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: wouter.dessein@chicagogsb.edu
"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.
1
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.
3
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.
4
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.
5
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.
6
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.
7
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.
8
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.
9
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.
10
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
k̂ ∈ (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 >
k̂. For k > k̂ the savings in communication costs under authoritative decision-making are
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
<0
decision-making by majority is preferred if and only if k/v > 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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28
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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
R≡
b
2k
2 + 1 + 2k/b − (1 + 2k/b)
We need to show that R is negative for any vector (α, b, v, k) satisfying α < b/v < 1 and k
such that
b b
b
1
km = α(1 − α)v < k < k au = (3 − ) (1 − )v
2
v v
v
(i) Note first that R is increasing in α. Indeed
µ∙
¸
¶
∂R
v
α(1 − α)v
v
=
1−
+ (1 − 2α)(1 − α)
∂α
b
2k
2k
which positive if and only if
(3α − 1)
(1 − α)v < 2
2
which is always verified since (3α − 1)/2 < α as long as α < 1.
(ii) Second, one can verify that ∂ 2 R/∂k2 > 0. Indeed both terms of R are strictly convex in k.
(iii)Third, for k = km , R < 0. Indeed, substituting k, then
p
1 − 2 1 + 2k/b + 1 + 2k/b
k
p
−
R = R̃ ≡
2αb
2 + 1 + 2k/b − (1 + 2k/b)
33
where k/b = (1 − α)αv/b < 1 − α. For k/b < 1 − α, one can verify that R̃ < 0.
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 − 2 1 + (1 − α)(3 − α) + 1 + (1 − α)(3 − α)
1
(1 − α)
p
1−
−
R = R̄ =
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<β<1
19
One can verify that there exists a unique solution for this for any α ∈ (0, 1)
34
Implicitly deriving the first expression, we find that
dβ/dα < 0 ⇔
(1 − β)3
+ (1 − 2α) > 0
(1 − α)2
It follows that dβ/dα < 0 for α < 1/2. Since β(0) = 1 this proves proposition 4. One can
numerically verify that minα∈(0,1) β(α) ∼
= 0.86186 > 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 k̃ ∈ (kau , km ) , such that majority decision-making
is strictly preferred over decision-making by authority if and only if k > k̃. Indeed, from lemma
4 decision-making by authority is always preferred for k ≤ kau , 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), k̃ 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
k̃ = 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
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