Leadership and Group Decision-Making. (preliminary and incomplete) Wouter Dessein∗ November 1, 2006

Abstract I develop a model of group decision-making, in which the group generates ideas and holds open discussions, but the ultimate decision is either taken by a leader (decision by authority) or by a majority vote. I endogenize the average quality of proposals, the amount of discussion they generate, and the average quality of the approved decision. Incentive conflicts arise as group members have ‘pet ideas’ whose quality is known only to them. Discussions are modeled as costly state-verifications of proposed ideas. By establishing a clear default, authoritative decision-making is shown to result in a higher average quality of proposed ideas and fewer discussions. Also the average quality of approved ideas is often higher. The proposed theory thus provides a rationale for strong leadership in group decision-making as opposed to more democratic alternatives. The model further highlights the cost and benefits of institutions which guarantee a "right to voice" by separating the roles of decision maker and discussion leader.

∗

University of Chicago, Graduate School of Business and CEPR. I have benefited from discussions with Luis

Garicano, Augustin Landier, Canice Prendergast, Tano Santos, Eric Van den Steen and seminar participants at the 1st MIT-NBER Organizational Economics Conference. A previous version of this paper was circulated under the title: “Hierarchies versus Committees." Email: [email protected]

"Rudi’s brilliant. He’s a tyrant; no, not a tyrant, a dictator. He has to be. You don’t have a leader if you don’t have a dictator. If you don’t have a dictator, you won’t be successful. Show me a company run by democracy, and I’ll show you a loser. There’s always got to be one chief and plenty of Indians" 1 (First violinist:) "I am a bit of dictator. It just seems logical that I decide. [...] I don’t think a democratic quartet can work. I think everybody recognizes that." His cellist concurred: "You must go with the first".2

1

Introduction

Freedom of speech and democracy are core values of modern societies. Most organizations operating in these societies, however, are far from democratic. While firms tend to encourage open discussions and debate — and set-up numerous committees and task-forces for this purpose — final decision authority often relies with a task leader, a chairman, the chief executive. In many firms, therefore, the term group decision-making simply refers to the ability of group members to generate proposals and voice their opinion on matters, not to a democratic process. Even in the political arena, democratic decision-making is not always a panacea. In April 2005, citing his "near imperial power", Time Magazine selected Chicago’s Richard W. Daley as one of America’s five best big city mayors. "Daley’s unchecked power sometimes short-circuits public debate," but "most of Chicago would have it no other way". Similarly, in many Asian countries, a strong leader is often preferred over a democracy. Why are most modern firms not run by democracy? When is decision authority best allocated to a leader, even if this leader has his own agenda? What is the role of procedures or institutions which guarantee a “right to voice"? To address these questions, we develop a model of group decision-making, in which the group generates ideas and holds open discussions, but the ultimate decision is either taken by a leader (decision by authority) or by a majority vote. At first sight, democratic decision-making is very attractive: a majority rule results in unbiased decision-making whereas potential leaders are self-interested and favor their own ideas. Yet, as we show, decision-making by authority is typically preferred over decision-making by majority. Not only does authoritative decision-making result in lower communication costs (that is, fewer 1

Senior executive quoted in "Rudi Gassner and the Executive Committee of BMG International", HBS Case

494-055, p12. 2 String quartet members quoted in "The Dynamics of Intense Work Groups: A Study of Britisch Spring Quartets", Murninghan and Conlon (1991), p 174.

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discussions), also the quality of decisions is often better. The intuition for this result is simple. A strong leader is pre-disposed towards his own ideas and has the final authority to implement these ideas. These ideas, therefore, constitute a default decision that only can be overturned by proposals which are clear improvements. As a result, only group members which are convinced of the merits of their alternative proposal are willing to challenge the leader. Under majority rule, in contrast, there is no such default decision and group members have strong incentives to lobby in favor of their idea, regardless of its merits. Relative to a democracy, a dictator therefore short-circuits debate, but this comes mainly at the expense of mediocre proposals. The debate is focussed on a smaller number of ideas of a higher average quality. In our model, a group of agents face a problem or opportunity and need to agree on a course of action (choice of restaurant, a new hire, a project, a policy or procedure). Information is dispersed in the sense that each group member may come up with a solution whose existence and value are unknown to its peers. Information aggregation occurs through the communication of soft information (‘proposals’) or hard information (‘discussions’). Agents propose their idea by issuing a ‘cheap talk’ statement about its quality. An agent, for example, can claim: “ I know a terrific restaurant.” Following a proposal, the group can engage in a discussion about the proposal. Discussions are modeled as a costly state verification of a proposed idea: the group can launch a time-consuming investigation in order to assess the true value of a proposal. Whereas proposing an idea involves neglegible communication costs, discussions are costly since they delay the implementation of a solution and waste the time of group members. Our model endogenizes the number and average quality of proposed ideas, and how much discussion a proposal tends to generate. In doing so, the model endogenizes communication costs in organizations: the only reason why communication is costly is because agents may have an incentive to misrepresent the value of their idea, inducing the group - or the leader to investigate the proposal in greater detail. As a key insight, we show that the average quality of proposals crucially depends on the decision-process: majority decision-making versus authoritative decision-making (dictatorship). Majority decision-making is very vulnerable to politicking: agents lobby in favor of their ideas regardless of its merits. Since proposals (soft information) contain little or no information, the group needs to engage in a full scale discussion (costly state verification) in order to select a proposal. In contrast, proposals are much more informative if one assigns authority to a leader who is pre-disposed towards her own solution. Intuitively, a leader applies a higher standard for adoption to alternative proposals: unless the leader is convinced 2

of the merits of an alternative proposal, she will implement her own idea. The establishment of the leader’s project as a default, in turn, discourages other group members from proposing mediocre ideas but not from advocating high-quality ones. Indeed, proposing a mediocre idea results in wasteful discussions, but the probability that this idea will be implemented by the leader is limited. Authoritative decision-making not only results in fewer discussions, it may also increase the average quality of decisions. Indeed, when problems are complex, discussions are often non-conclusive. Since the average quality of proposals is low under majority decision-making, mediocre ideas may then be selected more frequently than under authoritative decision-making. For the above two reasons, authoritative decision-making is always preferred for moderate incentive conflicts. If incentive conflicts are sufficiently large and problems sufficiently complex, however, the leader becomes dismissive of alternative proposals: she combines a suboptimally low level of discussion with a tendency to stick to her own mediocre ideas whenever a discussion is non-conclusive. Majority decision-making is then preferred unless one can ensure that alternative ideas receive sufficiently attention. In particular, if it is feasible to appoint an independent discussion leader who ensures sufficient debate prior to any decision, then authoritative decision-making is always preferred unless the leader’s bias is so strong that she never entertains alternative ideas. Separating discussion authority from decision authority, however, is not always recommended. If problems are sufficiently simple and the incentive conflict is moderate, the leader actually engages in a more intense discussion than an independent discussion leader would. The leader then optimally controls the discussion as this provides the necessary commitment that prevents other group members from proposing mediocre ideas. The remainder of the paper proceeds as follows. Section 2 discusses the related literature. The basic model is presented in Section 3. Section 4. and section 5 characterize the equilibrium respectively under majority decision-making and decision-making by authority. Section 6 then compares these two decision processes. Section 7 considers the option to separate discussion authority from decision authority. Section 7 concludes with a discussion of some testable implications of our model and points to some future avenues of research. Most proofs are relegated to the Appendix.

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2

Related Literature

This paper is related to and draws upon a number of literatures: Decision processes in organizations: The idea that decision-making by authority saves on communication costs has been put forward informally by Arrow (1974), Williamson (1975) and Chandler (1977). Williamson’s argument is exemplary and goes as follows: "Consider the problem of devising access rules for an indivisible physical asset which can be utilized by only one or a few members of the group at a time. (...) While a full group discussion may permit one of the efficient rules eventually to be selected, how much simpler if instrumental rules were to be “imposed" authoritatively. (...) Assigning the responsibility to specify access rules to whichever member occupies the position at the center avoids the need for full group discussion with little or no sacrifice in the quality of the decision. Economies of communication are thereby realized."3 Williamson’s reasoning, however, fails to explain why a group would engage in long and costly discussions whose informational benefits do not outweigh the costs. We consider a setup similar to the one proposed by Williamson, but endogenize the communication costs that each decision process generates. We find that decision-making by authority not only saves on communication costs, as assumed by Williamson, but also may result in decisions of higher quality. Williamson, in contrast, implicitly assumes a trade-off between communication savings and decision quality. One of the few papers to formalize the benefits of a central authority in terms of speedy decision-making is Bolton and Farrell (1990).4 In their model, two firms contemplate sinking costs to enter a natural monopoly market. Under decentralization, firms with a high cost structure postpone entry in order to avoid duplication. Under centralization, a central planner picks an entrant at random. Centralized decision-making therefore avoids delays, but makes no use of private information. Unlike our paper, Bolton and Farrell do not allow for communication. Decentralization is also not feasible in our model as group members must agree on a particular solution. Our rationale in favor of decision-making by authority is further reminiscent of the literature on influence costs (Milgrom (1988), Milgrom and Roberts (1988,1990), Meyer, Milgrom 3 4

Williamson (1975), Chapter 3: Peer Groups and Simple Hierarchies, pp 46-47. Also Segal (2001) formalizes the idea that authority saves on communication costs, but incentives conflicts

play no role. Communication problems arise because agents do not share a common labelling and need to describe potential actions, which is costly.

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and Roberts (1992)). This literature argues that members in organizations often spend considerable time and effort in attempting to influence decision-makers, time which could be otherwise used in more productive activities. Optimal decision processes should therefore try to limit these influence activities. Whereas most of this literature assumes that influence activities are a pure waste, in our model they take the form of agents proposing ideas and subjecting them to group discussion. While this improves the effectiveness of decisions if a proposal is high quality, it is a waste of time to the organization if ideas are mediocre. Using the terminology of the influence cost literature, our paper then argues that decision-making by authority is less vulnerable to rent-seeking activities than decision-making by majority. Strategic Communication: In modeling communication, our paper allows both for the transmission of soft information (Crawford and Sobel (1982)) and the disclosure of hard information (Milgrom (1981), Milgrom and Roberts (1986)). The classic cheap talk model by Crawford and Sobel has been applied to analyze the value of consulting multiple experts (Krishna and Morgan (2000), Battaglini (2002)), the impact of reputational concerns on communication (Ottaviani and Sorensen (2002)), delegation (Dessein 2002), the relative efficiency of vertical and horizontal communication (Alonso, Dessein, Matouschek (2006)), and most closely related to this paper, the optimal structure of collective decision-processes (Li, Rosen and Suen (2000)).5 Our paper differs from the above models in adding a discussion stage in which the decision-maker(s) have the option to verify cheap talk statements at a cost. The notion that soft information can be made "hard" at a cost is also present in Dewatripont and Tirole (2005), which emphasizes moral hazard problems in communication as well as different modes (issue-relevant and issue-irrelevant) of communication and in Caillaud and Tirole (2006), who study the strategies that the sponsor of a proposal may employ to convince a group to approve the proposal. Unlike our paper, the above papers do not provide normative results regarding decision processes such as majority decision-making or dictatorship. They take the authority structure as given and focus on the type of communication strategies used in equilibrium. Leadership and Vision: Finally, our paper contributes to a nascent literature which argues that firms may benefit from employing a CEO whose vision biases him in favor of certain projects (a strong leader), as opposed to a purely profit-maximizing CEO (a weak leader). In particular, a strong vision may improve incentives for employees or partners of the firm to undertake strategy-specific investments (Rotemberg and Saloner (2000)) and will attract, through sorting 5

The Crawford and Sobel (1982) setting has further been applied to the study legislative rules in congressional

committees (Gilligan and Krehbiel (1986, 1989)) and models of lobbying (for example Austen-Smith (1993)). Farrell and Rabin (1997) provide an overview of other cheap talk applications.

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in the labor market, employees with similar beliefs (Van den Steen (2001)).6 Unlike this paper, the above papers do not study alternative decision processes and group-decision making and communication play no role in their analysis.

3

The Model

3.1

Basic Structure

A committee consisting of three members, L, R and M, must formulate a response to a problem or an opportunity. Each group member may have an idea as how to solve the problem or exploit the opportunity, but only one idea can be implemented. Payoffs.– With a probability α, an idea is ‘high quality’ and yields benefits vH to all group members. With a probability 1 − α, it is ‘mediocre’ and yields benefits vL < vH . To reduce

notation, we denote v ≡ vH − vL and normalize vL = 0.

In addition to vL or vH , the ‘sponsor’ of the idea — that is the group member who

conceived the idea — also derives a private benefit b > 0 from his idea being implemented. This assumption is realistic: In developing ideas, group members will tend to focus on solutions which are self-serving or, in case of inter-divisional committees, have positive distributional consequences for their division.7 For example, group members may come up with solutions who exploit their human capital, skills or specific knowledge. Therefore, if adopted, they will probably play a leading role in the implementation of this solution or idea, resulting in additional opportunities for rent extraction, skill development, organizational influence and benefits of control. In order to make the analysis interesting, we will make the following assumption αv < b < v

(A1)

A1 implies that a group member prefers his own mediocre idea rather than a random idea from another agent. Group members with a mediocre idea, however, do prefer an alternative idea which is known to be high-quality. As shown further, A1 guarantees that communication is strategic: Agents cannot be trusted to truthfully reveal the quality of their idea.8 In contrast, 6

See also Ferreira and Rezende (2005), who endogenize commitment to a publicly announced strategy as the

result of carreer concerns rather than some exogenous bias or belief. 7 The latter assumes that an agent is subject to an (implicit or explict) incentive scheme which rewards positive performance by the division. 8 Note that if the quality of ideas were to be continuously distributed on [0, v],communication would always be strategic.

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if b < αv, majority decision-making would be able to implement the first best. Proposals and Discussions.– For simplicity, we assume that only L and R may have high quality ideas.9 The quality of an idea is privately known by its sponsor, L or R, but can be revealed in two ways: First, both L and R can propose their idea, that is describe it and make a statement about its quality. This communication of soft information is ‘free’: Committee members do not incur any costs by listening to these statements. Second, in order to assess the true value of the proposals – make the soft information hard – the group may decide to engage in a discussion (debate the problem or opportunity at hand, read numerous reports, order expert advice). In particular, by incurring a cost g(d) per group member, the group learns the true value of all proposals with probability d. We will refer to d as the discussion intensity and for tractability, we assume that g(d) ≡ kd2 /2 The discussion costs g(d) reflect the delay in the implementation of a solution and the opportunity cost of time of the group (as long as a particular problem is not solved, other problems or opportunities lack attention). The parameter k is best interpreted as a measure for the complexity or urgency of the problem. Since all members lose valuable time or suffer from a delay in the resolution of a problem, we assume that g(d) is incurred by each committee member.

3.2

Group Decision-making

It is instructive to think of the committee members as belonging to the same organization, headed by a principal who has no time to be actively involved in the decision-making process.10 Two decision processes are considered: • The principal delegates the decision-making authority to one of the agents. • The principal delegates the decision-making authority to the group, who then needs to agree on a common action by majority voting.

9

We assume that M has either no ideas, or his ideas have negative value. Hence, a mediocre idea is preferred

over M 0 s idea. 10 For many organizational problems, the opportunity cost of the principal’s time is likely to be very high relative to the importance of the problem. Furthermore, the principal may not be a physical person, but a board of directors or trustees, the share-holders, an electorate.

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Idea L Implemented

Idea L Implemented

L consults R R proposes idea R does not propose

L determines discussion intensity d

Discussion conclusive

Discussion non-conclusive

Idea R Implemented

Idea L Implemented

Idea L Implemented

Idea R Implemented

Figure 1: Decision-making by authority Decision-making by authority (Dictatorship).– Figure 1 illustrates decision-making by authority. For concreteness, we assume that group member L is allocated authority. L then can either directly implement her own idea or first consult R. If R does not propose an idea, L always implements her own project.11 If R proposes an idea, then L chooses the discussion intensity d. Depending on the outcome of the discussion, L then decides wether or not to accept the proposal. At each decision point, L maximizes his expected utility, both in discussing as in accepting proposals. No commitment as to future actions is possible. Decision-making by majority.– Figure 2 illustrates the majority decision-making. Both L and R simultaneously decide whether or not to propose their idea. Given monotone beliefs, a group member with a high quality idea will always propose this idea, implying that if a group member does not propose his idea, it must be mediocre.12 It follows that if only one group member proposes an idea, this idea is always implemented and no communication costs are incurred. Similarly, if neither L or R propose their idea, one of the ideas is implemented at 11

By proposing an idea, Y makes a statement about its value. We restrict the beliefs of X to be monotone,

that is when Y proposes an idea, the probability that X assigns to the event that Y 0 s idea is high-quality must be equal or higher than when Y would not have proposed an idea. 12 Beliefs are monontone when the probability which agents R and M assign to L0 s idea being high quality, does not decrease when L proposes his idea.

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Idea L Implemented

L and R decide whether to propose idea

M determines discussion intensity d

(propose, propose)

Discussion (Prob d) conclusive Discussion non-conclusive

(propose, not propose)

Idea L Implemented

Idea L Idea R Implemented

implemented

(not propose, propose)

Idea R Implemented

Idea R implemented

(not propose, not propose)

Idea L or R implemented at random

Figure 2: Decision-making by majority random and no communication costs are incurred.13 In contrast, if both L and R propose an idea, the group engages in a discussion, where the discussion intensity d is chosen in order to maximize expected surplus at that point in time. If both projects are revealed to be equal in value or if the discussion is non-informative, L or R0 s idea is chosen at random.14 Otherwise, the best project is selected. At each decision point, the group votes by majority, anticipating the subsequent game. No commitment as to future decisions is possible. The above decision processes establish a level playing field between authoritative and majority decision-making in terms of communication costs. In particular, our modeling of majority decision-making is such that there never occurs any wasteful communication. Com13

One might argue that also M 0 s mediocre idea is a valuable candidate. Allowing M 0 s mediocre idea to be

selected at this stage, however, would increase the incentives of L and R to propose a mediocre project. This would obviously reduce the quality of committee decision-making. In order not to bias our results in favor of authoritative decision-making, we therefore assume that L and R can commit not to vote in favor of M at this stage. Such a commitment is subgame perfect as both L and R weakly prefer R and L0 s idea over that of M. 14 In Appendix B, we endogenize the probability with which the group chooses a particular idea following an non-informative discussion or if both ideas are revealed to be mediocre. We consider a refinement in which, with neglegible probability, discussions produce false negatives and show that one can, without loss of generality, restrict attention to symmetric equilibria.

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munication costs are only incurred if the latter are justified by the expected informational benefits. If problems become arbitrarily complex (k goes to infinity), for example, the discussion intensity d will go to zero under both decision processes. Under majority decision-making, the group then simply votes to pick L or R0 s project at random. Similarly, under authoritative decision-making, the leader then simply chooses his own project. Decision-making by authority has thus no inherent advantage in terms of information processing. Our model, however, can be accused of drawing an overly rosy view of majority decisionmaking. When deciding on R0 s or L0 s proposal, M is always the medium voter and only cares about the efficiency consequences of the ideas of L and R. In this sense, M could be seen as a proxy for a large number of uninformed group members who take part in the decision process. Indeed, our model would be unaffected if there were N ≥ 3 committee members, but

only two of them are ‘inspired’. The virtue of majority decision-making is thus that it realizes the ‘democratic’ ideal of unbiased decision-making. Indeed, both the adoption decisions as

well as the information acquisition decisions are taken in an (ex post) welfare maximizing way. Despite this undoubtedly over-optimistic view on committee decision-making, we show that it is typically dominated by decision-making by authority. Theoretical Foundations: The above collective decision processes, majority voting versus dictatorship, arise naturally if, following Grossman and Hart (1985), one posits that actions (solutions) are not contractible, but the authority over who decides over a particular action (solution) is. No contracts in which one party agrees to implement an idea in return for a side-payment can then be enforced.15 In addition, our model presumes that during the communication stage, bargaining over decision-rights is impractical . This assumption is realistic if decision rights must be institutionalized ex ante (through company charters and procedures, allocated budgets, access to information and critical resources, reporting relationships with subordinates, ownership or control over assets) and cannot be easily or credibly transferred ‘on the spot’. While actions may not be contractible, the attentive reader may notice that it may be worthwhile to punish agents who ‘propose’ ideas. Proposing an idea, however, can be done in many different ways. As a result, such contracts may be extremely difficult to enforce.16 15

Majority decision-making can be interpreted as the focal equilibrium which prevails whenever decision

rights are distributed in such a way that the participation of all group members is required in order to execute a decision. To the extent that group members prefer some solution to no solution at all, majority decision-making and dictatorship are then two possible equilibria. 16 Such contracts would also punish agents who exert effort in order to develop high quality ideas. Therefore, in a more general model with endogenous information acquisitions, they would lose a lot of their appeal.

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4

Decision-making by Majority

We first analyze the decision process where L, R and M select an idea or solution by majority voting. The decision-making process then starts with L and R making a ‘cheap talk’ statement about the value of their idea. We say that an agent ‘proposes’ an idea if he claims to have a high quality idea. Given monotone beliefs, an agent with a high quality idea always proposes this idea, implying that if an idea is not proposed, it must be mediocre. Hence, if only one idea is proposed, the group implements this idea without further discussion. Similarly, if neither L nor R propose their idea, both L 0 s and R0 s idea are revealed to be mediocre. The group then selects one of them at random. The following lemma states the communication is always strategic in equilibrium: Lemma 1 (Communication is strategic) Given A1, no truthful equilibrium exists where agents only propose high quality ideas. To see this, note that discussions have no value in a truthful equilibrium. The group simply randomizes between selecting L or R0 s idea unless one and only idea is proposed, in which case this idea is chosen. For this to be an equilibrium, L should prefer not to propose a mediocre idea given truthtelling by R. The benefits to L of proposing a mediocre idea depend on the quality of R0 s idea. By proposing a mediocre idea, L increases the probability of adoption of this idea by 1/2. If also R0 s idea is mediocre (probability α), adoption increases L0 s pay-off with b. If instead R0 s idea is a high-quality (probability 1 − α), adoption decreases

L0 s pay-off by v − b. It follows that a truthful equilibrium exists if and only if 1 1 (1 − α) b − α [v − b] < 0 2 2

(1)

which will be satisfied if and only if b < αv, a violation of A1. Consider now equilibria that involve partial truthtelling. For the sake of the exposition, we only consider symmetric equilibria. In appendix, we show that no asymmetric equilibria exist. We denote by p ∈ [0, 1] the probability that a group member with a mediocre idea

proposes this idea, and by μ(p) the average quality of a proposal: μ(p) ≡

α ∈ [α, 1] . α + (1 − α)p

(2)

Conditional on two proposals being made, a discussion has value only if one proposal is highquality and the other one is mediocre, which occurs with probability 2(1 − μ(p))μ(p). With

probability d the group then finds out which proposal is high-quality, whereas with probability 11

(1 − d), it simply selects a proposal at random. It follows that following two proposals, the surplus maximizing discussion intensity d is given by n h o vi − kd2 /2 , d∗ = arg max 2(1 − μ(p))μ(p)d v − s 2

or still

d∗ = min {1, (1 − μ(p))μ(p)v/k} If both L and R have a mediocre idea, then regardless of p and d, a proposal by L raises the probability of adoption of L0 s idea with 1/2.17 If R has a high-quality idea, then a proposal by L reduces the probability of adoption of R0 s high-quality idea with (1 − d)/2. Finally, if

also R proposes his idea, a proposal by L results in communication costs kd2 /2 for all group

members. Thus, the expected value to L of proposing a mediocre idea equals 1 1 Vp ≡ (1 − α)b − α [1 − d] (v − b) − [α + (1 − α)p] kd2 /2. 2 2 Substituting d∗ , this yields 1 Vp ≡ (1 − α)b − [α + (1 − α)p] k/2. 2 if d∗ = 1, and Vp ≡

1 1 (1 − α)b − α [k − (1 − μ(p))μ(p)v] (v − b) 2 2k 1 − [α + (1 − α)p] (1 − μ(p))2 μ(p)2 v 2 2k

if d∗ < 1. By manipulating of the above expressions and the constraint b > αv, one can show that Vp > 0 for any p, yielding the following observation: Lemma 2 (Proposals are non-informative) Under majority decision-making, no equilibrium exists where p < 1.

Lemma 2 high-lights the inefficiency of majority decision-making: Agents always propose their own idea regardless of its quality (p = 1). In order to select an idea, the group then always needs to engage in time-consuming discussion whose intensity is given by d∗ = min {1, (1 − α)αv/k} 17

If R proposes his idea, then R0 s idea is definitely implemented if L does not proposes. In contrast, proposing

give L a 50% chance of adoption. Similarly, if R does not propose his idea, then by proposing, L is guaranteed of adoption. Not proposing only yields a 50% chance.

12

The fact that agents always lobby in favor of their own idea not only results in wasteful discussions, the group may also fail to implement an available high-quality idea. Indeed, whenever the problem at hand is sufficiently complex, that is k > kau with kau ≡ α(1 − α)v

(3)

then discussions are often inconclusive (d < 1) and with probability (1 − α)α(1 − d) the group

selects a mediocre idea even though a high-quality one is available. The following proposition summarizes the equilibrium under majority decision-making: Proposition 1 (Majority decision-making) There exists a unique equilibrium in which L and R always propose their idea (p∗ = 1). If k < kau , given by (3), then d∗ = 1 and the best idea is always selected. In contrast, if k > k au then d∗ < 1 and with probability 1 − d∗ > 0, a project is selected at random.

5

Decision-making by Authority

Consider now the decision process where L selects a solution after consulting R. Since it is common knowledge that M 0 s ideas are mediocre, L never consults M. We will refer to L as ‘leader ’ and R as the ‘advisor ’. Obviously, a leader always implements her own idea if it is high quality. Assume therefore that the leader’s idea is mediocre. Given monotone beliefs, an advisor with a high quality idea always proposes his idea. Abusing notation, we denote by p ∈ [0, 1] the probability that an advisor with a mediocre idea proposes his solution. Following an informative discussion, the leader then adopts a high quality proposal by R whenever b < v,

but rejects a mediocre one preferring to adopt her own mediocre idea instead. If a discussion is uninformative, the leader adopts a proposal by R only if μ(p)v ≥ b

(4)

where μ(p) denotes the average quality of a proposal and is given by (2). In equilibrium, p∗ must be such that μ(p∗ )v ≥ b implying that p∗ < 1. Indeed, if the advisor were always to proposes an idea (p∗ = 1) then μ(p∗ )v = αv < b and a the leader would only accept ideas

which were proven to be high quality. By proposing a mediocre idea, the advisor then only generates wasteful discussions, but never gets his proposal implemented. The following result follows: Lemma 3 (Proposals are informative) Under authoritative decision-making, no equilibrium exists where p = 1 and the advisor always proposes his idea. 13

Since μ(p∗ )v ≥ b in equilibrium, the leader weakly prefers to accept a proposal following

a non-conclusive discussion. It follows the leader optimally chooses a discussion intensity d∗ given by

½ ¾ k 2 d = arg max d(1 − μ(p)b − d d 2

(5)

where b is the opportunity cost of not identifying a mediocre proposal and 1−μ(p) the likelihood of a mediocre proposal. No equilibrium exists where d∗ = 1. Indeed, if discussions were always informative (d = 1), the leader would never select a mediocre idea and, hence, the advisor would never propose mediocre ideas (p = 0). But if the advisor only proposes high-quality ideas, there is no need for discussions (d∗ = 0). It follows that d∗ is given by d∗ = (1 − μ(p))b/k

(6)

It will be useful to denote d∗ = d(p), where from (6) d is increasing in p. We can now write down the value to the advisor of proposing a mediocre idea. Let a be the probability that the leader accepts a proposal if a discussion is uninformative. The value of proposing a mediocre idea is then given by Vp (p, a) ≡ [1 − d∗ (p)] ab − k(d∗ (p))2 /2

(7)

where [1 − d∗ ] a is the probability of a mediocre proposal being accepted and k(d∗ )2 /2 the

discussion costs resulting from proposing an idea. The advisor proposes a mediocre idea only if Vp (p, a) ≥ 0. Since no equilibrium exists where p = 1 and no equilibrium exists where p = 0, p∗ and a∗ must be such that must be such that Vp (p, a) = 0.

From (7) and (6), if problems are sufficiently simple (k small) and, hence, discussions likely to be informative (d∗ large), then a∗ = 0 and a leader with a mediocre idea only rejects mediocre proposals. Authoritative decision-making then always selects the best available idea. In contrast, if problems are sufficiently complex (k large) and, hence, discussions often non-conclusive (d∗ small), the leader sometimes rejects a high-quality proposal (a∗ < 1).and implements his own mediocre idea instead. The following proposition characterizes the unique equilibrium under authoritative decision-making: Proposition 2 (Authoritative decision-making) There exists a unique equilibrium where - The leader consults the advisor whenever his own idea is mediocre. - An advisor with a mediocre idea proposes this idea with probability 0 < p∗ < 1, where p∗ is weakly increasing in k with limk→0 p = 0 - A discussion is conclusive with probability 0 < d∗ < 1 where d∗ is decreasing in k 14

- There exists a kau such that whenever k < kau the best idea is always selected: The leader accepts any proposal unless proven to be mediocre. In contrast, if k > kau , the leader implements her own mediocre idea with probability 1 − a > 0 if a discussion is uninformative, where 1 − a is increasing in k and b.

A direct implication from proposition 2 is that decision-making by authority is more efficient at processing information than majority decision-making. Under majority rule, agents always claim to have a great idea (p = 1). Information processing then necessarily relies on ‘hard information’: discussions or investigations. In contrast, in a dictatorship, an advisor often shows restraint in advocating a mediocre idea (p < 1). The reason is that it is more difficult to get a mediocre idea ‘approved’ by a leader who is biased in favor of her own ideas than by a committee deciding in all objectivity. Advocating a mediocre idea then primarily results in wasteful discussions, but only rarely in this idea actually being adopted. Since many mediocre ideas are not brought forward for discussion, this yields considerable communication savings. In addition, a dictatorship has the obvious advantage that the leader can implement his own high quality ideas without any need for discussion. Communication savings are again obtained. While authoritative decision-making often avoids wasteful discussions, it is a priori ambiguous whether or not authoritative decision-making results in better or worse decisions. On the one hand, it is easy to verify that the leader’s bias results in his own idea being much more likely to be implemented than the advisor’s idea. On the other hand, at least for k small, this does not affect efficiency as the leader never selects a sub-optimal decision. When problems are complex (k is large), the leader’s bias does result in inefficient decisions, but also majority decision-making is then often selects the ‘wrong’ idea. In the next section we show that authoritative decision-making not only saves on communication costs, it also may result in better decisions (on average) than majority decision-making.

6

Authority versus Majority

We are now ready to compare the relative efficiency of majority decision-making and authoritative decision-making. From the above analysis, as long as proposals are relatively easy to evaluate (k is small), both authoritative and majority decision-making always select the best available idea. As problems become more complex, however, both decision-processes sacrifice some decision quality in order to save on the cost of information acquisition (costly discussions). Even when a high-quality alternative is available, a mediocre idea is then selected 15

with positive probability. Formally, relative to first best, majority decision-making results in a efficiency loss kd∗2 2 where 2α(1 − α) is the probability that the ideas of L and R vary in quality and (1 − d∗ ) is the Lm = α(1 − α)(1 − d∗ )v +

probability that a discussion is non-conclusive. Similarly, decision-making by majority yields an efficiency loss given by Lau = α(1 − α)(1 − d∗ )(1 − a∗ )v + (1 − α)p∗

kd∗2 2

(8)

where a∗ is the probability that a leader chooses his own mediocre project following a nonconclusive discussion. Consider first the case where k is small, that is k < min {km , kau }. Under majority

decision-making, a discussion then always reveals the best available idea (d∗ = 1). Under authoritative decision-making, discussions are often non-conclusive (d∗ < 1), but a leader only implements his own mediocre idea if a discussion reveals that his advisor’s idea is mediocre as well (a∗ < 1). It follows that for k small both decision-processes always select the best available idea. Authoritative decision-making, however, achieves this first best decision quality at a much lower communication cost. Indeed, whereas the group always engages in a full scale discussion under majority decision-making (d∗ = 1, p∗ = 1), discussions are often avoided under authoritative decision-making because • the advisor refrains from proposing a mediocre idea (p∗ < 1), or • the leader can implement a high-quality idea without any group discussion, and when discussions occur, they are less intense (d∗ < 1). Concretely, for k small, d∗ = 1 under majority decision-making and efficiency losses equal Lm = k/2, whereas under authoritative decision-making they amount to Lau = (1 − α)p∗ (d∗ )2 k/2 Since p∗ < 1 and d∗ < 1, then Lau < Lm and decision-making by authority is strictly preferred. Authoritative decision-making not only saves on discussion costs, however, it may also result in a higher average decision quality than majority decision-making. Indeed, from proposition 1, majority decision-making fails to implement an available high-quality idea with positive probability whenever k > km with km = 2α(1 − α)v 16

In contrast, from proposition 2, the leader always chooses the best available idea under authoritative decision-making as long as k < k au , where we show in Appendix that b b b k au ≡ (1 − ) (3 − )v v v v

(9)

We have that k au < k m if either the probability of having a high-quality idea α is small or incentive distortions, as measured by b, are small. For k ∈ (k m , kau ) , authoritative decision-

making then results in a strictly higher decision quality than majority decision-making. More generally, the following result holds: Lemma 4 (Authority versus majority: decision quality) Whenever k < kau , given by (9), authoritative decision-making is strictly preferred over majority decision-making and results in a weakly higher decision quality. We next show that wether or not authoritative decision-making is preferred over majority decision-making for a given k, crucially depends on the relative incentive conflict b/v. To see this, fix the complexity of a problem at a level k < k m such that the group always selects the best available idea under majority decision-making (d∗ = 1). If the relative incentive conflict as measured by b/v is sufficiently large, however, then k > k au under authoritative decisionmaking and the leader fails to select the best available idea with probability (1 − a∗ )(1 − d)α(1 − α)

(10)

where both a∗ < 1 and d∗ < 1. Authoritative decision-making then results in a lower decision quality than majority decision-making. Whereas from lemma 4 decision-making by authority is then still strictly preferred for k small, for b/v sufficiently large, there exists a treshold value kˆ ∈ (k au , k m ) which solves Lau = k/2,

(11)

where Lau is given by 8, such that majority decision-making is preferred if and only if k > ˆ For k > kˆ the savings in communication costs under authoritative decision-making are k. then outweighed by a better decision quality under majority decision-making. The following proposition characterizes the optimal decision process as a function of the relative incentive conflict (b/v) and the the relative complexity of the problem at hand (k/v).

17

Figure 3: Optimal decision process as a function of the relative complexity (k/v) and the relative incentive conflict (b/v), and this for α = 0.5, α = 0.6 and α = 0.25. Proposition 3 (Authority versus majority) There exists a cut-off value β(α) > max {α, 6/7} such that

(i) Whenever b/v ≤ β(α) decision-making by authority is preferred for any k.

(ii) Whenever b/v ≥ β(α), there exists a cut-off κ ∈ (kau /v, k m /v), solving (11), such that ∂κ κ. Furthermore ∂(b/v) Figure 3 illustrates proposition 3 for three value of α : α = 0.5, α = 0.6; α = 0.25 Proposition 3 is the central result of this paper. If the incentive conflicts as measured by b/v are only moderate, authoritative decision-making is always preferred for reasons highlighted in sections 4 and 5: • A leader never accepts proposals which a discussion has revealed to be mediocre. In contrast, such ideas are accepted with probability (1 − α)/2 under majority decision-

making. Authoritative decision-making, therefore, discourages agents from proposing mediocre ideas but not high-quality ones. This saves communication costs and increases the average quality of proposed and selected ideas. • For more complex problems (k large) discussions are often non-conclusive (d is small). 18

Since the average quality of proposals is low under majority decision-making, mediocre ideas are then often selected even though a high-quality one is available. If incentive conflicts (as measured by b/v) are sufficiently large and problems are sufficiently complex (k sufficiently large), however, decision-making by majority may be preferred. The reason is the leader under authoritative decision-making then becomes dismissive of alternative proposals. In particular, decision-making is then characterized by a destructive combination of • a leader who allows for only limited discussion of proposals (d is small) • a leader who tends to stick to his own mediocre idea whenever a discussion is nonconclusive (a is small).

To conclude, we discuss how the likelihood of high-quality ideas affects the optimal decision process. Figure 3 suggests that majority decision is optimal for the widest parameter range when the variance in the quality of ideas, as given by α(1 − α)v, is close to its maximum.

In Appendix, we show that the cut-off κ is indeed increasing in α for α < 0.50, whereas κ

is decreasing in α for α > 0.51, the exact cut-off depending on the value of b/v. Intuitively, the efficiency loss of not selecting the best available idea is proportional to the variance in the quality of ideas. Since for b/v large, the optimal decision-process involves a trade-off between better decision-making (under majority decision-making) and lower communication costs (under authoritative decision-making), it is then not surprising that κ is minimized when α(1 − α)v is close to its maximum.

The following proposition shows that b < v and for any k, authoritative decision-making

is preferred whenever the probability α of having a high-quality idea is sufficiently small: Proposition 4 (Authority versus majority when high-quality ideas are scarce.) . Given b < v, we can always find an α sufficiently small such that decision-making by authority is preferred for any k : ∂β(α)/∂α < 0 for α < 1/2 and limα→0 β(α) = 1. Intuitively, the value of constraining agents from proposing mediocre ideas is largest when high-quality ideas are scarce. Indeed, under majority decision-making, high-quality ideas then very often go undiscovered as the group is not willing to spend much time discussing ideas which are most likely to be of little value (d is very small). Instead, the group simply picks an idea at random after a short discussion. Under authoritative decision-making, in contrast, the 19

average quality of a proposal is bounded from below by b/v. Few ideas are then put forward, and when they are put forward, they are put to much more scrutiny (d is larger) than under majority decision-making. The smaller α the large the gap between the quality of proposals under majority decision-making and a dictatorship.

7

The Right to Voice: Decision versus Discussion Authority. (preliminary and incomplete)

An important disadvantage of decision-making by authority is that a leader may be too dismissive: Alternative proposals do not receive sufficient attention or scrutiny. As a result, they are often dismissed for not being convincing enough. A potential institutional response to a dismissive leader is to separate the control over the final decision from the right to decide on how much to debate to allow for. Bureaucrats or elected officials, for example, often need to organize "hearings" before they can make a decision. Similarly, in Congress, intricate procedures regulate how much discussion or debate must take place before an "up or down" vote can take place. The purpose of this section is to analyze the potential benefits of having a separate discussion leader, who decides on the amount of discussion to have before the decision-maker makes can make his choice. In particular, we model leadership with voice as a decision process where the leader still has the final say on what project is selected, but the discussion intensity d∗ is decided by the median voter M as under majority decision-making. Alternatively, one can think of M as an impartial "discussion leader". As a key result in this section, we first show that if a separate discussion leader is appointed, authoritative decision-making is preferred over majority decision-making for a much wider range. In particular, authoritative decision-making is then preferred as long as b < v and a leader does not always implements her own idea. We subsequently show that despite the above result, having is separate discussion leader is not always valuable: it is often preferred to let the final decision-maker also be the discussion leader.

7.1

Authority with voice versus majority

We first show that if a separate discussion leader is appointed, authoritative decision-making is preferred over majority decision-making as long as b < v. Note that whenever b > v, the decision-maker always chooses his own idea and majority decision-making is trivially optimal.

20

Two potential equilibria exist under authoritative decision-making with voice. In the first equilibrium, the decision-maker, L, is "credible" and asks for advice if she has a mediocre idea. The discussion leader, M, then refrains from engaging in a discussion whenever L claims to have a high-quality idea. In the second equilibrium, the decision-maker L is not credible: even if she has a mediocre idea, she prefers to avoid a discussion. As a result, the discussion leader M always engages in a discussion. Regardless of which equilibrium prevails, it will be characterized by a probability a∗ > 0 that the decision-maker accepts a proposal following a non-conclusive discussion and a probability p∗ < 1 that an advisor with a mediocre idea proposes this idea. The argument is identical as for the case of authoritative decision-making where the leader selects both d∗ and the final project. Whereas we subsequently will characterize these equilibria, the observation that a∗ > 0 and p∗ < 1 will be sufficient to proof the following result: Proposition 5 (Authority with voice versus majority) Whenever b < v, authoritative decision-making with voice is preferred over majority decision-making. The proof is instructive and is therefore provided in the text. Consider first the case where the discussion leader always engages in a discussion, as the decision-maker cannot be trusted to reveal the quality of his idea. Equilibrium profits are then given by © ª Uvoice = max αv + (1 − α)α [d + (1 − d)a∗ ] v − (α + (1 − α)p) kd2 /2 d

Indeed, with a probability α, L has a high-quality project and always implements this. With a probability (1 − α)α, L has a mediocre project but R has a high-quality project. R0 s project

is then implement if either a discussion is informative, or if the L accepts R0 s project following a non-conclusive discussion. Discussion costs, finally, will be incurred whenever R proposes an idea, which occurs with probability (α + (1 − α)p) . Moreover, since p < 1 and a > 0, we have

that

ª © Uvoice > Umajority = max αv + (1 − α)αdv − kd2 /2 d

Indeed, assume that probability α,

i0 s

i0 s

idea is chosen if a discussion is non-informative, with i ∈ {L, R} . With

project is high-quality and regardless of d, majority decision-making will

select a project of value v. With probability 1 − α, however, i0 s project is mediocre, in which case majority decision-making will select a high-quality project of value v with probability αd.

Discussion costs, finally, are always incurred. In sum, even if a leader cannot credibly reveal his idea, authoritative decision-making is strictly preferred over majority decision-making because 21

• There are less discussions (p < 1), and • If a discussion is non-informative, a high-quality project is selected with a larger prob-

ability: Under majority decision-making, a high-quality project is then selected with probability α, whereas under authoritative decision-making, a high-quality project is selected with probability α + (1 − α)a∗ > α The only reason why majority decision-making may be preferred is because d∗ is ineffi-

ciently low — which is avoided by having an impartial discussion leader — or because b > v and the leader always chooses his own idea. If the leader is credible, authoritative decision-making with voice is even more preferred, as wasteful discussion are then avoided when the leader has a high-quality idea. Indeed, we then have that ¤ª © £ Uvoice = max αv + (1 − α) α [d + (1 − d)a∗ ] v − (α + (1 − α)p) kd2 /2 d © ª > max αv + (1 − α)α [d + (1 − d)a∗ ] v − (α + (1 − α)p) kd2 /2 d © ª > Umajority = max αv + (1 − α)αdv − kd2 /2 d

7.2

Authoritative decision-making with voice: equilibrium

We now characterize in more detail the equilibrium under authoritative decision-making with voice, and show that having is separate discussion leader is not always valuable: it is often preferred to let the final decision-maker also be the discussion leader. As argued previously, two potential equilibria can exist: one in which the discussions take place regardless of the quality of the idea of the decision-maker and one in which the discussion leader only engages in a discussion if the decision-maker’s idea is mediocre. For the latter equilibrium to exist, the leader must be able to credibly reveal the quality of his idea. Since the leader my try to preempt a discussion by claiming to have a high-quality idea, such a claim is credible only if a leader with a mediocre idea prefers a discussion with intensity d∗ chosen by M over no discussion at all. Since one can show that in equilibrium, the leader must be indifferent between accepting a proposal or not following a non-conclusive discussion, a leader will be credible if and only if d∗ μ(p∗ )(v − b) > k(d∗ )2 /2

22

or since μ(p∗ ) = b/v,

k d∗ (12) b 2 where d∗ is the equilibrium level of discussion imposed by M. If condition (12) holds, then 1 − b/v >

M will only impose a discussion when the leader asks for advice. Let a∗ be, as before, the equilibrium probability that the leader accepts a proposal following a non-conclusive discussion, then d∗ is given by

ª © ∗ 2 )μ(p)v − kd /2 d(1 − a max ∗ d

where

(1 − a∗ )μ(v)

is the probability that, conditionally on a discussion being non-informative,

the leader implements his own mediocre idea even though the advisor’s project is high-quality. We will restrict attention to equilibria where the out-of-equilibrium beliefs of M about a∗ are independent of the choice of d.18 Since d∗ = 1 cannot be an equilibrium, as otherwise an advisor would never propose a mediocre idea, d∗ is given by the following first order condition (1 − a∗ )μ(p)v − kd∗ = 0 Note that a∗ = 1 cannot be an equilibrium, as otherwise d = 0. Similarly, a∗ = 0 cannot be an equilibrium as otherwise p = 0 and a∗ = 1. It follows that a∗ ∈ (0, 1) , from which p∗ is given

by μ(p∗ ) = b/v and, hence,

d∗ = (1 − a∗ )b/k

(13)

As before, a∗ must be such that an advisor is indifferent between proposing his idea or not, that is (1 − d)ab − kd2 /2 = 0 Substituting (13) this yields a=

1 (1 − a)2 b 2 k − (1 − a)b

Whenever k − (1 − a)b > 0, the RHS is decreasing in a and the LHS is increasing in a.

Moreover, if k > b, then for a = 0, the RHS is strictly positive. If k < b, then whenever a = 1 − k/b + ε, with ε small, the RHS is larger than 1. It follows that there exists a unique a∗ which satisfies the above inequality, given by a∗ = 18

p 1 + (k/b)2 − k/b

That is if (a∗ , d∗ ) is the equilibrium, then if the group were to choose d∗ 6= d, then it believes that the leader

still will play a = a∗ . In appendix we discuss how, by cleverly chosing out of equilibrium beliefs, M can commit to a first-best choice of d∗ , in which case it is trivially optimal to separate control over the discussion from the control over the final decision.

23

Substituting in (13) yields d∗ = 1 + b/k −

p 1 + (b/k)2

It follows that an equilibrium with a credible leader exists if and only if ´ p 1³ 1 − b/v > 1 + k/b − 1 + (k/b)2 2

or still

´ p 1³ (14) b/v + k/v − (b/v)2 + (k/v)2 2 Once can show that the above inequality is always satisfied whenever b/v < 1/2. For b/v > 1/2, b/v(1 − b/v) >

it will be satisfied whenever k/v is smaller than some cut-off value K, where K is decreasing in b/v and K = 0 in the limit where b/v = 1. If (14) does not hold, then there exists a unique equilibrium where the discussion leader always engages in a discussion. The equilibrium discussion intensity is then given by ª © d(1 − a∗ )(1 − α)μ(p)v − kd2 /2 max ∗ d

As above, one can then show that μ(p) = b/v, d∗ is given by the first order condition (1 − a∗ )(1 − α)b − kd∗ = 0 and a∗ is such that an advisor with a mediocre idea will be indifferent between proposing an idea or not if and only if (1 − d∗ )(1 − α)ab − kd2 /2 = 0, from which a∗ = and

p 1 + (k/(1 − α)b)2 − k/(1 − α)b,

d∗ = 1 + (1 − α)b/k −

7.3

p 1 + ((1 − α)b/k)2

Right to voice: when is a discussion leader valuable?

Since authoritative decision-making with an independent discussion leader always dominates majority decision-making for v < b, whereas this is not always the case without such a discussion leader, appointing an independent discussion leader often adds value. We now show, however, that when k is small, it may be preferred to let the leader L be both the decisionmaker and the discussion leader. For this purpose, it will be sufficient to focus on the case where the decision-maker L can credibly reveal the quality of his idea, that is condition (14) 24

holds. Recall that b/v < 1/2, is a sufficient condition for the decision-maker to be credible. We will denote by d∗ the discussion intensity in the latter case. Let us now compare d∗ with the profit maximizing level of discussion intensity df b . This profit maximizing discussion intensity is given by

ª © df b = arg max d(1 − a(d))b − kd2 /2 d

where a(d) is given by

(1 − d)ab − kd2 /2 = 0 or still a(d) =

1 k d2 2b1−d

It follows that df b is given by either the corner solution 1 k d2 =1 2b1−d or the interior solution (1 − a(d))b − kd +

1 k 2d(1 − d) + d2 =0 2b 1−d

or still (1 − a(d))b − kd + In contrast, recall that d∗ which is given by

1 k d(2 − d) =0 2b 1−d

(1 − a(d))b − kd = 0 Since a(d) is increasing in d, it follows that df b > d∗ : relative to first best, the discussion leader does not allow for enough discussion. Reason is that the discussion leader does not take into account the impact of d on a∗ : a higher discussion intensity increases the probability that decision-maker will accept a proposal following an uninformative discussion. Let us compare now d∗ to the level of discussion under leadership without voice, which we will refer to as dau . We have that for k > kau , dau is given by b dau = (1 − )b − kd = 0 v It follows that dau < d∗ < df b if and only if p b > a∗ = 1 + (k/b)2 − k/b v 25

It follows that for ´ p p 1³ b 1 + 1 + (k/b)2 − k/b 1 + (k/b)2 − k/b < < v 2

(15)

authoritative decision-making with voice is strictly preferred over decision-making without voice, where the second inequality is equivalent to condition (14) and guarantees the credibility of the discussion leader: Proposition 6 (right to voice is valuable) Whenever (15) holds, it is optimal to appoint an independent discussion leader under authoritative decision-making. In contrast if

or still

b p < 1 + (k/b)2 − k/b v

µ ¶2 p b < (b/v)2 + (k/v)2 − 1 v

(16)

the latter result in a suboptimal level of discussion intensity.

If dau > df b , however, there

then d∗ < dau . As long as dau < df b , an independent discussion leader then not desirable as is a trade-off: an independent discussion leader results in too little discussion, whereas no independent discussion leader results in too much discussion. In Appendix, however, we show that no discussion leader is then still preferred. Rewriting Proposition 7 (right to voice is not valuable) Whenever (16) holds, it is optimal to let the decision-maker also be the discussion leader.

8

Concluding remarks

To be added

26

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Gilligan, Thomas and Keith Krehbiel (1989) “Asymmetric Information and Legislative Rules with A Heterogenous Committee.” American Journal of Political Science 33: 459-490. Grossman, Sanford, and Oliver Hart (1986) "The Costs and Benefits of Ownership: A Theory of Vertical and Lateral Integration." Journal of Political Economy 94: 691-719 Li, Hao, Sherwin Rosen, and Wing Suen (2001). “Conflicts and Common Interests in Committees.” American Economic Review 91: 1478-97. Meyer, Margaret, Paul Milgrom and John Roberts (1992) "Organizational prospects, influence costs, and ownership changes." Journal of Economics and Management Strategy 1: 9-35 Murnighan, Keith and Conlon, Donald (1991) "The Dynamics of Intense Work Groups: A Study of British String Quartets." Administrative Science Quarterly 36: 165-87 Krishna, Vijay and John Morgan (2001) “A Model of Expertise.” Quarterly Journal of Economics 116: 747—75. Milgrom, Paul (1981). “Good News and Bad News: Representation Theorems and Applications.” Bell Journal of Economics 12: 380-91. Milgrom Paul (1988) "Employment contracts, influence activities, and efficient organizational design." Journal of Political Economy 96: 42-60 Milgrom, Paul and John Roberts (1986) "Relying on Information of Interested Parties.” Rand Journal of Economics 17: 18-32. Milgrom Paul and John Roberts (1988). "An Economic Approach to Influence Activities in Organizations." American Journal of Sociology 94 (Supplement), S154-S179. Milgrom Paul and John Roberts (1990). "The Efficiency of Equity in Organizational Decision Processes." American Economic Review 80: 154-159. Milgrom, Paul and John Roberts (1992) Economics, Organizations, and Management. Prentice Hall. Ottaviani, Marco and Peter Sorensen (2001) “Information Aggregation in Debate: Who Should Speak First?” Journal of Public Economics 81, 393-421. Rotemberg, Julio, and Garth Saloner (2000). "Visionaries, Managers, and Strategic Direction." Rand Journal of Economics 31, 693-716.

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Segal, Ilya (2001) "Communication Complexity and Coordination by Authority." Mimeo, Stanford. Van den Steen, Eric (2005) "Organizational Beliefs and Managerial Vision." Journal of Law, Economics, and Organization 21, 256-283. Williamson, Oliver (1975) Markets and Hierarchies. New York: Free Press

APPENDIX A Proof of Lemma 2. Assume first that d∗ = (1 − μ(p))μ(p)v/k, then Vp ≡

1 1 (1 − α)b − α [k − (1 − μ(p))μ(p)v] (v − b) 2 2k 1 − [α + (1 − α)p] (1 − μ(p))2 μ(p)2 v 2 2k

It follows that a equilibrium with p < 1 exists only if Vp ≤ 0, which is equivalent to k [b − αv] + (1 − μ(p))μ(p)v [α (v − b) − (1 − μ(p))αv] ≤ 0 Moreover, since d∗ is an interior solution, it must be that (1 − μ(p))μ(p)v ≤ k Hence a necessary condition for p < 1 is that k [b − αv] + k [α (v − b) − (1 − μ(p))αv] ≤ 0 implying k [b − αv] + k [α (v − b) − (1 − α)αv] ≤ 0 or still k [b − αv] (1 − α) ≤ 0 which is impossible given A1. It follows that no equilibrium exists where p < 1 and d < 1. Consider now a corner equilibrium where d = 1, then Vp ≤ 0 only if k 1 Vp ≡ (1 − α)b − [α + (1 − α)p] ≤ 0 2 2 or still (1 − α) which implies that (1 − α)

b ≤k [α + (1 − α)p]

α v = (1 − α)μ(p)v ≤ k [α + (1 − α)p] 29

which, in turn, implies (1 − μ(p)μ(p)v ≤ k but then d∗ < 1, a contradiction. It follows that no equilibrium exists where p < 1. Proof of proposition 2. The value of proposing a mediocre idea to and advisor is given by Vp ≡ [1 − d∗ (p)] ab − kd2 /2

(17)

Vp ≡ [1 − (1 − μ(p))b/k] ab − (1 − μ(p))2 b2 /2k.

(18)

or, substituting d∗ (p):

As argued in the text. no equilibrium exists where p = 1 or p = 0, hence in equilibrium the advisor must be indifferent between proposing or not. It follows that p∗ is given by Vp ≡ [1 − (1 − μ(p))b/k] ab − (1 − μ(p))2 b2 /2k = 0

(19)

We subsequently investigate the existence of equilibria where a = 1 and where a < 1. 1) Consider first potential equilibria where a = 1, then Vp = 0 implies [1 − (1 − μ(p))b/k] b − (1 − μ(p))2 b2 /2k = 0

(20)

2kb − 2(1 − μ(p))b2 − (1 − μ(p))2 b2 = 0

(21)

2k = (1 − μ(p∗ ))(3 − μ(p∗ ))b

(22)

or still from which p∗ is given by If 2k < (1 − μ(p))(3 − μ(p))b and μ(p)v ≥ b, there exists thus an equilibrium in which a∗ = 1 and where p∗ is given by 2k = (1 − μ(p))(3 − μ(p))b (23) Since we must have μ(p) ≤ b/v in equilibrium, an equilibrium with a∗ = 1 exists if and only if k < kau where b b k au ≡ (1 − )(3 − )b/2 v v ∗ From (23), if follows that p is increasing in k and decreasing in b. Moreover limk→0 p = 0. 2) Consider now candidate equilibria where a < 1. If a = 0, then whenever d > 0, it must be that p = 0, which cannot be an equilibrium. Hence, whenever a < 1 in equilibrium, the leader is indifferent between accepting or rejecting a proposal following an uninformative discussion. It follows that p∗ is given by μ(p∗ )v = b and a∗ given by

¸ ∙ b b 1 − (1 − )b/k α∗ b − (1 − )2 b2 /2k = 0 v v 30

or still

or still

∙ ¸ b b 2 k − (1 − )b a∗ − (1 − )2 b = 0 v v

(24)

b (1 − )2 b v ¸ a∗ = ∙ b 2 k − (1 − )b v

(25)

Thus, an equilibrium with a∗ < 1 exists if and only if ¸ ∙ b b (1 − )2 b < 2 k − (1 − )b v v or still b b 2k > (1 − )2 b + (1 − )b/2 v v b b = (1 − )(3 − )b/2 v v = kau From (25), if follows that 1 − a∗ is increasing in k and decreasing in b. Moreover limk→0 p = 0. Proof of Lemma 4. Preliminaries: We first calculate the expected efficiency losses under each decision process. (1) Under majority decision-making, the expected loss relative to first best decision-making is given by kd∗2 Lm = (1 − α)α(1 − d∗ )v + 2 ∗ (i) Whenever d = 1, then Lm = k/2. (ii) Whenever d∗ < 1, d∗ is given by (1 − α)αv/k and Lm can be rewritten as Lm = α(1 − α)(1 − or still

α2 (1 − α)2 v 2 α(1 − α)v )v + k 2k

¸ ∙ α(1 − α)v Lm = α(1 − α)v 1 − 2k

(26)

(2) Under decision-making by authority, the expected efficiency losses relative to first best is given by kd2 (27) Lau = α(1 − α)(1 − d∗ )(1 − a∗ )v + (1 − α)p 2

31

(i) Whenever k < kau , then a = 1 and Lau = (1 − α)p

(1 − μ(p))2 b2 2k

From (2) (1 − α)p =

(1 − μ(p))α μ(p)

and, hence Lau =

αb2 (1 − μ(p)) 3 2k μ(p)

(28)

Moreover, p is given by 2k = (1 − μ(p))(3 − μ(p))b from which p μ(p) = 2 − 1 + 2k/b p 1 − μ(p) = 1 + 2k/b − 1

Substituting in (28), we have that

Lau Alternatively, we can write Lau as Lau or still

p αb2 ( 1 + 2k/b − 1)3 p = 2k 2 − 1 + 2k/b

p ( 1 + 2k/b − 1)2 (1 − μ(p)) 2 αb p p = = αb (3 − μ(p)) μ(p) (1 + 1 + 2k/b)(2 − 1 + 2k/b) Lau

p 1 − 2 1 + 2k/b + 1 + 2k/b p = αb 2 + 1 + 2k/b − (1 + 2k/b)

(29)

where one can verify limk→0 Lau = 0 and also limk→0 ∂Lau /∂k = 0. (ii) Whenever k > kau , then a∗ < 1 is given by (24), p∗ is given by μ(p)v = b, and d∗ is given by b d∗ = (1 − )b/k v Substituting in (27), we have that ∙ ¸ b b ∗ b Lau = α(1 − α)(1 − (1 − )b/k)v − α(1 − α) 1 − (1 − ) a v v v k b +(1 − α)p(1 − )2 b2 /2k v b b b = α(1 − α)(1 − (1 − )b/k)v − α(1 − α)(1 − )2 bv/2k + (1 − α)p(1 − )2 b2 /2k v v v 32

Since p is given by

we can rewrite this as

α b α(1 − b/v)v = ⇔ (1 − α)p = α + (1 − α)p v b

b b Lau = α(1 − α)(1 − (1 − )b/k)v − α(1 − α)(1 − )2 bv/2k v v b b +α(1 − )(1 − )2 bv/2k v v b b b = α(1 − α)v − 2α(1 − α)(1 − )bv/2k − ( − α)α(1 − )2 bv/2k v v v or still Lau

∙ ¸ b b 1 2(1 − α) + (1 − )( − α) α(v − b)b = α(1 − α)v − 2k v v

(30)

Proof of Lemma 4: (1) Assume first that k < min {kau , k m } , then Lm =

k k > Lau = (1 − α)pd2 2 2

with p < 1 and d < 1, hence decision-making by authority is always strictly preferred over decision-making by majority (2) Assume next that km < kau and k ∈ (k m , k au ) . Then efficiency losses are respectively given by (26) under majority decision-making and by (29) under decision-making by authority. We have that Lau < Lm if and only if p ∙ ¸ 1 − 2 1 + 2k/b + 1 + 2k/b v α(1 − α)v p − (1 − α) 1 − 0. Indeed both terms of R are strictly convex in k. (iii)Third, for k = km , R < 0. Indeed, substituting k, then p 1 + 2k/b + 1 + 2k/b 1 − 2 k ˜≡ p − R=R 2αb 2 + 1 + 2k/b − (1 + 2k/b) 33

˜ < 0. where k/b = (1 − α)αv/b < 1 − α. For k/b < 1 − α, one can verify that R From observation (i), (ii) and (iii), it follows that if R < 0 for k = kau and α = b/v, then R < 0 for any (α, k) satisfying 0 < α < b/v, and km < k < k au . We now show that R < 0 for k = kau and α = b/v.Substituting α = b/v in R yields p ∙ ¸ 1 − 2 1 + 2k/(αv) + 1 + 2k/(αv) α(1 − α)v (1 − α) p 1− − α 2k 2 + 1 + 2k/(αv) − (1 + 2k/(αv)) Substituting k = kau yields p ∙ ¸ 1 + (1 − α)(3 − α) + 1 + (1 − α)(3 − α) 1 − 2 1 (1 − α) ¯= p 1− − R=R α (3 − α) 2 + 1 + (1 − α)(3 − α) − (1 + (1 − α)(3 − α))

One can verify that the above expression is always negative for α < 1. It follows that R = Lau − Lm < 0 for any k satisfying α < b/v < 1 and decision-making by authority is strictly preferred. Proof of Proposition 3. From lemma 4, we know that decision-making by authority is always preferred whenever k < kau . Assume therefore now that k > kau . We distinguish tow cases: (1) Consider first the case where k > max {k m , kau } , such that efficiency losses under majority decision-making are given by (26) and efficiency losses in a dictatorship are given by (30). Authority will be preferred over majority whenever ¸ ∙ b b b b 2 2 Lau < Lm ⇔ α (1 − α) < α 2(1 − α) + (1 − )( − α) (1 − ) v v v v or still

"

# b b ( vb − α) b (1 − ) α(1 − α) < 2 + (1 − ) v 1−α v v

Whenever

(31)

b b b 1 km < kau ⇔ α(1 − α)v < (3 − ) (1 − )v, 2 v v v the above condition is always satisfied. Hence, km < kau is a sufficient (but not necessary) condition for authority to be preferred, regardless of communication costs k. Note further that for b = αv, the above equation is always satisfied and for b = v it is always violated. It follows that given α, there exists a unique cut-off value β(α) such that authority is preferred for any k if and only if b/v < β(α) where β(α) is uniquely defined by the the following two conditions:19 ¸ ∙ β−α (1 − β)β α(1 − α) = 2 + (1 − β) 1−α and α 6/7 and arg minα∈(0,1) β(α) ∼ = (0.505) . au m au m (2) Consider finally the case where k < k and k ∈ (k , k ) , such that efficiency losses under majority decision-making equal k/2 and efficiency losses in a dictatorship are given by (30). Majority decision-making is then preferred if and only if ∙ ¸ b b k 1 2(1 − α) + (1 − )( − α) α(v − b)b > α(1 − α)v − 2k v v 2 or still

µ k 2−

k α(1 − α)v

¶

∙ > 2+

¸ 1 b b (1 − )( − α) (1 − b/v)b 1−α v v

(32)

Note that the LHS is increasing in k. Hence, a necessary condition for majority decision-making to be optimal for k ∈ (k au , k m ) is that it is optimal for k = k m . Substituting k = k m ≡ α(1−α)v into (32) yields (31). It follows that authority will be strictly preferred over majority regardless of k whenever (31) holds. If (31) is violated, then there exists a k˜ ∈ (kau , km ) , such that majority decision-making ˜ Indeed, from lemma is strictly preferred over decision-making by authority if and only if k > k. au 4 decision-making by authority is always preferred for k ≤ k , violation of (31) implies that decision-making by majority is preferred for k ≥ km and the LHS of (32) is strictly increasing in k. From (32), k˜ is given by " # µ ¶2 (1 − b/v) vb k k b vb − α 2 − − 2 + (1 − ) =0 α(1 − α)v α(1 − α)v (1 − α)α v 1−α from which where

´ ³ p k˜ = α(1 − α)v 1 − 1 − c(b/v, α)

# " (1 − b/v) vb (1 − vb ) b c(b/v, α) = ( − α) 2+ (1 − α)α 1−α v

There exists an Dα :

µ

(1 − x)x (1 − α)α

µ ¶¶ (1 − x) 2+ (x − α) 1−α

³ ´ p α(1 − α) 1 − 1 − c(b/v, α)

One can verify that whenever c < 1, which requires b/v > 0.861, c is increasing in b/v.

35

Proof of proposition 4. See proof of Proposition 3 Footnote 18: Out-of-equilibrium beliefs under authoritative decision-making with a separate discussion leader: Under authoritative decision-making with a separate discussion leader, one could specify the discussion leader’s out-of-equilibrium beliefs about a to be a function of d, that is a = a(d). In this case d∗ is given by ª © 2 max /2 d(1 − a(d))b − kd ∗ d

It is easy to see that any pair

(a0 , d0 )

then can be supported as an equilibrium as long as

(1 − d)ab − kd2 /2 = 0 and

ª © d(1 − a)b − kd2 /2 > max db − kd2 /2 d

a0

For example, one can specify beliefs a(d) = whenever d = d0 and a(d) = 0 otherwise. More generally, if d0 < d∗ , it will be sufficient to specify a(d) = a0 whenever d ≤ d0 and a(d) = 0 otherwise, or if d0 > d∗ , it will be sufficient to specify a(d) = a0 whenever d ≥ d0 and a(d) = 0. otherwise. Proof of proposition 7. Efficiency losses under voice are given by p p Lvoice = α(1 − α)( 1 + (b/k)2 − b/k)( 1 + (k/b)2 − k/b)v ´2 p α(1 − b/v)v k ³ 1 + b/k − 1 + (b/k)2 + b 2

(33) (34)

or still

p p Lvoice = α(1 − α)( 1 + (b/k)2 − b/k)( 1 + (k/b)2 − k/b)v ´2 p 1³ +α(1 − b/v)v 1 + k/b − 1 + (k/b)2 2

For k < kau , efficiency losses without voice are given by p ( 1 + 2k/b − 1)2 p p Lau = αb (1 + 1 + 2k/b)(2 − 1 + 2k/b)

For k < kau , no voice is then preferred whenever

p p 1 (Lvoice − Lau ) = (1 − α)( 1 + (b/k)2 − b/k)( 1 + (k/b)2 − k/b) αv ´2 p b 1³ +(1 − ) 1 + k/b − 1 + (k/b)2 v 2 p ( 1 + 2k/b − 1)2 b p p − v (1 + 1 + 2k/b)(2 − 1 + 2k/b) > 0 36

(35) (36)

A sufficient condition for the above condition to be positive is that, it is positive for α = b/v, or still ´2 p p p 1³ ( 1 + (b/k)2 − b/k)( 1 + (k/b)2 − k/b) + 1 + k/b − 1 + (k/b)2 2 p ( 1 + 2k/b − 1)2 b/v p p − 1 − b/v (1 + 1 + 2k/b)(2 − 1 + 2k/b) > 0 which can be rewritten as ´2 p p p 1³ ( 1 + (b/k)2 − b/k)( 1 + (k/b)2 − k/b) + 1 + k/b − 1 + (k/b)2 2 p p b ( ( 1 + 2k/b − 1) 1 + 2k/b − 1) p p − v b 1 − v (2 − 1 + 2k/b) ( 1 + 2k/b + 1) > 0 Since k < k au ⇔ 2 −

p 1 + 2k/b > b/v

a sufficient condition for this condition for Lvoice − Lau > 0 for any k < kau is that ´2 (p1 + 2k/b − 1) p p p 1³ 2 2 2 1 + k/b − 1 + (k/b) ( 1 + (b/k) −b/k)( 1 + (k/b) − k/b) + >0 − p 2 ( 1 + 2k/b + 1) which is always satisfied for any k/b > 0. QED

APPENDIX B: Asymmetric Equilibria under Majority Decision-Making In this Appendix, we show that under majority decision-making, no asymmetric equilibria exist where pL < pH . Lemma does this, maintaining the assumption that the group selects a proposal at random whenever a discussion reveals that both proposals are mediocre. Lemma considers equilibria where the group may favor one particular agent (L or R) in case a discussion reveals that both projects are mediocre. Under an equilibrium refinement where discussions reveal wrong information with an arbitrarily small probability, we show that also then no equilibrium exist where pL < pH . Symmetric Acceptance Equilibria The following lemma maintains the assumption the group selects a proposal at random whenever a discussion reveals that both proposals are mediocre. Lemma 5 (asymmetric proposal equilibria) No equilibrium exists in which L proposes with probability pL and R proposes with probability pR , where pR < pL .

37

Proof: Assume pR < pL , then whenever an investigation is non-informative, R0 s idea will be selected. It follows that the surplus maximizing discussion intensity d∗ is given by ª © d∗ = arg max (1 − μ(pR ))μ(pL )dv − kd2 /2 , d

or still

d∗ = min {1, (1 − μ(pR ))μ(pL )v/k} When is it optimal for L to propose a mediocre idea given an anticipated discussion intensity d and given that R proposes a mediocre idea with probability pR < pL ? If both L and R have a mediocre idea, then a proposal by L raises the probability of adoption of L0 s idea with 12 (1−pR )+ 12 pR d. If R has a high-quality idea, then R0 s idea will always be implemented. Finally, if also R proposes his idea, a proposal by L results in communication costs kd2 /2 for all group members. Thus, the expected value to L of proposing a mediocre idea equals 1 VpL ≡ [(1 − pR ) + pR d] (1 − α)b − [α + (1 − α)pR ] kd2 /2. 2 If d∗ = 1, this yields

1 VpL ≡ (1 − α)b − [α + (1 − α)pR ] k/2. 2 When is it optimal for R to propose a mediocre idea given an anticipated discussion intensity d and given that L proposes a mediocre idea with probability pL > pR ? If both L and R have a mediocre idea, then a ¢proposal by L raises the probability of adoption of L0 s ¡ 1 idea with 2 (1 − pL ) + pL 1 − d + 12 d . If L has a high-quality idea, then the latter will be implemented with probability d. Finally, if also R proposes his idea, a proposal by L results in communication costs kd2 /2 for all group members. Thus, the expected value to L of proposing a mediocre idea equals 1 VpR ≡ [(1 − pL ) + pL (2 − d)] (1 − α)b − α (v − b) (1 − d) − [α + (1 − α)pL ] kd2 /2. 2 If d∗ = 1 this yields

1 VpR ≡ (1 − α)b − [α + (1 − α)pL ] k/2. 2

Consider first the corner equilibrium where d = 1, then VpR ≤ 0 only if 1 k Vp ≡ (1 − α)b − [α + (1 − α)pL ] ≤ 0 2 2 or still (1 − α) which implies that (1 − α)

b ≤k [α + (1 − α)pL ]

α v = (1 − α)μ(pL )v ≤ k [α + (1 − α)pL ] 38

which, in turn, implies (1 − μ(pR )μ(pL )v ≤ k but then d∗ < 1, a contradiction. It follows that no equilibrium exists where pR < 1, and hence no equilibrium exists where pR < pL . Consider next that d∗ < 1. We first show that VpL > 0. Indeed, VpR ≤ 0 only if 1 [(1 − pR ) + pR d] (1 − α)b ≤ [α + (1 − α)pR ] kd2 /2. 2 A necessary condition for this to be true is that 1 1 d (1 − α)b ≤ k d2 [α + (1 − α)pR ] 2 2 or still (1 − α)b ≤ (1 − μ(pR ))μ(pL ) [α + (1 − α)pR ] v or still b ≤ pR μ(pL )v pR αv ≤ [α + (1 − α)pL ] pR ≤ αv [α + (1 − α)pR ] ≤ αv a contradiction. Hence, we must have that pL = 1. We now show that given pL = 1, also VpR > 0 and hence pR = 1. Indeed, if pL = 1, then VpR ≤ 0 only if 1 (2 − d) (1 − α)b − α (v − b) (1 − d) ≤ kd2 /2. 2 or still 1 (1 − d) (b − αv) + d(1 − α)b ≤ kd2 /2. 2 Since (1 − α)b > (1 − μ(pR )αv a necessary condition for VpR ≤ 0 is then (1 − d) (b − αv) + d(1 − μ(pR )αv

1 ≤ kd2 /2. 2

or still (1 − d) (b − αv) ≤ 0 which is never satisfied. It follows that pL = 1 implies that also pR = 1, an asymmetric equilibria exist. QED

39