup which, for most part, assumes uω (a) = −(a − ω)2 for the good type and uω (a) = a for the bad type.
The essence of the problem is still preserved with the current specification because the decision maker nevertakes an action that is too high for the bad type.

6

total utility of the bad type can then be expressed as3
X

yt u1 (at ),t=1

where yt > 0 is the weight attached to the payoff in period t. For expositional clarity, we
often refer to the good and bad types as strategic. In addition to these two strategic types,
we introduce an additional type, the honest, who commits to revealing the true informationin every opportunity he comes across.9 Let λ1θ denote the prior probability that the expert
is of type θ at the beginning of period 1. The prior type distribution is common knowledge.
Political and anti-political correctness: It is commonly understood that the bad type
always has a current incentive to announce 1 regardless of the true state. To separate fromthe bad type, there arises a reputational incentive to announce 0 even when the true state is
1. This intention is the so-called “political correctness,” where the expert intentionally lies
and takes the politically correct stance (announces 0), and is analyzed extensively in Morris(2001). In this analysis, on the other hand, we focus on the possibility, which we call “antipolitical correctness,” where the expert announces 1 in state 0 purely out of reputational
concerns. Note that the good type falsely announces 1 only when there is a reputational gain
because he derives no current benefit from lying in any state. The emergence of anti-politicalcorrectness is then equivalent to the following phenomenon.
Definition 1 An equilibrium with anti-political correctness is the one in which the good typechooses mt = 1 when ωt = 0 with any strictly positive probability.

3

The general model

In this section, we consider the model with general payoff functions. The goal of this section
is to establish generally that anti-political correctness cannot arise in a two-period model,
including the original setup, and to illustrate why. In the course of this analysis, we also obtainnecessary conditions for anti-political correctness to be sustained in equilibrium, which will
be used to show its existence in the following section.

3.1

Preliminary: strategies and beliefs

The expert’s strategy: The expert’s strategy in each period is a mapping from the observed
current state ωt into a probability distribution over messages. Let µωtθ denote the expert’s
9

There are at least two distinct ways to interpret this honest type: in one way, the honest type can be
regarded as the myopic version of the good expert with no weight on the future payoffs; in the other, it canalso be the one with a sufficiently large cost of lying, be it psychological or reputational.

7

(type-contingent) strategy, which is defined as the probability that the type θ expert tells
the truth in period t given the state ωt = ω. By assumption, µωtH = 1 for ω = 0, 1. Letµtθ := (µ0tθ , µ1tθ ) and Mt := (µtG , µtB ).
The decision maker’s strategy: At the beginning of each period t, the decision makercomes in with a certain belief about the expert’s type. We denote this belief by λtθ , whichindicates the probability that the expert is of type θ at the beginning of period t. SinceλtG + λtB + λtH = 1, we let Λt := (λtG , λtH ) represent the belief about the expert’s type,which thoroughly captures his “reputation.”
Without loss of generality, we focus on a class of equilibria in which the decision makerchooses a (weakly) higher action after receiving mt = 1. Upon receiving a message mt , thedecision maker forms a belief about the current state defined as
pt := prob(ωt = 1 | mt ; Mt , Λt ).
for a given set of strategies Mt and the current belief Λt . We in general write the belief as
pt = πtmt (Mt , Λt ), or simply πtmt , where
πt0 =

λtG (1 − µ1tG ) + (1 − λtH − λtG )(1 − µ1tB ),λtH + λtG (µ0tG + (1 − µ1tG )) + (1 − λtH − λtG )(µ0tB + (1 − µ1tB ))

(1)

πt1 =

λtH + λtG µ1tG + (1 − λtH − λtG )µ1tB.
λtH + λtG (µ1tG + (1 − µ0tG )) + (1 − λtH − λtG )(µ1tB + (1 − µ0tB ))(2)

The decision maker chooses an action at that is optimal for her based on this belief pt .
We let a∗ (pt ) represent the decision maker’s optimal action, which is given bya∗ (p) = arg max (1 − p)u0 (a) + pu1 (a).
aUnder the maintained assumptions, one can easily show that a∗ is continuous and increasing∗ min p.10 To save notation, we often write amt := a (π (Mt , Λt )). In addition, we defineuω := uω (a∗ (ω)) as the payoff under complete information, and uω := uω (a∗ ( 21 )) as the payoff
under no information so that any feasible payoff is bounded in [uω , uω ].
The next-period belief: At the end of period t = 1, 2, the decision maker updates her belief
about the expert’s type after observing the realized state ωt . The decision maker’s next-period
belief is then determined by the realized state ωt , the message mt , the set of strategies Mt
t ,mt
and the current belief Λt . We thus denote the next-period belief by λωt+1θ
(Mt , Λt ), or simply
t ,mt
λωt+1θ
.

The fact that u (at , 0) and u (at , 1) are concave implies that a∗ (p) ∈ [a∗ (0) , a∗ (1)] is increasing. To ensurethat a∗ (p) increases continuously, we assume that at least one of the two functions, u (at , 0) and u (at , 1), isstrictly concave.
10

8

3.2

The third period

The problem in the final period is quite straightforward with no future interactions. Following
Morris (2001), we restrict our attention to the unique informative equilibrium. In the absenceof any reputational concerns, it is easy to see that the bad type always announces 1 while
the good type always announces the true state. Under this set of strategies and the belief
Λ3 , we can compute
π30 = 0, π31 =1,2 − λ3where λt := λtG + λtH . From this, we obtain the value functions for the two strategic types
as follows:
v3G (λ3G , λ3H ) =x3 0 ∗1(u (a (0)) + u1 (a∗ ( 2−λ
))),
3
2

1
v3B (λ3G , λ3H ) = y3 (u1 (a∗ ( 2−λ)).
3There are two key properties of the continuation payoffs that prove to be crucial. First,
λ3G and λ3H are perfect substitutes in that v3θ (`, λ3 − `) is constant for any 1 ≥ λ3 ≥ ` ≥ 0.
Second, because the honest type is always credible, both value functions are monotonicallyincreasing in λt ∈ [0, 1], thereby giving rise to the incentive to stay on the good side of thedecision maker in the earlier stages. As we will see in section 3.4, these two properties play
an important role in ruling out the existence of anti-political correctness when the model has
only two periods.

3.3

The second period

The second-period problem is clearly more complicated because the good type, with reputational concerns, may now behave differently from the honest type.

ω,m:=Define vtθ

ω,mvtθ (λω,mt+1G , λt+1H ), which thoroughly captures the expert’s reputational concerns. Taking

the expert’s and decision maker’s strategies as given, the good type chooses m2 = 1 withpositive probability only if
ω2 ,1
ω2 ,0
x2 uω2 (a12 ) + v3G≥ x2 uω2 (a02 ) + v3G.

(3)

Similarly, the bad type chooses m2 = 1 with positive probability only if
ω2 ,1
ω2 ,0
y2 u1 (a12 ) + v3B≥ y2 u1 (a02 ) + v3B.

(4)

Each type’s payoff in period t = 1, 2 can be decomposed into two parts. The first part is thecurrent payoff, uωt (at ), which is determined by the match between the current state and theωt ,mtaction chosen by the decision maker. The second part is the continuation payoff, vt+1θ
, which9

is determined by the match between the current state and the message sent by the expert.
The tradeoff, if any, is clear in this environment. In state 1, the good type unambiguously
derives a lower current payoff by announcing 0 in any informative equilibrium, but that may
improve his reputation (a higher continuation payoff). If the increase in the continuation
payoff is large enough, the good type may lie and announce 0 to protect his reputation. Now
that the reputational cost of political incorrectness is even higher, the bad type may also
follow the good type by announcing 0 more frequently. As argued in Morris (2001), thisentire process is self-defeating, and no information may be conveyed in the worst case.
The following proposition shows that even in this extended setup with the honest type andwith more general (though slightly altered) payoff functions, any informative (nonbabbling)equilibrium in period 2 still has the properties characterized in Morris (2001).
Proposition 1 Any nonbabbling equilibrium satisfies the following three properties:
1. µ02G = 1, that is, the good type always tells the truth when the state is 0.
2. µ12B ≥ µ12G and µ02B ≤ µ02G , that is, the bad type announces 1 more often than the good
type.
1,1
1,0
0,1
1,1
0,0
3. λ1,03 ≥ λ3 ≥ λ3 ≥ λ3 with λ3 > λ3 for any Λ2 .

Proof: See Appendix A.
The most important result of the proposition is the first property which immediately
implies the following.
Corollary 1 Anti-political correctness never arises when there are only two periods. In otherwords, an expert can only tarnish his reputation by falsely announcing 1 in state 0 in a twoperiod model.

3.4

Necessary conditions for anti-political correctness

An equilibrium with anti-political correctness inherently requires that the bad type announce
0 in state 0 more frequently than the good type, so that there is a reputational gain for thegood type to deviate from the politically correct message. At a glance, the presence of thehonest type appears to give an additional incentive for the bad type to announce 0 in state
0 because that allows the bad type to pool with the honest. If the bad type chooses to do
so with sufficient frequency, the good type may be able to separate from the bad type byannouncing 1. Proposition 1 suggests, however, that it is generally infeasible to have this type10

of separating equilibrium when there are only two periods. Before we proceed to analyze thefirst-period problem, we would like to discuss briefly why anti-political correctness cannot be
supported as an equilibrium outcome in a two-period framework, even with the addition of
the honest type.
To see why the presence of the honest type alone is not sufficient, we first establish thefollowing statement.
Lemma 1 Suppose that from period t on, (i) the good type always reports truthfully, and (ii)the bad type’s period t strategy is independent of the belief Λt . Then,
ω,1
ω,0
ω,1
ω,0
(vtG
− vtG
)(vtB
− vtB
) ≥ 0,(5)

0,1
1,1 1,1
for any (λ0,1tG , λtH ) and (λtG , λtH ).

Proof:

It suffices to show that vtG (a, b) − vtG (a0 , b0 ) always has the same sign as

vtB (a, b) − vtB (a0 , b0 ). First, condition (i) directly implies that the good type is strategicallyequivalent to the honest type, so that vtθ (a, b) = vtθ (a + b, 0).11 Second, conditions (i)and (ii) together imply that vtθ (a + b, 0) is monotonically (and weakly) increasing in a + b(vtθ (a + b, 0) is constant with respect to a + b if the bad type also always reports truthfully).
Then, if vtθ (a + b, 0) > vtθ (a0 + b0 , 0), it must be a + b > a0 + b0 because of the monotonicity.
We thus have vtθ (a, b) − vtθ (a0 , b0 ) > (<)0 for both θ = G, B if and only if a + b > (<)a0 + b0 .
The property (5) means that if the reputational gain from switching to a different message
is positive for one type, it must also be positive for the other type, thereby always inducing
the two strategic types to move in the same direction. The intuition behind this result is asfollows. First, under condition (i), the good type is strategically equivalent to the honest type,
and the value function hence depends only on λt = λtG + λtH . The model is thus effectively
reduced to the one with two expert types where the expert’s reputation is thoroughly captured
by a single-dimensional variable λt . Given this property, condition (ii) then suggests that thevalue function is (weakly) increasing in λt , i.e., a higher λt is always more desirable. These
two properties together imply that there is a reputational gain from announcing 1 in state ωfor either type if and only if λω,1> λω,0tt .
It is fairly straightforward to verify that it is generally not feasible to have anti-politicalcorrectness when (5) is satisfied.
Proposition 2 Anti-political correctness cannot be supported as an equilibrium outcome inperiod t − 1 if the two conditions in Lemma 1 are satisfied.
11
Note that this strategic equivalence is irrelevant in the original setup where the expert’s reputation is byconstruction captured by a single-dimensional variable.

11

Proof: Note first that a necessary condition for anti-political correctness is that given theset of strategies and beliefs, we have
0,1
0,0
v3G
− v3G
> 0.

If this holds in equilibrium, however, it follows from (5) that the bad type must have a0,00,0strict incentive to announce m2 = 0, so that µ02B = 0 and λ3G+ λ3H= 1. This results in

a contradiction because the good type must then have a strict incentive to deviate to m2 = 0.
This proposition illustrates why anti-political correctness cannot arise in a two-period
environment, including the original setup. First, the good type always reports truthfully inthe final period. Second, because the bad type always sends message 1 regardless of thecurrent state, his strategy is also independent of his reputation. In the unique informativeequilibrium which prevails in the final period, therefore, the two conditions in the lemma arenecessarily satisfied, so that the reputational gain from announcing 1 always has the same
sign between the two strategic types. This means that if there is ever a reputational incentive
for the good type to announce 1 in state 0, the incentive must be even stronger for the bad
type to do so because he can also derive a current benefit by doing so. This totally eliminates
any reputational gain, thereby leaving no room for anti-political correctness when there are
only two periods, regardless of whether we have the honest type or not.

3.5

The first period

To focus on the possibility of anti-political correctness, we restrict our attention to the casewhere the true state happens to be ω1 = 0. Taking the expert’s and decision maker’s strategies
as given, the good type chooses m1 = 1 with positive probability (anti-political correctness)
only if0,10,0.
x1 u0 (a11 ) + v2G≥ x1 u0 (a01 ) + v2G

Similarly, taking the expert’s and decision maker’s strategies as given, the bad type chooses
m1 = 1 with positive probability only if
0,10,0
y1 u1 (a11 ) + v2B≥ y1 u1 (a01 ) + v2B.

The following result summarizes a necessary condition for anti-political correctness.
Proposition 3 Given the set of strategies and beliefs, there exists an equilibrium with antipolitical correctness only if
ω,1
ω,0
ω,1
ω,0
(v2G
− v2G
)(v2B
− v2B
) < 0,

12(6)

or, equivalently, either one of the two conditions in Lemma 1 fails to hold in period 2.
Proof: Since u0 (a∗ (0)) > u0 (a∗ (1)), a necessary condition for anti-political correctness is
0,1
0,0
v2G
− v2G
> 0,

that is, lying must result in a higher continuation payoff for the good type. Now, if
0,1
0,0
v2B
− v2B
≥ 0,
0,1
the bad type has a strict incentive to send m1 = 1. Then, for any µ1G> 0, we have0,0λ0,02G + λ2H = 1, in which case the expert’s continuation payoff is maximized. Since this is
0,10,0
a contradiction, this means that we must have v2B− v2B< 0, so that the bad expert does

not have a strict incentive to lie when ω1 = 0. The second condition follows directly from
Lemma 1.
Note that, because the good type may now behave differently from the honest type inperiod 2, the relative values of being perceived as good and honest in general differ acrossthe two strategic types. To see this more clearly, given some set of equilibrium strategies, we
can compute the value functions as follows:
v2G (λ2G , λ2H ) =1

1 X ω
ω,1−ω
ω,ω
) ,
) + (1 − µω2G )(x2 uω (a∗ (π21−ω )) + v3Gµ2G (x2 uω (a∗ (π2ω )) + v3G2
ω=0v2B (λ2G , λ2H ) =1

1 X ω
ω,1−ω
ω,ω
) ,
µ2B (y2 u1 (a∗ (π2ω )) + v3B) + (1 − µω2B )(x2 u1 (a∗ (π21−ω )) + v3B2
ω=0where π20 and π21 are given by (1) and (2), respectively. It is clear from these that λ2G and
λ2H are no longer perfect substitutes when µω2G < 1 for some ω. As a consequence, we may
have a situation in which the bad type prefers to pool with the honest type by announcing
0, while the good type prefers to separate from them by announcing 1. We dedicate the next
section to exploring this possibility.

4

The model with asymmetric states

4.1

Existence of anti-political correctness

The previous section has identified a necessary condition for anti-political correctness toemerge, but it is not necessarily clear from the argument whether or not an equilibrium with
anti-political correctness actually exists. Since adding one more period makes the analysis
13substantially more complicated, and it is highly tedious, if possible at all, to obtain a fullcharacterization of our general three-period setup, we here introduce an additional assumption which makes the model substantially more tractable. With this additional assumption,
we obtain a sufficient condition for anti-political correctness and show that anti-politicalcorrectness can indeed be supported as an equilibrium outcome in our extended setup.
Assumption 1 ū1 → ∞, i.e., the maximum attainable payoff in state 1 is unbounded.
Lemma 2 Suppose that λ2H = 0 and λ2G ∈ [0, 1). Then, no information is revealed inperiod 2 under Assumption 1.
Proof: See the Appendix.
Assumption 1 clarifies our focus in two different ways. First, the uniqueness of the continuation equilibrium when λ2H = 0 substantially simplifies the analysis and sharpens our
predictions, which is especially important in a dynamic setup like ours. Second, and more
importantly, with this assumption, the good type now has a stronger incentive to lie in thesecond period following m1 = 1. This feature is very crucial because, as we have noted
above, anti-political correctness requires (6) which is satisfied when the good type lies insome contingency and hence behaves differently from the honest type. This feature of themodel immensely helps in illustrating this underlying logic.
The assumption also illuminates the essential role played by the presence of the honest
type. The following proposition shows that the presence of the honest type is indeed necessary
for the emergence of anti-political correctness in this setup with asymmetric states.
Proposition 4 Under Assumption 1, there exists no equilibrium with anti-political correctness if λ1H = 0.
Proof: Note first that it is not possible to achieve full separation in the model with asymmetric states because if µ0,m2G = 1 for some m1 = m, the bad expert can obtain an unbounded
payoff by deviating to m1 = m. Given that the continuation equilibrium in period 2 must bebabbling when λ2G ∈ (0, 1), the value functions for both types must be strictly increasing in
λ2G . We then have
0,1
0,0
0,1
0,0
(v2G
− v2G
)(v2B
− v2B
) ≥ 0,

It thus follows from Proposition 3 that anti-political correctness is not feasible in the modelwith asymmetric states when λ1H = 0.

14

It should be noted here that this result relies heavily on the uniqueness of the continuation
equilibrium guaranteed by Lemma 2, but does not hold in general when there are multiple
continuation equilibria. We show in Appendix B that it is possible to construct an equilibriumwith anti-political correctness without the honest type when there are multiple continuation
equilibria, because the value function v2θ (λ2G , 0) need not be monotonic in this case. We do
not place much emphasis on this possibility, however, because the constructed equilibrium
in Appendix B is sustained by strategies that are highly arbitrary and hence does not carrymuch economic insight.
We are now ready to obtain a sufficient condition under which the equilibrium with antipolitical correctness exists. The following is the main result of the paper.
Proposition 5 Under Assumption 1, given x1 , y1 , x2 , y2 , and y3 , there exist λ̄ and x̄ suchthat an equilibrium with anti-political correctness exists if λ1H > λ̄ and x3 > x̄. In theequilibrium, the good type always announces 1 in period 1 while the bad type randomizes
between the two messages.
Proof: See the Appendix.
The proposition shows that there exists at least one route though which anti-politicalcorrectness can be supported in equilibrium. To see what happens in this equilibrium, supposethat the good type always announces 1 (anti-political correctness) in period 1. This leaves
the bad type with two alternatives, either to announce 0 and pool with the honest type or toannounce 1 and pool with the good type. The reputational gain from the former generally
dominates the latter because the honest type is always credible, while the good type is notin period 2. The bad type thus has an incentive to announce 0 with a higher probability(while he still randomizes) not to separate from the honest type. The clustering of the bad
type on message 0 then provides the good type with a different incentive because he can
now separate from the bad type by announcing 1. Although this generally results in a lower
payoff in period 2 because of the ensuing babbling equilibrium, the good type can capitalize
on his good reputation in period 3. It can hence be optimal for the good type to announce
1 in period 1 if he cares enough about his “long-run” reputation.

4.2

WelfareAs in the original setup, reputational concerns lead to the loss of socially valuable information
which by itself must be welfare-reducing. However, anti-political correctness also necessarily
entails a positive sorting effect because it is a type of separating equilibrium which generates
15

some information about the expert’s predispositions. The overall welfare effect is hence
ambiguous, depending on numerous environmental factors in a complicated manner. Here,
we show through a more specific example that anti-political correctness can indeed enhance
welfare, despite the loss of information in period 1.
We evaluate the welfare properties of the model by the decision maker’s expected payoff.
Following Morris (2001), the welfare benchmark is the equilibrium with no reputational updating, i.e., the one in which the decision maker’s belief remains at Λ1 . One can think of thisas a case where there is a different decision maker in each period who is unable to observe
the expert’s past choices. Without reputational concerns, the equilibrium of this game takes
a very simple form: the bad type always claims mt = 1, while the good type always tells thetruth. Since there is no reputational updating, the decision maker’s expected payoff is thesame across all three periods.
The situation draws a clear contrast to the equilibrium with anti-political correctness inwhich the good type separates from the bad type in period 1. Moreover, in the continuation
game following m1 = 0, the bad type is only pooled with the honest type. Because the bad
type is perceived to be more credible and the decision maker responds more to his (possibly
misrepresented) message in period 2, he tends to be more aggressive and more frequently
announces 1 in state 0. Roughly speaking, therefore, the equilibrium with anti-politicalcorrectness induces early separation of types, so that the decision maker could end up witha clearer idea about the expert’s type and realize a larger payoff in the end. In general,anti-political correctness is more beneficial when the decision maker’s payoff depends on λ3
in a convex manner.
We need to consider a more tightly specified model to evaluate the welfare impact of antipolitical correctness more precisely. Consider an environment in which the payoff functions
are specified as follows:u0 (a) = −a2 and u1 (a) = a.

(7)

Note that the maximum payoff is unbounded in state 1, so that this specification satisfies
Assumption 1. Under these payoff functions, the optimal action choice in any given period
is computed asa∗ (pt ) =pt.
2(1 − pt )1
In particular, since π30 = 0 and π31 = 2−λ, the decision maker’s expected payoff in period 3
316

as a function of (λ3G , λ3H ) can be obtained as u0 (a∗ (0)) + u1 (a∗ (π 1 ))u0 (a∗ (π31 )) + u1 (a∗ (π31 )) 3
v3G (λ3G , λ3H ) = x3 λ3+ (1 − λ3 )
2
2
x3
=
,
(8)
8(1 − λ3 )The third-period payoff is hence convex in λ3 and diverges to infinity as λ3 → 1, meaningthat early separation of types can be highly beneficial in this type of environment.
Intuitively, anti-political correctness is welfare-improving when the relative weight for thethird-period payoff is sufficiently large. To make this point, consider the limit case where
both x1 and x2 approach zero, so that welfare is determined entirely by the third-period
payoff. In addition, let y1 → 0 for simplicity. In this example, we can show that anti-politicalcorrectness can be supported as an equilibrium outcome if yy32 is sufficiently close to zero.
In the equilibrium, the good type always claims m1 = 1, while the bad type randomizes inperiod 1. Moreover, as yy32 approaches zero, the probability of the bad type announcing 0
in period 1 converges to one, allowing the good type to almost fully separate from the bad
type. Then, as can be seen from (8), this constitutes a sufficient condition for anti-politicalcorrectness to be welfare-improving. We summarize this finding as follows.
Proposition 6 Suppose that the payoff functions are given by (7) and let x1 , x2 , y1 → 0.
Then, the equilibrium with anti-political correctness exists and is welfare-improving if yy23 issufficiently close to zero.
Proof: See the Appendix.

4.3

Anti-political correctness as a form of contrarianism

Our notion of anti-political correctness can be regarded as a form of contrarian behavior or“anti-herding,” and in this sense, the current analysis has some connection to the literature
on conformism/contrarianism. In this section, we interpret our analysis from this perspective
and briefly discuss its relation to the existing literature.
The existing literature offers at least two approaches to analyzing conformism. One approach, which is perhaps more conventional in economics, is informational, where an agent
imitates the decisions of his predecessors even when his private information indicates otherwise, because actions taken by other agents provide relevant information (Banerjee, 1992;
Bikhchandani et al., 1992). By contrast, the other approach assumes that agents somehow
benefit from adhering to a code of behavior, or a “social norm,” and emphasizes its role ininducing convergence of actions (Akerlof, 1980; Bernheim, 1994). The current analysis is
more closely related to this latter notion of conformism.
17

As a prime example of analyses of conformism, Bernheim (1994) considers a situation
where agents care about status as well as intrinsic utility. An agent’s status is assumed todepend on public perceptions of his predispositions rather than on his actions themselves.
In this setup, it is shown that agents with moderate preferences converge to a homogeneous
standard of behavior, which can be regarded as a social norm, in order to avoid an inference
that they have undesirable extreme preferences. The key aspects of his analysis are:
• There are some preference types or “predispositions” that are deemed undesirable.
• The preference types are not directly observable, which must instead be inferred fromactions.
Under this setting, a social norm is defined as a set of actions which would not be taken by
those undesirable types in the absence of any reputational concerns. Conformism then arises
when agents abide by a particular norm and abstain from those extreme actions in order toavoid an inference that they possess undesirable predispositions.12 According to this view,
Morris (2001) and Ely and Valimaki (2003) are also models of conformism which share thisfeature, although they endogenize the value of reputation through repeated interactions, as
we do here.13
In the current analysis as well as in Morris (2001), by this definition, a norm is to announce
0 because: (i) it is unambiguously undesirable to be perceived as bad; (ii) the bad type would
never announce 0 in the absence of any reputational concerns. In clear contrast to Bernheim(1994) and Morris (2001), our analysis can then be seen as a model of contrarianism because
agents deliberately deviate from the norm, by taking an action which would be intrinsically
preferred by the bad type, i.e., announcing 1. What is interesting here is that contrarianismin this context is sustained by conformism on the part of the bad type: more forward-looking
agents deviate from a social norm to differentiate from more myopic counterparts who follow
the norm to secure “short-run” reputational gains. We argue that the current analysis departs
from the previous literature which focuses on the case where agents attempt to signal theircompetence and provides a new rationale for a different form of contrarianism through which
agents attempt to signal their payoff congruence.
12

To be more precise, Bernheim (1994) considers a setting in which the type space is [0, 2], which represents
an agent’s bliss point, where types closer to either end are deemed undesirable by assumption. It is further
assumed that there is some exogenous esteem function which has a unique maximum at 1. Agents then havean incentive to abstain from actions close to either end, inducing those with moderate preferences to converge
towards some middle point.
13
In Morris (2001) and Ely and Valimaki (2003), biased-preference types are deemed undesirable although
the reputation cost is derived endogenously.

18

5

Conclusion

In his seminal analysis of political correctness, Morris (2001) argues that reputational concerns may induce informed experts to take a politically correct stance even when it is not
the right thing to do. In this example, therefore, reputational concerns are the source of
information manipulation and eventually result in the loss of socially valuable information.
By the same token, one can conjecture that reputational concerns can discipline experts tobe more truthful when they are supposed to take a politically correct stance. If this is indeed
the case, political incorrectness should be regarded as a sign of honesty which can be taken
at (almost) face value.
In this paper, to take a step towards disentangling and better understanding the generalnature of political incorrectness, we provide an analytical framework which scrutinizes thisrough intuition. We show that the intuition does not always hold true, as unbiased experts
may deliberately take a politically incorrect stance to prove their good intentions when they
are sufficiently concerned about their long-run reputations. In light of this result, we argue
that political incorrectness is not necessarily a sign of honesty in long-term relationships
with sufficiently strong career concerns. Moreover, we also show that a form of political
incorrectness can be welfare-improving because it induces early separation of types.
As a final remark, it is important to note that our analysis sheds light on only one aspectof political incorrectness, and there are many different sides of this issue which cannot be
captured by the current framework. In the future, it is of interest to see more works in thisdirection.

References
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Avery, C.N. and J. Chevalier, 1999, Herding over the Career, Economics Letters, 63, 327-33.
Banerjee, A.J., 1992, A Simple Model of Herd Behavior, Quarterly Journal of Economics,
107, 797-817.
Benabou, R. and G. Laroque, 1992, Using Privileged Information to Manipulate Markets:
Insiders, Gurus, and Credibility, Quarterly Journal of Economics, 107, 921-58.
Bernheim, B.D., 1994, A Theory of Conformity, Journal of Political Economy, 102, 841-77.
Bikhchandani, S, D. Hirshleifer and I. Welch, 1992, A Theory of Fads, Fashion, Custom, andCultural Change as Informational Cascades, Journal of Political Economy, 100, 992-1026.
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Effinger, M.R. and M.K. Polborn, 2001, Herding and Anti-Herding: A Model of Reputational
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Levy, G., 2004. Anti-Herding and Strategic Consultation, European Economic Review, 48,
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Prendergast, C., 1993, A Theory of ‘Yes Men,’ American Economic Review, 83, 757-70.
Prendergast, C. and L. Stole, 1996, Inpetuous Youngsters and Jaded Old Timers: Acquring
a Reputation for Learning, Journal of Political Economy, 104, 1105-34.
Scharfstein, D.S. and J.C. Stein, 1990, Herd Behavior and Investment, American Economic
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20

Appendix A: the proofs
2 ,m2+ λω3B2 ,m2 . It is straightforward to
Proof of Proposition 1: Define λω3 2 ,m2 := λω3Gcompute
λ1,1
=
3
λ1,0
=
3
λ0,1
=
3
λ0,0
=
3

λ2H + λ2G µ12G,
λ2H + λ2G µ12G + (1 − λ2H − λ2G ) µ12B

λ2G 1 − µ12G
,
λ2G 1 − µ12G + (1 − λ2H − λ2G ) 1 − µ12B

λ2G 1 − µ02G,
λ2G 1 − µ02G + (1 − λ2H − λ2G ) 1 − µ02B

(10)

λ2H + λ2G µ02G.
λ2H + λ2G µ02G + (1 − λ2H − λ2G ) µ02B

(12)

(9)

(11)

0,1
0
0
0
First, we show that λ0,03 ≥ λ3 , µ2G = 1, and µ2B ≤ µ2G . Suppose on the contrary that0,1λ0,03 < λ3 . Then, when ω2 = 0, the bad type earns a higher payoff by announcing 1, andhence µ02B = 0. However, by (11) and (12), λ30,0 < λ30,1 implies thatλ2H (1 − λ2H − λ2G )(1 − µ02B ) < λ2G (1 − λ2H − λ2G )(µ02B − µ02G ),0,1
and hence, if µ02B = 0, µ02G < 0, a contradiction. Therefore, λ0,03 ≥ λ3 . It follows from (3)14
and (4) that µ02G = 1 and µ02B ≤ µ02G , which implies λ0,13 = 0.
1,1
1
1
Next, we show that λ1,03 ≥ λ3 and µ2B ≥ µ2G . Again, suppose on the contrary that1,1λ1,03 < λ3 . If this is the case, both the good and bad types can obtain a higher payoff byannouncing 1 when ω2 = 1, so that µ12B = µ12G = 1 and λ31,1 = λ2H + λ2G . To compute λ31,0 ,we follow Morris (2001) and consider the limit of a sequence of perturbed games, in which
the expert is misinformed about the state with probability ε. Given µ0G = 1, as ε → 0, we
have
λ1,0
3



ε λ2H + λ2G µ0G= lim ε→0 ε λ2H + λ2G µ0 + (1 − λ2H − λ2G ) µ0GB
λ2H + λ2G
> λ1,1=3 ,λ2H + λ2G + (1 − λ2H − λ2G ) µ0B

which is a contradiction. Given this, we have λ31,0 ≥ λ31,1 . By (9) and (10),
λ2H (1 − λ2H − λ2G ) 1 − µ12B ≤ λ2G (1 − λ2H − λ2G ) µ12B − µ12G ,11which implies µ12B ≥ µ12G .15 Therefore, λ31,0 ≥ λ1,13 and µ2B ≥ µ2G .
0,0
1,1
0,1
1,0
1,1
Finally, we show that λ1,03 ≥ λ3 ≥ λ3 ≥ λ3 and λ3 > λ3 through the following four

steps:(ω2 = 0, m2 = 1) is off the equilibrium path when µ02G = µ02B = 1, and we assume λ0,1
= 0 in this case.
3
As shown above, (ω2 = 1, m2 = 0) is off the equilibrium path when µ12G = µ12B = 1, and we assume
λ1,0= 1 in this case.
3
14

15

21

1,1
0,1
0
0
1. If µ02B < 1, λ0,13 = 0, meaning that λ3 ≥ 0 = λ3 . If µ2B = 1, µ2G = 1 too, so we have

λ31,1 ≥ λ30,1 = 0,1,10,1
where λ0,1
3 is as defined in footnote 14. Therefore, λ3 ≥ λ3 .

2. Let qω denote the probability of observing m2 = 1 in state ω2 = ω. Since
0,0
1,1
1,0
q0 λ0,1
3 + (1 − q0 )λ3 = q1 λ3 + (1 − q1 )λ3 ,

(13)

0,0
it must be the case that λ1,13 ≤ λ3 .
1,0
1,1
0,1
3. We show that we cannot have λ0,03 > λ3 ≥ λ3 ≥ λ3 . Suppose on the contrary that

this is the case. Then, by (4), the bad type has a stronger incentive to announce 0
when ω2 = 0. This means that µ02B ≥ 1 − µ12B , and more specifically either µ02B = 1 or
µ12B = 1. If µ02B = 1, it is implied that q0 in (13) is 0. But then, (13) can never hold
0,0under the presupposed condition. If µ12B = 1, λ1,03 = 1 ≥ λ3 , and therefore we have acontradiction. This shows that λ31,0 ≥ λ30,0 ≥ λ31,1 ≥ λ0,1
3 .
1,0
1,1
1,0
1,1
4. Lastly, we prove that λ1,03 > λ3 . Suppose that λ3 = λ3 , which implies that λ3 =
1,1
1,0
1,1
1
1
λ0,0
3 = λ3 . If µ2G = µ2B = 1, then as defined in footnote 15, λ3 = 1 > λ3 = λ2H +0,01,1
λ2G , a contradiction. Next consider the case where µ12G < 1. Given λ1,03 = λ3 = λ3 ,0,00(13) implies that either λ0,13 = λ3 or q0 = 0, which can only be satisfied when µ2B = 1,
1,1
0,0
1,0
and therefore, λ0,13 = 0 by footnote 14, and λ3 = λ3 = λ3 = λ2H + λ2G . Sinceµ12G < 1 and µ02G = µ02B = 1, π21 > π20 and a∗ (π21 ) > a∗ (π20 ) . But then, when ω2 = 1,
the good type has a strict incentive to choose m2 = 1, so µ12G = 1, a contradiction. We
1,1thus have λ1,03 > λ3 .

Proof of Lemma 2: Suppose that ω2 = 1. The necessary conditions for truth telling are
1,1
1,0
x2 u1 (a∗ (π21 )) + v3G≥ x2 u1 (a∗ (π20 )) + v3G
,

1,1
1,0
y2 u1 (a∗ (π21 )) + v3B≥ y2 u1 (a∗ (π20 )) + v3B
,

where
1,m
v3G
=

x3 0
1,m
(uG + u1G (a∗ ( 11,m ))), v3B= y3 uB (a∗ ( 11,m )).
2−λ
2−λ3
2
3

22

These conditions can thus be written as
2x2 1 ∗ 1
(u (a (π2 )) − u1 (a∗ (π20 ))) ≥ u1 (a∗ ( 1 1,0 )) − u1 (a∗ ( 1 1,1 )),
2−λ3
2−λ3
x3
y2 1 ∗ 1(u (a (π2 )) − u1 (a∗ (π20 ))) ≥ u1 (a∗ ( 1 1,0 )) − u1 (a∗ ( 1 1,1 )).
2−λ3
2−λ3
y3
1,1
1
1
Since λ1,03 > λ3 in any informative equilibrium, when λ2H = 0, we must have µ2B = µ2G = 1

or µ12B > µ12G . When µ12B > µ12G , we generically have either µ12B = 1 or µ12G = 0.16
It follows from Proposition 1 that if the continuation equilibrium in the second period isinformative, µ02G = 1. Suppose first that µ12G > 0 which also implies µ12B = 1. Note that if
0,1µ02B = 1, then π 1 = 1, π 0 < 1, λ0,03 < 1 and λ3 = 0. The bad type then has a strict incentive

to announce 1 if ū1 is sufficiently large. Since this is a contradiction, we must have µ02B < 1and π 1 < 1. However, since µ12B = 1, λ31,0 = 1, which means that the bad type can ensure
a payoff of ū1 in the third period by announcing m2 = 0. We must therefore have µ12B < 1
if ū1 is sufficiently large, a contradiction. This shows that we cannot have µ12G > 0 in any
informative equilibrium.
The only remaining possibility is µ02G = 1 and µ12G = 0, but we can also rule out thispossibility. Given that µ02G = 1 and µ12G = 0, any informative equilibrium requires that
µ12B > 1 − µ02B , i.e., the bad type must have a stronger incentive to announce 1 in state 1.
Since the bad type’s incentive compatibility constraint can be written asy2 1 ∗ 1(u (a (π2 )) − u1 (a∗ (π20 ))) ≥ u1 (a∗ ( 1ω,0 )) − u1 (a∗ ( 1ω,1 )),
2−λ3
2−λ3
y3
for ω = 0, 1, this condition implies thatu1 (a∗ (1)) − u1 (a∗ ( 1 0,1 )) ≥ u1 (a∗ ( 1 1,0 )) − u1 (a∗ ( 1 1,1 )),2−λ0,0
2−λ3
2−λ3
2−λ3
3

17 and this condition is reduced toGiven that µ02G = 1 and µ12G = 0, λω,13 = 0 for ω = 0, 1,

u1 (a∗ (1)) ≥ u1 (a∗ ( 1 1,0 )).
2−λ30,02−λ3This is a contradiction, however, because µ12B > 1 − µ02B implies that λ31,0 > λ30,0 . This meansthat the equilibrium must be babbling if ū1 is sufficiently large.
2
When yy23 = 2x
, both of the strategic types may adopt mixed strategies where any combination ofx3
1
1(µ2B , µ2G ) is feasible. We will ignore this non-generic case throughout the proof.
17(ω2 = 0, m2 = 1) may be off the equilibrium path, but we define λ0,1
= 0 in this case (see footnote 14).
3
Alternatively, we can also show that λ0,1
= 0 by considering the limit of a sequence of perturbed games (as in316

the proof of Proposition 1) in which the expert is misinformed about the current state with probability ε.

23This result proves that there only exists the babbling equilibrium when λ2H = 0 andλ2G ∈ [0, 1), if the attainable payoff in state 1 is unbounded.
Proof of Proposition 5: We construct an equilibrium in which the good type always
chooses m1 = 1 in state ω1 = 0. Note that, given this strategy, the bad type must randomize
between the two messages because there would always be a profitable deviation otherwise.
Let µ̂ω2θ2 denote the strategy in the continuation game following m1 = 0, and µ̂2θ := (µ̂02θ , µ̂12θ ).
ω2 ,m2
denote the third-period belief and third-period value function,
Similarly, let λ̂ω3θ2 ,m2 and v̂3θ

respectively, contingent on (ω2 , m2 ) in the continuation game following m1 = 0, and λ̂ω3 2 ,m2 :=
2 ,m2
2 ,m2
λ̂ω3G
+ λ̂ω3H.

Given that λ1G > 0, µ01G = 0, and µ01B ∈ (0, 1), both π10 and π11 are continuous inµ01B ∈ (0, 1) and, moreover, are bounded by a number less than 1. Let p̄ be the upper bound.
Moreover there exists λ0 such thatπ11 >1
> π10 ,
2

0,0
0,1
0,1
λ1H
λ1G
for λ1H > λ0 . Note also that λ0,02G = 0, λ2H > 1−λ1G , λ2G > 1−λ1H , and λ2H = 0. We now

need to consider two possibilities, depending on m1 .
1. The continuation game following m1 = 0.
Note first that the continuation equilibrium in period 2 following m1 = 0 must be informative.18 Given that there are only the honest and bad types, the bad type has no incentive
to lie in state 1, so that µ̂12B = 1.19 Furthermore, µ̂02B ∈ (0, 1]; otherwise, there exists a
profitable deviation for the bad type if µ̂02B = 0. We then obtain
π20 = 0, π21 =

1

.
0,0
0 )
λ0,0
+
(1
−
λ
)(2
−
µ̂
2H
2H
2B

The beliefs regarding the expert’s type are given
1,1
0,0
0,0
λ̂1,0
3 = 0, λ̂3 = λ2H , λ̂3 =

0,0
λ2H
0,0
0,0
λ2H
+ (1 − λ2H)µ̂02B, λ̂0,1
3 = 0.

The value function for the bad type is now computed as
1 X  ω
ω,ω
ω,1−ω
v2B =
µ2B (y2 u1 (a∗ (π̂2ω )) + v̂3B
) + (1 − µω2B )(y2 u1 (a∗ (π̂21−ω )) + v̂3B
) ,
2

(14)

ω∈{0,1}18

Suppose that the continuation equilibrium following m1 = 0 is also babbling. If that is the case, then thesecond-period payoff is constant regardless of m1 for both types since the continuation equilibrium following
m1 = 1 is also babbling (see Lemma 2). Anti-political correctness cannot then arise because it is effectively
reduced to a two-period model.
19
We assume that since the honest type never lies, λ1,03H = 0 even though it is off the equilibrium path.

24

0,0
ω2 ,m22 ,m2, λ̂ω3H). Since the bad type feels
where π̂2ω2 := π2ω2 (µ̂2G , µ̂2B ; λ0,02G , λ2H ) and v̂3θ := v3θ (λ̂3Gindifferent between the two messages in state 0 if µ̂12B ∈ (0, 1), (14) can be written as
1 X  1 ∗ ω
ω,ω
v2B =
y2 u (a (π̂2 )) + v̂3B
.
2

(15)

ω∈{0,1}

Now suppose that the good type deviates and sends m1 = 0 in the first period. Given thecontinuation equilibrium following m1 = 0, µ̂12G = µ̂02G = 1. The value function for the goodtype is then obtained as

1 X 
ω,ω
x2 uω (a∗ (π̂2ω )) + v̂3G
.
2

v2G =

ω={0,1}

2. The continuation game following m1 = 1.
By Lemma 2, we know that the continuation equilibrium in period 2 must be babbling.
We thus have
10,1.
π20 = π21 = , λ3 = λ2G2
The value functions are given, respectively, by
v2B = y2 u1 + v3B (λ0,12G , 0),

v2G =

(16)


1 X 
0,1
, 0) ,
x2 uω + v3G (λ2G2ω∈{0,1}

where uω is the minimum payoff in state ω as defined earlier (the payoff under no information).
We are now ready to prove the proposition. We first compare the bad type’s expected
second-period payoffs in (15) and (16):
1  1 ∗1
1 (a∗ (0)) and y u1 .
))+u
y2 u (a ( 0,020λ2H +(1−λ0,022H )(2−µ̂2B )Given that u1 (a∗ (λ)) increases unboundedly in λ, there exists a λ̄ ≥ λ0 such that if λ0,02H >λ̄1−λ1G ,

1  1 ∗1
1 (a∗ (0)) > y u1 (a∗ (p̄)) + y u1 .
y2 u (a ( 0,0
))
+
u
1
2
0
λ2H +(1−λ0,022H )(2−µ̂2B )0,1
λ1HNext, note that if λ1H > λ̄, then λ0,02H > 1−λ1G and λ2G =

λ1G
.
λ
1− 1H
0,0
λ
2H

(17)

Therefore, as λ0,0
2H

λ1H
λ1G
increases from 1−λto 1, λ0,1
2G decreases from 1 to 1−λ1H , the value of (15) increases to ∞,
1G

25

0,0
λ1H
and the value of (16) decreases from ∞. This means that there exist λ2H∈ ( 1−λ, 1) and
1G

λ0,1
2G =

λ1G

0,0
λ

such that the bad type is indifferent between the two messages. That is,1−λ1H − λ2H1H
given λ0,0
and
thecorresponding µ01B ,2H

y1 u1 (a∗ (π10 )) +
1 X  1 ∗ ωω,ω
y2 u (a (π̂2 )) + v̂3B
= y1 u1 (a∗ (π11 )) + y2 u1 + v3B (λ0,12G , 0).
2
ω∈{0,1}

It follows from (17) that
0,1
v3B (λ2G, 0) >

1 X ω,ω
v̂3B .
2ω∈{0,1}This in turn implies thatv3G (λ0,12G , 0) >1 X ω,ω
v̂3G .
2ω∈{0,1}Furthermore, if x3 is large enough,


1 X 
1 X 
ω,ω
0,1
< x1 u0 (a∗ (π11 )) +, 0) .
x1 u0 (a∗ (π10 )) +x2 uω (a∗ (π̂2ω )) + v̂3G
x2 uω + v3G (λ2G22ω∈{0,1}

ω∈{0,1}

Therefore, when λ1H and x3 are large enough, the good type chooses m1 = 1 while the bad
type randomizes between the two messages in state ω1 = 0.
Proof of Proposition 6: We first explicitly construct an equilibrium in which, given
ω1 = 0, the good type always announces 1 while the bad type announces 0 with probabilityµ01B ∈ (0, 1) in period 1. We then derive a sufficient condition under which anti-politicalcorrectness is indeed welfare-improving.
Period 3: As we have seen, the equilibrium in the final period is straightforward: the goodtype always tells the truth while the bad type always announces 1. We then obtain
v3G (λ3G , λ3H ) =x3
y3, v3B (λ3G , λ3H ) =,8(1 − λ3 )2(1 − λ3 )for any prior belief Λ3 .
Period 2: There only exists the babbling equilibrium in the continuation game following
m1 = 1, so we focus on the continuation game following m1 = 0.
We first consider the problem for the bad type. It is clear that the bad type always
announces 1 in state 1 because that can raise both the current payoff and the continuation
payoff. In state 0, on the other hand, the bad type tells the truth only if
0,0y2 u1 (a02 ) + v3B (0, λ3H) ≥ y2 u1 (a12 ) + v3B (0, 0),26

wherea12 =102(1 − λ0,02H )(1 − µ2B )

0,0
, λ3H
=

λ0,0
2H
0,0
0
λ0,0
2H + (1 − λ2H )µ2B.

Since u1 (0) = 0 and v3B (0, 0) = 12 , this condition can be written as0,00y3 (λ0,02H + (1 − λ2H )µ2B )02(1 − λ0,02H )µ2B

≥

y2
y3
+ ,
0,0
0
2
2(1 − λ2H )(1 − µ2B )which is further simplified to0,0y3 λ2H(1 − µ02B ) ≥ y2 µ02B .

It is easy to see that this condition must hold with equality in equilibrium. The bad typetherefore always randomizes in this contingency whereµ02B =

y3 λ0,02H
y2 + y3 λ0,02H.

Now suppose that the good type deviates and announces 0 in period 1. In this case, it isclear that the good type always tells the truth because that can raise both the current payoffand the continuation payoff.
Period 1: We are now ready to check when anti-political correctness can be supported inequilibrium. First, the incentive compatibility constraint for the good type is given by
0,0
0,1
,
≥ v2G
v2G

which can be written asv3G ( λλ1G
0
1G +(1−λ1G −λ1H )(1−µ1B )v3G (0,, 0) ≥

λ0,0
0,0
2H
0,0
0 ) + v3G (0, λ2H )λ2H+(1−λ0,02H )µ2B

2

,

where
λ0,0
2H =

λ1H
.
λ1H + (1 − λ1G − λ1H )µ01B

A sufficient condition for this is
0,0
λ2H
λ1H
λ1G
≥
=
,
0
0,0
0,0
λ1G + (1 − λ1G − λ1H )(1 − µ1B )λ1H + (1 − λ1G − λ1H )µ01B µ02B
λ2H + (1 − λ2H )µ02B

which holds if µ01B is sufficiently close to one.
Second, the incentive compatibility constraint for the bad type is given by
0,1
0,0
v2B
= v2B
,

27

which can be written as
y2
1G, 0) = v3B (0, λ +(1−λ λ1H+ v3B ( λ +(1−λ λ−λ
0
0
0 ).
1G
1G
1H )(1−µ1B )1H
1G −λ1H )µ1B µ2B
2This is further reduced toy2 +y3 (λ1G + (1 − λ1G − λ1H )(1 − µ01B ))y3 (λ1H + (1 − λ1G − λ1H )µ01B µ02B ),=(1 − λ1G − λ1H )(1 − µ01B )(1 − λ1G − λ1H )µ01B µ02B

(18)

where
µ02B =y3 λ1H.
y2 (λ1H + (1 − λ1G − λ1H )µ01B ) + y3 λ1H(19)

Substituting (19) into (18) and rearranging, we obtainy2 +

y2 (λ1H + (1 − λ1G − λ1H )µ01B ) + y3 λ1Hy3 λ1G=.
(1 − λ1G − λ1H )(1 − µ01B )(1 − λ1G − λ1H )µ01B(20)

Note that the left-hand side is strictly increasing in µ01B while the right-hand side is strictly
decreasing, which implies that, for any given (y2 , y3 ), there exists a unique solution whichsolves (20).
We next show that the equilibrium with anti-political correctness exists for any given(y2 , y3 ) if yy32 → 0. To see this, we rewrite (20) as
y3
y2 λ1G

1 − λ1G − λ1H=

λ1H (1 − µ01B ) + yy32 λ1H(1 − λ1G − λ1H )µ01B.

We can then see that µ01B → 1 and the equilibrium with anti-political correctness exists if
y3
y2 → 0.

Given this explicit solution, it is straightforward to compare the expected payoffs. First,
the expected payoff without reputational updating is
v3G (λ1G , λ1H ) =x3
8(1 − λ1G − λ1H ).

On the other hand, the expected payoff with anti-political correctness when the expert happens to be good is
v3G ( λx3 (λ1G + (1 − λ1G − λ1H )(1 − µ01B ))
λ1G
,
0)
=
.
0
1G +(1−λ1G −λ1H )(1−µ1B )8(1 − λ1G − λ1H )(1 − µ01B )It is clear that the expected payoff with anti-political correctness becomes unbounded as
y30y2 → 0 and hence µ1B → 1, so that anti-political correctness is welfare-improving.

28

Appendix B: the emergence of anti-political correctness with
two types
In this appendix, we show by construction that it is indeed possible to have anti-politicalcorrectness in equilibrium even in the absence of the honest type. Consider the following
example.
Example: Suppose thatu0 (a) = −a2 and u1 (a) = −(1 − a)2 ,Let λ3H = 0, λ3G ∈

1 7
2, 8



49
5
, xx12 < 162, and yy12 < 18.

We can show that both types randomize in period 1 in this specified example. Now
suppose that
7
5
+ εG
− εB
x3
y3
= 18 7= 18 11 ,
and
x2
y2
81
36

where
εG :=

x1 0 ∗ 1
x1(u (a (π1 )) − u0 (a∗ (π10 ))) <
,
x2
x2

εB :=

y1 0 ∗ 1y1(u (a (π1 )) − u0 (a∗ (π10 ))) < .
y2y2Then there exists an equilibrium with anti-political correctness in which: (i) in the continuation game following m1 = 0, the good type tells the truth in period 2 while the bad type
always claims m2 = 1; (ii) in the continuation game following m1 = 1, the second-period
equilibrium is babbling.
One can readily check that neither type has an incentive to deviate. The constructedequilibrium fully exploits the fact that there exist multiple continuation equilibria in period 2,
in particular that the babbling equilibrium always exists under any circumstance in cheap-talkmodels. For this reason, we can always construct a situation in which different equilibria (areexpected to) occur after different messages. To be more concrete, anti-political correctness issustained in this particular case because the expert (somehow) holds a pessimistic expectationthat an informative babbling equilibrium ensues when he chooses m1 = 1. The same tradeoff
then arises, in which the expert must trade off the short-run (second-period) reputation
against the long-run (third-period) reputation.

29