In this appendix, I generalize the truth-telling result of the paper in three directions. First, we assume a …nite state space and a …nite action space. Second, the expert has imperfect information about the state!:Third, we consider strictly concave utility functions for all the agents. This shows that the main mechanism through the receiver’s accountability is robust, and relevant, in a more general context.
This also constitutes a novel theoretical result; in fact, while Ottaviani and Sorensen (2006b) shows that truth-telling in the presence of expert’s career concerns is impossible, we show that the decision maker’s reputation is a more e¤ective incentive. The framework is similar to Olszewski (2004), with the main di¤erence being the focus on the role played by the receiver’s accountability. I will embed the model of Section III in a game where the expert receives a signal s from
S = fs1; :::; smg; and the decision maker receives a signal r from R = fr1; :::; rng about state
!2 . The probability that the sender receives s=si and the decision maker receives r=rk is
de…ned aspki; and the prior is given by
P = 2 6 6 4 p1 1 p21 ::: pn1 p1 2 p22 ::: pn2 ::: ::: ::: p1m p2m ::: pn m 3 7 7 5
The sender’s type is de…ned by ti (si; pi). He sends a message s 2S and the receiver after
observing the message, chooses an action a 2 A where A is a …nite set. It is essential that the set of messages contains all of the signals from the sender; the assumption that there are no other messages is made only for the sake of simplicity. Players are allowed to use mixed strategies. A mixed strategy for the sender is denoted by : S S ! [0;1] where (sj; si) stands for the
probability that the sender with signalsi sends message sj. A mixed strategy for the receiver is
: A R S ! [0;1]; where (a; sj; rk) stands for the probability of taking action a by the
receiver, with signalrk when the sender sends messages=sj:
There are two types of receivers. There is a probability"that the receiver is honest (a behavioral type i.e. > ) and a probability1 "that the receiver is corrupt and strategic. As in the text, the sender receives a signal about the receiver’s type and updates the probability of the incumbent’s being honest to : The honest receivers always implement the best action given the available information, and the strategic receivers maximize their payo¤. Moreover, honest politicians receive an informative signalrk;while corrupt receivers do not have any valuable information besides the
sender’s message, that is,pk
i =pzi for every kandz. The decision payo¤ for the corrupt receiver is
represented byucr a real-valued function of the action and the state. Thus, the expected payo¤ is given by: E[ucrjItc] Urc= 2 X t=1 X at2A m X j=1 2 t t(at; sj; rk) t(sj; si)ucr(at; si)
where we assume, for simplicity, that the signals are independently distributed across periods. The payo¤ for an honest receiver is instead given by
E uhr j Ith Urh= 2 X t=1 X at2A m X j=1 2 t t(at; sj; rk) t(sj; si) t(sijrk)uhr(a; si; rk)
where t(sijrk)stands for the probability assigned by the receiver with signalrk to the event that
the sender has obtained signalsi: Notice that
t(sijrk) =
pk i
Pm j=1pkj
I assume that the voter’s preferences coincide with those of the honest type, that is,
E[uv jIv] U v= 2 X t=1 X at2A m X j=1 n X k=1 t(at; sj; rk) t(sj; si)uhr(a; si; rk)
Finally, we have to de…ne the expert’s expected payo¤:
Ue= 2 X t=1 X at2A 2 t 2 4 m X j=1 t(at; sj; rk) t(sj; si)ue(at; si) + (1 )ue(at; si) 3 5
where >1:We make the following assumption:
Assumption 1. There exists in each state a unique maximizer ofUe, that is,sj = arg maxsjE[Ue(si; ai; sj)]
whereai is the receiver’s best response. De…neUe(sj) = maxsjUe(si; sj; ai):
Note that sj may well be di¤erent from the true state; in this caseUe(sj) can be interpreted
as the payo¤ from lying.
I assume that having in o¢ce a corrupt politician in the second period gives the sender, in expectation, a higher payo¤ than when a challenger chooses the action:
Assumption 2. We assume thatUe(sj)> Eunre and we normalize the utility range to the unit
interval.
This assumption ensures that the problem is interesting and supposes that the expert has an incentive to deal with a corrupt incumbent.
Assumption 3. We suppose that the re-election probabilityG is increasing in the incumbent’s reputation.
This assumption re‡ects a more primitive assumption on the matrixP, analogous to the condi- tion > in the text.
First, we can characterize the second-period equilibrium:
Proof. Suppose that in some equilibrium the sender depending on the state revealed to him sends at least two di¤erent reports, s1 and s2; and a1 and a2 are the actions in response to those two
reports. It must be thata1=a2:Presume that this is not the case, note thatue(a1)6=ue(a2)due
to Assumption 1;and take the caseue(a1)> ue(a2):Observe that it is not rational to send report
s2; ass1 induces a more favorable decision. Then, this contradicts that boths1 ands2 are sent in
equilibrium.
We can now state the main result:
Proposition 17 If the voter’s beliefs are continuous in the pair(si; ai);we can …nd a sequence of
thresholdsf gmi=1;such that if < the expert truthfully reports his signal in the …rst period.
The intuition for this result is that, if it is true that the expert must receive the same expected payo¤ from all the messages sent in equilibrium, and if the decision payo¤ has a unique maximizer for each state, the di¤erence in payo¤ generated by the optimal action given a truthful revelation and the action chosen in the case of a lie may be compensated by the di¤erence in the probability of the corrupt incumbent’s re-election. This means that the expert faces a trade-o¤ between being truthful in the …rst period -increasing the corrupt chances of being reelected- or lying but decreasing in this way the probability of being in‡uential in the second period.
The existence of a separating equilibrium does not require that every typet can be separated from all other types, but rather that it be only separated from types that strictly prefer to be misidenti…ed as typet:Assumption 1 simpli…es the analysis, restricting attention for the sender to one incentive compatibility.
Proof. De…ne Ue(sj) as the maximum payo¤ he can achieve by truthfully revealing the state.
Since for each state there is only one message that maximizes the expert’s payo¤, he will compare the expected payo¤ of lying - sending that message- with the expected payo¤ in the case of truthful disclosure. Suppose that the state issiandsj = arg maxsjUe(si; ai; sj) ;in general; the expert will
truthfully reveal the state if the following incentive compatibility constraint holds:
E[Uejs=si] E[Uejs=sj]
Let us suppose that the receiver will follow the sender’s suggestions (remember that the honest incumbent is a behavioral type) and note that we can rewrite the previous condition as:
U1 +G U2 + (1 G )Eunre U1+GdU2+ 1 Gd Eunre , G Gd U2 U1 U1 G Gd Eunre where U1 = (at; sj; rk) (sj; si)ue(at; si) + (1 )ue(at; si) and U1= (at; sj; rk) (sj; si)ue(at; si) + (1 )ue(at; si)
are the expected payo¤ of the sender, respectively, in the case of truthful revelation and when he sends his most preferred message. In the …rst period, assume that the receiver is in‡uenced by the
expert’s message. He will then choose an actiona ;which, by assumption 1, is less preferred than action a; which would be implemented by a corrupt politician in an uninformative equilibrium. Consequently, U1 > U1 for every : However, the second-period expected payo¤ is the same,
ceteris paribus, for any message sj sent in the …rst period. This is a consequence of lemma A.1:
in the second period the honest will choose the best policy given his signal rk; while the corrupt
decision maker chooses the expert’s most preferred policy. The only di¤erence is in the re-election probability G ;which is greater in the case of truth-telling in the …rst period than in the case of babbling, as G is increasing in by assumption 3. However, the expected payo¤ in the second period is a continuous function of :This means that we can …nd a threshold (s)such that for every < (s) the equilibrium is separating and the receiver optimally follows his suggestions. The cuto¤ for each state is de…ned as:
U2( ) =Eunre +
U1( )
(G Gd) (2)
Assumptions 1 and 2 assure his existence, as the right-hand side of equation (2) belongs to the unit interval:If the di¤erence in the re-election probability continuously changes with the state and the action implemented; i.e., if the voters assign to each pair(si; ai)a di¤erent belief i, we can easily
…nd a sequence of thresholds,f (si)gmi=1-one for each signal, that satis…es (2). The normalization
ensures that each of these cuto¤s is less than one.
The result then crucially depends on the voter’s updating process: if the voter after each pair of action and state assigns di¤erent beliefs to the incumbent, this disciplines the expert’s behavior. On the other hand, if voters interpret a set of actions in the same way, the expert is induced to send the message that corresponds to the implementation of his most preferred policy among these actions.
The proposition highlights, in a more general context, the role played by political accountability on the expert’s incentives to be truthful. This is a central and novel result in the cheap talk literature, which may be applied to a variety of settings.