2008-35
Leadership by Confidence in Teams
Hajime Kobayashi Hideo Suehiro

Leadership by Confidence in Teams∗
Hajime Kobayashi

Hideo Suehiro

School of Economics,
Osaka Prefecture University

Graduate School ofBusiness Administration,
Kobe University

July 7, 2008

Abstract
We study endogenous signaling by analyzing a team production problem
with endogenous timing. Each agent of the team is privately endowed withsome level of confidence about team productivity. Each of them must thencommit a level of effort in one of two periods. At the end of each period, eachagent observes his partner’s move in this period. Both agents are rewarded
by a team output determined by team productivity and total invested effort.
Each agent must personally incur the cost of effort that he invested. We
show a set of sufficient conditions under which an agent chooses to become aleader or a follower depending on his confidence about reward from the teamin a stable equilibrium. This means that a player endogenously becomes asignal sender or a signal receiver depending only on the cost-benefits frombecoming a sender or a receiver.

JEL Classification: C72, D82
Keywords: Endogenous signaling, Endogenous timing, Signaling, Valueof information, Leadership, Team production∗

We thank Shinsuke Kambe for his helpful comments.

1

Introduction

1.1

Overview

The study of endogenous timing games has advanced our understanding ofmany economic phenomena through explaining when an agent takes an action. It has successfully covered many fields such as oligopoly, bargaining,
and investment decisions. Yet, one field has remained unexplored to date.
Leadership in teams is a typical phenomenon in which the choice of action
timing becomes a central issue. It has been widely recognized that leadership is a key to understand teams. Nevertheless, no successful explanation
has been made about when leadership emerges and who takes leadership inteams.
This paper investigates leadership in teams by analyzing the model ofteam production with endogenous timing. Leadership emerges in a team if
one member moves voluntarily prior to others and causes favorable changes
in the effort choices of the other members. We show in this paper that
under a set of sufficient conditions, leadership emerges as an equilibrium of
the team production with endogenous timing.
The failure to address leadership in teams in economics is mainly due tothe lack of an appropriate model of the leadership mechanism. The focus of
the studies of endogenous timing games is on mechanisms by which players
are sorted to move at different points of time. By the types of mechanisms,
the existing studies of endogenous timing games are broadly classified into
two groups. The first group mainly adopts private information and time
preference as driving forces. This type of models explains sorting by the
degree to which players face trade-off between the cost of waiting by itself
and the benefit of learning the rival’s private information by waiting (orsignaling the player’s own private information by waiting). These models
have been applied to areas such as investment decisions (Hendricks andKovenock (1989), Chamley and Gale (1994), and Gul and Lundholm (1995)),
bargaining (Kambe (1999) and Abreu and Gul (2000)), and war of attrition(Fudenberg and Tirole (1986)).
The second group explains the sorting of the timings of moves by the advantage or disadvantage of commitment that is determined by strategic relations such as strategic complementarity or substitution. This type of models
has been mainly applied to oligopolistic situations such as Cournot competition and Bertrand competition (Gal-Or (1987), Hamilton and Slutsky(1990), Mailath (1993), Normann (1997), van-Damme and Hurkens (1999),
Normann (2002), van Damme and Hurkens (2004), Amir and Stepanova(2006), and so on).
Emergence of leadership in teams cannot be fully explained by those
types of combinations of the forces. Either time preference or strategic complementarity or substitution is not necessarily present in team production.

2

Rather, team output and contribution cost do not usually depend on timingof investing efforts. Then, the team production problem is free from time discounting. Furthermore, as suggested by Holmstrom (1982), although there
are possibly many types of team production functions, the natural specification of the function is the product of total efforts and the productivity
parameter. This function is the prototype of the team production function, as has appeared in many applications (e.g., Hermalin (1998)). Observethat the function implies neither strategic complementarity nor substitution. Therefore, the above two existing groups are not sufficient to explain
the emergence of leadership in teams.
We adopt a new combination of driving forces in order to explain emergence of leadership. The driving forces are multi-sided private informationand simple payoff externality. Multi-sided private information means thatevery player has private information about team productivity. Simple payoff
externality is that a player’s payoff depends on the action choices of otherplayers, but a player’s optimal action does not depend on the action choices
by other players. The production function that is the product of total efforts
and the productivity parameter implies simple payoff externality.
The combination of multi-sided private information and simple payoff
externality makes leadership emerge in teams as follows. Every agent holds
his personal expectation about team productivity based on his private information. The expectation pushes an agent to voluntarily move before others
do when he is more confident and it induces him to wait for action at alater timing when he is less confident. When an agent becomes a leader
or a follower according to his private information, signaling from the leader
to the follower is realized. Then, the sender enjoys a benefit due to thesimple payoff externality from increased effort investment by the receiver.
The receiver benefits from learning the sender’s private information. The
expectation of this signaling is the cause that sorts some agents into leadersand others into followers. We say that endogenous signaling is realized when(i) an agent chooses to be a sender or a receiver solely by comparing thebenefit of sending a signal with the benefit of learning a signal; and (ii) a
sender-receiver relation is endogenously realized by these choices.
We formalize the situation above into a simple team production game.
There are two members and each agent is privately endowed with some levelof confidence about team productivity. Each member must then commit alevel of effort in one of two periods. At the end of each period, each agent
observes his partner’s move in this period. Both agents are rewarded bya team output determined by team productivity and total invested effort.
Each agent must personally incur the cost of the effort that he invested.
In this model, leadership by confidence prevails when (i) an agent, whoholds an optimistic belief about the reward from his effort investment for
the team, moves voluntarily and commits a certain level of effort in thefirst period; and (ii) the other agent, seeing the first mover’s action, is3induced to invest levels of efforts in the second period higher than he would
invest if the first move action did not take place. We show a sufficientcondition under which leadership by confidence is supported as a stableoutcome of sequential equilibria in the team production game. The sufficientcondition consists of four assumptions on the environment. First, confidence
is independent across agents. Second, agents are symmetric. Third, theincrease in the conditional expectations of team productivity when an agent
learns his partner is more confident is smaller when the agent himself ismore confident than when he is less confident. Fourth, effort cost is subject
to strong convexity.
The intuition is as follows. In order for leadership by confidence toemerge, two properties must hold: (i) when an agent is more confident
about team productivity, he prefers moving first and signaling his confidence
to moving second and learning his partner’s confidence; and (ii) when an
agent is less confident about team productivity, he is unwilling to mimic the
behavior he would take if he were more confident.
The essential part of (i) is the trade-off between signaling and learning.
If an agent chooses to move second, he can make a better decision about his
effort by learning his partner’s confidence about team productivity. If hemoves first, he must give up the value of this information; however, he canenjoy the increase in his partner’s effort. When agents are symmetric (thesecond assumption above), the size of one’s own effort adjustment in moving
second and the size of an increase in his partner’s effort become comparable.
When an agent is more confident about team productivity, the impact onutility is smaller for the former factor by the third assumption. Therefore,
he prefers signaling to learning.
Property (ii) is a single crossing property about a choice between moving
first and moving second. When an agent is more confident about teamproductivity, he expects a larger impact on team output from an increasein his partner’s effort by the first assumption. He also expects a smaller
loss of value of information by the third and fourth assumptions, because
his expectation about team productivity is less influenced by his partner’s
information, and he foresees that effort adjustment in moving second isrelatively negligible. Therefore, an agent is more willing to move first when
he is more confident about team productivity.

1.2

The Literature

We model endogenous signaling under multi-sided private information as thedriving force for leadership in teams. We relate our model to the literatureof endogenous timing games in the light of signaling.
There are broadly two groups of studies of endogenous timing games, asmentioned above. The studies in the first group are exactly the ones that
paid attention to private information and signaling in endogenous timing
4

games. However, once we remove time discounting from the models, every
player prefers to be a receiver and never desires to be a sender. The mere
private information is not enough to explain endogenous timings of moves
in teams.
On the other hand, the studies in the second group simply extend staticgames such as Cournot competition into dynamic complete information
games by allowing players to choose the timings of moves. Signaling has
no role to play in explaining endogenous timings of moves.
Some papers in the second group studied the endogenous timing mechanism of strategic substitution in the presence of private information. Mailath(1993) is the first attempt. He studied a Cournout market in which only one
firm has private information and has a choice of timing for move in one of two
periods while the other firms have no private information and must move inthe second period. He showed that the informed firm endogenously becomesa Stackelberg leader in equilibrium. Normann (1997) and Normann (2002)
extended Mailath (1993) to a model in which all the firms have choices oftiming for move while keeping private information restricted to one firm.
Signaling takes place in equilibrium in these models. However, since only
one firm is privately informed, the potential leader is determined a priori
and strategic substitution continues to play a crucial role for endogenous
timings of moves.
There is one previous study that explores endogenous choice of timing
solely by signaling and without assuming time discounting or strategic relations such as strategic substitution or complementarity. Kobayashi and
Suehiro (2005) studied the team production problem in which every player
has their own private information and every player can choose timings ofmoves at their own discretion.1 This team production problem is precisely
the one that this paper investigates. They found three possible types ofleadership in teams: leadership by confidence, leadership by identity, andleadership by identity with confidence. Leadership by confidence is the outcome that we defined above. Leadership by identity means that a particular
player moves first irrespective of his private information. Leadership byidentity with confidence is a hybrid of the first two. They showed that if
private information is independent across players: (i) leadership by identity
always exists as a stable equilibrium; and (ii) no stable equilibria exist otherthan the ones by the three types of leadership.
An equilibrium with a particular leader in leadership by identity may
emerge only depending on an outside context about the team beyond thedescription of the game. In contrast, leadership by confidence is independent
1
This is an extension of Hermalin (1998) to the case of endogenous choice of timing.
Hermalin (1998) studied a signaling game with specific rules: (i) one member exclusively
holds information about team productivity; (ii) the member must commit a level of effort
in the first period; and (iii) the other members choose their levels of efforts in the second
period after observing the choice by the leader.

5

of an outside context. Therefore, unless there is a specific outside mechanism
at work to induce a particular member to become a leader, leadership byconfidence should naturally realize in the environment in which every playerchooses his timings of moves. Hence, leadership by confidence is the mostnatural solution of the team game. Precisely the issue of under what condition leadership by confidence emerges as a stable outcome has remained
unsolved. This paper explains the natural mode of leadership in teams as anequilibrium of endogenous timing games by identifying a sufficient condition
for the emergence of leadership by confidence.
The rest of this paper is organized as follows. We describe our team production game in Section 2. In Section 3, we present the sufficient condition
for leadership by confidence and prove our main result. Section 4 explains
the role of each condition in realizing leadership by confidence. We conclude
in Section 5.

2

The Model

We study the team production game of Kobayashi and Suehiro (2005). A
team with two agents i = 1, 2 is engaged in production. Each agent iinvests an effort ei ∈ RP
+ with cost ci (ei ). The output level x of the teamis determined by x = θ i ei , where θ is a productivity parameter. Agent i
receives a benefit vi (x) from the output x. When agent i invests ei and theteam output is x, his payoff is ui = vi (x) − ci (ei ).
We assume agent i is risk neutral in x, that is, vi (x) = si x for somesi > 0. We assume the cost function ci satisfies the following conditions.
A1 ci is differentiable and c0i (ei ) > 0 for any ei > 0 with limei →0 c0i (ei ) = 0
and limei →∞ c0i (ei ) = ∞.
A2 ci is twice-differentiable and c00i (ei ) > 0 for any ei > 0.
The cost is strictly increasing in ei and is subject to increasing marginalcost.
The parameter θ is a realization of some random variable θ. Each agent
i receives private information about θ before production. According to theinformation, he is in one of the two states of confidence about team productivity. He may be more confident or less confident. Agent i is called H-type
in the former case and L-type in the latter. Let p(ti , tj ) denote a probability
for an event that agent i is ti -type and agent j is tj -type. Let E[θ|ti , tj ]denote the conditional expectation of θ given the event. The expectation
by agent i of type ti is expressed by:p(ti , H)p(ti , L)E[θ|ti ] = P
E[θ|ti , H] + P
E[θ|ti , L].
p(t
,
t
)
i j
tj
tj p(ti , tj )6The interpretation that H-type is more confident than L-type is modeled
without loss of generality as:
E[θ|L, L] < E[θ|H, L], E[θ|L, H] < E[θ|H, H].
This implies:
E[θ|ti = H] > E[θ|ti = L].
Agents must choose levels of their efforts according to the following time
sequence. There are two periods, 1 and 2. In period 1, each agent i may
exert an effort level ei or may choose to do nothing (denoted as ∅). If he
implements his effort in period 1, then he cannot do anything in period 2.
On the other hand, if he chooses to do nothing in period 1, then he must
implement his effort in period 2.
In this sequence of moves, the two agents must move (taking some ei or∅) independently and simultaneously in period 1. Each agent i immediately
observes the behavior that the other agent j has taken. Agent i can thenutilize this information for his choice in period 2 if he has chosen to do
nothing in period 1. If both agents have chosen to do nothing in period 1,
both must invest some level of effort independently and simultaneously inperiod 2.
1 , σ2 )Agent i’s strategy σi is a profile (σi,ti,ti ti =H,L of Bayesian strategies.
i
1
The part σi,ti prescribes his behavior in period 1 for ti -type and it takes a
1 = a1 in R ∪ {∅}. The part σ 2 prescribes his behavior in period
value σi,t
+
i
i,ti
i
2 (a1 ) = a2 in R for each possible
2 for ti -type and it assigns a value σi,t
+
j
i
i
1
value aj of agent j’s choice from R+ ∪ {∅} in period 1.
1 ∈A strategy profile (σ1 , σ2 ) is called leadership by confidence when σi,H
1 = ∅ for i = 1, 2. In words, leadership by confidence means each
R+ and σi,Lagent moves first if and only if he is more confident about team productivity.

3

A Sufficient Condition for Leadership by Confidence to be a Stable Equilibrium

Let us develop a sufficient condition for leadership by confidence to be astable equilibrium. Consider the following assumptions.
A3 (Type independence): Let ρj ≡ Prob(tj = H) = p(H, H) + p(L, H).
i ,H)Then, Pp(tp(t= ρj for ti = L, H.
i ,tj )tj

A4 (Payoff symmetry): si = sj ≡ s and ci (e) = cj (e) ≡ c(e) for alle ∈ R+ .
A5 (Informational symmetry): E[θ|H, L] = E[θ|L, H].
7A6 (Increasing convexity): ci is differentiable three times and c000i (ei ) ≥0 for any ei > 0.
A7 (Decreasing returns to good news):
E[θ|H, H] − E[θ|H, L] ≤ E[θ|L, H] − E[θ|L, L]
E[θ|H, H] − E[θ|L, H] ≤ E[θ|H, L] − E[θ|L, L].
A3 means agent types are independently distributed. A4 means the payoff
functions are symmetric. A5 means the conditional expectations of teamproductivity are symmetric with respect to the realization of types. In
other words, good news and bad news have the same informational contents
irrespective of who holds the news. A6 means the rate at which the marginal
effort cost increases is higher for higher levels of effort. A7 means the increasein the conditional expectations of team productivity when an agent learns
his partner is more confident is smaller when the agent himself is more
confident than when he is less confident.
To state the stable outcome of leadership by confidence under these conditions, let us develop formulae for players’ payoffs from a play in leadershipby confidence. Suppose leadership by confidence (σ1 , σ2 ) prevails. Let µj,tj
be a belief of player j of type tj that agent i is H-type. Then, the expectationof his own contribution to his payoff by investing ej is given by:³´
s µj,tj E[θ|H, tj ] + (1 − µj,tj )E[θ|L, tj ] ej − c(ej ).
Let e∗ (tj , µj,tj ) be the maximizer of this expectation.2 The first order condition for the maximization problem is:³´
s µj,tj E[θ|H, tj ] + (1 − µj,tj )E[θ|L, tj ] = c0 (ej ).
Assumptions A1 and A2 guarantee that the first order condition uniquelydetermines the interior solution for the maximization problem.
Suppose agent i of type ti moves first with ei . In the equilibrium, he1
expects that agent j of H-type moves first with σj,Hand agent j of L-type
moves second. Suppose agent i believes the following: by observing ei , agent
j of L-type holds a belief µj,L and responds with the sequentially rational
choice e∗ (L, µj,L ). Then, under assumption A3, agent i’s expected payofffrom his first move is:³1
Ui (ti , ei , µj,L ) = sE[θ|ti ]ei − c(ei ) + s ρj E[θ|ti , H]σj,H´+ (1 − ρj )E[θ|ti , L]e∗ (L, µj,L ) .
2

Note that we write e∗ rather than e∗j because the maximizer depends on j only throughtj and µj,tj because of assumptions A4 and A5.

8

1
The payoff Ui also depends on player j’s behavior σj,Hand a parameter ρj ,but we drop them from its domain because our analysis is valid irrespectiveof their values.
Similarly, suppose agent i of type ti moves second. In the equilibrium,1
and he invests thehe expects that agent j of H-type moves first with σj,H∗sequentially rational choice e (ti , 1) under the belief µi,ti = 1 after observing
this move. He also expects that agent j of L-type moves second and heinvests the sequentially rational choice e∗ (ti , 0) under the belief µi,ti = 0
after observing this move while agent j responds with e∗ (L, 0) under the
belief µj,L = 0 after observing that agent i moves second. Then, underassumption A3, agent i’s expected payoff from moving second is:³´1
Ui (ti , ∅, 0) = ρj sE[θ|ti , H](e∗ (ti , 1) + σj,H
) − c(e∗ (ti , 1))³´+ (1 − ρj ) sE[θ|ti , L](e∗ (ti , 0) + e∗ (L, 0)) − c(e∗ (ti , 0)) .
1 , ρ for the same reason as for U (t , e , µHere we do not explicitly write σj,Hj
i i i j,L ).
Our candidate for stable outcome of leadership by confidence is the quasiRiley outcome defined as follows.
1 , σ 1 ) is the quasi-Riley outcome of leadership
Definition. A pair of (σ1,H2,H
by confidence if for i = 1, 2:
1
σi,H=arg max Ui (H, ei , 1)ei ∈R+(1)subject to Ui (L, ∅, 0) ≥ Ui (L, ei , 1).
1 , whether σ 1
Note that although Ui depends on σj,Hi,H satisfies (1) does not1
1
depend on σj,H . Therefore, a value of σi,H that constitutes a quasi-Riley
1 .
outcome is determined independent of σj,HFirst, we will show that there uniquely exists the quasi-Riley outcomein the described environment. Let:Mi ≡ {ei ∈ R+ |Ui (L, ∅, 0) ≤ Ui (L, ei , 1)}.
Consider the equation of ei :Ui (L, ∅, 0) = Ui (L, ei , 1).
+−+
If Mi 6= ∅, then there exists solutions e−
i , ei (ei ≤ ei ) to this equation and− +Mi = [ei , ei ], because Ui (L, ei , 1) is strictly concave in ei under assumption
A2. With such a e+
i , the quasi-Riley outcome is characterized as follows.

Proposition 1. Under assumptions A1 through A5, there exists the unique
1 , σ̂ 1 ). Namely, if
quasi-Riley outcome of leadership by confidence, (σ̂i,Hj,H
1 = e∗ (H, ρ ). If e∗ (H, ρ ) ∈ M , then σ̂ 1 = e+ .
e∗ (H, ρj ) 6∈ Mi , then σ̂i,H
j
j
i
i,H
i
9

Proof. Suppose e∗ (H, ρj ) 6∈ Mi . Then, e∗ (H, ρj ) is the unique solution to:max

ei ∈R+ \MiUi (H, ei , 1).

This means e∗ (H, ρj ) constitutes the unique quasi-Riley outcome.
+Suppose e∗ (H, ρj ) ∈ Mi . Then, either e−i or ei constitutes the quasiRiley outcome, because Ui (H, ei , 1) is strictly concave in ei under assumption+
A2. From the definition of e−i and ei , we have:−
Ui (L, e+i , 1) − Ui (L, ei , 1) = 0.

This reduces to:
³
´ ³
´
+
−
−
sE[θ|L]e+−c(e)−
sE[θ|L]e
−
c(e
)
= 0.
i
i
i
i
Then, we have:
³
´ ³
´
+
−
−
sE[θ|H]e+−c(e)−
sE[θ|H]e
−
c(e
)
≥ 0,
i
i
i
i
because E[θ|H] > E[θ|L]. This inequality implies:−
Ui (H, e+i , 1) ≥ Ui (H, ei , 1).
++
The equality holds for the case of e−
i = ei . Therefore, ei constitutes theunique quasi-Riley outcome.

The intuition is as follows. Let us consider the case e∗ (H, ρj ) ∈ Mi .
In the definition of the quasi-Riley outcome, the maximization of agent
i’s payoff is equivalent to the maximization of agent i’s own contribution,1
because his partner j is assumed to move first with σj,H
if he is H-type and∗to move second with response e (L, 1) if he is L-type. Agent i is more willingto invest higher effort for his own contribution when he is H-type than when
he is L-type. Therefore, when agent i of L-type is indifferent between e−i+−+and e+
i , agent i of H-type strictly prefers ei to ei . This selects ei as theunique quasi-Riley outcome in this case.
Next, we will show that the quasi-Riley outcome is supported by a sequential equilibrium. We will prepare two basic lemmata that characterize
the strength of agents’ incentives to move first for the purpose of convincing partners that they are H-type. The first lemma shows that if an agent
is H-type, the benefit of moving first with the interim optimal action andsuccessfully signaling his type exceeds the value of information he will enjoy
in moving second.
Lemma 1. Assume A1 through A7. Then, Ui (H, e∗ (H, ρj ), 1) > Ui (H, ∅, 0).
Proof. See Appendix.
10

The intuition is as follows. The difference between the payoff from moving first and moving second is decomposed as follows:
Ui (H, e∗ (H, ρj ), 1) − Ui (H, ∅, 0)= (1 − ρj )sE[θ|H, L](e∗ (L, 1) − e∗ (L, 0))
·n ³
´
³
´o
− ρj sE[θ|H, H]e∗ (H, 1) − c(e∗ (H, 1) + (1 − ρj ) sE[θ|H, L]e∗ (H, 0) − c(e∗ (H, 0))no¸∗∗− s(ρj E[θ|H, H] + (1 − ρj )E[θ|H, L])e (H, ρj ) − c(e (H, ρj )) .
The first term represents the benefit of signaling by moving first, that is,
the increased contribution by the partner. The second term is the value of
information by moving second. The benefit of signaling is linear in ρj andvanishes at ρj = 1. The value of information is concave in ρj and vanishes
at ρj = 0 and ρj = 1. Therefore, if it is the case that the benefit of signaling
exceeds the value of information at ρj = 1 − ² for small ² > 0, it is the casefor any ρj ∈ (0, 1).
Consider the case of ρj = 1 − ² for small ² > 0. In this case, the benefitof signaling is:²sE[θ|H, L](e∗ (L, 1) − e∗ (L, 0)).

(2)

On the other hand, the value of information is approximated by:³´³´² sE[θ|H, L]e∗ (H, 0) − c(e∗ (H, 0)) − ² sE[θ|H, L]e∗ (H, 1) − c(e∗ (H, 1)) ,because e∗ (H, ρj ) ≈ e∗ (H, 1) for small ² > 0. The approximated form of thevalue of information represents the improvement in payoff of choosing thecorrect action e∗ (H, 0) instead of the wrong action e∗ (H, 1) for E[θ|H, L].
Consider the parallel formula for the improvement in payoff of choosing thecorrect action e∗ (H, 1) instead of the wrong action e∗ (H, 0) for E[θ|H, H].
The following identity holds for the sum of the two improvements:
·³
´ ³
´¸
∗
∗
∗
∗
sE[θ|H, L]e (H, 0) − c(e (H, 0)) − sE[θ|H, L]e (H, 1) − c(e (H, 1))
·³
´ ³
´¸
+ sE[θ|H, H]e∗ (H, 1) − c(e∗ (H, 1)) − sE[θ|H, H]e∗ (H, 0) − c(e∗ (H, 0))= s(E[θ|H, H] − E[θ|H, L])(e∗ (H, 1) − e∗ (H, 0)).
Therefore, we know the value of information is bounded from above by:²s(E[θ|H, H] − E[θ|H, L])(e∗ (H, 1) − e∗ (H, 0)),(3)that is, the increment of team productivity times the increase in own optimal
effort corresponding to the good news that the partner is H-type rather than11

L-type. This is smaller than the benefit of signaling for the following reasons.
The partner’s effort adjustment e∗ (L, 1) − e∗ (L, 0) in formula (2) is larger
than the agent’s own adjustment e∗ (H, 1) − e∗ (H, 0) in formula (3) becauseof the increased convexity of effort cost (assumption A6) and the decreasing
returns to good news (assumption A7). The team productivity E[θ|H, L]
as the impact coefficient of the increase in partner’s effort in formula (2) islarger than the increment in team productivity in formula (3) because of thedecreasing returns to good news (assumption A7).
The second lemma shows that the relative benefit of signaling by moving
first over the value of information by moving second is greater for H-type
than for L-type. In other words, the single crossing property holds for the
choice between moving first and moving second.
Lemma 2. Assume A1 through A7. Then, Ui (H, e, 1) − Ui (H, ∅, 0) ≥Ui (L, e, 1) − Ui (L, ∅, 0) for any e ≥ e∗ (H, ρj ).
Proof. See Appendix.
The intuition is as follows. Consider the payoff difference Ui (ti , e, 1) −Ui (ti , ∅, 0). The difference can be decomposed into three parts: (i) the lossof moving first with e instead of e∗ (ti , ρj ); (ii) the loss of moving first withe∗ (ti , ρj ) irrespective of the partner’s type, that is, the loss from giving up
the value of information from moving second; and (iii) the benefit from thechange in partner’s effort choice from e∗ (L, 0) to e∗ (L, 1). The benefit of(iii) is larger for H-type than L-type, because the expectation of θ as themultiplier of partner’s effort is higher for H-type. The loss of (i) is smallerfor H-type than L-type because the interim optimal level e∗ (ti , ρj ) is higher
for H-type.
The loss of (ii) is rather complicated. Suppose hypothetically that player
i did not know his type and believed with probability δ he is H-type. Consider the case in which assumption 7 holds with equality. In this case, when
δ increases slightly, the increase in expected team productivity given tj = L
is the same as the increase in expected team productivity given tj = H.
Call them the increase in expected team productivity simply. Then, using
the envelope theorem, the change in the value of information from moving
second is approximated by:s × the increase in expected team productivity
× [ the expected value of player i’s effort in moving second−the interim optimal effort level in moving first ].
Player i’s effort choice e∗ (ti , ρj ) is concave in ρj under assumption 6. Thisimplies the term in brackets is negative. Therefore, the value of information
is decreasing in δ, and the value of information is smaller for H-type thanL-type. In the case in which assumption 7 holds with strict inequality, Htype requires a smaller effort adjustment to partner’s type in moving second.
12

Therefore, the value of information is even smaller for H-type in this caseand the loss of (ii) is smaller for H-type than L-type.
Now we will establish our main result based on the prepared lemmata.
Proposition 2. Under assumptions A1 through A7, the unique quasi-Riley
outcome of leadership by confidence is supported by a sequential equilibrium.
Proof. By Proposition 1, the quasi-Riley outcome uniquely exists. Denote
1 , σ̂ 1 ). Consider the following strategy profile. For i = 1, 2:
it as (σ̂i,H
j,H
1
1
σi,H
= σ̂i,H
1
σi,L
=∅
2
σi,t
(a1j ) =i½1e∗ (ti , 1) if a1j = σ̂j,He∗ (ti , 0) otherwise

for ti = H, L.

Consider the belief system:½
1
1 if a1j = ej = σ̂j,Hµi (a1j ) ≡ µi,ti (a1j ) =1
1
1
0 if aj = ∅ or aj = ej 6= σ̂j,Hfor ti = H, L.

The constructed belief system is consistent with the strategy profile, that1 ) = 1, µ (∅) = 0 along the equilibrium paths satisfy
is, the beliefs µi (σ̂j,Hi
Bayes rule.
Let us verify sequential rationality of the strategies. Consider the incentive for L-type to mimic the behavior of H-type and to deviate to moving
1 . If σ̂ 1
first with σ̂i,Hi,H 6∈ Mi , obviously L-type has no incentive to deviate.
1
1
If σ̂i,H ∈ Mi , then σ̂i,H= e+i by Proposition 1 and L-type is indifferent
between moving first with e+i and moving second. Consider the incentive
1 . By the construction
for L-type to deviate to moving first with ei 6= σ̂i,Hof the belief system, L-type must expect his partner of L-type to hold thesame belief µj (ei ) = 0 as the one his partner will hold when he moves second. Therefore, the expectation of his partner’s contribution to his payoffremains the same when he deviates to the first move. On the other hand,the expectation of his own contribution to his payoff is strictly greater when
he moves second than when he deviates to the first move, because he learns
his partner’s type and makes a better decision about his effort choice.
Consider the incentive for H-type to mimic the behavior of L-type and1
to deviate to moving second. In the case of σ̂i,H= e∗ (H, ρj ), H-type does1
not deviate to moving second by Lemma 1. In the case of σ̂i,H= e+i , Htype does not deviate to moving second either, because Lemma 2 and thedefinition of e+i mean:1
1
Ui (H, σ̂i,H, 1) − Ui (H, ∅, 0) ≥ Ui (L, σ̂i,H, 1) − Ui (L, ∅, 0) = 0.
1 .
Consider the incentive for H-type to deviate to moving first with ei 6= σ̂i,HFor the same argument made above for L-type, H-type prefers moving second

131 . Because H-type was proved to prefer moving
to moving first with ei 6= σ̂i,H1
first with σ̂i,Hto moving second, he has no incentive to deviate to moving
1 .
first with ei 6= σ̂i,HThe quasi-Riley outcome of leadership by confidence is not only an equilibrium outcome, but a stable outcome.
Proposition 3. Under assumptions A1 through A7, the unique quasi-Riley
outcome of leadership by confidence is stable.
Proof. Consider the sequential equilibrium constructed in the proof of
Proposition 2. In order to see that the belief system µi of player i satisfies the Cho–Kreps criterion, let us examine an incentive for player j todeviate from the equilibrium. Consider the case of e∗ (H, ρi ) 6∈ Mj . Because1σ̂j,H= e∗ (H, ρi ) by Proposition 1, H-type never has an incentive to deviate
1 . If M = ∅, then L-type does not have an incentive to devito ej 6= σ̂j,Hj
1
ate to moving first with ej 6= σ̂j,Heither. Hence, the Cho–Kreps criterion
1 . If M 6= ∅, then L-type isputs no restriction on µi (ej ) for any ej 6= σ̂j,Hjwilling to deviate to moving first with ej ∈ Mj for some µi ∈ [0, 1]. Therefore, the Cho–Kreps criterion requires µi (ej ) = 0 for ej ∈ Mj . On the other
hand, the Cho–Kreps criterion puts no restriction on µi (ej ) for any ej 6∈ Mj .
Therefore, the constructed belief system satisfies the Cho–Kreps criterion.
Consider the case of e∗ (H, ρi ) ∈ Mj . Define ẽ−j as a solution to theequation of ej :Uj (H, e+j , 1) = Uj (H, ej , 1).
The inequality:+
Uj (H, e−j , 1) ≤ Uj (H, ej , 1)−
− −in the proof of Proposition 1 implies e−
j ≤ ẽj . For ej ∈ [ej , ẽj ), L-type iswilling to deviate to moving first with the ej for some µi ∈ [0, 1] while Htype never has an incentive to deviate to the ej . Therefore, the Cho–Kreps+
criterion requires µi (ej ) = 0 for such ej . For ej ∈ [ẽ−j , ej ), both L-type andH-type are willing to deviate to moving first with the ej for some µi ∈ [0, 1].
Therefore, the Cho–Kreps criterion puts no restriction on µi (ej ) for such ej .
+For ej 6∈ [e−j , ej ], neither L-type nor H-type has an incentive to deviate to
the ej . Therefore, the Cho–Kreps criterion puts no restriction on µi (ej ) forsuch ej , and the constructed belief system satisfies the Cho–Kreps criterion.

4

Are the Assumptions Indispensable?

We showed that leadership by confidence is a stable equilibrium under theset of assumptions A3 to A7. Are these assumptions indispensable for emergence of leadership by confidence? The answer is no. However, leadership14

by confidence fails to emerge if any of the assumptions are violated severely.
We will show where leadership by confidence fails to emerge for each of theassumptions when violated severely, and will explain the causes of the failure. The role of assumption A3 is explored in Kobayashi and Suehiro (2005),
and it is shown that the assumption is important for the emergence of leadership in various patterns including leadership by confidence. Therefore, wewill examine the remaining assumptions from A4 to A7 below. We appliedthese assumptions in various steps for establishment of the emergence ofleadership by confidence. The most basic step is to prove Lemma 1, forwhich we employed all assumptions. We will show that Lemma 1 ceases tohold without the assumptions.
Lemma 1 asserts that H-type has an incentive to move first if he isbelieved to be H-type when he invests the interim optimal level of efforte∗ (H, ρj ). After Lemma 1, we showed a decomposition of the difference
between the payoff from moving first and the payoff from moving second. If
we reproduce it for agent i of H-type without applying the assumptions ofsymmetry, the decomposition can be written as follows:
Ui (H, e∗i (H, ρj ), 1) − Ui (H, ∅, 0) = Si − Ii ,where e∗i (ti , µi,ti ) satisfies the first order condition:³´
si µi,ti E[θ|ti , H] + (1 − µi,ti )E[θ|ti , L] = c0i (ei ),

(4)

and:
Si = (1 − ρj )si E[θ|H, L](e∗j (L, 1) − e∗j (L, 0)),n ³´
Ii = ρj si E[θ|H, H]e∗i (H, 1) − ci (e∗i (H, 1))³´o∗∗+(1 − ρj ) si E[θ|H, L]ei (H, 0) − ci (ei (H, 0))no− si (ρj E[θ|H, H] + (1 − ρj )E[θ|H, L])e∗i (H, ρj ) − ci (e∗i (H, ρj )) .
Now we write e∗i rather than e∗ to show explicitly that it depends not only
on (ti , µi,ti ), but also on si , ci (·). The term Si represents the benefit ofsignaling by moving first and the term Ii is the value of information by
moving second. Agent i of H-type has an incentive to move first when Si
dominates Ii , that is, Si ≥ Ii .
Now we examine assumption A4 with the rest of the assumptions inplace. Starting from the case of payoff symmetry, imagine we change sj or
cj (·) for agent j. This affects e∗j (L, 1) − e∗j (L, 0) in the benefit of signaling
only. The first order conditions that determine the effort levels e∗j (L, 1),e∗j (L, 0) give:³´
sj E[θ|H, L] − E[θ|L, L] = c0j (e∗j (L, 1)) − c0j (e∗j (L, 0)).
(5)15This shows that the increase in partner’s effort e∗j (L, 1) − e∗j (L, 0) will vanish
if sj falls to zero or if the slope of cj (·) becomes steeper, that is, the slopeof c−1j (·) becomes flat. Therefore, Si approaches zero while Ii remains unchanged. Therefore, the condition for the first move incentive is no longer
satisfied.
When agent i of H-type signals his type by moving first, the partner isreluctant to increase his level of effort if the reward rate sj from the teamproduction is negligibly small or if the cost of increasing effort is extremely
high. If the partner will increase his level of effort negligibly, it is not
worthwhile for agent i to give up the value of information and signal his
type. The same argument applies when we change the role of agent i andj and we change si or ci (·). Therefore, the equilibrium in leadership byconfidence requires that payoff symmetry is not violated severely.
Let us show the tightness of the payoff symmetry requirement by an
example. Consider the case of ρi = ρj = 21 , E[θ|L, L] = 2, E[θ|H, L] =E[θ|L, H] = 3, E[θ|H, H] = 4, and ci (e) = cj (e) = e2 in which all assumptions except assumption A4 with respect to si and sj are satisfied. Then,1
the first move incentive Si ≥ Ii for agent i of H-type reduces to sj ≥ 12
si .
The first move incentive Sj ≥ Ij for agent j of H-type is sj ≤ 12si . The
parameters si and sj must be in the symmetric fan along the line si = sj .
Next, we examine assumption A5 with the rest of the assumptions inplace. Starting from the case of informational symmetry, imagine we change
E[θ|H, L]. Consider how the first move incentive for agent i of H-type will beaffected. The benefit of signaling Si will vanish as E[θ|H, L] falls to E[θ|L, L]
for the same reason as in the analysis of assumption A4. That is, equation (5)indicates that when E[θ|H, L] − E[θ|L, L] falls to zero, e∗j (L, 1) − e∗j (L, 0)also falls to zero. On the other hand, the value of information Ii increases
as E[θ|H, L] decreases because agent i of H-type needs to adjust his levelof effort downward more drastically when he learns his partner is L-type.
Formally, we have by the envelope theorem:³´∂Ii= si (1 − ρj ) e∗i (H, 0) − e∗i (H, ρj ) < 0.
∂E[θ|H, L]
Therefore, the first move incentive is no longer satisfied when E[θ|H, L] is
close to E[θ|L, L] rather than equal to E[θ|L, H]. The same argument applies when we change the role of agents i and j and we change E[θ|L, H].
Therefore, the equilibrium in leadership by confidence requires that informational symmetry is not violated severely.
Let us examine the tightness of the informational symmetry requirementby an example. Consider the case of ρi = ρj = 12 , si = sj = 1, and ci (e) =cj (e) = e2 . Suppose E[θ|L, L] = 2, E[θ|H, L] = 3−², E[θ|L, H] = 3+², andE[θ|H, H] = 4 for a parameter ² ∈ (−1, 1). Then, all assumptions except
assumption A5 are satisfied. The case of ² = 0 corresponds to informational
symmetry. The first move incentive Si ≥ Ii for agent i of H-type reduces16to −1 < ² ≤ 23 . The first move incentive Sj ≥ Ij for agent j of H-type
is − 32 ≤ ² < 1. The parameter ² must be in the interval [− 23 , 23 ], which issymmetric around zero, so that informational symmetry is required.
Next, we examine assumption A6 with the rest of the assumptions inplace. We claim there exists an effort cost function c(·) for which:e∗ (H, 0) < e∗ (H, ρj ) < e∗ (H, 1),∗∗∗e (L, 0) ≈ e (L, ρj ) ≈ e (L, 1),

(6)
(7)

if we do not require the function c(·) to satisfy assumption 6. The inequality(6) means an agent of H-type adjusts his level of effort according to his
expectation about his partner’s type. When the inequality (6) holds, wehave a positive value of information Ii > 0. The near-equality (7) means
an agent of L-type chooses almost the same level of effort irrespective of his
expectation about his partner’s type. When the near-equality (7) holds, thebenefit of signaling is almost zero, that is, Si ≈ 0. Therefore, the first move
incentive for agent i of H-type is no longer satisfied.
The near-equality (7) occurs for a cost function c(·), which is almost
linear below e∗ (L, 1) and is very convex at e∗ (L, 1). Without assumption
A6, this property may be compatible with the regular convexity property ofc(·), which guarantees inequality (6). An example of such a function is:½
0 if 0 ≤ e < 1(8)c(e) =
2e − 1 if e ≥ 1.
Consider the case of ρi = ρj = 21 , E[θ|L, L] = 1, E[θ|H, L] = E[θ|L, H] = 2,
and E[θ|H, H] = 3. Then:53e∗ (H, 0) = 1, e∗ (H, ρj ) = , e∗ (H, 1) = ,42e∗ (L, 0) = e∗ (L, ρj ) = e∗ (L, 1) = 1.
Note that this function c(·) does not satisfy assumptions A1 and A2. One
can smooth the function so as to satisfy these assumptions and continue tohave the inequality (6) and near-equality (7).3
Finally, we examine assumption A7 with the rest of the assumptions
in place. Assume E[θ|H, L] = E[θ|L, H] holds and these values fall to
3
For example, consider an arbitrarily given a > 1 and a function c(e) = αeβ for0 ≤ e < a. Then, connect this function to the function c(e) = e2 − 1 at a “smoothly.”
Namely, find a set of values for parameters α and β such that the two functions crossat a, that is, αaβ = a2 − 1, and the two functions have the same slope at a, that is,22αβaβ−1 = 2a. These values are α = a 2a−1
and β = a2a2
2 −1 . The second derivative ofa a2 −1

the connected function is discontinuous and decreasing when a approaches 1. Therefore,
assumption A6 is violated. This connected function approaches the cost function (8) whena approaches 1.

17E[θ|L, L]. When E[θ|H, L] approaches E[θ|L, L], the analysis of assumption
A5 holds with respect to the first move incentive for agent i of H-type. Thatis, the first move incentive is no longer satisfied. Therefore, the decreasing
returns to good news should not be violated severely for agent i. The sameis true for agent j.

5Concluding Remarks

We studied a team production game in which each member holds his personal
judgment about the team productivity and chooses a level of effort at a time
he prefers. We examined the emergence of leadership by confidence, andderived a set of sufficient conditions for this leadership.
The mechanism of emergence of leadership by confidence is endogenous
signaling. A sender-receiver relation is not determined by the rule of thegame, but is rather an outcome of equilibrium play. The expectation of playof this sort induces a player to choose to be a leader or a follower depending
only on the cost-benefits from becoming a sender or a receiver in teams withmulti-sided private information and simple payoff externality.
The endogenous signaling explains endogenous choice of timings of moves
in an environment without time discounting and without strategic substitution or complementarity. There are many possible applications for endogenous choice of timings of moves.
On the other hand, the sufficient condition for leadership by confidence
to be a stable equilibrium also guarantees that leadership by identity is stable. We argued that leadership by confidence should naturally emerge unlessthere is a specific outside mechanism at work to induce a particular member to become a leader. It is a worthy challenge to identify theoreticallyand empirically under what kinds of play environment leadership by confidence prevails and under what other kinds of play environment leadershipby identity emerges.
[2008.7.7 880]

18

Appendix
In this Appendix, we prove Lemma 1 and Lemma 2. To prove Lemma
1, let us prepare the following lemma.
Lemma A. Assume A1, A2, A6, and A7. Let e∗i (ti , µi ) satisfy the firstorder condition (4). Then, e∗i (H, 1) − e∗i (H, 0) ≤ e∗i (L, 1) − e∗i (L, 0).
Proof. By differentiating the first order conditions (4), we have:∂ ∗e (H, µi ) =
∂µi i
∂ ∗
e (L, µi ) =∂µi i

si (E[θ|H, H] − E[θ|H, L])00
ci (e∗i (H, µi ))si (E[θ|L, H] − E[θ|L, L]).
00
ci (e∗i (L, µi ))Assumption A7 implies:si (E[θ|H, H] − E[θ|H, L]) ≤ si (E[θ|L, H] − E[θ|L, L]).
On the other hand, assumption A2 implies e∗i (H, µi ) > e∗i (L, µi ). Together
with assumption A6, this in turn implies c00i (e∗i (H, µi )) ≥ c00i (e∗i (L, µi )).
∂ ∗∂ ∗Therefore, we have ∂µe (H, µi ) ≤ ∂µe (L, µi ) and we have:
i i
i i
Z 1

∂ ∗
ei (H, µi )dµi
0 ∂µi
Z 1
∂ ∗
ei (L, µi )dµi≤
0 ∂µi
= e∗i (L, 1) − e∗i (L, 0).

e∗i (H, 1) − e∗i (H, 0) =Proof of Lemma 1. Define:³´∆(ρj ) ≡ s ρj E[θ|H, H] + (1 − ρj )E[θ|H, L] e∗ (H, ρj ) − c(e∗ (H, ρj ))+(1 − ρj )sE[θ|H, L](e∗ (H, 1) − e∗ (H, 0))h− ρj {sE[θ|H, H]e∗ (H, 1) − c(e∗ (H, 1))}i+(1 − ρj ){sE[θ|H, L]e∗ (H, 0) − c(e∗ (H, 0))} .

Then, because of Lemma A, Ui (H, e∗ (H, ρj ), 1) − Ui (H, ∅, 0) ≥ ∆(ρj ) is
obtained for any ρj ∈ (0, 1) by replacing e∗ (H, 1) − e∗ (H, 0) in the definition of ∆(ρj ) with e∗ (L, 1) − e∗ (L, 0). Therefore, our proof is complete
if we show ∆(ρj ) > 0. Observe ∆(1) = 0 and ∆00 (ρj ) = s(E[θ|H, H] −E[θ|H, L]) ∂ρ∂ j e∗ (H, ρj ) > 0. Examine:³´∆0 (1) = s E[θ|H, H] − E[θ|H, L] − E[θ|H, L] (e∗ (H, 1) − e∗ (H, 0))hi− {sE[θ|H, H]e∗ (H, 1) − c(e∗ (H, 1))} − {sE[θ|H, H]e∗ (H, 0) − c(e∗ (H, 0))} .
19

Assumptions A5 and A7 imply:
E[θ|H, H] − E[θ|H, L] − E[θ|H, L]≤ E[θ|L, H] − E[θ|L, L] − E[θ|H, L]= −E[θ|L, L] < 0.
Furthermore, sE[θ|H, H]e∗ (H, 1)−c(e∗ (H, 1)) > sE[θ|H, H]e∗ (H, 0)−c(e∗ (H, 0))follows from the optimality of e∗ (H, 1). Therefore, ∆0 (1) < 0. From thethree facts combined, we conclude ∆(ρj ) > 0 for any ρj ∈ (0, 1).
Proof of Lemma 2. Define:
·³
´ ³
´¸
Ai ≡
sE[θ|ti = H]e − c(e) − sE[θ|ti = H]e∗ (H, ρj ) − c(e∗ (H, ρj ))
·³
´ ³
´¸
∗
∗
− sE[θ|ti = L]e − c(e) − sE[θ|ti = L]e (L, ρj ) − c(e (L, ρj ))
³
´ n
Bi ≡
sE[θ|ti = H]e∗ (H, ρj ) − c(e∗ (H, ρj )) − ρj (sE[θ|H, H]e∗ (H, 1) − c(e∗ (H, 1)))o+(1 − ρj )(sE[θ|H, L]e∗ (H, 0) − c(e∗ (H, 0)))
³
´ n
∗
∗
Ci ≡
sE[θ|ti = L]e (L, ρj ) − c(e (L, ρj )) − ρj (sE[θ|L, H]e∗ (L, 1) − c(e∗ (L, 1)))o+(1 − ρj )(sE[θ|L, L]e∗ (L, 0) − c(e∗ (L, 0))) .
Then:
h
i h
i
Ui (H, e, 1) − Ui (H, ∅, 0) − Ui (L, e, 1) − Ui (L, ∅, 0)= Ai + (Bi − Ci ) + (1 − ρj )s(E[θ|H, L] − E[θ|L, L])(e∗ (L, 1) − e∗ (L, 0)).
The last term is obviously positive. The first term can be rewritten as:
Ai = s(E[θ|ti = H] − E[θ|ti = L])(e − e∗ (H, ρj ))
h¡
¢ ¡
¢i
+ sE[θ|ti = L]e∗ (L, ρj ) − c(e∗ (L, ρj )) − sE[θ|ti = L]e∗ (H, ρj ) − c(e∗ (H, ρj )) .
The first term is nonnegative for any e ≥ e∗ (H, ρj ). The second term is
positive by the optimality of e∗ (L, ρj ). Therefore, Ai > 0 for any e ≥e∗ (H, ρj ), and the proof is complete if we show (Bi − Ci ) is positive.
For δ ∈ [0, 1], define:³´³´Di (δ) = ρj sE[θ|δ, H]e∗ (δ, 1) − c(e∗ (δ, 1)) + (1 − ρj ) sE[θ|δ, L]e∗ (δ, 0) − c(e∗ (δ, 0))hi− s(ρj E[θ|δ, H] + (1 − ρj )E[θ|δ, L])e∗ (δ, ρj ) − c(e∗ (δ, ρj )) ,where we denote:E[θ|δ, H] = δE[θ|H, H] + (1 − δ)E[θ|L, H],E[θ|δ, L] = δE[θ|H, L] + (1 − δ)E[θ|L, L],20

and we define e∗ (δ, µ) in a parallel way to the definition of e∗ (t, µ) by usingE[θ|δ, H] and E[θ|δ, L]. Then, Bi − Ci = Di (0) − Di (1). Examine:³´³´
Di0 (δ) = sρj E[θ|H, H] − E[θ|L, H] e∗ (δ, 1) + s(1 − ρj ) E[θ|H, L] − E[θ|L, L] e∗ (δ, 0)
h ³
´
³
´i
−s ρj E[θ|H, H] − E[θ|L, H] + (1 − ρj ) E[θ|H, L] − E[θ|L, L] e∗ (δ, ρj ).
Assumption A7 and e∗ (δ, 0) < e∗ (δ, ρj ) imply:³´³´(1 − ρj ) E[θ|H, L] − E[θ|L, L] e∗ (δ, 0) − (1 − ρj ) E[θ|H, L] − E[θ|L, L] e∗ (δ, ρj )³´³´
≤ (1 − ρj ) E[θ|H, H] − E[θ|L, H] e∗ (δ, 0) − (1 − ρj ) E[θ|H, H] − E[θ|L, H] e∗ (δ, ρj ).
Therefore:hi
Di0 (δ) ≤ s ρj e∗ (δ, 1) + (1 − ρj )e∗ (δ, 0) − e∗ (δ, ρj ) (E[θ|H, H] − E[θ|L, H]).
The first order conditions for e∗ (δ, 1), e∗ (δ, 0), and e∗ (δ, ρj ) are:sE[θ|δ, H] = c0 (e∗ (δ, 1)),sE[θ|δ, L] = c0 (e∗ (δ, 0)),³´
s ρj E[θ|δ, H] + (1 − ρj )E[θ|δ, L]= c0 (e∗ (δ, ρj )).
From this system of equations follows:ρj c0 (e∗ (δ, 1)) + (1 − ρj )c0 (e∗ (δ, 0)) = c0 (e∗ (δ, ρj )).

(9)

On the other hand, assumption A6 implies:c0 (ρj e∗ (δ, 1) + (1 − ρj )e∗ (δ, 0)) ≤ ρj c0 (e∗ (δ, 1)) + (1 − ρj )c0 (e∗ (δ, 0)). (10)Given (9) and (10), assumption A2 guarantees:ρj e∗ (δ, 1) + (1 − ρj )e∗ (δ, 0) ≤ e∗ (δ, ρj ).
This establishes Di0 (δ) ≤ 0. Therefore, Bi − Ci = Di (0) − Di (1) ≥ 0. This
completes the proof.

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