A Folk Theorem for Contract Games with
Multiple Principals and Agents
Siyang Xiong†
March 27, 2013
Abstract
We fully characterize the set of deterministic equilibrium allocations in a competing contract game with multiple principals and agents. Compared to similar folk
theorems in the literature, our main theorem relaxes an indispensable assumption on
equilibrium existence in Yamashita [8], and does not require a particular communication protocol as required in Peters and Troncoso-Valverde [6].
 I thank Eddie Dekel, Songying Fang, Mike Peters, Balázs Szentes, Stephen Wolff, Takuro Yamashita and
anonymous referees for helpful comments. I thank the National Science Foundation (grant SES-1227620)
for financial support. All remaining errors are my own.
† Department of Economics, Rice University, [email protected]
1
1
Introduction
In the presence of asymmetric information, economic players may write contracts to elicit
other players’ information, in order to make optimal decisions. Most of the large literature
on contract theory focuses on the classical principal-agent (i.e., one principal and one or
multiple agents) model and the common-agency (i.e., one agent and multiple principals)
model.1 However, both multiple principals and multiple agents are usually involved in
many economic situations. For example, there are many competing insurance companies that offer different policies, and there are many consumers trying to find the best
insurance plans.
To understand such situations, we study a two-stage multi-principal-multi-agent
game in this paper. In this contract game, the payoffs of the players2 depend on both
the states and the actions. States are privately observed by the agents, while actions are
chosen by the principals. In stage 1, the principals offer contracts, which specify their
committed actions contingent on the messages reported by the agents. In stage 2, the
agents report messages and the contracts are executed. Our objective is to fully characterize the set of deterministic equilibrium allocations3 in this contract game.
Our main result (Theorem 1) says that the equilibrium allocations are fully characterized by two incentive compatibility (IC) conditions for the agents and the principals.


truthful reporting by the agents forms a Bayes Nash equilibrium
;
(I) agents: 
in the direct mechanisms defined by the allocation


(II) principals: 

every principal is endowed with a min-max-min value
(as defined in (5)); under the allocation,
every principal achieves utility larger than the min-max-min value.


.

Condition (I) must hold in every equilibrium; the intuition is the same as the revelation principle in the classical principal-agent model. However, the “revelation principle”
1 See
Bolton and Dewatripont [1] and Martimort [4] for surveys on the principal-agent model and the
comon-agency model, respectively.
2 Throughout the paper, we use “she” and “he” to refer to a principal and an agent, respectively. We use
“players” to refer to both principals and agents.
3 We discuss stochastic equilibrium allocations in Section 5.2.
2
ignores IC for the principals, so we need condition (II). The min-max-min value in condition (II) has the same role as the min-max value in a usual 2-player normal-form game —
given other principals’ and agents’ strategies, principal j can always find a contract that
guarantees her min-max-min value. As a result, in any equilibrium, all principals must
achieve utility larger than the min-max-min values, i.e., condition (II) must hold in every
equilibrium.
Conversely, for any allocation satisfying conditions (I) and (II), we construct an equilibrium to implement the allocation. Specifically, for any principal k, principals k can
find punishing contracts ckk under which principal k achieves utility less than her min-
max-min value.4 Then, the intuition of the equilibrium construction is clear: in stage 1,
each principal offers a set of contracts, which includes the equilibrium-allocation contract and the punishing contracts; in stage 2, the agents choose the equilibrium-allocation
contract on the equilibrium path and they choose the punishing contracts ckk whenever
principal k deviates from the equilibrium path.
Yamashita [8]’s theorem shares a similar intuition. However, his proof requires the
agents play pure strategies in the subgames in stage 2, which poses a technical problem
— a pure-strategy equilibrium does not exist in many games. To eliminate this problem,
Yamashita assumes the equilibrium existence for all the possible subgames in stage 2.
Equivalently, any contracts that induce no pure-strategy Nash equilibrium in stage 2 are
automatically excluded from the principals’ choice sets in stage 1. This limits the scope
of Yamashita’s theorem, as illustrated by Example 1 in Section 2. In this example, market
information is observed only by two agents who have diametrically opposed preferences.
Thus, any contract offered by the principal defines a zero-sum game for the agents. Since
pure-strategy Nash equilibria exist only in degenerate zero-sum games, no valid contract
can extract the market information. As a result, the unique equilibrium allocation is inefficient.
In this paper, we allow for mixed strategies in stage 2. Under some usual technical conditions (i.e., compact message spaces and continuous contracts), a mixed-strategy
Nash equilibrium always exists in stage 2 (Glicksberg’s Theorem, see [2]). Hence, we do
not need the equilibrium existence assumption in [8]. Equivalently, we do not place any
restrictions on the set of contracts offered by the principals in stage 1. As a result, our
4 We
adopt the standard notation of letting k denote all members of a group other than individual k.
3
folk theorem expands the scope of Yamashita’s theorem. For instance, if we allow for
mixed-strategy equilibria in Example 1, the principals can find a contract to elicit market
information. Consequently, an efficient equilibrium allocation exists.
It is worth noting that each principal in our contract game is allowed to offer only
a single contract, while we let each principal offer a set of contracts, as suggested by
the equilibrium construction above. One methodological contribution of this paper is to
introduce a way to embed such a set of contracts into one single valid contract.
One caveat of our equilibrium characterization is that it is complicated, because
the min-max-min values are hard to compute. Peters and Troncoso-Valverde [6] provide
a much simpler characterization, but they require a particular complicated protocol for
communication among the players.5 We discuss these issues in Sections 5.4 and 5.5.
The remainder of the paper is organized as follows. An illustrating example is provided in Section 2. The model is defined in Section 3. We present the main result in Section
4. Section 5 concludes with discussions. Technical proofs can be found in Appendix A.
2
Pure-strategy implementation versus mixed-strategy implementation: an example
We use the following simple example to illustrate the major difference between our setup
and that in Yamashita [8]: mixed strategies are allowed in our setup, while only pure
strategies are valid in [8].
Example 1 There are two agents i1 , i2 , and one principal j.6 The state is θ  0, 1  Θ, which
is uniformly distributed. Suppose i1 and i2 observe θ, but j does not. Principal j has to choose an
action a  1, 0, 1  A. The players are expected-utility maximizers, with Bernoulli utility
5 The complicated communication protocol is rarely observed in most economic situations.
In such cases,
our folk theorem applies.
6 For simplicity, we consider only two agents and one principal. The example remains valid if we add additional payoff-irrelevant agents and principals (i.e., these agents do not observe the states, these principals
do not take payoff-relevant actions, and their payoffs are always constant).
4
functions defined as follows:
u j  a, θ  



 0,



2,
if a  0;
if  a  1  θ;
3, if  a  1 and θ  0;
ui1  a, θ   a;
ui2  a, θ    a.
Following Yamashita [8], a valid contract for principal j is a function c j : Mi1  Mi2  A,
where Mi1 and Mi2 are some exogenously given message spaces for i1 and i2 , respectively. Upon
receiving the contract, the agents send their messages and then the contract is executed.
Pure-strategy implementation As in [8], the agents are required to play pure strategies,
and we assume the existence of pure-strategy equilibria in the subgames defined by all the
valid contracts of the principal. Equivalently, any contract that induces no pure-strategy
Nash equilibria in stage 2 is not available to j.
For every a  A, let z a denote the constant allocation, in which a is taken at both
states by principal j. It is easy to draw the following observations.
(1) Only constant allocations can be implemented by pure-strategy equilibria.7
(2) Principal j strictly prefers z a0 to z a1 or z a1 .
By 1 and 2, z a0 is the unique equilibrium allocation, under which the players
achieve a payoff of 0. We show below that z a0 is not ex-ante pareto efficient.
7 Suppose otherwise.
I.e.,  a at θ  0 and a  a at θ  1 is implemented by some contract c j and some




pure-strategy Nash equilibrium m1 , m2  at θ  0 and m1 , m2 at θ  1 . Suppose c j m1 , m2   a .
Without loss of generality, suppose a i1 a and a i2 a. By the IC of i2 at state θ  0, we have a i2 a .
Hence, we have a i2 a i2 a . Furthermore, by the IC of i1 at state θ  1, we have a i1 a , which implies
a i2 a , contradicting a i2 a i2 a .
5
Mixed-strategy implementation Suppose we allow the agents to play mixed strategies.
Consider the contract ζ j for principal j defined as follows:8
m2
m2
m2
ζ j : m1
a0
a1
m1
a0
a  1
a  1
a1
The following strategy profile of the agents forms a mixed-strategy Nash equilibrium in
the game defined by ζ j .

agent i1 :

 agent i :
2

at both states, report m1 and m1 with equal probability (i.e., 21 );
at state θ  0, report m2 with probability 1
at state θ  1, report m2 and m2 with equal probability (i.e., 12 ).




Under this equilibrium, the
 agents fully reveal
 the true states, and principal j achieves the
maximal ex-ante payoff 1 
1
2
 0  12  2 , while both agents get an ex-ante payoff of 0.
Therefore, this equilibrium allocation is pareto efficient, and it pareto dominates z a0 .
3
Setup
3.1
The contract game
There are two sets of players: the principals   1, ..., J  and the agents   1, ..., I .
Each agent i   privately observes his type θ i  Θi , and each principal j   has to
take an action y j   j . Let Θ  ∏ Θi and   ∏  j . The Bernoulli utility functions for
i 
j
principal j and agent i are denoted by v j :   Θ  R and ui :   Θ  R, respectively.
We assume    1,   2, Θ  ∞,    ∞,     , and θ  Θ is distributed
according to a common prior p   Θ.
Let  ji denote the set of messages that agent i can send to principal j. Throughout
this paper, we assume  ji   j  i    0, 1  R for every  j, i      .
8 The
contract ζ j is not valid for prinicipal j in [8], because no pure-strategy Nash equilibrium exists in
the game defined by ζ j .
6
A strategy of each principal j   is a valid contract, i.e., a continuous function
 
c j : ∏  ji   Yj .9 Let  j denote j’s strategy set and   ∏  j . A strategy of
i 
j
each agent i 
  is a (type,
contract)-contingent signalling scheme, i.e., a function Si :

∏  ji .10 Let i denote i’s strategy set and   ∏ i . That is, in this
j
i 
 
contract game, the principals first choose contracts c  c j j   in stage 1. Upon
Θi    
observing c and θ i , each agent i sends (possibly stochastic) messages to the principals in
stage 2, and then the contracts are implemented. I.e., this is a 2-stage dynamic game. Each
c   chosen by the principals in stage 1 defines a subgame for the agentsin stage 2.
 In
every such subgame, each agent i chooses a behavior strategy si : Θi  

the set isub  

∏  ji
j
Θi
∏  ji
j
in
. Define  sub  ∏ isub .
i 
We use Si and si to denote generic elements in i and isub , respectively. Given c   ,
we use Si c  isub to denote the behavior strategy induced by Si and c, i.e., Si c θ i  
Si  θ i , c  .
Let φ s, θ   

∏
j , i 
 ji

denote the independent11 distribution over message
profiles induced by any s, θ    sub  Θ, i.e.,


φ s, θ   Ei i 
∏ si θ i  Ei  ,  measurable
i 
 Ei i  ∏
i 

∏  ji
j

.
Let ψ c, m     denote the independent distribution over action profiles induced by
9 We
 
require the codomain of c j be  Yj rather than Yj , so that c j can be made continuous and Glicks-
berg’s Theorem can be applied later.
10 We implicitly embed mixed strategies of the agents into our model, i.e., a pure strategy for agent i is a


function from Θi   to ∏  ji , and a mixed strategy is a function from Θi   to 
j


11 Given
each θ  Θ, every agent i takes his strategies si θ i   

the distribution φ s, θ  induced by s, θ  is a distribution over ∏
i  I.
i 
7
∏  ji

j
∏  ji
j
∏  ji .
j
independently. As a result,
which is independent across
any c, m 
 

 
c j j , m ji j , i   
ψ c, m
 

y j j 
∏
j
cj

∏
j , i 
m ji

 ji , i.e.,
 
 
y j ,  y j j   .
i 
Given c, s, θ , Γ c, s, θ, y defined below calculates the probability that the action profile
y is taken by the principals:
Γ c, s, θ, y 
m

∏
j , i 
ψ c, m y d φ s, θ  .
 ji
Define Vj :    sub  R and Ui :    sub  Θi  R as follows, where, given
c, s, Vj c, s and Ui c, s, θ i  denote the expected payoffs of principal j and agent i who
observes θ i , respectively:
Vj c, s 
Ui c, s, θ i  
∑
θ i  Θ i





∑
θ Θ
∑
y

∑ v j y,
y
θ   Γ c, s, θ, y

v j y, θ i , θ i   Γ c, s, θ i , θ i , y

 p θ  ;




p  θ i , θ i 
.




p
θ
,
θ
i i
∑
θ i Θi
 
In the subgame defined by any c   , the continuity of c j j and the compact-
ness of ∏  ji ensure that a (mixed-strategy) Nash equilibrium (NE) always exists by
j
Glicksberg’s Theorem (see [2]). Let  c   sub denote the set of Nash equilibria in the
subgame defined by c, i.e.,


 
 


sub
sub
.
 c  s   : Ui c, si , si , θ i   Ui c, si , si , θ i ,  i, θ, s    Θ  
Hence,  c   for any c   . Throughout the paper, for every c   , fix some
e c  ei ci   c   sub .
(1)
We adopt the solution concept of subgame perfect Nash equilibrium (SPNE)12 for
this 2-stage contract game. Specifically, it is defined as follows.
12 Throughout
the paper, we use “SPNE” and “NE” to represent the equilibrium in the 2-stage contract
game and the subgame in stage 2, respectively.
8
 

c j j , Si i     is a SPNE if


  





Vj c j , c j , S c j , c j  Vj c j , c j , S c j , c j ,  j, c     ,
  
 
and Si c i   c , c   .
Definition 1 c, S 
3.2
(2)
Transformation of a mixed strategy through a homeomorphism
Let A and B denote two subsets of the message spaces such that A is homeomorphic to
B. Fix any homeomorphism ξ : A  B. In our proofs below, we need to transform a
mixed strategy in A (i.e., α    A) to a mixed strategy in B (i.e., β    B) via the
homeomorphism ξ. Specifically, for any α    A, define ξ  α    B as follows:
ξ  α  E  α  a  A : ξ  a  E ,  measurable E  B.
That is, ξ renames the elements in A to the homeomorphic elements in B, and ξ  α is just
α, respecting this renaming.
4
Main result
4.1
Equilibrium allocations
A (deterministic) allocation is a function z : Θ   ; let z j : Θ   j denote the jth
projection of z. As in [8], we say z is incentive compatible iff



∑ ui z j θ i , θ i  j , θ i , θ i  p θ i , θ i 
θ i  Θ i

∑
θ i  Θ i
ui
(3)
 

  


z j θ ji , θ i j , θ i , θ i  p θ i , θ i  , i   ,  θ i , θ ji j  Θi  Θi   .
We say an allocation z is induced by c, s     sub iff
z θ   y  Γ c, s, θ, y  1,  θ, y  Θ  Y.
Note that if z is induced by c, s, then
Vj c, s 
∑ v j z θ  , θ   p θ  ,
θ Θ
9
j   .
(4)
Let Z IC denote the set of incentive compatible allocations. Let Z  denote the set of
allocations induced by SPNEs.13 It is easy to show Z   Z IC .14
Define
inf
vj 
c j  j

sup
c j  j


min Vj c j , c j , s
s c


, j   ;
(5)
note that v j is well-defined by Lemma 3 in Appendix A.1.
However, min
c j  j
 


j :
sup
c j  j


min Vj c j , c j , s
s c
min
c j  j

sup
c j  j




may or may not exist. Define
min Vj c j , c j , s
s c



is well-defined .
The following is our main result.
Theorem 1
Z








z
Z IC
:
∑
θ Θ

v j z θ  , θ   p θ   v j for every j  ; 


∑ v j z θ  , θ   p θ   v j for every j / .
θ Θ



Consider the following thought experiment: We ask each agent i to report his type


θ i , and require the principals carry out their actions according to z θ i i , i.e., j takes
the action z j θ . Theorem 1 says that we can implement z by a SPNE in the contract game
defined in Section 3.1 iff in the thought experiment, no agent finds it profitable to deviate
from the truthful report and every principal j achieves at least an expected payoff of v j .15
The “only if (i.e., )” direction of the proof of Theorem 1, which is similar to that in
Yamashita [8], can be found in Appendix A.2. We focus on the “if (i.e., )” direction in
Section 4.2.
13 I.e.,
Z   allocation z : there exists a SPNE c, S such that z is induced by c, S c.
intuition is just the revelation principle. See also the proof of Lemma 1 in Yamashita [8, p.795-796].
15 Different from [8], no principal in    can achieve the payoff v in any SPNE, due to the technical
j
14 The
difficulty associated with the infinite-message mechanisms that we consider.
10
4.2
The proof of Theorem 1: the “if” direction
Throughout this subsection, we fix any z  Z IC such that
every j   and
∑
θ Θ
∑ v j z θ  , θ   p θ   v j for
θ Θ
v j z θ  , θ   p θ   v j for every j 
/ . We will construct a SPNE
c , S      such that z is induced by c , S c .
The following two lemmas will be used later, and their proofs can be found in Appendix A.3 and A.4.
s .
s     sub such that 
s   c and z is induced by c, 
Lemma 1 There exists c, 
 
Lemma 2 For any k   , there exists ckk  ckj
 k such that for any ck  k , we can
j


k




find s ck , ckk   ck , ckk satisfying



Vk ck , ckk , s ck , ckk

∑ vk z θ  , θ   p θ  .
(6)
θ Θ
Lemma 1 says that z is induced by c, 
s and 
s is a NE in the subgame defined by
c. Lemma 2 says that for any principal k,
k have the punishing contracts ckk
 principals

and the agents have the punishing NE s ck , ckk to deter principal k from deviating from
the allocation z . It is worth noting that the principals’ punishing contracts ckk does not
depend on the deviating

 strategy (i.e., ck  ck ) chosen by principal k, while the agents’
punishing NE s ck , ckk does.
4.2.1
A sketch of the proof
This contract game has two stages. In stage 1, the principals choose c, and in stage 2, the
agents choose s. Ignoring the IC of the principals, Lemma 1 says that z can be implemented by c, 
s, and the IC of the agents is satisfied, i.e., 
s   c.
To ensure the IC of the principals, we modify the rules of the game, hypothetically.
Suppose there is an extra stage between the two stages, which we call stage 1.5. In stage
1.5, every principal j observes all the contracts chosen in stage 1, and then she revises
11
her contract according to the following protocol16 : revise to ckj if principal k  j is the
unique principal who deviates from the strategy profile c defined in Lemma 1; do not
revise otherwise. Lastly, based on the final contracts offered in stage 1.5, the agents choose
s in stage 2.
Given the extra stage and the described protocol, the following strategy profile
forms a SPNE which implements z .
A) Equilibrium Path: in stage 1, the principals offer c; in stage 1.5, no revision occurs;
in stage 2, the agents choose 
s;
B) Off-Equilibrium Path - case i): in stage 1, the principals offer ck , ck   c; in stage
1.5, principals k revise their contractsto ckk and principal k does not revise her
contract; in stage 2, the agents choose s ck , ckk ;
C) Off-Equilibrium Path - other cases: in stage 1, the principals offer c  c and two or
more principals deviate from c; in stage 1.5, no revision occurs; in stage 2, the agents
choose e c   c fixed in (1).
Lemma 1 implies that z is implemented on the equilibrium path. Furthermore,
the IC of the principals is satisfied: if principal k deviates unilaterally from c in stage 1,
principals k would choose
contracts ckk in stage 1.5 and the agents would
 the punishing

play the punishing NE s ck , ckk in stage 2. By (6) in Lemma 2, principal k does not find
it profitable to deviate from the equilibrium path in stage 1.
The problem for the scheme above is that stage 1.5 does not exist in our original
contract game. In particular, the principals cannot observe who deviates in stage 1, even
though they have the punishing contracts to deter deviation. Instead, it is the agents
(in stage 2) who can observe the deviating principal. Furthermore, in stage 2, even if the
agents inform the principals about the deviating one in stage 1, the principals do not have
a chance to revise their contracts after stage 1. Hence, it should be the agents that revise
the contracts for the principals in stage 2.
To achieve this without stage 1.5, roughly, in stage 1 on the equilibrium path, we
let each principal j offer a set of   contracts including cj and ckj for every k     j
16 In
stage 1.5, the principals follow the protocol mechanically, i.e., they do not behave strategically.
12
and delegate the choice of contracts to the agents in stage 2: on the equilibrium path in
 
s to
which no principal deviates, the agents choose cj j for the principals and play 
implement z ; on the off-equilibrium path in whichprincipal
 k deviates unilaterally to ck ,
the agents choose ckk for principals k and play s ck , ckk to punish principal k.
Still, we are facing one problem. The “set of   contracts” described above is not
a “valid” contract in the contract game defined in Section 3.1. We embed the set of  
contracts into one single valid contract in the next subsection.
4.2.2
The embedding and the extension
 
To define each cj : ∏  ji    j , we first break each ∏  ji into several parts, and
i 
i 
then define cj on these parts separately. For any E  ∏  ji , we use cj  E to denote cj
E.17
with the restricted domain


Fix any a0 , a1 , ..., a2  , a2 1 such that
i 
0  a0  a1  ...  a2   a2 1  1.
Recall    ji  0, 1. Define k  kji   a2k , a2k1  for every  j, i      and
every k  0, 1, 2, ...,  . That is, we fix    1 disjoint closed intervals in , and the
agents use messages in these disjoint intervals to tell the principals about who deviates
from the equilibrium path in stage 1: m ji  0ji means no one deviates in stage 1, and
m ji  kji (with k  0) means principal k deviates in stage 1.
For every k  0, 1, 2, ...,  , k is homeomorphic to ; let hk : k   be a
  1
homeomorphism. We use hk
to denote the inverse function of hk .
Furthermore, let ID k : k  k be the identity function, i.e., ID k m  m for
every m  k . Since k is closed, by the Tietze Extension Theorem (see [5, p.219]), there
exists a continuous function gk :   k such that gk m  ID k m  m for every
m  k . That is, gk , while preserving the definition of ID k on k , extends the domain
continuously to .
We describe the embedding in three cases.
17 I.e., c 
j E
 
: E    j satisfies cj  E m  cj m, m  E.
13
Recall 
s   c in Lemma
A) The equilibrium path: the principals offer c in stage 1.
1, i.e., for every i, θ     Θ, we have




Ui c, 
si , 
si , θ i   Ui c, si , 
si , θ i , si  
∏  ji
j
Θi
.
(7)
 
For every j   , define cj  ∏ 0 : ∏ 0ji    j as follows:
cj

m ji

i 
i 

ji
i 
 cj

 
h0 m ji
i 

 
,  m ji i 
∏ 0ji .
i 
I.e., on the equilibrium path, the agents reports messages in 0ji . Upon receiving the
messages, the principals translate them to messages in  ji via the homeomorphism h0 ,
and then implement c.
For every i   , consider the homeomorphism H 0 : ∏  ji  ∏ 0ji defined as
follows:
H
0

m ji

j

 

1 

0
 h
m ji
For every θ i  Θi , note that 
si  θ i   

j
j
∏  ji
j

j
 
,  m ji j 
∏  ji .
j
and H 0  
si  θ i   


∏ 0ji . That
j
  1
is, each agent i encodes the messages via h0
and transforms the strategy 
si to H 0  
si ,
while each principal j decodes messages via h0 and transforms the strategy H 0  
si back
to 
si before executing the contract cj . As implied by (7), for every i, θ     Θ, we have
 
Θi




si , θ i , si   ∏ 0ji
. (8)
Ui c , H 0  
si , H 0  
si , θ i  Ui c , si , H 0  
j
On the equilibrium path, if we hypothetically require agents report messages in 0ji , (8)
would imply that H 0  
s is a NE in the subgame defined by c . However, agent i may devi-
ate to report messages outside 0ji . To make H 0  
s remain a NE without the hypothetical
requirement, we need to extend cj  ∏ 0 to cj   
ji
i 

cj   
 ji 0j, i
i 

m ji

i 

i 

 ji 0j, i
cj  ∏ 0
ji
i 


g
0
as follows:

m ji


 
 m ji , m j, i 
 ji  0j, i .

i 

,
i 
I.e., any unilateral deviation to a message outside 0ji is first translated to a message
s remains a NE in the subgame defined by cj .
inside 0ji via g0 . As implied by (8), H 0  
14
B) The off-equilibrium path with unilateral deviation In stage 1, suppose
principal


k


offers ck  ck and principals k offer ck . By Lemma 2, s  s ck , ckk   ck , ckk ,
i.e., for every i, θ     Θ, we have
Ui




ck , ckk , si , si , θ i  Ui




ck , ckk , si , si , θ i , si  
∏  ji
j
 
For every j    k , define cj  ∏ k : ∏ kji    j as follows:
cj

m ji

i 

i 
 ckj
ji

i 
 
hk m ji
i 

 
,  m ji i 
Θi
.
(9)
∏ kji .
i 
I.e., the agents reports messages in kji , and principals k translate them to messages in
 ji via hk before implementing ckk .


For every i   , consider the homeomorphism H k :  ji j  ki  ∏ kji
defined as follows:

 



1 



k
k
H mki , m ji jk  mki ,
h
m ji
For every θ i  Θi , note that si θ i   

jk
jk
∏  ji
j



  
,  mki , m ji jk  ki  ∏ kji .
and H k  si θ i   

jk

ki  ∏ kji .
jk
  1
That is, each agent i encodes the messages to principals k via hk
and transforms
the strategy si to H k  si , while principals k decode messages via hk and transform the
strategy H k  si back to si before executing the contract ckk . As implied by (9), for every
i, θ     Θ, we have





 
k

k
k

Ui ck , ck , H  si , H  si , θ i  Ui ck , ck , si , H  si , θ i ,
 
Θi
si   ki  ∏ kji
(10)
.
jk
If we hypothetically require agents report messages in kji to each principal j  k, (10)


would imply that H k  s is a NE in the subgame defined by ck , ck . However, agent i
may deviate to report messages outside kji . To make H k  s remain a NE without the
hypothetical requirement, we need to extend cj  ∏ k to cj   
cj   
i 
 ji kj, i



m ji


i 
 m ji , m j, i 

i 
ji
 cj  ∏ k

i 
15
i 
ji

i 
 ji kj, i
 
gk m ji
 ji  kj, i

.
i 

,

as follows:
I.e., any unilateral deviation to a message outside kji is first translated to a message
inside kji via gk . As implied by (10), H k  s remains a NE in the subgame defined by


ck , ck .
C) The other off-equilibrium paths In the above, we have defined each cj on the re
  
stricted domain
 ji  kj, i . Since the restricted domain is closed and cj is
k  j i 
continuous on it, by the Tietze Extension Theorem, there exists a continuous function
 
cj : ∏  ji    j such that
i 
cj m  cj  

k j i 
m 
 
k  j i 
 ji kj, i

m ,

 ji  kj, i ,
which completes the definition of cj .
4.2.3
The definition of S
Finally, we define S as follows:
S c   H 0  
s;






k
k
S ck , ck  H  s ck , ck , k   , ck  k  ck  ;
S c  e c otherwise;
where e c   c is fixed for every c in (1).
This completes the definition of c , S . Clearly, c , S  is a SPNE as argued above,
and z is induced by c , S c .
5
Discussions
We conclude the paper with several discussions.
16
5.1
Finite messages versus infinite messages
Unlike Yamashita [8], we have to consider infinite-message spaces. In our proof of the
“if” direction of Theorem 1, a message m ji sent from agent i to principal j plays two roles:
i) it recommends a contract c j for principal j, and ii) it reports i’s “message” when c j is
adopted. That is, we have to establish a surjective function from the message set M ji to
L  M ji , where L denotes the set of potential contracts recommended by i. Clearly, such a
surjective function exists only if M ji is an infinite set.
Nevertheless, it is without loss of generality to focus on infinite-message mechanisms. Let Q j denote the set of all finite-message mechanisms for principal j, i.e.,


 
   
 j  q j : Wji i I    j  Wji   ∞,   j, i      .
Although  j   j  , every element in  j can be represented by some element in  j :
For any allocation z that is induced by some q  Π j J  j and s   q, there exist c 
Π j J  j and s   c such that z is induced by c and s .18 Thus, any allocation, which is
implementable via finite-message mechanisms, can be implemented via infinite-message
mechanisms.
5.2
Stochastic equilibrium allocations
For notational ease, we focus on deterministic equilibrium allocations throughout the paper. With some modification, our result also applies to stochastic equilibrium allocations.
A stochastic allocation is a function β : Θ    . We say a stochastic allocation
β is independent if β θ  is an independent distribution over Y for every θ; otherwise, we
say that β is correlated.
Recall that we use a “direct mechanism” on the equilibrium path to implement a
potential equilibrium allocation. However, an implicit assumption of our model is that
the principals take actions independently for each profile of messages received. Consequently, a stochastic allocation can be implemented on the equilibrium path if and only if
it is independent. Hence, Theorem 1 remains true for independent stochastic equilibrium
allocations.
18 The
proof is similar to the proof of Lemma 1 and is omitted.
17
For correlated stochastic equilibrium allocations, our characterization is true if we
endow the principals with the correlation-generating device used in Kalai, Kalai, Lehrer
and Samet [3] and Peters and Troncoso-Valverde [6], so that, for each profile of message
received, the principals are induced by the correlation device to take correlated actions.
5.3
Szentes’ critique
Following the setup of Yamashita [8], Szentes [7] constructs a complete-information example, in which a principal cannot even achieve her min-max utility. The problem, according
to [7], is that the principals delegate their actions to the agents in [8]. Szentes then modifies Yamashita’s model to a no-delegation model, and proves a folk theorem under the
complete-information setup.19
In Yamashita’s setup, the principals are restricted to choose deterministic contracts.
If we allow for random contracts, every principal in Szentes’ example is able to achieve
her min-max value. Hence, our model, which allows for random contracts, is immune to
the critique raised by Szentes [7].
5.4
A caveat and a re-interpretation
A caveat of our theorem is that v j defined in (5) is difficult to compute, which seems to
make Theorem 1 hard to use. However, the following result shows that we can apply
Theorem 1 without knowing v j .
Theorem 2 For any z  Z  , we have



z  Z IC : ∑ v j z θ  , θ  p θ  
θ Θ
∑ v j z θ  , θ   p θ  ,  j  
θ Θ

 Z .
Theorem 2 says that any IC allocation, which dominates another SPNE allocation
regarding the principals’ payoffs, can be implemented by a SPNE. Though Theorem 2 is
immediately implied by Theorem 1, the arguments in Section 4.2 provide a constructive
19 It
remains an open question to extend Szentes’ folk theorem to incomplete-information setups.
18
way to prove Theorem 2 directly. Suppose z  Z  is implemented by a SPNE c, S.
Then, ck can be used as punishing contracts for principals k to deter principal k from
deviating from the equilibrium path for z , i.e.,
∑ vk
θ Θ


z θ  , θ  p θ  
∑ vk z θ  , θ   p θ   Vk c, S c  Vk
θ Θ



ck , ck , S ck , ck , ck  k ,
where the last inequality follows from the fact that c, S is a SPNE. Therefore, with ckk
replaced by ck , the argument in Section 4.2 proves Theorem 2. We use the following
example to illustrate the idea.
Example 2 (A Public Good Project) Multiple principals decide whether to participate in a publicgood project and how much to invest if they participate. The project can be completed if and only
if every principal participates. Furthermore, multiple agents privately observe the states which
determines the payoffs of the project to all the players.
Clearly, it is a SPNE that no principal participates, and all players get 0 in this equilibrium.
Furthermore, by Theorem 2, any IC allocation, in which the principals gets non-negative payoffs,
can be implemented by a SPNE.20
5.5
Comparison to Peters and Troncoso-Valverde [6]
We differ from Yamashita [8] only on the strategy spaces: we allow for mixed strategies,
but [8] does not. We differ from Peters and Troncoso-Valverde[6] on three respects.
First, as in Yamashita [8], we follow the classical setup in contract theory, in which
minimal communication is required: the principals first offer the contracts, i.e., functions
from (exogenously given) message spaces to action spaces; the agents then send their
messages, and the contracts are executed. However, Peters and Troncoso-Valverde [6] requires a particular procedure of two-stage communication among all players. Due to this
difference, our equilibrium characterization is more complicated than that in [6]. Hence,
an alternative way to interpret [6] is that it finds a communication protocol which induces
a simple and elegant characterization of equilibrium allocations.
20 To
implement such an allocation, the principals use the direct mechanism on the equilibrium path; if
any principal unilaterally deviates, the agents recommed non-participation for the other principals.
19
Second, we adopt the solution concept of SPNE, while Bayesian Nash equilibrium
is adopted in [6]. That is, different from [6], we require the agents’ equilibrium strategy
profile remains a continuation equilibrium in the subgame defined by any contract profile
offered by the principals.21 Due to this difference, we define v j as a min-max-min value,
while [6] defines v j as a min-max value, where the extra “min” takes care of all potential
equilibria in subgames.
Finally, in Peters and Troncoso-Valverde [6], every player is both a principal and
an agent. Through the two-stage communication procedure, all players observe anyone
who deviates from the equilibrium path, and they can revise their “equilibrium contracts”
to “punishing contracts” so as to punish the deviator. However, in Yamashita [8] and
this paper, though the principals are endowed with “punishing contracts,” they have
to offer their contracts before observing the deviating principal, and they cannot revise
their contracts later. Furthermore, the agents, who observe the deviating principal, are
not endowed with “punishing contracts.” Hence, it is the agents who tell the principals
how to revise their “equilibrium contracts” to “punishing contracts.” Thus, different from
[6], we overcome a conceptual and technical obstacle: the transmission of information
regarding the deviating principal from the agents (who observe the deviator but are not
endowed with “punishing contracts”) to the principals (who do not observe the deviator
but are endowed with “punishing contracts”).
A
Appendix
A.1
Lemma 3
Lemma 3 min Vj c, s is well defined for every  j, c     .
s c
Proof. Fix any  j, c     . Suppose inf Vj c, s  α. Then, for any positive ins c
  c such that Vj c, sn   α  n1 . Since  sub is compact, sn 
has a convergent subsequence. With abuse of notation, let sn  denote this convergent
subsequence, i.e., sn   
s for some 
s   sub .
teger n, there exists
21 See
sn
more discussion in Troncoso-Valverde [6, Section 7].
20
Note that 

∏  ji
j


is endowed with weak topology and isub  

∏  ji
j
Θi
is endowed with product topology. Furthermore, for any  j, i, θ i , both Vj c, s and Ui c, s, θ i 
are continuous on Ssub .22 Consequently, the NE is upper hemi-continuous, i.e., sn   c
s imply 
s   c. Hence,
for all n and sn   
Vj c, 
s  inf Vj c, s  α.
(11)
s c
1
n
s imply
for all n and sn   


1
n
Vj c, 
s  lim Vj c, s   lim α 
 α.
n∞
n∞
n
Furthermore, Vj c, sn   α 
(12)
(11) and (12) imply Vj c, 
s  α  inf Vj c, s. Finally, 
s   c implies that min Vj c, s
s c
is well defined.
A.2
s c
Proof of Theorem 1: the “only if” direction


Fix any allocation z  Z  , i.e., there exists a SPNE c, S     such that z is induced





by c, S c . For notational ease, let 
s denote S c. Hence, by (4), we have
Vj c, 
s 
We now show
∑ v j z θ  , θ   p θ  ,
θ Θ
∑ v j z θ  , θ   p θ 
θ Θ
∑ v j z θ  , θ   p θ 
θ Θ
j   .
 v j if j  ;
 v j if j 
/ .
Suppose otherwise. I.e., there are two cases: 1) there exists j   such that
p θ   v j ; 2) there exists j    such that
∑ v j z θ  , θ   p θ   v j .
∑ v j z θ  , θ  
θ Θ
θ Θ
In case 1),
Vj c, 
s 
∑ v j z θ  , θ   p θ 
θ Θ
 vj 
22 This
inf
c j  j

sup
c j  j


min Vj c j , c j , s
s c


 sup
c j  j

min
s c j ,
c j 

Vj c j , c j , s

is immediately implied by the definition of weak topology and the fact that c is continuous.
21

.
In case 2),
s 
Vj c, 
∑ v j z θ  , θ   p θ 
θ Θ
 vj 
inf
c j  j

sup
c j  j


min Vj c j , c j , s
s c


 sup
c j  j
where the last inequality follows from j 
/ , i.e., arg min
c j  j
does not exist.


min
s c j ,
c j 
sup
c j  j




Vj c j , c j , s
min Vj c j , c j , s
s c


,

Therefore, in both cases, we have
Vj c, 
s  sup
c j  j

min
s c j ,
c j 
which implies that there exists cj   j s.t.
Vj c, 
s 
s cj ,
c j
 Vj
I.e.,
min



Vj c j , c j , s





V
c
,
c
,
s
 j
j j
cj , c j , S

cj , c j


,
.









Vj cj , c j , S cj , c j  Vj c j , c j , S c j , c j ,



contradicting the fact that c, S is an equilibrium (i.e., (2)).

A.3
Proof of Lemma 1
Lemma 1 There exists c, 
s     sub such that 
s   c and z is induced by c, 
s .
Proof. Recall z  Z IC . Suppose every principal j offers the direct mechanism zj (where
 
z  zj
). Since z is incentive compatible (see (3)), truthful reporting by the agents
j
is a NE and z is implemented. That is, (3) is equivalent to



u
z
θ
,
θ
,
θ
,
θ
∑ i j  i i  j i i  p θ i , θ i 
θ i  Θ i


θ ji  j Θi  

∑
θ i  Θ i
 


ui z j θ ji , θ i j , θ i , θ i  p θ i , θ i 
22
(13)



dρ, ρ   Θi   .
However, zj is not a “valid” contract, i.e., it is not a continuous function from ∏  ji
i 
 
to   j . We thus need to transform every zj to a valid contract.
First, based on zj : ∏ Θi   j , consider the continuous function Γzj : ∏  Θi  
i 
i 
 
  j defined as follows:

 
Γzj µi i y j 

∑

θ  θ Θ:
zj

θ y j

∏ µi  θ i  ,
i 
 µi i  ∏  Θi  , y j   j .
i 
That is, under the direct mechanism zj , the agents report deterministic types, while Γzj
is the stochastic direct mechanism induced by zj , in which the agents report stochastic
types. Clearly, truthful reporting remains a NE under Γzj .
Second, for any a, we use δ a to denote the Dirac measure on the set  a. For every
i   , fix any injective function i : Θi   ji . We use i θ i  to represent θ i , i.e., by
reporting i θ i    ji , agent i informs principal j of i’s type θ i . Consider the continuous
function


Λj : i θ i i : θ  Θ 


  
Λj i θ i i  δθ i

i 
∏   Θi  ,
i 
, θ  Θ.


Since i θ i i : θ  Θ is a closed subset of  ji i , by the Tietze Extension Theo-
rem, there exists a continuous function


Λ j :  ji i 
∏  Θi  , s.t.
i 




Λ j i θ i i  Λj i θ i i , θ  Θ.
That is, i θ i    ji is used by agent i to inform principal j about i’s type θ i , and any
other message in  ji reported by i would suggest a stochastic type in  Θi .
Third, define c, 
s as follows:
cj  Γzj  Λ j ,  j   ;
23



s i  θ i   δ i  θ i 
j
,  i, θ     Θ.
I.e., cj is the generalized stochastic “direct mechanism,” and 
si is the “truthful reporting”
s  .
strategy. By (13), 
s   c and z is induced by c, 
A.4
Proof of Lemma 2
ckk
Lemma 2 For any k   , there exists




find s ck , ckk   ck , ckk satisfying
 
 ckj



Vk ck , ckk , s ck , ckk

jk
 k such that for any ck  k , we can
∑ vk z θ  , θ   p θ  .
(14)
θ Θ
Proof. First, for any k   , we will show below that there exists ckk  k such that



 
(15)
sup
min Vk ck , ckk , s
 ∑ vk z θ  , θ   p θ  .
 ,ck

s

c
 k k 
ck k
θ Θ


Second, fix any k   and ckk described above. For any ck  k , pick some s ck , ckk 

 
arg min Vk ck , ckk , s . Hence,
s ck ,ckk 


 

 
Vk ck , ckk , s ck , ckk

min Vk ck , ckk , s
(16)
s ck ,ckk 



 
 sup
min Vk ck , ckk , s .
 k
ck k s ck ,ck 
Therefore, (14) is implied by (16) and (15).
Finally, we prove (15) by considering two cases: k   and k 
/ .






For any k  , pick ckk  arg min  sup
min Vk ck , ck , s . Then,

ck k
ck k s ck ,ck 
sup
ck k

min Vk
s ck ,ckk 


ck , ckk , s




min  sup
ck k
 vk 
24
ck k

min
s ck ,ck 

 

Vk ck , ck , s 
∑ vk z θ  , θ   p θ  ,
θ Θ
i.e., (15) holds. For any k 
/ , since



 
 
inf  sup
min Vk ck , ck , s   vk  ∑ vk z θ  , θ   p θ  ,

ck k
ck k s ck ,ck 
θ Θ
by the definition of "inf," we have

ckk  k s.t. sup
ck k
min Vk
s ck , ckk 


ck , ckk , s



∑ vk z θ  , θ   p θ  ,
θ Θ
i.e., (15) holds.
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