Leadership with Trustworthy Associates the highest expected payoffs to all politicians are such with bias bi of distance within 1/10 and 1/8 to bi will that i's information di(mi)is maximal.That is,we con- be truthful to j if and only if i has no other trustworthy sider the most information that jcould obtain when any associate.10 politician who could communicate truthfully in equi- librium will in fact do so.This allows us to define d;as SELECTING THE LEADER the maximal size of the group of politician who form i's trustworthy associates.Straightforwardly,we can relate Having defined the size of a leader's network of trust- the maximal size of this group to a leader's equilibrium worthy associates,we now turn to the question of lead- judgment. ership selection.We define the optimal leader as one Next we derive this leadership characteristic from who maximizes group welfare.In the absence of a first principles.In particular,we can define it as a con- mechanism that ensures the first best choice,it is natu- sequence of fs ideological position relative to that ral to ask which leader would be chosen by the group of other politicians in her party.To do so,we define when each casts a vote with the outcome determined the function Ni:R>N as the ideological "neighbor- by majority rule.Using the result of the previous sec- hood"function of politician j.For any real number b, tion,we show that the characteristics of optimal and the quantity N(b)is the number of politicians whose majority-preferred leadership can be derived from first ideology is within distance b of her own;that is,the principles. number of politicians whose ideology is in i's ideologi- cal neighborhood of size b.To formally define the func The Optimal Leader N tion Ni,we exploit the fact that politicians are ordered according to their bias,so that We first show that optimal leader selection involves trading off a politician's ideological moderation and N;(b)=maxi∈N:lb:-bl≤b} her judgment.To formalize this insight,we denote 4号 -min{i∈N:lb:-bl≤bl, politician i's moderation as-bi -Eie Nbiln,the con- trary of the distance between b;and the average ideol- for any real number b.For example,if the group of play- ogyb=ieN bi/n.We have definedd as the maximal ers who are truthful to leader j=5 is (3,4,6,71,then size of a leader's network of trustworthy associates.It is Ni(b)=7-3=4.We use the function Ni(.)combined but a small step to relate this number to her judgment, with the equilibrium condition Equation(2)to calcu- the second critical and endogenous leadership char- late the maximal size of any politician i's network of acteristic.When combining the information obtained trustworthy associates: from others with her own view.a leader forms an inde- pendent judgment of the best course of action.Thus a leader's judgment is(strictly)increasing in the number =max :(3) of informative signals she obtains from her trustworthy associates. In fact,and armed with these definitions,we can This provides a simple rule to calculate d by count- prove that the equilibrium ex ante sum of players'pay- ing the number of politicians other than j that are ideo- offs W*(j)can be rewritten as logically close to her.For example,suppose that b;=0 and the three politicians closest to jhave bias distance 1 W*()=-∑(b:-b2-n (4 ⑧ less than 1/12 from bi;that is,they have a bias in the (d+3) interval (-1/12,1/12).These politicians would provide truthful advice to j were she to be selected as leader. For j to have one more trustworthy associate,it must Equation (4)decomposes the welfare function into two be that no member of that circle has a bias further from elements:the aggregate ideological loss EiN(bi-bi)2 bi=0 than 1/14.Interestingly,the size of the ideological associated with the decision taken by j,and the ag neighborhood of leader j(to which a politician needs to gregate residual variance of her decisionEv- belong to be trustworthy to j)decreases in the number of associates truthful to j.For example,a politician i idently,a more moderate leader,whose bias is closer This equilibrium selection can also be motivated by the concept of 10 Costly information acquisition does not change our main insights focal leadership as in Calvert (1995):a leader provides a focal mech- Although it is possible that the set of associates of leader j is differ- anism allowing followers to coordinate on truthful communication, ent and so leadership choice may differ in this scenario,nevertheless, with the consequence that information is aggregated optimally.Ex- with costly acquisition,our first main result-that a leader can rely tensions could consider equilibria in which a politician threatens to on truthful advice only from those whose ideological preferences are babble(thus not transmit information to the elected leader)as a way similar to her own-continues to hold.If a player anticipates that to try and force her own election as leader.In the extreme case,all he will be unable to convey any information to the leader,then he players could commit to babble,communication would break down will not acquire costly information in the first place.Conversely those in equilibrium,and the most moderate politician would be always who do acquire information are those able to convey this informa- elected as leader.More interestingly,early voters could have an ad- tion:they are ideologically close to the leader.As a consequence of vantage in sequential voting because of a forward induction argu this finding,we can then use the same technique to derive the size of ment:by voting for herself,an early voter would "signal"to the others a leader's network of trustworthy associates and the tradeoffs iden- that she plans to babble if not elected. tified in our key results to follow shall continue to apply. 849
Leadership with Trustworthy Associates the highest expected payoffs to all politicians are such that j’s information dj(m−j) is maximal.That is,we consider the most information that j could obtain when any politician who could communicate truthfully in equilibrium will in fact do so.9 This allows us to define d∗ j as the maximal size of the group of politician who form j’s trustworthy associates. Straightforwardly, we can relate the maximal size of this group to a leader’s equilibrium judgment. Next we derive this leadership characteristic from first principles. In particular, we can define it as a consequence of j’s ideological position relative to that of other politicians in her party. To do so, we define the function Nj : R → N as the ideological “neighborhood” function of politician j. For any real number b, the quantity Nj(b) is the number of politicians whose ideology is within distance b of her own; that is, the number of politicians whose ideology is in j’s ideological neighborhood of size b. To formally define the function Nj, we exploit the fact that politicians are ordered according to their bias, so that Nj (b) = max{i ∈ N : |bi − bj| ≤ b} − min{i ∈ N : |bi − bj| ≤ b}, for any real number b.For example,if the group of players who are truthful to leader j = 5 is {3, 4, 6, 7}, then Nj(b) = 7 − 3 = 4. We use the function Nj(·) combined with the equilibrium condition Equation (2) to calculate the maximal size of any politician j’s network of trustworthy associates: d∗ j = max d ∈ N : Nj 1 2 (d + 3) ≤ d . (3) This provides a simple rule to calculate d∗ j by counting the number of politicians other than j that are ideologically close to her. For example, suppose that bj = 0 and the three politicians closest to j have bias distance less than 1/12 from bj; that is, they have a bias in the interval (− 1/12, 1/12). These politicians would provide truthful advice to j were she to be selected as leader. For j to have one more trustworthy associate, it must be that no member of that circle has a bias further from bj = 0 than 1/14. Interestingly, the size of the ideological neighborhood of leaderj(to which a politician needs to belong to be trustworthy to j) decreases in the number of associates truthful to j. For example, a politician i 9 This equilibrium selection can also be motivated by the concept of focal leadership as in Calvert (1995): a leader provides a focal mechanism allowing followers to coordinate on truthful communication, with the consequence that information is aggregated optimally. Extensions could consider equilibria in which a politician threatens to babble (thus not transmit information to the elected leader) as a way to try and force her own election as leader. In the extreme case, all players could commit to babble, communication would break down in equilibrium, and the most moderate politician would be always elected as leader. More interestingly, early voters could have an advantage in sequential voting because of a forward induction argument: by voting for herself, an early voter would “signal” to the others that she plans to babble if not elected. with bias bi of distance within 1/10 and 1/8 to bj will be truthful to j if and only if j has no other trustworthy associate.10 SELECTING THE LEADER Having defined the size of a leader’s network of trustworthy associates, we now turn to the question of leadership selection. We define the optimal leader as one who maximizes group welfare. In the absence of a mechanism that ensures the first best choice, it is natural to ask which leader would be chosen by the group when each casts a vote with the outcome determined by majority rule. Using the result of the previous section, we show that the characteristics of optimal and majority-preferred leadership can be derived from first principles. The Optimal Leader We first show that optimal leader selection involves trading off a politician’s ideological moderation and her judgment. To formalize this insight, we denote politician j’s moderation as −|bj − i ∈ Nbi/n|, the contrary of the distance between bj and the average ideology b¯ ≡ i∈N bi/n. We have defined d∗ j as the maximal size of a leader’s network of trustworthy associates. It is but a small step to relate this number to her judgment, the second critical and endogenous leadership characteristic. When combining the information obtained from others with her own view, a leader forms an independent judgment of the best course of action. Thus a leader’s judgment is (strictly) increasing in the number of informative signals she obtains from her trustworthy associates. In fact, and armed with these definitions, we can prove that the equilibrium ex ante sum of players’ payoffs W∗(j) can be rewritten as W∗ (j) = − i∈N (bi − bj) 2 − n 1 6(d∗ j + 3). (4) Equation (4) decomposes the welfare function into two elements: the aggregate ideological loss i ∈ N(bi − bj)2 associated with the decision taken by j, and the aggregate residual variance of her decision n 1 6(d∗ j +3). Evidently, a more moderate leader, whose bias is closer 10 Costly information acquisition does not change our main insights. Although it is possible that the set of associates of leader j is different and so leadership choice may differ in this scenario, nevertheless, with costly acquisition, our first main result—that a leader can rely on truthful advice only from those whose ideological preferences are similar to her own—continues to hold. If a player anticipates that he will be unable to convey any information to the leader, then he will not acquire costly information in the first place. Conversely those who do acquire information are those able to convey this information: they are ideologically close to the leader. As a consequence of this finding, we can then use the same technique to derive the size of a leader’s network of trustworthy associates and the tradeoffs identified in our key results to follow shall continue to apply. 849 Downloaded from https://www.cambridge.org/core. Shanghai JiaoTong University, on 26 Oct 2018 at 03:53:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0003055418000229
Torun Dewan and Francesco Squintani to average ideology b,makes the aggregate ideolog- straightforward.we can.nevertheless,make progress ical loss ieN(bi-bi)2 smaller.Further,the resid- by showing that utility functions are single-crossing. ual variance is inversely related to the size of Lemma 1.The utility functions U(j)are single cross- the leader's maximal informant set d;and hence to ing in iandj:ifi<i andj<j,then U(j)>U ( her judgment.11 The optimal leader jmaximizes W*() implies U*()>U();and ifizi andi>i,then Thus,optimal leader selection takes into account each U李(i)>U李()implies U()>U*(i). politician's moderation and her endogenous judgment that are related to the core primitives of our model. As a consequence of this result,we appeal to The- orem 2 by Gans and Smart (1996)to show that the namely the ideologies of members of the group. op//s Leader i's moderation can be understood spatially player with median ideology will determine the out- as the relative position of i's bias bi with respect to come of the election.The unique Condorcet winner of the whole ideology distribution b=(b1,...,bn)in the the election game is the politician j who maximizes the group.In fact,every element of b,even extreme ones, expected payoff of the median player. matters for the determination of the average ideol- Proposition 2.The group elects as leader the player j ogy b.In this sense,moderation is a "global"prop- who maximizes the utility U(j)of the median politi- erty of i's ideology bi with respect to the distribution cian m =(n +1)/2.The collective choice considers the b={b1,..,bn}. ideological proximity of any player j to m,as well as j's On the other hand,judgment is a "local"property judgment d that is determined by her number of close- of i's ideology bi within b=(b1,...,bn):it depends minded associates. only on how many other politicians are ideologically close to j,in the sense defined by Equation (3).The Having established the outcome of a majority elec- leader's understanding is thus defined by those close to tion,we can compare it with the optimal leader selec- the leader,or adopting Machiavelli's text,by "the men tion by inspecting Equations(4)and(5),the latter for he has around him."This analysis of the role played by i=m.As in the earlier case,there is a tradeoff be- the local ideological distribution is,to our knowledge, tween moderation and judgment:the Condorcet win- novel in the large contemporary and formal literature ner j keeps both the ideological loss (bm-bi)2 and on collective choice:though it echoes the insights of Machiavelli made in Il Principe 500 years ago. the residual variance as low as possible.Just as We summarize our findings as follows. with optimal leadership,the majority choice involves a tradeoff between the desire for a moderate leader and Proposition 1.The optimal leader j is determined by that for a leader with good judgment,which,in turn, ideological moderation,the proximity of bi to the aver- stems from having a large group of close-minded asso- age group ideology b,and by good judgment,her num- ciates.Beyond this similarity.there is a critical differ- 是 ber d of close-minded associates. ence and it is this:whereas a majority preferred leader makes this tradeoff decision by considering only her own payoff,by contrast,an optimally selected leader Electing the Leader would consider the preferences of the entire group. We now determine which politician is elected as leader Straightforwardly,and as the weights placed on these by a simple majority decision taken within the group. two features of good leadership are different in our key Each player i's utility as a function of the leader's expressions,the majority choice of leader may not be identity jis optimal.As we shall see,the implications are surprising in that we identify instances in which the median politi- cian's utility U places less weight on moderation(and U(U)=-(b-b;)2 (5 more on judgment)than the group's welfare W.Thus, 6(d+3) majority choice may be inefficient because it places too much weight on the leader's judgment. As in Equation (4),the first term on the right-hand side illustrates the ideological loss-(bi-bi)2 suffered by WHAT MAKES A GOOD LEADER? each member of the group i when jis chosen as leader. The second term illustrates player i's preference for an Our analysis relates the characteristics that define good informed leader j,as it increases in the judgment d:.We leadership-moderation and judgment-to the com- note that player utilities are not single-peaked with re- munication structure that emerges in the equilibrium spect to a leader's identity:a politician who is ideologi- of our model.The importance of the former is well cally distant may in fact be better informed,and so have known.Indeed,it is easy to see that if there were no better judgment,than one who is ideologically similar informative signals(or just no communication)in this While Black's theorem does not apply in this setting, game,then the chosen leader would be the median L so leadership choice under majority rule is far from politician m,while the optimal one would be the one whose bias is the closest to the average bias b.On the other hand,the role played by judgment,that in turn Mathematically,the residual variancecorresponds to the is related to a leader's trustworthy associates,is novel inverse of the precision of the leader's decision. and central to the results that follow. 850
Torun Dewan and Francesco Squintani to average ideology b¯, makes the aggregate ideological loss i ∈ N(bi − bj)2 smaller. Further, the residual variance 1 6(d∗ j +3) is inversely related to the size of the leader’s maximal informant set d∗ j and hence to her judgment.11 The optimal leader j maximizes W∗(j). Thus, optimal leader selection takes into account each politician’s moderation and her endogenous judgment that are related to the core primitives of our model, namely the ideologies of members of the group. Leader j’s moderation can be understood spatially as the relative position of j’s bias bj with respect to the whole ideology distribution b = {b1,..., bn} in the group. In fact, every element of b, even extreme ones, matters for the determination of the average ideology b¯. In this sense, moderation is a “global” property of j’s ideology bj with respect to the distribution b = {b1,..., bn}. On the other hand, judgment is a “local” property of j’s ideology bj within b = {b1,..., bn}: it depends only on how many other politicians are ideologically close to j, in the sense defined by Equation (3). The leader’s understanding is thus defined by those close to the leader, or adopting Machiavelli’s text, by “the men he has around him.” This analysis of the role played by the local ideological distribution is, to our knowledge, novel in the large contemporary and formal literature on collective choice; though it echoes the insights of Machiavelli made in Il Principe 500 years ago. We summarize our findings as follows. Proposition 1. The optimal leader j is determined by ideological moderation, the proximity of bj to the average group ideology b, and by good judgment, her num- ¯ ber d∗ j of close-minded associates. Electing the Leader We now determine which politician is elected as leader by a simple majority decision taken within the group. Each player i’s utility as a function of the leader’s identity j is U∗ i (j) = −(bi − bj) 2 − 1 6(d∗ j + 3). (5) As in Equation (4), the first term on the right-hand side illustrates the ideological loss −(bi − bj)2 suffered by each member of the group i when j is chosen as leader. The second term illustrates player i’s preference for an informed leaderj, as it increases in the judgment d∗ j .We note that player utilities are not single-peaked with respect to a leader’s identity: a politician who is ideologically distant may in fact be better informed, and so have better judgment, than one who is ideologically similar. While Black’s theorem does not apply in this setting, so leadership choice under majority rule is far from 11 Mathematically, the residual variance 1 6(d∗ j +3) corresponds to the inverse of the precision of the leader’s decision. straightforward, we can, nevertheless, make progress by showing that utility functions are single-crossing. Lemma 1. The utility functions U∗ i (j) are single crossing in i and j : if i < i and j < j , then U∗ i (j) > U∗ i (j ) implies U∗ i (j) > U∗ i (j ); and if i > i and j > j , then U∗ i (j) > U∗ i (j ) implies U∗ i (j) > U∗ i (j ). As a consequence of this result, we appeal to Theorem 2 by Gans and Smart (1996) to show that the player with median ideology will determine the outcome of the election. The unique Condorcet winner of the election game is the politician j who maximizes the expected payoff of the median player. Proposition 2. The group elects as leader the player j who maximizes the utility U∗ m(j) of the median politician m ≡ (n + 1)/2. The collective choice considers the ideological proximity of any player j to m, as well as j’s judgment d∗ j that is determined by her number of closeminded associates. Having established the outcome of a majority election, we can compare it with the optimal leader selection by inspecting Equations (4) and (5), the latter for i = m. As in the earlier case, there is a tradeoff between moderation and judgment: the Condorcet winner j keeps both the ideological loss (bm − bj)2 and the residual variance 1 6(d∗ j +3) as low as possible. Just as with optimal leadership, the majority choice involves a tradeoff between the desire for a moderate leader and that for a leader with good judgment, which, in turn, stems from having a large group of close-minded associates. Beyond this similarity, there is a critical difference and it is this: whereas a majority preferred leader makes this tradeoff decision by considering only her own payoff, by contrast, an optimally selected leader would consider the preferences of the entire group. Straightforwardly, and as the weights placed on these two features of good leadership are different in our key expressions, the majority choice of leader may not be optimal.As we shall see, the implications are surprising in that we identify instances in which the median politician’s utility U∗ m places less weight on moderation (and more on judgment) than the group’s welfare W∗. Thus, majority choice may be inefficient because it places too much weight on the leader’s judgment. WHAT MAKES A GOOD LEADER? Our analysis relates the characteristics that define good leadership—moderation and judgment—to the communication structure that emerges in the equilibrium of our model. The importance of the former is well known. Indeed, it is easy to see that if there were no informative signals (or just no communication) in this game, then the chosen leader would be the median politician m, while the optimal one would be the one whose bias is the closest to the average bias b¯. On the other hand, the role played by judgment, that in turn is related to a leader’s trustworthy associates, is novel and central to the results that follow. 850 Downloaded from https://www.cambridge.org/core. Shanghai JiaoTong University, on 26 Oct 2018 at 03:53:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0003055418000229