This paper analyzed truthtelling incentives of experts competing for in‡uence. On the one hand competition for in‡uence determines a “reduced in‡uence” e¤ect both in the current and in the future period: a biased sender knows he is less likely to in‡uence future decisions, so that he is less willing to sacri…ce current payo¤s to build a reputation for providing sound advice; however a biased sender is not able to enjoy the full value of a current deviation, thus the opportunity cost of maintaining a reputation is reduced. On the other hand, competition for in‡uence determines a lost reputation e¤ect: biased senders fear that if they deplete their reputation, other senders will in‡uence future decisions. Finally, competition allows the decision maker to aggregate more information and this may enable, in some circumstances, the decision maker to distinguish cor-rect reports from wrong reports. The interplay among these e¤ects generates interesting results and o¤ers novel insights for organizational design. The …rst is that the decision-making process can be less prone to errors if only one sender reports information, as competition may harm decision-making. The second result is that the quality of decision-making can be improved if one sender is delegated authority to make decisions, becoming an “in‡uential sender”. The third is that decision-making could be further improved if the decision maker biases the competition for in‡uence: this shows favoritism can arise as an optimal way to foster truthtelling incentives.
Thus, this paper provides a new theory for the allocation of authority and for the use of favoritism in organizations: they arise endogenously as rational organizational responses to the incentives created by competition to in‡uence decision-making. The leading application of these results is the analysis of resources allocation among divisions within an organization, but the insights of the model can be applied to investigate a variety of economic interactions: politicians competing to be elected, lobbies willing to in‡uence politicians, …nancial analysts providing information to investors, investment banks providing advice to corporate clients.
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Appendix - Proofs
Proof of Lemma 1
Part 1: In the last period the sender has no reputational concerns. By reporting his preferred decision he can enjoy a positive payo¤, while his payo¤ is non-positive if he does not report his preferred decision. When he does not have enough credibility, he randomizes and the decision maker puts zero weight on the message provided.
Part 2: this is obvious as by reporting the true state of the world he enjoys a current gain without incurring any loss in reputation.25
Part 3: state C is less likely than the other two states when > 13:Any strategy that attaches positive weight to this state, when the decision maker is uninformed, is strictly dominated by a strategy that randomizes between states L and R: Such strategy yields an expected payo¤ of zero.
Part 4: the sender will be able to credibly transmit information in period 2 if and only if A[p2+(1 p2 2)( (1 2 )) +(1 p22)( (1 2 ))] > 0;which holds if p2 > 2(11 2 ):This follows because the sender is honest with probability p2 and then reports the truth. With probability 1 p2 he is left-biased, and with probability the true state is L; so he is reporting the truth, while with probability (1 2 ) + the state is either C, or R; and the left-biased sender lies. The same reasoning describes the behavior of a right-biased sender. In period 1 the sender is able to credibly transmit information if and only if Afp +(1 p)2 [ + (1 )(qs1 (1 qs1)] +(1 p)2 [ + (1 )(zs1 (1 zs1))]g > 0: In order to ensure the existence of truthtelling equilibria in pure strategies, it is necessary that p > 2(11 2 ). In fact, in such a case, both types of biased senders report the truth in the …rst period setting qs1 = z1s = 1, so that p2 = p and information can be credibly transmitted if and only if p2 > 2(11 2 ):
Proof of Lemma 2
Part 1: This follows from the fact that when senders have enough credibility, the expected payo¤ from following their advice is larger than that from making decisions without information.
25Furthermore, it never pays to lie by falsely reporting the true state is the unbiased state C. This follows because the sender would su¤er both a current period loss, and a reputational loss. The latter is implied by the assumptions that the true state is observed perfectly. Otherwise, it could happen that a biased sender lied in order to gain a reputation for being unbiased. This mechanism would be similar to that unveiled in Morris (2001).
Thus the decision d coincides either with mi or with m i: Moreover, when this is true, as the decision maker has a linear payo¤ function, it is optimal to set i;mi;m i + i;mi;m i = 1:
Part 2: The proof is in the text.
Proof of Lemma 3
Part 1: The expected payo¤ by randomizing is
4p1 p
2 (2( A A)) + 2(1 p
2 )2(2( (A A)) (1 2 )A) = (1 2 )(1 p)2
2 A
In fact, con‡icting messages L and R can occur when the decision maker faces an honest sender and a biased sender (this occurs with probability 4p1 p2 ; or when both sender are biased, but one is left-biased and the other right-biased (this occurs with probability 2(1 p2 )2):The decision maker might use a strategy that implements action k 2 fL; C; Rg when observing disagreeing messages Land R: In such a case, suppose the true state is L and the strategy is “implement state R when messages disagree”: a left-biased sender will report the truth because he has no way to in‡uence the decision maker. A right-biased sender, on the contrary, can decide to ensure getting the current period payo¤ by lying. When observing con‡icting messages L and R; the decision maker knows the true state is L and will want to deviate from the proposed strategy. The same applies to strategies prescribing to choose C when observing messages L and R: The decision maker gets (1 2 )(1 p)2 2A by randomizing while she gets 4p1 p2 ( A) + 2(1 p)4 2(1 4 )A by choosing C:
The latter follows because if there is at least one honest sender, and messages are L and R; by choosing decision C, the decision maker surely implements a wrong action. If both senders are biased, and messages are con‡icting, expected payo¤ by choosing action C is (1 2 )A 2 A:It can be seen that (1 2 )(1 p)2 2A > 4p1 p2 ( A) + (1 p)2 2(1 4 )A is always veri…ed when there is information transmission (i.e. p > 2(11 2 )):It can be veri…ed that this is also true for the lowest prior probability consistent with the existence of a mixed strategy equilibrium. With the same reasoning it is possible to rule out mixed strategies that implement action k with asymmetric probabilities.
Part 2: suppose not and suppose that when there is disagreement the action of sender i is implemented. This cannot be true if sender i suggests action L and sender i suggests action C.
In general, sender i will prefer to tell the truth as she will not be able to in‡uence the current decision, but then, in case of disagreement, the decision maker knows sender i is lying and she will prefer not to abide by the proposed equilibrium strategy.
Proof of Proposition 1
The …rst part was proved in Lemma 1, part 2. For the second part, the payo¤ of a biased sender, when the true state is di¤erent from the one he prefers, is given by
VT = B + E(B)
if he tells the truth in the …rst period, and
VL= B
if he lies in the …rst period. The necessary condition for a pure strategy equilibrium with truthtelling is VT > VL, which is veri…ed when
B < E(B) 2
The model has a continuum of mixed strategy equilibria. When the true state is C; both a left and a right-biased senders lie. As payo¤s are the same, the equilibrium is symmetric and qC = zC = q; therefore, p2 = p
p + (1 p)q . The posterior probability that an agent is honest should be high enough in the second period, in particular p2 = p
p + (1 p)q > 1 2
2(1 ) which is veri…ed as long as q < p
(1 p)(1 2 ) which is a necessary condition for a mixed strategy equilibrium to exist. When instead the true state is R; zR = 1; and p2 = p
p + (1 p)2 +(1 p)2 q; the condition p2 > 1 2
2(1 ) is veri…ed as long as q < p(3 2 ) (1 2 )
(1 p)(1 2 ) : Such mixed strategy equilibria occur over a set of measure zero. In fact, it is a measure zero event that parameters are exactly such that the …rst period importance happens to be
B = E(B) 2 Proof of Proposition 2
The …rst part was proved in Lemma 1, part 2. The second part regarding the behavior of a biased sender when the true state is state C was proved in Lemma 2. The rest of the second part can be proved as follows: in a pure strategy equilibrium, by de…nition, qi;1R = qRi = 1: Also, as proved by Lemma 3, i;mi;m i = i;mi;m i = 12 where mi; m i = L; R: The proof of the second part follows by comparing payo¤s from lying and telling the truth and imposing the condition VT > VL. Now consider mixed strategy equilibria. When the sender is left-biased, he is willing to randomize if the true state in the …rst period is R; otherwise when the true state is C, or L; there is truthtelling in pure strategies. Then qRi has to be such that VTi = VLi, and to ease notation, drop the dependence of q on the observed state and on the time period. Then, by rearranging
equations 1 and 2, it follows that
[p( i;R;R1 1i;R;R i;L;R1 + 1i;L;R)+
(1 p
2 )( i;R;R1 1i;R;R i;R;L1 + 1i;R;L i;L;R1 + 1i;L;R i;L;L1 1i;L;L)]B+
E(B)[p( ( i;L;L2 + 2i;L;L) (1 2 )+ ( i;L;R2 2i;L;R))+
(1 p)
2 ( i;L;R2 2i;L;R) + (1 p)
2 +(1 p)
2 + p(1 2 )]
=
q i[( i;L;R1 1i;L;R i;L;L1 1i;L;L+ i;R;R1 + 1i;R;R+ i;R;L1 1i;R;L)1 p 2 B+
E(B)(1 p
2 + 1 p
2
1 p
2 ( i;L;L2 + 2i;L;L))]
Plugging the equilibrium values of i;mi;m i :
q i = 2[ E(B)(1 (1 )p) B]
E(B)(1 p)
the equilibrium is clearly symmetric and therefore q i = qi = q: In order for this to be an equilibrium, two additional conditions have to be met. Firstly, q has to be a well de…ned probability, hence 0 < q < 1; secondly p2 > 1 2
2(1 ); i.e., second period reputation must be high enough for senders to exert in‡uence. This implies p
p + (1 p)2 + (1 p)2 q > 1 2 2(1 ); or q < p(3 2 ) (1 2 )
(1 p)(1 2 ) : For this to hold, it must be that p(3 2 ) (1 2 )
(1 p)(1 2 ) > 0 which implies p > 1 2
3 2 (it can easily veri…ed that 1 2
3 2 < 1 2
2(1 )). The fact that the symmet-ric mixed strategy equilibrium is unique follows by the non-existence of asymmetsymmet-ric equilibria, established by Lemma 3. Thus, necessary conditions for a mixed strategy equilibrium are
p > 1 2
3 2
B < E(B)[1 (1 )p]
B > E(B)[1 2 p(1
2 )]
Proof of Proposition 3
When the true state is C, the “unbiased”state, there is always truthtelling with two senders, while there is truthtelling with one sender only if period importance is not too large. When the true state is di¤erent from C, in the one sender case there is truthtelling (in pure strategies) if
and only if
B < E(B)
2 B1
In the two senders case, truthtelling in pure strategies in the …rst period occurs i¤
B < E(B)[1 2 p(1
2 )] B2
Truthtelling in pure strategies occurs over a set of parameters of larger measure when there is only one sender, i¤
E(B)
2 > E(B)
2 [1 p(1 2 )]
which is always veri…ed as < 12. There is truthtelling in mixed strategies with two senders if and only if
B < E(B)[1 (1 )p] Bmix2
and E(B)
2 > E(B)[1 (1 )p]
if and only if
p > 1
2(1 )
Therefore there can be parameter values for which there is truthtelling in pure strategies with one sender and truthtelling in mixed strategies with two senders and, if the probability the opponent is honest is large enough, there can even be a region of parameters such that there is truthtelling in pure strategies with one sender and no truthtelling with two senders.
Proof of Proposition 4
It can be seen that delegation generates stronger truthtelling incentives than communication in both the one and the two senders cases. In fact, it is easy to see that
E(B)[1 + p(1 2 )]
2 > E(B)
2 > E(B)
2 [1 p(1 2 )]
Moreover,
E(B)[1 + p(1 2 )]
2 > E(B)[1 (1 )p]
if and only if p > 3 41 so that, when the true state in the …rst period is di¤erent from state C, and the probability the opponent is honest is relatively large there are values of period importance for which there is truthtelling in pure strategies under delegation and not even truthtelling in mixed strategies under communication.
Proof of Proposition 5
I …rstly derive conditions sustaining truthtelling for the in‡uential and for the non-in‡uential
sender. I assume sender i is left-biased. When the true state is L he trivially reports the truth.
When the true state is not L; it is important to distinguish the case when the true state is C, from that when the true state is R: In fact, in the latter case, a right-biased opponent surely reports the truth, while, if the true state is C, a right-biased sender might prefer to lie. Therefore, the expected payo¤ of a left-biased in‡uential sender is given by:
VTi(C) = i1B + (1 i1)[ 1 p
2 z 1 p
2 (1 z) p 1 p
2 q + 1 p
2 (1 q)]B + E(B) (5)
VLi(C) = + i1B + (1 i1)[ 1 p
2 z 1 p
2 (1 z) p 1 p
2 q + 1 p
2 (1 q)]]B + (6) E(B)[1 p
2 q + p (1 2 )p p 1 p
2 z]
when the true state is C, and by
VTi(R) = i1B + (1 i1)[ 1 p
2 p 1 p
2 q + 1 p
2 (1 q)]B + E(B) (7)
VLi(R) = + i1B+(1 i1)[ 1 p
2 p 1 p
2 q+1 p
2 (1 q)]B+ E(B)[1 p
2 q+ p (1 2 )p p 1 p 2 ] (8) when the true state is R. When the true state is C, the right-biased sender reports the truth with probability z; and reports his preferred state otherwise. Therefore, he retains his credibility with probability z: On the contrary, when the true state is R; a right-biased sender always reports the truth and retains his credibility. The intuition for these equations is essentially analogous to that for equations 1 and 2.
A biased non-in‡uential sender always lies in a truthtelling equilibrium, as he will not have any chance to in‡uence future decisions, unless the true state coincides with his preferences. Thus, when the true state is C biased non-in‡uential senders lie (and q = z = 0) and a left-biased in‡uential sender reports the truth in the …rst period if and only if
B < E(B)[1 + (1 2 )p]
2 i1
When the true state is R and the in‡uential sender reports the truth, a right-biased non-in‡uential sender reports the truth (so that z = 1); while a left-biased non-in‡uential sender lies (so that q = 0)and a left-biased in‡uential sender reports the truth if and only if
B < 3 + (1 4 )p
4 i1 E(B)
If the in‡uential sender lies, a biased non-in‡uential sender might prefer to report the truth.
Again it is important to distinguish the case when the true state is C from the case when the true state is R: In the former situation, both a left and a right-biased non-in‡uential senders behave analogously. Expected payo¤s for such senders are given by
VT i(C) = i1Bp (1 i1)B + E(B)(1 p) (1 2 )p E(B) VLi(C) = i1Bp + (1 i1)B (1 2 )p E(B)
The continuation payo¤ follows because with probability (1 p)2 + (1 p)2 = (1 p) the in‡uential sender is either left, or right-biased, and lies in the …rst period, so that the non-in‡uential sender can in‡uence second period decision. On the contrary, with probability p; the in‡uential sender is honest, reports the truth in the …rst period and in‡uences second period decision leading to an expected payo¤ of (1 2 )p E(B): The payo¤ from lying can be understood analogously.
Therefore, biased non-in‡uential senders are willing to report the truth if and only if
B < E(B)(1 p) 2(1 i1)
If the true state is R a right-biased sender always reports the truth, while a left-biased sender has the following expected payo¤ functions:
VT i(R) = i1Bp (1 i1)B +1 p
2 E(B) (1 2 )p E(B) 1 p
2 E(B) VLi(R) = i1Bp + (1 i1)B (1 2 )p E(B) 1 p
2 E(B) and he is willing to report the truth if and only if
B < E(B)(1 p) 4(1 i1)
According to the degree of favoritism (the parameter i1); there can exist two equilibria. When the true state is C, a biased in‡uential sender reports the truth in the …rst period as long as
B < E(B)[1 + (1 2 )p]
2 i1
and lies otherwise, while non-in‡uential senders always lie if E(B)(1 p)
2(1 i1) < E(B)[1 + (1 2 )p]
2 i1 which is veri…ed as long as
i
1 < 1 + p 2 p 2(1 p)
When this condition is not veri…ed, in equilibrium the biased in‡uential sender reports the truth as long as
B < E(B)[1 + (1 2 )p]
2 i1
and lies otherwise, while biased non-in‡uential senders lie when
B < E(B)[1 + (1 2 )p]
2 i1 and report the truth for
E(B)[1 + (1 2 )p]
2 i1 < B < E(B)(1 p) 2(1 i1)
Similarly, when the true state is R; there exists an equilibrium where the in‡uential sender reports the truth as long as
B < 3 + (1 4 )p
4 i1 E(B)
and lies otherwise, and a left-biased non-in‡uential sender always lies. Such equilibrium occurs
when E(B)(1 p)
4(1 i1) < 3 + (1 4 )p
4 i1 E(B) which is veri…ed when
i
1 < 3 + (1 4 )p 4(1 p)
When this condition is not satis…ed, in equilibrium the biased in‡uential sender reports the truth as long as
B < 3 + (1 4 )p
4 i1 E(B) and lies otherwise. A left-biased non-in‡uential sender lies when
B < 3 + (1 4 )p
4 i1 E(B) and reports the truth for
3 + (1 4 )p
4 i1 E(B) < B < E(B)(1 p) 4(1 i1)
A biased in‡uential sender has stronger incentives to report the truth than under either delegation or communication. In fact,
E(B)[1 + (1 2 )p]
2 i1 > E(B)[1 + p(1 2 )]
2 > E(B)
2 [1 p(1 2 )]
as i1 1;and
3 + (1 4 )p
4 i1 E(B) > E(B)[1 + p(1 2 )]
2 > E(B)
2 [1 p(1 2 )]
Moreover, the non-in‡uential sender reports the truth if he is honest, and biased non-in‡uential senders may report the truth according to the whether the state of the world coincides with their preferences and to the degree of favoritism.
Proof of Proposition 6
The proposition can be proved by comparing payo¤s for the decision maker for di¤erent real-izations of period importance.
1. If B 2 [B; B2];there is truthtelling in pure strategies both under communication and under delegation. In this case expected payo¤ for the decision maker under communication is
UCommDM = p2[A + E(A)] + 4p(1 p)
2 [A + (1 ) E(A)] + (1 p
2 )2[4A 2(1 2 ) E(A))]
while that under delegation is
UDelDM = p2[A + E(A)] + 2p(1 p)
2 [A + E(A)] + 2p(1 p)
2 [A (1 2 ) E(A)] + (1 p
2 )2[4A 4(1 2 ) E(A))]
The intuition for these expressions is as follows: with probability p2 both senders are honest and report the truth no matter the state and period importance. Then, the decision maker implements the correct decision ensuring a payo¤ of A in the …rst period, and an expected payo¤ of E(A) in the second. With probability p1 p2 ; one sender is honest, and the other is biased and as the latter can be left or right-biased, the total number of such cases is four. Period importance is low enough so that there is truthtelling in pure strategies in the …rst period, and payo¤ is A both under delegation and under communication. In the latter case expected second period payo¤ is (1 ) E(A); because with probability (1 2 ) the true state is C, and the decision maker observes a C message from the honest sender and a non C message from the biased sender, and learns the true state is C. With probability the true state accords with the preferences of the biased sender and the decision maker observes two agreeing messages and implements the correct decision. Finally, with probability the true state is opposed to the preferences of the biased sender and the decision maker observes con‡icting messages and randomizes, so that expected payo¤ is zero. Under delegation, if the honest sender is delegated decision powers, second period decision is made correctly, otherwise, it is correct only when the true state is the one preferred by the biased sender, and this happens with probability : In the other cases, the biased sender implements a wrong decision yielding an expected payo¤ of E(A): Finally, with
probability (1 p2 )2 both senders are biased, either left or right. They report the truth in the
…rst period as period importance is lower than B2; while they lie in the second period. Under communication, there can be 2 cases: both senders have the same bias, or they have opposed biases. In the latter case, which occurs with probability 2(1 p2 )2; the decision maker observes con‡icting messages and randomizes. In the former case, which occurs with probability 2(1 p2 )2;
…rst period as period importance is lower than B2; while they lie in the second period. Under communication, there can be 2 cases: both senders have the same bias, or they have opposed biases. In the latter case, which occurs with probability 2(1 p2 )2; the decision maker observes con‡icting messages and randomizes. In the former case, which occurs with probability 2(1 p2 )2;