HIAS-E-18

Asymmetric information allocation
to avoid coordination failure
Fumitoshi Moriya
Kobe City University of Foreign Studies

Takuro Yamashita
Toulouse School of Economics
Hitotsubashi Institute for Advanced Study, Hitotsubashi University

December 14, 2015

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
tel:+81 42 580 8604
http://hias.ad.hit-u.ac.jp/
HIAS discussion papers can be downloaded without charge from:http://hdl.handle.net/10086/27202
https://ideas.repec.org/s/hit/hiasdp.html

All rights reserved.

Asymmetric information allocation
to avoid coordination failure ∗Fumitoshi Moriya†

Takuro Yamashita‡§

December 14, 2015

Abstract
This study addresses optimal information allocation in team production. We present a unique implementation problem of desirable
eﬀort levels and show that, under certain conditions, it is optimal toasymmetrically inform the agents even if they are ex ante symmetric. The main intuition is that the asymmetric information allocation
is eﬀective in avoiding “bad” equilibria, that is, equilibria with coordination failure. This analysis provides an explanation as to why
informing agents asymmetrically might be beneficial in improving theagents’ coordination behaviors.
KEYWORDS: Moral hazard, Unique implementation, Asymmetric
information
JEL Classification: D21, D23, D86
We are grateful to Hideshi Itoh, Shintaro Miura, Shinsuke Kambe, Benjamin Hermalin, Satoru Tanaka, Marco Celentani,and seminar participants at Hitotsubashi University,
Contract Theory Workshop (CTW), EEA-ESEM 2015, Universidad Carlos III de Madrid,and ESSET 2015. This work is supported by JSPS KAKENHI Grant Number 25780189.
†
Kobe City University of Foreign Studies, moriyafumitoshi@inst.kobe-cufs.ac.jp‡
Toulouse School of Economics, takuro.yamashita@tse-fr.eu§
Hitotsubashi Institute for Advanced Study, Hitotsubashi University
∗

1

1

Introduction

This paper examines an optimal information allocation among agents in an
organization. In many types of organizations (such as firms, universities,
and communities), agents are often asymmetrically informed. For example,
certain information is often retained within a subset of managers in a company even if the information is relevant to the entire company. Managersof company-owned chain stores are typically exposed to more information
through periodic meetings compared to franchisees. Explaining such phenomena requires an understanding of the key channels through which information structure aﬀects organizational performance.
If agents engage in diﬀerent tasks or have diﬀerent characteristics, trivial situations would occur where an asymmetric information structure arises
naturally. However, this paper establishes a novel channel between information allocation and implementation cost that implies that an asymmetric
information structure among the agents is optimal, even when the agents arecompletely symmetric in terms of their characteristics and the tasks in which
they engage. We interpret this channel as a source of “intrinsic” motivation
for asymmetric information allocation in an organization.
Specifically, we consider the following team production model based on
Holmstrom (1982).1 Two agents engage in a single project, and each of
the agents chooses a binary eﬀort (high/low or work/shirk). The principal, aresidual claimant, oﬀers a bonus contract contingent on the binary outcome ofthe project (success/failure). The success probability of the project depends
on the agents’ total eﬀorts and the binary state of the world (good/bad). We
assume that the failure of the project is extremely hazardous (e.g., accidents
in a nuclear power plant or loss of a brand’s long-term reputation). In such asituation, we believe it is reasonable to require that the optimal contract admits a unique (Bayesian) equilibrium such that every agent chooses a desired1

We make simple assumptions to highlight the key insights, but we believe that similar
results should hold in more general environments.

2

level of eﬀort in every state.2 As a direct predecessor, in the team production context but without uncertain state variables, Winter (2004) studies
optimal bonus contracts that uniquely implement the desired eﬀort choices
and shows the optimality of an asymmetric bonus contract even if the agentsare symmetric. This paper considers a similar model but with uncertain parameters in the production function. This introduces a novel dimension to
the principal’s design problem, namely, the allocation of information among
the agents.3
The key observation is that asymmetric information allocation can significantly mitigate the agents’ coordination failure. To provide a rough intuition,
consider the following numerical example.
Example 1. Each agent earns, regardless of the state of the world, an expected payoﬀ ”3” if both agents work, ”2” if a specific agent shirks but theother works, and ”1” if he works but the other shirks. In the case where both
agents shirk, the agent’s expected payoﬀ is ”0” if the state is good, and ”3”
if the state is bad (see Table 1 below).4(“good”) work shirk
work(5, 5) (1, 2)
shirk(2, 1) (0, 0)(“bad”) work shirk
work(5, 5) (1, 2)
shirk(2, 1) (3, 3)

Each state is equally likely and, hence, if no agent is informed about
the state, then “both work” and “both shirk” are equilibria, which is notdesirable for the principal.5 Informing both agents about the state is not
2

It is a common practice in the literature to focus on the implementation of such a“high-eﬀort” equilibrium to simplify the analysis. We consider the unique implementationof such a desired equilibrium given the concerns in the literature for multiple-equilibrium
problems (e.g., Mookherjee (1984)). We discuss this in more detail in Section 2.
3
We discuss how the asymmetry of bonus contracts and the asymmetry of information
allocation interact in Section 5.1.
4
One can interpret these numbers as the agents’ expected utilities given an arbitrary
bonus contract fixed.
5
Of course, if the principal employs wishful thinking and believes that both will work,3

desirable either because “both shirk” is again an equilibrium in the badstate.
Nevertheless, informing just one agent can eliminate this bad coordination. Specifically, suppose that only agent 2 is informed about the state,
whereas agent 1 is not (but agent 1 knows that agent 2 knows the state, andso on). First, if the state is good, then it is dominant for agent 2 to work.
Given this, it is now (iteratively) dominant for agent 1 to work, as illustrated
in the following table:

work
shirk

work in both states(5, 5)(2, 1)

work only in good state, 5+2))(3, 25 ) (= ( 5+1
2
2
2+3 1+3
5( 2 , 2) (= ( 2 , 2 ))Finally, given that agent 1 works (in any state), it is (iteratively) dominant
for agent 2 to work even in the bad state. Therefore, the desired outcomethat “both work in every state” is the unique strategy profile that survives
iterative elimination of dominated strategies.
The conclusion in this example may seem somewhat ad hoc. For example,
a bonus contract that generates these numbers may not be the optimal contract. Nevertheless, the example illustrates an important observation thatthe bad coordination outcome (i.e., both shirking) becomes less sustainabale in an equilibrium by asymmetrically informing the agents.6 The aim ofthen the principal does not foresee a problem. However, in case project failure is hazardous,
the principal may adopts a more pessimistic view.
6
The problem of information disclosure by the principal is also studied in mechanism
design, such as in Bergemann and Pesendorfer (2007) and Esó and Szentes (2007). Particularly, Bergemann and Pesendorfer (2007) show that asymmetrically informing bidders(agents) in auction may be optimal for the revenue-maximizing seller (principal). In theirpaper, more information implies that the auction outcome becomes more eﬃcient, whichitself induces more revenue through bidders’ bidding behaviors, but it also implies thatthe bidders enjoy more information rent. Our paper shows that providing information
asymmetrically can be useful in eliminating undesirable equilibria in a team production
setting, and in this sense, our paper features a diﬀerent aspect of providing information
asymmetrically to the agents.

4this paper is to show that this novel intuition holds in more general teamproduction environments.
The paper is structured as follows. Section 2 introduces the model, andSection 3 studies the optimal bonus contract and information structure ina simple two-agent case with anonymous contracts. Sections 4 and 5 examine several extensions and generalizations of the base model in Section 3
with respect to the number of agents (Section 4.1), asymmetry in agents’ efforts costs (Section 4.2), and elaborate contracts such as non-anonymous (orindividual-specific) bonus contracts (Section 5.1) and stochastic information
allocation (Section 5.2). Section 6 concludes the paper. Except for those in
Section 3, all proofs are shown in the Appendix.

2

Model

We consider a team production model with one manager (a principal) andn workers (agents) who engage in a project. Each worker i ∈ {1, . . . , n}simultaneously chooses an eﬀort level ei ∈ {0, 1}, which costs cei for c > 0.
The profit of the project is y ∈ {S, F } (S > F ), and pθ (x) denotes theprobability of success (y = S), which depends on the agents’ total eﬀort!
x =i ei and task environment θ ∈ {H, L}. We assume that pθ (x) is
increasing in x for any θ. The probability that θ = H is f ∈ (0, 1), and theprobability that θ = L is 1 − f . Let pφ (x) = f pH (x) + (1 − f )pL (x) denotethe mean success probability given x.
The marginal productivity of eﬀort, denoted by ∆pθ (x) ≡ pθ (x) − pθ (x −1), satisfies (i) ∆pH (x) > ∆pL (x) for all x and (ii) ∆pθ (x) > ∆pθ (x−1) for all
x and θ. The first condition requires that the marginal productivity is always
higher in state H than in state L, and the second condition requires that theagents’ eﬀorts are complementary and, therefore, the agents’ collaboration isimportant for success.7
7This convexity in the production function is essential for the multiple-equilibrium
problem, as observed by Winter (2004).

5

Information structure The principal can organize the information structure of the team. Except for Section 5.2, this simply refers to how many
agents are informed of θ when they make eﬀort choices.8 Specifically, we
compare the following three types of information structure.
• No information (NI): no agent observes θ,• Asymmetric information (AI): agent i ∈ {1, . . . , m} does not observeθ, but agent j ∈ {m + 1, . . . , n} observes θ,• Full information (FI): all agents observe θ.
Let si ∈ Si = {H, L, φ} represent agent i’s information about the state.
More specifically, (i) if he is informed, then si = θ for each θ, and (ii) if heis uninformed, then si = φ for each θ. Hence, agent i’s strategy is to chooseei (si ) ∈ {0, 1} for each si .
Once the principal chooses an information structure, we assume that thisstructure itself becomes common knowledge among the agents. We also assume that the principal commits to the information structure of the teamwithout knowing the realization of θ.9
Contract Except for Section 5.1, we consider anonymous bonus contracts
where all the agents are paid symmetrically. Specifically, an anonymousbonus contract is represented by b ≥ 0, where each agent is paid 0 if y = F ,8An essentially equivalent formulation is that the principal controls the cost of acquiring
information on θ. Specifically, the cost is set as zero for an agent whom the principal wants
to inform, and it is set to infinity for an agent whom the principal does not want to inform.
This assumption contrasts with the literature on information acquisition in moral hazard
environments. For example, Andreoni (2006) and Kobayashi and Suehiro (2005) study
models of endogenous leadership or leadership battles, where each agent chooses whether
to acquire certain information on the environment to influence other agents as a leader.
9
The results of the paper would not qualitatively change even if the bonus contract is
contingent on the realized θ (i.e., b becomes a function of θ rather than a scalar invariantin θ), as long as the principal can commit to the information structure and the (possibly
contingent) bonus contract before observing θ.

6and each agent is paid a bonus b if y = S.10 In Section 5.1, we examine
how the results would change if the principal can oﬀer diﬀerent contracts todiﬀerent agents.
Given bonus b and the agents’ eﬀort profile e = (e1 , . . . , en ) ∈ {0, 1}N ,!
!N
the principal’s payoﬀ in state θ is pθ ( N
i=1 ei )(S − nb) + (1 − pθ (i=1 ei ))F .
We assume that S is much larger than F so that the principal’s objective isto implement the full eﬀort strategy of the agents, that is, (ei (si ))i,si suchthat ei (si ) = 1 for every i and si .
Given bonus b and the agents’ eﬀort profile e = (e1 , . . . , en ) ∈ {0, 1}N ,!
agent i’s payoﬀ in state θ is ui (e, θ; b) = bpθ ( N
j=1 ej ) − cei . Thus, given theinformation structure, the agents’ strategy profile e = (ei (si ))i,si is a (purestrategy Bayesian) equilibrium if (i) for each i who is informed: for eachθ ∈ {H, L} and e′i ∈ {0, 1},ui (ei (θ), e−i (s−i ), θ; b) ≥ ui (e′i , e−i (s−i ), θ; b),and (ii) for each i who is uninformed: for each e′i ∈ {0, 1},E[ui (ei (φ), e−i (s−i ), θ; b)] ≥ E[ui (e′i , e−i (s−i ), θ; b)].
Benchmark: Full-eﬀort strategy profile as one of the equilibria We
first derive the optimal contract that makes the full-eﬀort strategy profile oneof the equilibria and observe that the no-information scenario is the optimal
information structure.
Under the no-information scenario, the full-eﬀort strategy profile is anequilibrium if bpφ (n) − c ≥ bpφ (n − 1) or, equivalently, b ≥ ∆pφc (n) . Thus, theoptimal contract is b = ∆pφc (n) . Under any other information structure, thebonus must be suﬃciently high so that an informed agent works in the low
10

Alternatively, we define an anonymous bonus contract by a pair (b, w) where eachagent is paid w if y = F , and each agent is paid b + w if y = S. When the agents areprotected by limited liability so that a feasible contract must satisfy w ≥ 0 and b + w ≥ 0,
then, as in the standard argument, we focus on bonus contracts with w = 0 without lossof generality to seek optimal contracts.

7state, that is, bpL (n) − c ≥ bpL (n − 1) or, equivalently, b ≥ ∆pLc (n) . Thus,
the optimal contract is b = ∆pLc (n) > ∆pφ1(n) . Therefore, the no-information
scenario dominates any other information structure. Intuitively, this is because we must incentivize the agents based on the average state under theno-information structure, whereas under any other information structure wemust incentivize informed agents for every state.
Incentive-inducing contract The contract that implements the full-eﬀort
strategy profile only as one of the equilibria implicitly assumes that the agentswould play the best equilibrium in view of the principal, even if there are
multiple equilibria given the contract. However, in case the failure of the
project is extremely hazardous (e.g., accidents in a nuclear power plant, orloss of a brand’s long-term reputation), the principal may not want to follow
such a wishful thinking. Rather, it may be more reasonable to require thatthe full-eﬀort strategy profile is a unique equilibrium.
Potential multiple equilibria have been an important topic in the literature.11 According to Kreps (1990), the equilibrium played is determined by
the “corporate culture.”12 Unless the principal can fully control the culture
within an organization, undesirable equilibria may be selected. Given thisconcern, we consider the problem of making the desired equilibria a uniqueequilibrium, so that the desired outcome is “guaranteed”, independent of thecorporate culture.
There are several studies that derive the optimal mechanism that uniquely
implements the desired eﬀort choice. Ma (1988) shows that a mechanismwith communication can eliminate undesirable equilibria without any additional cost if the agents’ outputs are individually observed. However, in
11See, for example, Mookherjee (1984) for general moral hazard environments. See
also Baliga and Sjostrom (1998) for the study of collusive behaviors in moral hazard
environments.
12
See Hermalin (2012) for a survey of the literature of corporate culture. In empiricalstudies, Cronqvist, Low, and Nilsson (2007) argue that corporate culture may be a key
determinant of the long-term tendency of corporate policies and performance.

8

team-production models as in Holmstrom (1982), the agents’ outputs are
not individually observed (instead, only the aggregate output is observed).
Therefore, we cannot directly apply the mechanism in Ma (1988).13 Winter (2004) considers a team-production model without state uncertainty andderives the optimal bonus contracts that uniquely implement the full-eﬀort
strategy profile. In this sense, our approach is closest to that of Winter(2004), although we allow for state uncertainty.
Following Winter (2004), we assume that b is an incentive-inducing contract (for the full-eﬀort strategy profile) if (i) the full-eﬀort strategy profile
is an equilibrium given b, and (ii) any other strategy profile is not an equilibrium. Our goal is to identify the minimum (or the infimum) bonus thatuniquely implements the full-eﬀort strategy profile and to study how it varies
among diﬀerent information structures.
Remark. In some applications, the entity that engages in information allocation (principal) may have limited ability to design b. For example, consider
research collaboration between two junior researchers (agents) working on a
joint project, and a senior researcher (principal) engages in information allocation. In such a case, the senior researcher may not fully control b, theprizes for successful outcomes (such as publication or future promotion). As
another example, consider multiple investors (agents) and a startup company (principal), where the startup has information concerning its potential
for success. The startup may have limited ability to control b through thedesign of financial contracts. In such cases, our results can be interpreted
as providing the region of b where unique implementation is possible (given
each information structure). Our comparison of various information structures is completely valid if we say that one information structure is better
than another if and only if the region of b, where unique implementation is13

Arya, Glover, and Hughes (1997) consider a variant of a team-production model where
each agent has an option to quit, and an agent’s output can be individually observed if
the other agent quits. In this case, the authors obtain a mechanism that can approximate
the second-best outcome.

9possible in the first information structure, is larger than the region of b in
the second structure.

3

Baseline caseWe first study the case where there are only two agents. In Sections 3.1, 3.2,
and 3.3, we identify the cost of uniquely implementing the full-eﬀort strategy
profile under each information structure. Then, in Section 3.4, we examine
the optimal information structure and perform comparative statics.

3.1

No information

In the no-information scenario where neither agent is informed, the optimal
bonus contract that implements e = (1, 1) as one of the equilibria is b =c. However, under this contract, not only e = (1, 1), but also other∆pφ (2)eﬀort choices could be an equilibrium. In fact, if i chooses ei = 0, then itis (strictly) optimal for j to choose ej = 0 and, thus, e = (0, 0) is anotherequilibrium given this contract.14
Therefore, the optimal bonus level that uniquely implements e = (1, 1)
must be strictly greater than ∆pφc (2) . Specifically, for e = (0, 0) to not be an
equilibrium, we must have b > ∆pφc (1) (> ∆pφc (2) ) so that an agent works even
if the other agent does not work.
Now, given b > ∆pφc (1) , a high eﬀort is dominant for each agent and, hence,e = (0, 0) and any other eﬀort choice (except for e = (1, 1)) cannot be an
equilibrium. Therefore, the optimal bonus level that uniquely implementse = (1, 1) is ∆pφc (1) .15 Thus, we obtain the following.
14

Note that the convexity of the success probability function, p, plays a key role in thisargument.
15
To be rigorous, for e = (0, 0) not to be an equilibrium, we must set b > ∆pφc (1) rather
than b = ∆pφc (1) . For this openness issue, we follow Winter (2004) by defining the optimal
bonus level as the infimum of those that uniquely implement the desired eﬀort level.

10Proposition 1. Under the no-information scenario, the optimal bonus isbN I = ∆pφc (1) .

3.2

Asymmetric information

We now consider the asymmetric-information scenario where only one of theagents (for example, agent 2) is informed and the other agent (agent 1) isnot. The optimal bonus level is given in the following proposition.
Proposition 2. Under
the asymmetric-information #scenario, the optimal"cc.
bonus is bAI = max ∆pH (1) , ∆pLc (2) , f ∆pH (2)+(1−f)∆pL (1)Proof. We first show the following lemma.
Lemma 1. Let b be any bonus contract such that" neither (0, (0, 0)), (0, (1, 0)),#cnor (1, (1, 0)) are equilibria. Then, b > b̄ = max ∆pHc (1) , ∆pLc (2) , f ∆pH (2)+(1−f.
)∆pL (1)Proof. First, to prevent (0, (0, 0)) from being an equilibrium, we must have
either b > ∆pφc (1) , b > ∆pHc (1) , or b > ∆pLc (1) . Because ∆pHc (1) < ∆pφc (1) , ∆pLc (1) ,we obtain b > ∆pHc (1) as its necessary condition.
Given b > ∆pHc (1) , to prevent (0, (1, 0)) from being an equilibrium, we mustccor b > ∆pLc (1) . Because f ∆pH (2)+(1−f<have either b > f ∆pH (2)+(1−f)∆pL (1))∆pL (1)cc, we obtain b > f ∆pH (2)+(1−f )∆pL (1) as its necessary condition.
∆pL (1)c}, to prevent (1, (1, 0)) from
Given b > max{ ∆pHc (1) , f ∆pH (2)+(1−f)∆pL (1)being an equilibrium, we must have b > ∆pLc (2) .
Note that b > b̄ is a necessary condition for uniquely implementing
the full-eﬀort profile (1, (1, 1)). Now we show that, conversely, (1, (1, 1))
is uniquely implemented by any b such that b > b̄. First, because b > ∆pHc (1) ,it is strictly dominant for the informed agent to make a high eﬀort in state H.
c, it is (iteratively) strictly domGiven this, because b > f ∆pH (2)+(1−f)∆pL (1)inant for the uninformed agent to make a high eﬀort. Given this, becauseb > ∆pLc (2) , it is (iteratively) strictly dominant for the informed agent to make11

a high eﬀort even in state L. Therefore, (1, (1, 1)) is the unique strategy profile that survives iterative elimination of strictly dominated strategies and,hence, it is a unique equilibrium.

3.3

Full information

Finally, we consider the case where both of the agents are informed. In thiscase, it is necessary to incentivize an agent to make a high eﬀort in any state,even if the other agent does not work. Therefore, the bonus must be at leastc. Given such a bonus level, it is dominant for each agent to choose a∆pL (1)high eﬀort in any state. Hence, the full-eﬀort strategy profile becomes theunique equilibrium. Thus, we obtain the following.
Proposition 3. Under the full-information scenario, the optimal bonus isbF I = ∆pLc (1) .

3.4

Optimal information structure

In this subsection, we compare the three information structures discussed
above. Recall that bN I = ∆pφc (1) is the bonus level that incentivizes an agent
to work in the average state even if the other agent does not work, whereas
bF I = ∆pLc (1) is the bonus level that incentivizes an agent to work in any stateeven if the other agent does not work. We have bN I < bF I and, hence, theno-information scenario is optimal compared to the full-information scenario.
Whether the no-information scenario is better than the asymmetric-information
scenario, or vice versa, depends on the parameter values .
Theorem 1. The no-information scenario is always better than the fullinformation scenario. The asymmetric-information scenario is better than
the no-information scenario if and only if∆pL (2) − ∆pL (1)≥ f.
∆pH (1) − ∆pL (1)12cProof. We only prove the second statement. If bAI = f ∆pH (2)+(1−f,)∆pL (1)then the no-information scenario is never optimal becausef ∆pH (2) + (1 − f )∆pL (1) − ∆pφ (1) = f [∆pH (2) − ∆pH (1)] ≥ 0.
Similarly, if bAI = ∆pHc (1) , then the no-information scenario is never optimal because∆pH (1) − ∆pφ (1) = (1 − f )[∆pH (1) − ∆pL (1)] ≥ 0.
If bAI = ∆pLc (2) , we have bN I ≥ bAI if and only if∆pL (2) − ∆pL (1)≥ f.
∆pH (1) − ∆pL (1)The proposition shows that, first, allocating information asymmetrically
can be better than no information (and than full information). It also showsthat no information can be better than asymmetric information only if bAI =cc. If bAI ̸= ∆pLc (2) , bAI is the greater of ∆pHc (1) and f ∆pH (2)+(1−f,∆pL (2))∆pL (1)cbut both are smaller than bN I = ∆pφ (1) .
In case bAI = ∆pLc (2) , bAI becomes higher than bN I (i.e., the no-information
scenario becomes better) as (i) the “eﬀort eﬀect” on the production function
measured by ∆pL (2) − ∆pL (1) becomes smaller, and (ii) the “state eﬀect”
measured by ∆pH (1)−∆pL (1) becomes larger (see Figure 1). This is because(i) if the eﬀort eﬀect is smaller, which means that the complementarity of theagents’ eﬀorts in success probability is smaller, then the concern of potential coordination failure is smaller, and therefore, the benefit of asymmetric
information allocation is smaller. On the other hand, (ii) if the state eﬀect
is larger, then it is costly to incentivize an informed agent in the low state,
and therefore, informing no agent is likely to be better.

13

Figure 1: Optimal Information Structure(pθ (x) = βex/2+θ )4

Extensions

The basic model examined in the previous section is extended in two directions. In Section 4.1, we assume there are n agents. The main conclusion
remains similar, but we argue that with more agents, asymmetric information tends to be better than no information, in a certain sense. The mainquestions are: How many agents should be informed, and how does this optimal number of informed agents vary with the parameters? In Section 4.2, weconsider a case where the agents have diﬀerent eﬀort costs. The analysis provides some insights for which agents should be informed under asymmetric
information allocation. The analysis also shows how the optimal information
structure varies with the cost diﬀerence across agents.

14

4.1

n-agent caseThis subsection considers the model where a team consists of n agents. Let
bm denote the optimal bonus when m(≤ n) agents are uninformed and m − nagents are informed.
First, we observe that under the no-information and full-information scenarios, the optimal bonus has a similar property as in Section 3 in that the
optimal bonus level is at exactly the level above which an agent is incentivized to work even if no agent works (in the average state and in every
state, respectively).
Proposition 4. Under the no-information scenario, the optimal bonus isbn = ∆pφc (1) .
Similarly, under the full-information scenario, the optimal bonus is b0 =.
∆pL (1)
Next, we provide the optimal bonus under the asymmetric-information
scenario, i.e., with 1 ≤ m ≤ n − 1.
cProposition 5. Under the asymmetric-information scenario where m agentsare uninformed and n − m agents are informed, the optimal bonus is bm =cc
, f ∆pH (n−m+1)+(1−f, c }.
max{ ∆pL (m+1))∆pL (1) ∆pH (1)Now, we consider how many agents should be informed or uninformed
among n agents. Let m∗ ∈ {0, . . . , n} denote the optimal number of uninformed agents (i.e., bm∗ ≤ bm for any m ̸= m∗ ). We provide some generalproperties concerning m∗ .
Theorem 2. (i) m∗ ̸= 0. (ii) bm is quasi-convex in m for m = 1, . . . , n. (iii)
The no-information scenario is optimal if and only if∆pL (n) − ∆pL (1)≤ f.
∆pH (1) − ∆pL (1)The first statement shows that full information is never optimal. The second statement implies that the minimizer of bm , m∗ , smoothly varies with the15parameter values. The second statement also implies that the no-information
scenario (i.e., m = 0) is optimal if and only if bm is (globally) decreasing in
m. Therefore, as a corollary of the second statement, we obtain the thirdstatement, which is analogous to the condition in Theorem 1 in the basemodel. As the eﬀort eﬀect ∆pL (n) − ∆pL (1) increases relative to the stateeﬀect ∆pH (1) − ∆pL (1), asymmetric information becomes more advantageous. Notice that the eﬀort-eﬀect term ∆pL (n) − ∆pL (1) is increasing in
n, while the state-eﬀect term ∆pH (1) − ∆pL (1) is not. Therefore, adding
more agents in the team (without changing the success probability function)
makes asymmetric information more likely be optimal.

4.2

Asymmetric eﬀort cost

In this subsection, we allow the agents to have diﬀerent eﬀort costs. The
main objective of the analysis is to study which agent should be informed if
the agents have diﬀerent characteristics.
We consider the same model as that of the base model in Section 2, except that each agent i incurs cost ci for ei = 1. Assume c1 < c2 , whereby
agent 1 is more “productive” than agent 2. Because of this change, under
the asymmetric-information scenario, we consider two cases separately: one
where the productive agent 1 is informed, and the other where the unproductive agent 2 is informed.
We first consider the no-information scenario.
Proposition 6. Under the no-information scenario, the optimal bonus isbN I = max{

c1
c2
,
}.
∆pφ (1) ∆pφ (2)As long as c1 and c2 are close to each other, we have bN I = ∆pcφ1(1) , thatis, the condition of avoiding the low eﬀort equilibrium binds. However, when
the cost diﬀerence becomes suﬃciently large, the incentive condition for the
non-productive agent starts to bind.

16

We omit the full-information case because the intuition is similar to theno-information case.
Next, we consider the asymmetric-information scenario. An interestingquestion is which agent (productive or unproductive) should be informed.
The answer depends on the parameter values. Intuitively, it is better toinform the productive agent if the optimal bonus is determined by the informed agent’s incentive conditions, and vice versa. Specifically, first assume
that c1 and c2 are close to each other. Recall that, in the base model (withc1 = c2 = c), the optimal bonus under the asymmetric-information scenariocbAI is the greatest of ∆pHc (1) , ∆pLc (2) , and f ∆pH (2)+(1−f.
)∆pL (1)cc
If bAI = ∆pH (1) or ∆pL (2) , this implies that the optimal bonus is determinedby one of the incentive conditions of the informed agent. Therefore, if c1 is(slightly) smaller than c2 , it is optimal to inform the productive agent, agent
1.
c, this implies that the optimal bonus is deterIf bAI = f ∆pH (2)+(1−f)∆pL (1)mined by the incentive condition of the uninformed agent. Therefore, if c1
is (slightly) smaller than c2 , it is optimal to inform the unproductive agent,
agent 2.
Therefore, we obtain the following result.
Proposition 7. Assume that the cost diﬀerence is suﬃciently small so that∆pφ (1)∆pφ (1)c1≥ γ ∗ , where γ ∗ is the smaller of f ∆pH (2)+(1−fand min{∆pH (1),∆p.
c2)∆pL (1)L (2)}
Then, under the asymmetric-information scenario, bAI is the smaller of
c1
c2
c1
,
,
},
b1AI = max{
∆pH (1) ∆pL (2) f ∆pH (2) + (1 − f )∆pL (1)where the productive agent 1 is informed, andc2c2c1b2AI = max{,,},
∆pH (1) ∆pL (2) f ∆pH (2) + (1 − f )∆pL (1)where the unproductive agent 2 is informed.
When the cost diﬀerence is suﬃciently large to satisfy cc21 < γ ∗ , the noinformation scenario outperforms the asymmetric-information scenario (regardless of who is informed). Therefore, we do not derive the optimal bonus
17

contract under asymmetric information in such a case.16 The following result
characterizes the optimal information structure.
Theorem 3. When cc12 < γ ∗ , the no-information scenario is better than theasymmetric-information scenario.
When cc21 ≥ γ ∗ ,17 the no-information scenario is better than the asymmetricinformation scenario if and only if
c1≥ min{b1AI , b2AI }.
∆pφ (1)
Again, when the cost diﬀerence is small, then the result is qualitatively
similar to the base case (although the expressions are more complicated),
whereas when the cost diﬀerence is large, then the potential advantage ofasymmetric information is lost and, hence, a symmetric information structure
is better.

5

Elaborate contracts

The previous sections only consider symmetric bonus contracts and deterministic information allocation. Although we believe that both features arereasonable restrictions in practice, it is theoretically interesting to investigate
whether the main results under those restrictions still hold even if they arerelaxed. The full characterization of the optimal contract and information
allocation without any restriction is beyond the scope of the paper, and weleave this to future research.
16

The intuition is roughly as follows. When the cost diﬀerence is large, the optimal bonus
becomes one whereby (regardless of who is informed) the productive agent is incentivized
to work in any state even if the unproductive agent does not work. In such a case,however, the no-information scenario becomes better than the asymmetric-information
scenario (regardless of who is informed).
17
Note that this implies ∆pcφ1(1) ≥ ∆pcφ2(2) and, hence, the optimal bonus under theno-information scenario is bN I = ∆pcφ1(1) .

18

5.1

Individual BonusIn this subsection, we allow for the contracts that are not necessarily anonymous (which we refer to as individual bonus contracts). The main qualitative
result stays the same: informing the agents asymmetrically can be optimal in
minimizing the agency cost. However, the interaction of individual bonuses
and information allocation adds a new dimension for the analysis. For example, we observe that asymmetric information allocation and asymmetricbonus schemes are “(imperfect) substitutes” for the principal in the sensethat the set of parameters under which asymmetric information allocation is
optimal becomes smaller under individual bonus contracts than under symmetric bonus contracts. The main objective of this subsection is to provide
the intuition for this substitution eﬀect.
An individual bonus contract is represented by a pair (b1 , b2 ) ∈ R2+ suchthat each agent i is paid zero if y = F , and each agent is paid bi if y = S. We
aim to characterize the optimal incentive-inducing contract by minimizing
b1 + b2 among all contracts that uniquely implement the full-eﬀort strategy
profile.
If we allow for individual-specific bonus contracts, the optimal contract
is typically asymmetric, as found by Winter (2004) in the context of teamproduction without the state uncertainty. In the no-information scenario andfull-information scenario in our model, the result directly applies as follows.
Proposition 8. Under the no-information scenario, the optimal bonus contract isccccbN I = (,) or (,).
∆pφ (2) ∆pφ (1)∆pφ (1) ∆pφ (2)Similarly, under the full-information scenario, the optimal bonus contractisbF I = (c,c∆pL (2) ∆pL (1)) or (c,c∆pL (1) ∆pL (2)).

To provide some intuition for the optimal bonus contract under the noinformation scenario (and a similar intuition applies to the full-information19

case), consider a bonus contract that uniquely implements e = (1, 1). To
prevent e = (0, 0) from being an equilibrium, such a contract must givecto one of the agents, for example, agent 1, so that agent 1 would work∆pφ (1)even if agent 2 does not work. As opposed to anonymous contracts, we do
not need to give ∆pφc (1) to both of the agents. In fact, for agent 2, we only
need to give ∆pφc (2) so that he would work if agent 1 works. Therefore, theoptimal contract is as given in the proposition.
In the asymmetric-information scenario, the optimal individual-bonus
contract is given as follows.
Proposition 9. Under the asymmetric-information scenario, the optimal
bonus contract bAI is the one that minimizes the total bonus payment amongcc,),∆pφ (1) ∆pL (2)ccc(, max{,}), andf ∆pH (2) + (1 − f )∆pL (1)∆pH (1) ∆pL (2)
c
c
(
,
).
∆pφ (2) ∆pL (1)(Consider a bonus contract that uniquely implements e = (1, (1, 1)). To
prevent e = (0, (0, 0)) from being an equilibrium, such a contract must giveeither (i) ∆pφc (1) to the uninformed agent or (ii) ∆pHc (1) to the informed agent.
In Case (i), working becomes dominant for the uninformed agent and,hence, we must give ∆pLc (2) to uniquely implement e = (1, (1, 1)) because,then, the informed agent has an incentive to work in any state given thatthe uninformed agent works. The total bonus in such a contract is thusc
+ ∆pLc (2) .
∆pφ (1)In Case (ii), working becomes dominant for the informed agent in state
H. To prevent (0, (1, 0)) from being an equilibrium, we must give either (iia) ∆pLc (1) to the informed agent so that working becomes dominant for thecinformed agent in any state, or (ii-b) f ∆pH (2)+(1−fto the uninformed)∆pL (1)agent so that he has an incentive to work given that the informed agent works
in state H.
20

In Case (ii-a), we must give ∆pφc (2) to uniquely implement e = (1, (1, 1))
because, then, the uninformed agent has an incentive to work given that the
informed agent works in any state. The total bonus in such a contract is,thus, ∆pφc (2) + ∆pLc (1) .
In Case (ii-b), we must give ∆pLc (2) to the informed agent so that theagent works in state L. Thus, in Case (ii-b), the bonus to the informed agent
is max{ ∆pHc (1) , ∆pLc (2) } and, therefore, the total bonus in such a contract isc+ max{ ∆pHc (1) , ∆pLc (2) }.
f ∆pH (2)+(1−f )∆pL (1)Now, we compare the three information structures.
Theorem 4. The no-information scenario is better than the full-information
scenario. The asymmetric-information scenario is better than the no-information
scenario if and only if
1
1
1
1
+
≥
+
.
∆pφ (2) ∆pφ (1)f ∆pH (2) + (1 − f )∆pL (1) min{∆pL (2), ∆pH (1)}As in the anonymous-contract case, the no-information scenario is better
than the asymmetric-information scenario when (i) the eﬀort eﬀect, measuredby ∆pθ (2) − ∆pθ (1), θ = H, L, is suﬃciently small (so that the concern forpotential coordination failure is small) and (ii) the state eﬀect, measured by∆pH (x) − ∆pL (x), x = 1, 2, is suﬃciently large (so that the incentive cost
for an informed agent is large).
However, with individual contracts, the no-information scenario is better in the opposite case as well, that is, when (i) the eﬀort eﬀect is suﬃciently large and (ii) the state eﬀect is suﬃciently small (see Figure 2). The
main reason for this diﬀerence is that, with individual contracts, we can savethe implementation cost by making the bonus payment asymmetric. Whenthe eﬀort eﬀect is large and the state eﬀect is small, the amount of thiscost saving is more significant under the no-information scenario than under
the asymmetric-information scenario, which makes the total bonus payment
smaller under the no-information scenario.
To see this more formally, recall that, with anonymous contracts, the optimal bonus level under the no-information scenario is ∆pφc (1) , whereas under21

Figure 2: Optimal Information Structure(pθ (x) = βex/2+θ )c, c , cthe asymmetric-information scenario, the optimal bonus level is max{ f ∆pH (2)+(1−f)∆pL (1) ∆pH (1) ∆pL (2)Thus, the diﬀerence of the two terms converges to zero as the state eﬀect goesto zero, that is, as ∆pH (x) − ∆pL (x) goes to zero for each x = 1, 2. Now, if
individual contracts are allowed, then, under the no-information scenario, we
can save the total bonus payment compared to the anonymous-bonus case
by

2

c
c
c
c
c
−(
+
)=
−
> 0,
∆pφ (1)∆pφ (1) ∆pφ (2)∆pφ (1) ∆pφ (2)which does not vanish even if the state eﬀect goes to zero (and it is larger
when the eﬀort eﬀect is larger). On the other hand, under the asymmetricinformation scenario, we can save the total bonus payment compared to theanonymous-bonus case by
c
c
c
,
,
}
f ∆pH (2) + (1 − f )∆pL (1) ∆pH (1) ∆pL (2)ccc−− max{,},f ∆pH (2) + (1 − f )∆pL (1)∆pH (1) ∆pL (2)2 max{

22

which converges to zero as the state eﬀect goes to zero. Therefore, when thestate eﬀect is suﬃciently small and the eﬀort eﬀect is suﬃciently large, theno-information scenario becomes better.

5.2

Stochastic information allocation

In the previous sections, only “deterministic” information allocation is considered, that is, each agent is either perfectly informed of θ or not informed atall. However, at least theoretically, we can consider more general “stochastic”
information allocations.
The complete characterization of the optimal stochastic information allocation is beyond the scope of this paper.18 Instead, this subsection shows
two partial characterization results. First, as soon as stochastic information allocation is allowed, the no-information scenario (in fact, any (possibly
stochastic) “symmetric” information allocation) becomes suboptimal for any
parameter values. Therefore, the optimal information allocation is necessarily asymmetric. Second, under certain parameter values, deterministic
asymmetric information allocation (as in Section 3) is optimal, even if arbitrary stochastic information allocation is allowed.
In the two-agent setting considered in Section 3, a stochastic information allocation is denoted by (S1 , S2 , µ) where Si is an arbitrary finite set fori = 1, 2 and denotes the set of the signals agent i receives (before choosing
his eﬀort level), and µ : Θ → ∆(S1 × S2 ) is such that, for each θ ∈ Θ and(s1 , s2 ) ∈ S1 × S2 , µ(s1 , s2 |θ) represents the probability that each agent i
receives signal si when the state is θ. and µ is a joint probability distribution over Θ × S1 × S2 . In the most general setting, the principal can choose18

A potential challenge for the complete characterization is that the standard techniquein the Bayesian persuasion literature (e.g., Kamenica and Gentzkow (2011)) is not directly
applicable. Some results in Bayesian persuasion crucially depend on the assumption thatreceivers (agents) play the best equilibrium with respect to the sender (principal) if there
are multiple equilibria, whereas in our paper the key driving force for optimal information
allocation is the concern that the agents may play a “bad” equilibrium if there are multiple
equilibria.

23

an arbitrary Si for each i and arbitrary µ under the feasibility constraintthat the marginal of µ over Θ coincides with the common prior. As special cases, the deterministic allocations considered in the previous sections
are in this class. For example, the no-information scheme is identified bya singleton Si = {∅} for every i, and µ coincides with the common prior.
(Deterministic) asymmetric information is identified by S1 = {∅}, S2 = Θ,µ(∅, H|H) = µ(∅, L|L) = 1. However, in general, more complicated information allocations are allowed here.
Agent i’s strategy is given by ei : Si → {0, 1}, and the full-eﬀort strategy
profile is e = (e1 , e2 ) such that ei (si ) = 1 for all i, si .
Given a (symmetric) bonus contract b ∈ R+ , a strategy profile e is a(pure-strategy Bayesian) equilibrium if, for each i, si ∈ Si and ei ∈ {0, 1},E[ui (ei (si ), e−i (s−i ), θ; b)|si ] ≥ E[ui (ei , e−i (s−i ), θ; b)|si ],where E[·|si ] represents the conditional (on si ) expected-value operator withrespect to s−i and θ induced by µ. We say that b uniquely implements
the full-eﬀort strategy profile if the full-eﬀort strategy profile is the uniqueequilibrium given b.
First, as soon as stochastic information allocation is allowed, any symmetric information allocation (including no information as a special case)becomes suboptimal for any parameter values. We say that (S1 , S2 , µ) is symmetric if (i) S1 = S2 and (ii) s1 ̸= s2 implies µ(s1 , s2 |H) = µ(s1 , s2 |L) = 0.
That is, the agents do not have asymmetric information in such an information allocation rule. The following result implies that the optimal information
structure is necessarily asymmetric among the class of stochastic information
allocation.
Theorem 5. There exist sn (asymmetric) stochastic information allocationthat is strictly better than any symmetric information allocation.
Next, under certain parameter values, deterministic asymmetric information allocation (as in Section 3) is optimal, even if arbitrary stochastic
information allocation is allowed.
24

Theorem 6. If ∆pH (1) < ∆pL (2) and ∆pH (1) < f ∆pH (2) + (1 − f )pL (1),then (deterministic) asymmetric information allocation is optimal.
Note that under the conditions in the statement, bAI = ∆pHc (1) . The conditions are satisfied if the eﬀort eﬀect is suﬃciently large, that is, when theconcern for potential coordination failure is large. In this case, we have shownin Section 3 that asymmetric information is better than no information (andfull information). The result is strengthened because asymmetric information is proven to be better than any other, possibly stochastic, information
allocation.
To provide some intuition, suppose, contrarily, that there exists a stochastic information allocation (S1 , S2 , µ) with a strictly smaller bonus b < bAI
that uniquely implements the full-eﬀort profile. As a necessary condition forthis unique implementation, there must exist some agent i and some signal
si ∈ Si such that, even if the other agent works with probability zero, anagent still prefers to work. Therefore,b(qpH (1) + (1 − q)pL (1)) − c ≥ b(qpL (0) + (1 − q)pL (0)),
where q is the conditional probability that the state is H given i observes si .
Or equivalently,b≥c.
q∆pH (1) + (1 − q)∆pL (1)Note that the right-hand side is decreasing in q and coincides with bAI
when q = 1. This contradicts b < bAI . Therefore, we conclude that there is no
stochastic information allocation that is strictly better than the deterministic
asymmetric information allocation.

6

Conclusion

This paper considers a team-production model with state uncertainty. Whenthe principal’s goal is to uniquely implement desired eﬀort choices, we show25

that, under certain conditions, asymmetrically informing the agents is the
optimal information allocation. The main intuition is that by allocating
information asymmetrically, it becomes less costly to avoid badly coordinatedequilibria. As the degree of eﬀort complementarity (called the eﬀort eﬀect inthis paper) increases, asymmetric information allocation tends to improve.
On the other hand, informing an agent is always costly in the sense that
this agent must be incentivized even in a low state. As the state eﬀect on
the success probability function increases, asymmetric information allocation
tends to worsen. Therefore, the optimal information allocation is determinedby the relative magnitude of the two eﬀects.
While we show the robustness of this main intuition in a number of extensions and generalizations, further research is necessary for a more comprehensive understanding of desirable information allocation in organizations.
We believe that the analysis in this paper can serve as a useful benchmark
for future research.

A

Proof of Proposition 4

Let b represent any bonus contract such that (0, . . . , 0) is not an equilibrium.
Then, we have b > ∆pφc (1) . Note that this is a necessary condition for uniquely
implementing the full-eﬀort profile (1, . . . , 1).
Conversely, given any b such that b > ∆pφc (1) , it is strictly dominant foreach agent to make a high eﬀort. Therefore, the full-eﬀort profile is theunique equilibrium.

B

Proof of Proposition 5

Let (eU , (eIH , eIL )) represent a strategy profile such that all the uninformed
agents play eU , and all the informed agents play eIH in state H and play eILin state L. We first show the following lemma.

26Lemma 2. Let b be any bonus contract such that neither (0, (0, 0)), (0, (1, 0)),ccnor (1, (1, 0)) are equilibria. Then, b > b̂ = max{ ∆pL(m+1), f ∆pH (n−m+1)+(1−f, c }.
)∆pL (1) ∆pH (1)Proof. First, to prevent (0, (0, 0)) from being an equilibrium, we must have
either b > ∆pφc (1) , b > ∆pHc (1) , or b > ∆pLc (1) . Because ∆pHc (1) < ∆pφc (1) , ∆pLc (1) ,we obtain b > ∆pHc (1) as its necessary condition.
Given b > ∆pHc (1) , to prevent (0, (1, 0)) from being an equilibrium, we mustcchave either b > f ∆pH (n−m+1)+(1−for b > ∆pLc (1) . Because f ∆pH (n−m+1)+(1−f<)∆pL (1))∆pL (1)cc, we obtain b > f ∆pH (n−m+1)+(1−f )∆pL (1) as its necessary condition.
∆pL (1)c}, to prevent (1, (1, 0))Given b > max{ ∆pHc (1) , f ∆pH (n−m+1)+(1−f)∆pL (1)c.
from being an equilibrium, we must have b > ∆pL (m+1)
Note that b > b̄ is a necessary condition for uniquely implementing
the full-eﬀort profile (1, (1, 1)). Now, we show that, conversely, (1, (1, 1))
is uniquely implemented by any b such that b > b̄. First, because b > ∆pHc (1) ,it is strictly dominant for each informed agent to make a high eﬀort in statec
H. Given this, because b > f ∆pH (n−m+1)+(1−f, it is (iteratively) strictly)∆pL (1)dominant for each uninformed agent to make a high eﬀort. Given this, bec, it is (iteratively) strictly dominant for each informed
cause b > ∆pL (m+1)
agent to make a high eﬀort even in state L. Therefore, (1, (1, 1)) is theunique strategy profile that survives iterative elimination of strictly dominated strategies and, hence, it is a unique equilibrium.

C

Proof of Theorem 2

The statement (i) is trivial. For the statement (ii), recall first that, forccm = 1, . . . , n−1, bm = max{ ∆pL(m+1), f ∆pH (n−m+1)+(1−f, c }. As m)∆pL (1) ∆pH (1)ccdecreases, f ∆pH (n−m+1)+(1−fincreases, and ∆pHc (1)increases, ∆pL (m+1))∆pL (1)stays constant. Therefore, bm is quasi-convex in m for m = 1, . . . , n − 1.
For the statement (iii), observe first that, for any m = 1, . . . , n − 1, wec. Therefore, we have
have bn > ∆pHc (1) and bn > f ∆pH (n−m+1)+(1−f)∆pL (1)ccbn < bm if and only if bn < ∆pL (m+1) . Because ∆pL (m+1) is decreasing in m,
27

we have bn < bm for all m > 1 if and only if bn < bn−1 , or equivalently,cc<,∆pφ (1)∆pL (n)and hence we obtain the inequality as in the statement.

D

Proof of Proposition 6

First, for e = (0, 0) not to be an equilibrium, we must have b > ∆pcφi(1) atleast for some i so that the agent will make a high eﬀort even if the otheragent does not. Given c1 < c2 , we obtain b > ∆pcφ1(1) .
Also, for e = (1, 1) to be an equilibrium, we must have b ≥ maxi ∆pcφi(2) .
Given c1 < c2 , we obtain b ≥ ∆pcφ2(2) . Therefore, we obtain b > bN I as anecessary condition for uniquely implementing e = (1, 1).
Conversely, b > bN I is also suﬃcient for unique implementation. First,
e1 = 1 is strictly dominant for agent 1 because b > ∆pcφ1(1) . Given this, e2 = 1
is (iteratively) strictly dominant for agent 2 because b > ∆pcφ2(2) .

E

Proof of Proposition 7

1≥ max{ ∆pH1 (1) , ∆pL1 (2) }.
We only consider the case with f ∆pH (2)+(1−f)∆pL (1)1
The other case with f ∆pH (2)+(1−f< max{ ∆pH1 (1) , ∆pL1 (2) } is similar and,)∆pL (1)hence, the proof is omitted.
We havec2
b1AI =≥ b2AI .
f ∆pH (2) + (1 − f )∆pL (1)As in the base case in Section 3, if b > b2AI , then the full-eﬀort strategy
profile is the unique equilibrium when agent 2 is informed: first, it is strictly
dominant for (informed) agent 2 to make a high eﬀort in the high state,even if agent 1 does not work; given this, a high eﬀort is (iteratively) strictly
dominant for (uninformed) agent 1; given this, a high eﬀort in any state is(iteratively) strictly dominant for agent 2.
28

Conversely, b > b2AI is also necessary to uniquely implement the full-eﬀort
strategy profile when agent 2 is informed. First, to prevent e = (0, (0, 0))from being an equilibrium, the least costly way is to incentivize (informed)
agent 2 in the high state as long as ∆pcφ1(1) ≥ ∆pcH2(1) . This last inequality is∆p (1)∆p (1)

satisfied because cc21 ≥ γ ∗ = max{ ∆pHφ (1) , ∆pLφ (2) }. Therefore, we must haveb > ∆pcH2(1) .
Second, to prevent e = (0, (1, 0)) from being an equilibrium, the least
costly way is to incentivize (uninformed) agent 1. Therefore, we must havec1.
b > f ∆pH (2)+(1−f)∆pL (1)Finally, to prevent e = (1, (1, 0)) from being an equilibrium, agent 2
must be incentivized in the low state. Therefore, b > ∆pcL2(2) . These threeinequalities amount to b > b2AI .

F

Proof of Theorem 3

The case with cc21 ≥ γ ∗ is straightforward and, hence, we omit the proof.
Therefore, in the following, we assume cc12 < γ ∗ .
1≥ max{ ∆pH1 (1) , ∆pL1 (2) },Moreover, we only consider the case with f ∆pH (2)+(1−f)∆pL (1)∆p (1)∆p (1)

which implies cc12 < γ ∗ = max{ ∆pHφ (1) , ∆pLφ (2) }. The other case is similar and,hence, the proof for that case is omitted.
Lemma 3. Whenever agent 1 is informed, any bonus contract b that uniquely
implements the full-eﬀort strategy profile e = ((1, 1), 1) must satisfy b >bN I = max{ ∆pcφ1(1) , ∆pcφ2(2) }, and, therefore, the no-information scenario isbetter than informing agent 1.
c2.
Proof. (of the lemma) First, we consider the case with ∆pcL1(1) ≥ f ∆pH (2)+(1−f)∆pL (1)In this case, to prevent e = ((1, 0), 0) from being an equilibrium, thec2least costly way is to incentivize agent 2 by setting b > f ∆pH (2)+(1−f.
)∆pL (1)As in the base case, to prevent e = ((0, 0), 0) and e = ((1, 0), 1) frombeing equilibria, the least costly way is to incentivize agent 1 by setting

29

b > max{ ∆pcH1(1) , ∆pcL1(2) }. These inequalities imply
b > b1AI

c2
f ∆pH (2) + (1 − f )∆pL (1)
c2
c2
,
},
> max{∆pH (1) ∆pL (2)=where the equality and the last inequality is because of our current assump1
tion that f ∆pH (2)+(1−f≥ max{ ∆pH1 (1) , ∆pL1 (2) }. This immediately im)∆pL (1)plies b > ∆pcφ2(2) and, moreover, together with cc12 < γ ∗ , implies b > ∆pcφ1(1) .
Therefore, we have b > max{ ∆pcφ2(2) , ∆pcφ1(1) } = bN I .
c2.
Next, we consider the opposite case with ∆pcL1(1) < f ∆pH (2)+(1−f)∆pL (1)In this case, to prevent e = ((1, 0), 0) from being an equilibrium, the least
costly way is to incentivize agent 1 in the low state by setting b > ∆pcL1(1) . To
prevent e = ((1, 1), 0) from being an equilibrium, we must have b > ∆pcφ2(2) .
These two inequalities imply b > max{ ∆pcφ2(2) , ∆pcφ1(1) } = bN I .
Lemma 4. Whenever agent 2 is informed, any bonus contract b that uniquely
implements the full-eﬀort strategy profile e = (1, (1, 1)) must satisfy b >bN I = max{ ∆pcφ1(1) , ∆pcφ2(2) }, and, therefore, the no-information scenario isbetter than informing agent 2.
Proof. (of the lemma)
To prevent e = (1, (1, 0)) from being an equilibrium, we must have b >c2.
∆pL (2)∆p (1)First, we consider the case with ∆pH1 (1) < ∆pL1 (2) (hence, γ ∗ = ∆pLφ (2) ). In
this case, cc12 < γ ∗ implies ∆pcL2(2) > ∆pcφ1(1) . Therefore, b > ∆pcL2(2) directly
implies b > max{ ∆pcφ1(1) , ∆pcφ2(2) } = bN I .
∆p (1)

Second, we consider the case with ∆pH1 (1) ≥ ∆pL1 (2) (hence, γ ∗ = ∆pHφ (1) ).
Because b > ∆pcL2(2) implies b > ∆pcφ2(2) , it suﬃces to show b > ∆pcφ1(1) . To
show this, note that cc12 < γ ∗ implies ∆pcφ1(1) < ∆pcH2(1) . Therefore, to prevent
e = (0, (0, 0)) from being an equilibrium, the least costly way is to incentivize
agent 1 by setting b > ∆pcφ1(1) , which is our desired inequality.
30

G

Proof of Proposition 8

We first show the following lemma.
Lemma 5. Let b be any bonus contract such that neither (0, 0), (1, 0), nor(0, 1) are equilibria. Then, either (i) b1 > ∆pφc (1) and b2 > ∆pφc (2) , or (ii)b1 > ∆pφc (2) and b2 > ∆pφc (1) .
Proof. Let b be any bonus contract such that (0, 0) are not equilibria. Then,we must have bi > ∆pφc (1) for some agent i. Given any b = (bi , bj ) such thatbi > ∆pφc (1) , to prevent e = (ei , ej ) = (1, 0) from being an equilibrium, wemust have bi > ∆pφc (1) for j ̸= i.
Note that this is a necessary condition for uniquely implementing thefull-eﬀort profile (1, 1). Conversely, consider any b such that b1 > ∆pφc (1) andb2 > ∆pφc (2) . First, it is strictly dominant for agent 1 to make a high eﬀort.
Then, given this, it is (iteratively) strictly dominant for agent 2 to make ahigh eﬀort. Therefore, the full-eﬀort profile is the unique equilibrium. The
case where b satisfies b1 > ∆pφc (2) and b2 > ∆pφc (1) is analogous, and so weomit this case.

H

Proof of Proposition 9

We first show the following lemma.
Lemma 6. Let b be any bonus contract such that neither (0, (0, 0)), (0, (1, 0)),(1, (1, 0)), nor (0, (1, 1)) are equilibria. Then, either (I) b1 > ∆pφc (1) andcand b2 > max{ ∆pHc (1) , ∆pLc (2) }, orb2 > ∆pLc (2) , (II) b1 > f ∆pH (2)+(1−f)∆pL (1)(III) b1 > ∆pφc (2) and b2 > ∆pLc (1) .
Proof. For any b such that (0, (0, 0)) is not an equilibrium, we must have
either (i) b1 > ∆pφc (1) , or (ii) b2 > ∆pHc (1) .
First, consider any b such that b1 > ∆pφc (1) . To prevent (1, (1, 0)) frombeing an equilibrium, we must have b2 > ∆pLc (2) . Hence, we obtain Case (I).
31

Second, consider any b such that b2 > ∆pHc (1) . To prevent (0, (1, 0)) fromcbeing an equilibrium, we must have either (ii-a) b1 > f ∆pH (2)+(1−for)∆pL (1)c(ii-b) b2 > ∆pL (1) .
cConsider any b such that b2 > ∆pHc (1) and b1 > f ∆pH (2)+(1−f. To)∆pL (1)prevent (1, (1, 0)) from being an equilibrium, we must have b2 > ∆pLc (2) .
Hence, we obtain Case (II).
Finally, consider any b such that (b2 > ∆pHc (1) and) b2 > ∆pLc (1) . To prevent(0, (1, 1)) from being an equilibrium, we must have b1 > ∆pφc (2) . Hence, we
obtain Case (III).
Note that this is a necessary condition for uniquely implementing thefull-eﬀort profile (1, 1).
We now show that the converse is true. First, consider any b such thatb1 > ∆pφc (1) and b2 > ∆pLc (2) . It is strictly dominant for the uninformed agent
to make a high eﬀort. Given this, it is (iteratively) strictly dominant for the
informed agent to make a high eﬀort for any state. Therefore, (1, (1, 1)) is aunique equilibrium.
cand b2 >Second, consider any b such that b1 > f ∆pH (2)+(1−f)∆pL (1)cccmax{ ∆pH (1) , ∆pL (2) }. Because b2 > ∆pH (1) , it is strictly dominant for the
informed agent to make a high eﬀort in state H. Given this, it is (iteratively) strictly dominant for the uninformed agent to make a high eﬀort.
Given this, because b2 > ∆pLc (2) , it is (iteratively) strictly dominant for the
informed agent to make a high eﬀort in state L. Therefore, (1, (1, 1)) is aunique equilibrium.
Third, consider any b such that b1 > ∆pφc (2) and b2 > ∆pLc (1) . It is strictly
dominant for the informed agent to make a high eﬀort in any state. Given
this, it is (iteratively) strictly dominant for the uninformed agent to make ahigh eﬀort. Therefore, (1, (1, 1)) is a unique equilibrium.

32

I

Proof of Theorem 4

The first statement is trivial, hence, we omit its proof. To prove the second
statement, first, assume
1
1
1
1
+
≥
+
.
∆pφ (2) ∆pφ (1)f ∆pH (2) + (1 − f )∆pL (1) min{∆pL (2), ∆pH (1)}
Then,
c
c
+
f ∆pH (2) + (1 − f )∆pL (1) min{∆pL (2), ∆pH (1)}cc+≤∆pφ (2) ∆pφ (1)= b1,N I + b2,N I .

b1,AI + b2,AI ≤Conversely, assume
1
1
1
1
+
<
+
.
∆pφ (2) ∆pφ (1)f ∆pH (2) + (1 − f )∆pL (1) min{∆pL (2), ∆pH (1)}c
If bAI = ( f ∆pH (2)+(1−f, max{ ∆pHc (1) , ∆pLc (2) }), then the no-information)∆pL (1)scenario is better.
If bAI = ( ∆pφc (1) , ∆pLc (2) ), then the no-information scenario is better becausef (∆pH (2) − ∆pL (2))(b1,AI + b2,AI − (b1,N I + b1,N I ) =≥ 0.
∆pφ (2)∆pL (2)Finally, if bAI = ( ∆pφc (2) , ∆pLc (1) ), then the no-information scenario is better
becausef (∆pH (1) − ∆pL (1))≥ 0.
(b1,AI + b2,AI ) − (b1,N I + b2,N I ) =∆pφ (1)∆pL (1)J

Proof of Theorem 5

The proof is composed of two steps. First, we show that the no-information
scenario is better than any (possibly stochastic) symmetric information allocation. Next, we show that there is a stochastic (asymmetric) information
allocation that is strictly better than the no-information scenario.
33

Lemma 7. Let b < bN I = ∆pφc (1) . Given the bonus contract b and any symmetric information allocation (S, S, µ), there exists a Bayesian equilibriume = ((e1 (s1 ))s1 ∈S , (e2 (s2 ))s2 ∈S ), s ∈ S and θ such that (i) e1 (s) = e2 (s) = 0
and (ii) µ(s, s, θ) > 0 (i.e., unique implementation is not achieved).
Proof. Suppose not. Without loss of generality, we assume that for each s,
there is some θ such that µ(s, s, θ) > 0. Then, for any s ∈ S, we havebf µ(s, s|H)pH (0) + (1 − f )µ(s, s|L)pL (0)f µ(s, s|H)pH (1) + (1 − f )µ(s, s|L)pL (1)−c≥b,f µ(s, s|H) + (1 − f )µ(s, s|L)f µ(s, s|H) + (1 − f )µ(s, s|L)or equivalently,b(f µ(s, s|H)∆pH (1) + (1 − f )µ(s, s|L)∆pL (1)) > c(f µ(s, s|H) + (1 − f )µ(s, s|L)).
Summing up each side across all s, because
1, we have!

s µ(s, s|H) =!

s µ(s, s|L) =b(f ∆pH (1) + (1 − f )∆pL (1)) > c,or equivalently, b > ∆pHc (1) = bN I , which contradicts b < bN I .
Lemma 8. There exists a stochastic information allocation (S1 , S2 , µ) thatis strictly better than the no-information scenario.
Proof. Recall that under the no-information scenario, the optimal bonus
is bN I = ∆pφc (1) . We now construct a stochastic information allocation(S1 , S2 , µ) that achieves a strictly lower bonus level.
Let ε ∈ (0, 1). Define S1 = {∅}, S2 = {Leak, Not}, µ(∅, Leak|H) = ε,µ(∅, Not|H) = f (1 − ε), and µ(∅, Not|L) = 1 − f . The idea is that agent 2
would be informed (“leaked”) with some probability ε if the state is H.
Let b∗ denote the optimal bonus under this (S1 , S2 , µ). First, to prevent
e = (0, (0, 0)) from being an equilibrium, b∗ must be at least as high asc∆pH (1)(< bN I ),34

so that agent 2, given the leak, has an incentive to work even if agent 1 does
not work. Next, to prevent e = (0, (1, 0)) from being an equilibrium, b∗ mustbe at least as high asc(< bN I ),f ε∆pH (2) + f (1 − ε)∆pH (1) + (1 − f )∆pL (2)so that agent 1 has an incentive to work if agent 2 works at least when theleak occurs. Finally, to prevent e = (1, (1, 0)) from being an equilibrium, b∗
must be at least as high ascf (1−ε)1−f∆pH (2) − 1−f∆pL (2)1−f εε,so that agent 2 without the leak has an incentive to work if agent 1 definitely
works. Observe that when ε = 0, this reduces to ∆pφc (2) (< bN I ). Therefore,by continuity, there exists some ε∗ > 0 such that, for any ε ∈ (0, ε∗),cf (1−ε)1−f∆pH (2) − 1−f∆pL (2)1−f ε
ε< bN I .

We fix one such ε in the following.
Conversely, as in the deterministic asymmetric information allocation, as
long as b∗ is higher than any of these three terms, the full-eﬀort strategy
profile is the only equilibrium outcome (obtained by iterative elimination of
dominated strategies). Therefore, we conclude b∗ < bN I .

KProof of Theorem 6

Suppose that under information allocation (S1 , S2 , µ), bonus b uniquely implements the full-eﬀort strategy profile. Then, because the “no eﬀort” strategy profile e0 = (e01 , e02 ) (i.e., e0i (si ) = 0 for all i, si ) is not an equilibrium,
there exists some i and si such that agent i has an incentive to work even if
the other agent works with probability zero. Hence,b(hpH (2) + (1 − h)pL (2)) − c ≥ b(hpH (1) + (1 − h)pL (1)),
35where h is agent i’s conditional belief on θ = H given the signal si . Becauseh ≤ 1, this implies that b ≥ ∆pHc (1) .
Recall that under the deterministic asymmetric-information allocation,
the optimal bonus level that uniquely implements the full-eﬀort strategy proc}, which reduces to bAI =file is bAI = max{ ∆pHc (1) , ∆pLc (2) , f ∆pH (2)+(1−f)pL (1)cby the assumption in the statement. This implies that the deter∆pH (1)ministic asymmetric-information allocation is optimal among all stochastic
information allocations.

References
Andreoni, J. (2006): “Leadership Giving in Charitable Fund-Raising,”
Journal of Public Economic Theory, 8(1), 1–22.
Arya, A., J. Glover, and J. S. Hughes (1997): “Implementing Coordinated Team Play,” Journal of Economic Theory, 74(1), 218–232.
Baliga, S., and T. Sjostrom (1998): “Decentralization and Collusion,”
Journal of Economic Theory, 83(2), 196–232.
Bergemann, D., and M. Pesendorfer (2007): “Information structuresin optimal auctions,” Journal of Economic Theory, 137(1), 580 – 609.
Cronqvist, H., A. Low, and M. Nilsson (2007): “Does Corporate Culture Matter for Firm Policies?,” Working Paper (Ohio State University).
Esó, P., and B. Szentes (2007): “Optimal Information Disclosure in Auctions and the Handicap Auction,” Review of Economic Studies, 74(3), 705–
731.
Hermalin, B. E. (2012): “Leadership and Corporate Culture,” in The
Handbook of Organizational Economics, ed. by R. Gibbons, and J. Roberts.
Princeton University Press.

36Holmstrom, B. (1982): “Moral Hazard in Teams,” Bell Journal of Economics, 13(2), 324–340.
Kamenica, E., and M. Gentzkow (2011): “Bayesian Persuasion,” American Economic Review, 101, 2590–2615.
Kobayashi, H., and H. Suehiro (2005): “Emergence of Leadership inTeams,” Japanese Economic Review, 56(3), 295–316.
Kreps, D. M. (1990): “Corporate Culture and Economic Theory,” in Perspectives on Positive Political Economy, ed. by J. E. Alt, and K. A. Shelpsle. Cambridge University Press, Cambridge.
Ma, C.-T. (1988): “Unique Implementation of Incentive Contracts with
Many Agents,” Review of Economic Studies, 55(4), 555–572.
Mookherjee, D. (1984): “Optimal Incentive Schemes with Many Agents,”
Review of Economic Studies, 51(3), 433–446.
Winter, E. (2004): “Incentives and Discrimination,” American Economic
Review, 94, 764–773.

37