An Outcome Mechanism for Partially Honest Nash
Implementation
Makoto Hagiwara, Hirofumi Yamamuray, Takehiko Yamatoz
February 5, 2016
Abstract
Dutta and Sen [6] show that if there exists at least one partially honest agent, then
no veto power is su cient for partially honest implementation in Nash equilibria. Also,
Kimya [8] shows that if there are at least two partially honest agents, then unanimity
is su cient for partially honest implementation in Nash equilibria. While in Dutta
and Sen [6]'s mechanism and Kimya [8]'s mechanism, each agent reports a preference
pro le, an outcome, and a positive integer, it is in fact unnecessary to ask agents to
reveal a preference pro le. We introduce an outcome mechanism in which each agent
only reports an outcome and a positive integer from 1 to n. We show that the results
of Dutta and Sen [6] and Kimya [8] are still valid by our mechanism with a smaller
strategy space.
JEL Classi cation: C72, D71, D78
Key words: Social choice correspondence, Partial honesty, Nash implementation,
No veto power, Unanimity, Outcome mechanism, Strategy space reduction
1
Introduction
The theory of mechanism design aims to identify a mechanism achieving a socially goal across
a domain of preferences of the agents. The literature on mechanism design usually assumes
that each agent only cares about the outcome(s) obtained from the mechanism.
However, some recent studies assume that at least some agents may have an intrinsic
preference for honesty (Dutta and Sen [6], Lombardi and Yoshihara [10] [12]).1 While there
Department of Social Engineering, Graduate School of Decision Science and Technology, Tokyo Institute
of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552 Japan; E-mail: [email protected]
y
Department of Social Engineering, Graduate School of Decision Science and Technology, Tokyo Institute
of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552 Japan; E-mail: [email protected]
z
Department of Social Engineering, Graduate School of Decision Science and Technology, Tokyo Institute
of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552 Japan; E-mail: [email protected]
1
There are other papers about intrinsic preference for honest. For example, Diss et al. [2], Doghmi [3],
Doghmi and Ziad [4] [5], Kartik et al. [7], Kimya [8], Korpela [9], Lombardi and Yoshihara [11] [13] [14],
Matsushima [15] [16], Mukherjee and Muto [18], Ortner [19], Saporiti [21].
1
are various options about how to model such a preference for honesty, we follow the de nition
of partially honest agents which we generalize the original de nition of partially honest agents
introduced by Dutta and Sen [6]. We assume the following: Suppose the mechanism requires
each agent to announce an outcome not a preference pro le. Then, an agent is said to be
partially honest if she prefers to lie whenever the outcome obtained by lying is preferred to
the outcome obtained by telling the socially desirable outcome.
Maskin [17] shows that if there are at least three agents and all agents are self-interested,
then Maskin monotonicity and no veto power are su cient for implementation in Nash equilibria. Also, Dutta and Sen [6] show that if there are at least three agents and there exists at
least one partially honest agent, then no veto power is su cient for partially honest implementation in Nash equilibria. Hence, we no longer need Maskin monotonicity as a necessary
condition of implementation in Nash equilibria as discussed in Maskin [17]. Moreover, Kimya
[8] shows that if there are at least three agents and all agents are partially honest, then unanimity is su cient for partially honest implementation in Nash equilibria. Kimya [8] also
mentions that the result is still valid if there are at least two agents.
In Maskin [17]'s mechanism, Dutta and Sen [6]'s mechanism and Kimya [8]' mechanism,
each agent reports a preference pro le, an outcome, and a positive integer. But it is in fact
unnecessary to ask agents to reveal a preference pro le by constructing a mechanism in which
each agent reports an outcome and a positive integer from 1 to n. We call this an outcome
mechanism. We show that if there are at least three agents and there exists at least one
partially honest agent, then no veto power is su cient for partially honest implementation
in Nash equilibria by an outcome mechanism. Also, we show that if there are at least three
agents and there are at least two partially honest agents, then unanimity is su cient for
partially honest implementation in Nash equilibria by an outcome mechanism.
This paper is organized as follows. Section 2 presents the theoretical framework and
outlines the basic model. Section 3 reports assumptions on partially honest agents. Section
4 reports our results about partially honest implementation in Nash Equilibria. Section 5
provides concluding remarks.
2
Notation
Let A be the arbitrary set of outcomes, and N = f1; :::; ng be the set of agents, with generic
element i. Let Ri be a preference ordering for agent i 2 N over A and <i be the set of
all preference orderings for agent i 2 N .2 Let Pi and Ii be the asymmetric and symmetric
components of Ri 2 <i , respectively. Let R = (R1 ; :::; Rn ) be a preference pro le and
< = i2N <i be the set of all preference pro les. Let D = i2N Di
< where Di
<i for
each i 2 N be a domain.
A social choice correspondence (SCC) is a mapping F : D
A that speci es a non-empty
set F (R)
A for each R 2 D. Denote the class of admissible SCCs by F. Given a SCC
2
A preference ordering Ri 2 Di over A is a complete and transitive binary ordering over A. A ordering
Ri 2 Di over A is complete if for each a; b 2 A, aRi b or bRi a; transitive if for each a; b; c 2 A, if aRi b and
bRi c, then aRi c.
2
F 2 F, an outcome a 2 A is F -optimal at R 2 D if a 2 F (R).
A mechanism
consists of a pair (M; g) where M = i2N Mi , Mi is the message (or
strategy) space of agent i 2 N , and g : M ! A is the outcome function mapping each
message pro le m 2 M into an outcome g(m) 2 A. For each i 2 N and each m 2 M;
let m i 2 M i = j6=i Mj be the message pro le except agent i 2 N . That is, m i =
(m1 ; :::mi 1 ; mi+1 ; :::; mn ). The message pro le m 2 M is also written as (mi ; m i ). Given
m 2 M and m0i 2 Mi ; (m0i ; m i ) is the message pro le (m1 ; :::mi 1 ; m0i ; mi+1 ; :::; mn ) obtained
after the replacement of mi 2 Mi by m0i 2 Mi .
Let ( ; R) be a game induced by a mechanism and a preference pro le R 2 D. A
message pro le m 2 M is a Nash equilibrium in the game ( ; R) if for each i 2 N and
each m0i 2 Mi , g(mi ; m i )Ri g(m0i ; m i ). The set of Nash Equilibria in ( ; R) is denoted by
N E( ; R). Also, the set of Nash equilibrium outcomes in ( ; R) is denoted by N EA ( ; R) =
fa 2 Aj9m 2 N E( ; R) with g(m) = ag.
A mechanism implements a SCC F in Nash equilibria if for each R 2 D, F (R) =
N EA ( ; R).
3
Assumptions on Partially Honest Agents
The literature on mechanism design usually assumes that each agent only cares about the
outcome(s) obtained from the mechanism. However, some recent studies assume that at least
some agents may have an intrinsic preference for honesty. While there are various options
about how to model such a preference for honesty, we follow the de nition of partially honest
agents which we generalize the original de nition of partially honest agents introduced by
Dutta and Sen [6].
For each i 2 N and each mechanism , a truth-telling correspondence Ti is a mapping
Mi that speci es a non-empty set of truth-telling messages Ti (R; F ) Mi
Ti : D F
for each (R; F ) 2 D F. Given a mechanism , a truth-telling correspondence Ti , and a
pair (R; F ) 2 D F, we say that agent i 2 N behaves truthfully at m 2 M if and only if
mi 2 Ti (R; F ).
We focus on following mechanisms and truth-telling correspondences.
Example 1. Preference-outcome mechanisms. Dutta and Sen [6] focus on mechanisms
in which each agent reports a preference pro le, an outcome, and a supplemental message.
The message space of agent i 2 N consists of Mi = D A Si , where Si denotes the set of
supplemental messages.
For each i 2 N , mi = (Ri ; ai ; si ) is a truth-telling message if and only if Ri = R. In this
case, for each i 2 N and each (R; F ) 2 D F, a truth-telling correspondence is de ned by
Ti (R; F ) = fRg F (R) Si .
Example 2. Outcome mechanisms. In this paper, we focus on mechanisms in which
each agent reports an outcome and a supplemental message but not a preference pro le. The
message space of agent i 2 N consists of Mi = A Si .
3
If the mechanism designer asks agents which outcome is socially desirable, then for each
i 2 N , mi = (ai ; si ) is a truth-telling message if and only if ai 2 F (R). In this case,
for each i 2 N and each (R; F ) 2 D F, a truth-telling correspondence is de ned by
Ti (R; F ) = F (R) Si .
For each i 2 N , each R 2 D, each mechanism , and each truth-telling correspondence
3
Ti , agent i's preference ordering %R
i over M at R 2 D is de ned below.
De nition 1. An agent i 2 N is partially honest if for each (R; F ) 2 D F and each
(mi ; m i ); (m0i ; m i ) 2 M , the following properties hold:
= Ti (R; F ) and if g(mi ; m i )Ri g(m0i ; m i ); then (mi ; m i ) R
(1) If mi 2 Ti (R; F ) and m0i 2
i
(m0i ; m i ).
0
0
(2) In all other cases, (mi ; m i ) %R
i (mi ; m i ) if and only if g(mi ; m i )Ri g(mi ; m i ).
The rst part of the de nition captures the agent's limited preference for honesty - she
strictly prefers (mi ; m i ) to (m0i ; m i ) when she reports truthfully in (mi ; m i ) but not in
(m0i ; m i ) if g(mi ; m i ) is at least as good as g(m0i ; m i ).
De nition 2. An agent i 2 N is self-interested if for each (R; F ) 2 D
(mi ; m i ); (m0i ; m i ) 2 M ,
F and each
0
0
(mi ; m i ) %R
i (mi ; m i ) if and only if g(mi ; m i )Ri g(mi ; m i ).
Since agents who are self-interested care only about the outcomes obtained from the
mechanism, their preference orderings over M are straightforward to de ne.
The traditional literature on mechanism design usually assumes the following:
Assumption 0. There is no partially honest agent in N . That is, all agents are selfinterested.
In contract to the traditional literature, Diss et al. [2], Doghmi [3], Doghmi and Ziad
[4] [5], Dutta and Sen [6], and Lombardi and Yoshihara [10] [12] consider the following
assumption:
Assumption 1. There exists at least one partially honest agent in N . The mechanism
designer knows that there exists at least one partially honest agent in N , though she does
not know their identities or their exact number.
Moreover, Korpela [9], Matsushima [16], and Saporiti [21] consider the following assumption:
3
Let
R
i
and
R
i
be the asymmetric and symmetric components of %R
i , respectively.
4
Assumption n. There are n partially honest agents in N . That is, all agents are partially
honest.
We introduce the following new assumption:
Assumption 2. There are at least two partially honest agents in N . The mechanism
designer knows that there are at least two partially honest agents in N , though she does not
know their identities or their exact number.
Clearly, Assumption 2 is stronger than Assumption 1, but signi cantly weaker than Assumption n.
We introduce our formal models with partially honest agents under Assumptions k 2
f1; 2; ng. For each k 2 f1; 2; ng, let Hk = fS
N j jSj
kg. For each R 2 D and each
R;H
k
R;H
R;H
H 2 H , let % = (%1 ; :::; %n ) be the preference pro le over M such that for each
i 2 H, %R;H
is de ned by De nition 1 and for each i 2 N nH, %R;H
is de ned by De nition
i
i
2.
Let ( ; (Ti )i2N ; %R;H ) be a game with partially honest agents induced by a mechanism ,
a truth-telling correspondence Ti for each i 2 N , and a preference pro le %R;H . A message
pro le m 2 M is a Nash equilibrium with partially honest agents in ( ; (Ti )i2N ; %R;H ) if
for each i 2 N and each m0i 2 Mi , (mi ; m i ) %R;H
(m0i ; m i ). The set of Nash equilibria
i
with partially honest agents in ( ; (Ti )i2N ; %R;H ) is denoted by N E( ; (Ti )i2N ; %R;H ). Note
N E( ; R). Also, the set of Nash equilibrium outcomes with
that N E( ; (Ti )i2N ; %R;H )
partially honest agents in ( ; (Ti )i2N ; %R;H ) is denoted by N EA ( ; (Ti )i2N ; %R;H ) = fa 2 A
j 9m 2 N E( ; (Ti )i2N ; %R;H ) with g(m) = ag:
Under Assumptions 1 and 2, the mechanism designer knows that there are partially honest agents in N but does not know who these agents are. Hence, the mechanism designer
needs to cover all feasible cases of partially honest agents to her knowledge. To enable the
mechanism designer to implement a SCC with partially honest agents, we amend the standard de nition of implementation as follows:
De nition 3. Under Assumptions k 2 f1; 2; ng, a mechanism
partially honest implements a SCC F in Nash equilibria if for each R 2 D and each H 2 Hk , F (R) =
N EA ( ; (Ti )i2N ; %R;H ).
4
Main Results
First, we review previous results on implementation in Nash equilibria under Assumption 0
and partially honest implementation in Nash equilibria under Assumption 1.
Maskin [17] introduces the following properties of SCC's.
For each i 2 N , each Ri 2 Di , and each a 2 A, let L(Ri ; a) = fb 2 A j aRi bg be the lower
contour set of a 2 A for i 2 N at Ri 2 Di .
A SCC F satis es Maskin monotonicity if for each R,R0 2 D and each a 2 F (R), if for
each i 2 N , L(Ri ; a) L(Ri0 ; a), then a 2 F (R0 ). Maskin monotonicity requires that if an
5
outcome a 2 A is F -optimal at some preference pro le and the pro le is then altered so that,
in each agent's ordering, the outcome a does not fall below any outcome that was not below
before, then the outcome a remains F -optimal at the new pro le.
A SCC F satis es no veto power if for each i 2 N , each R 2 D, and each a 2 A if for
each j 6= i, L(Rj ; a) = A, then a 2 F (R). No veto power says that if an outcome a 2 A
is at the top of (n 1) agents' preference orderings, then the last agent cannot prevent the
outcome a from being F -optimal.
The following property is important for partially honest implementation in Nash equilibria
under Assumption 2.
A SCC F satis es unanimity if for each R 2 D and each a 2 A, if for each i 2 N ,
L(Ri ; a) = A, then a 2 F (R). Unanimity says that if an outcome a 2 A is at the top of all
agents' preference orderings, then the outcome a is F -optimal at the preference pro le.
Maskin [17] provides a su cient condition for implementation in Nash equilibria under
Assumption 0 by a preference-outcome mechanism.
Theorem 1. (Maskin [17]) Let n 3 and suppose Assumption 0 holds. Then, every SCC
F satisfying Maskin monotonicity and no veto power can be implemented in Nash equilibria
by a preference-outcome mechanism.4
Dutta and Sen [6] provide a su cient condition for partially honest implementation in
Nash equilibria under Assumption 1 by a preference-outcome mechanism.
Theorem 2. (Dutta and Sen [6]) Let n 3 and suppose Assumption 1 holds. Then, every
SCC F satisfying no veto power can be partially honest implemented in Nash equilibria by a
preference-outcome mechanism.5
We succeed in designing a simpler mechanism for partially honest implementation in Nash
equilibria under Assumption 1.
Theorem 3. Let n 3 and suppose Assumption 1 holds. Then, every SCC F satisfying no
veto power can be partially honest implemented in Nash equilibria by an outcome mechanism.
Proof: Let F be a SCC satisfying no veto power. We construct an outcome mechanism
= (M; g). The message space of agent i 2 N consists of Mi = A N . Denote an element
of Mi by mi = (ai ; k i ). The outcome function g : M ! A is de ned as follows:
Rule 1 : If there is i 2 N such that for each j 6= i, mj = (a; k j ), then g(m) = a.
Rule 2 : In all other cases, g(m) = ai , where i = ( i2N k i )(mod n) + 1:
For each i 2 N , a truth-telling correspondence is de ned by Ti (R; F ) = F (R)
The proof consists of two lemmata.
4
N.
Maskin [17] also shows that if a SCC F does not satisfy Maskin monotonicity, it cannot be implemented
in Nash equilibria.
5
Lombardi and Yoshihara [12] extend Theorem 2 by providing a necessary and su cient condition for
partially honest implementation in Nash equilibria under Assumption 1. Also, Doghmi [3] provides a necessary
condition for partially honest implementation in Nash equilibria under Assumption 1.
6
Lemma 1. For each R 2 D and each H 2 H1 , F (R)
N EA ( ; (Ti )i2N ; %R;H ).
Proof: For each i 2 N and each a 2 F (R), let mi = (a; k i ). By Rule 1, g(m) = a. No
unilateral deviation can change the outcome and mi 2 Ti (R; F ) for each i 2 N . Hence,
m 2 N E( ; (Ti )i2N ; %R;H ) and a = g(m) 2 N EA ( ; (Ti )i2N ; %R;H ).
Lemma 2. For each R 2 D and each H 2 H1 , N EA ( ; (Ti )i2N ; %R;H )
F (R).
Proof: There are two cases to consider.
Case 1. For each i 2 N , mi = (a; k i ) such that a 2
= F (R).
We show that if g(m) 2
= F (R), then m 2
= N E( ; (Ti )i2N ; %R;H ). By Rule 1, g(m) =
a 2
= F (R). Under Assumption 1, there exists a partially honest agent h 2 H. Let m0h =
(a0h ; k 0h ) be such that a0h 2 F (R). By the de nition of the truth-telling correspondence,
mh 2
= Th (R; F ) and m0h 2 Th (R; F ). By Rule 1, g(m0h ; m h ) = a so that g(m0h ; m h ) = g(m).
Since h 2 H, (m0h ; m h ) R;H
(mh ; m h ). Hence, m 2
= N E( ; (Ti )i2N ; %R;H ).
h
Case 2. There are i; j 2 N (i 6= j) such that ai 6= aj .
Let the outcome be some b 2 A. Then, any one of (n 1) agents can deviate, precipitate
the modulo game, and be the winner of the modulo game. Clearly, if the original announcement is to be a Nash equilibrium, then it must be the case that L(Rj ; b) = A for (n 1)
agents. Then since F satis es no veto power, b 2 F (R).
Kimya [8] shows that unanimity is su cient for partially honest implementation in Nash
equilibria under Assumption n by a preference-outcome mechanism. Kimya [8] also mentions
that the result is still valid under Assumption 2.
Theorem 4. (Kimya [8]) Let n 3 and suppose Assumption 2 holds. Then, every SCC F
satisfying unanimity can be partially honest implemented in Nash equilibria by a preferenceoutcome mechanism.
We succeed in designing a simpler mechanism for partially honest implementation in Nash
equilibria under Assumption 2.
Theorem 5. Let n
3 and suppose Assumption 2 holds. Then, every SCC F satisfying
unanimity can be partially honest implemented in Nash equilibria by an outcome mechanism.
Remark 1. Let us give an example of a SCC which satis es unanimity but violates Maskin
monotonicity and no veto power if D = <:
Strong Pareto correspondence(SP): SP(R) = fa 2 Aj9b
/ 2 A such that for each i 2 N ,
bi Ri ai , and for some i 2 N; bi Pi ai g
Since Maskin [17] shows that Maskin monotonicity is necessary for implementation in Nash
equilibria under Assumption 0, SP cannot be implemented in Nash equilibria under Assumption 0.
7
On the other hand, since SP satis es unanimity, SP can be partially honest implemented
in Nash equilibria under Assumption 2.
Proof of Theorem 5. Let F be a SCC satisfying unanimity. First, we consider the case
that n 4. We construct an outcome mechanism = (M; g). The message space of agent
i 2 N consists of Mi = A N . Denote an element of Mi by mi = (ai ; k i ). The outcome
function g : M ! A is de ned as follows:
Rule 1 : If there is i 2 N such that for each j 6= i, mj = (a; k j ), then g(m) = a.
Rule 2 : In all other cases, g(m) = ai , where i = ( i2N k i )(mod n) + 1:
For each i 2 N , a truth-telling correspondence is de ned by Ti (R; F ) = F (R)
This case consists of two lemmata.
Lemma 3. For each R 2 D and each H 2 H2 , F (R)
N.
N EA ( ; (Ti )i2N ; %R;H ).
The proof of Lemma 3 is omitted. It follows from the same reasoning as Lemma 1.
Lemma 4. For each R 2 D and each H 2 H2 , N EA ( ; (Ti )i2N ; %R;H )
F (R).
Proof: We show that if g(m) 2
= F (R), then m 2
= N E( ; (Ti )i2N ; %R;H ). There are three
cases to consider.
Case 1. For each i 2 N , mi = (a; k i ) such that a 2
= F (R).
By Rule 1, g(m) = a 2
= F (R). Under Assumption 2, there exists a partially honest
agent h 2 H. Let m0h = (a0h ; k 0h ) be such that a0h 2 F (R). By the de nition of the truthtelling correspondence, mh 2
= Th (R; F ) and m0h 2 Th (R; F ). By Rule 1, g(m0h ; m h ) =
0
a so that g(mh ; m h ) = g(m). Since h 2 H, (m0h ; m h ) R;H
(mh ; m h ). Hence, m 2
=
h
R;H
N E( ; (Ti )i2N ; % ).
Case 2. There is i 2 N such that for each j 6= i, mj = (a; k j ) such that a 2
= F (R) and
mi = (b; k i ) such that b 6= a.
By Rule 1, g(m) = a 2
= F (R). Under Assumption 2, since jHj 2 there exists a partially
honest agent h 2 Hnfig.6 Without loss of generality, let i = 1 and h = 2. Let m02 = (a02 ; k 02 )
be such that a02 2 F (R) and ( j6=2 k j + k 02 )(mod n) + 1 = 3.7 By the de nition of the truthtelling correspondence, m2 2
= T2 (R; F ) and m02 2 T2 (R; F ). By Rule 2, g(m02 ; m 2 ) = a3 = a
so that g(m02 ; m 2 ) = g(m). Since agent 2 is partially honest, (m02 ; m 2 ) R;H
(m2 ; m 2 ).
2
R;H
Hence, m 2
= N E( ; (Ti )i2N ; % ).
Case 3. In all other cases, Rule 2 is applied.
Suppose g(m) 2
= F (R). Since F satis es unanimity, there is i 2 N and b 2 A such that
0
bPi g(m). Let mi = (b; k 0i ) 6= mi be such that ( j6=i k j + k 0i )(mod n) + 1 = i. By Rule
6
Note that under Assumption 1, there is no partially honest agent in N nfig when jHj = 1 and agent i is
partially honest.
7
While in the modulo game, any agent can exactly designate the winner from all agents, in the integer
game, any agent can designate the winner from only two agents. Since a partially honest agent h 2 Hnfig
needs to make agent j 2 N nfi; hg the winner of the modulo game, we must use the modulo game instead of
the integer game.
8
2, g(m0i ; m i ) = b so that g(m0i ; m i )Pi g(m). Whether agent i is partially honest or not,
(m0i ; m i ) R;H
(mi ; m i ). Hence, m 2
= N E( ; (Ti )i2N ; %R;H ).
i
Next, we consider the case that n = 3. In this case, we construct an another outcome
mechanism = (M; g). The message space of agent i 2 N consists of Mi = A f0; 1g N .
Denote an element of Mi by mi = (ai ; f i ; k i ). The outcome function g : M ! A is de ned
as follows:
Rule 1 : If there is i 2 N such that for each j 6= i, mj = (a; 0; k j ), then g(m) = a.
Rule 2 : In all other cases, g(m) = ai , where i = ( i2N k i )(mod n) + 1:
N.
For each i 2 N , a truth-telling correspondence is de ned by Ti (R; F ) = F (R)
f0; 1g
This case consists of two lemmata.
Lemma 5. For each R 2 D and each H 2 H1 , F (R)
N EA ( ; (Ti )i2N ; %R;H ).
Proof: For each i 2 N and a 2 F (R), let mi = (a; 0; k i ). By the same argument as Lemma
1, g(m) 2 N EA ( ; (Ti )i2N ; %R;H ).
Lemma 6. For each R 2 D and each H 2 H2 , N EA ( ; (Ti )i2N ; %R;H )
F (R).
Proof: We show that if g(m) 2
= F (R), then m 2
= N E( ; (Ti )i2N ; %R;H ). There are four
cases to consider.
Case 1. For each i 2 N , mi = (a; 0; k i ) such that a 2
= F (R).
By Rule 1, g(m) = a 2
= F (R). Under Assumption 2, there exists a partially honest
agent h 2 H. Let m0h = (a0h ; ; k 0h ) be such that a0h 2 F (R). By the de nition of the
truth-telling correspondence, mh 2
= Th (R; F ) and m0h 2 Th (R; F ). By Rule 1, g(m0h ; m h ) =
0
a so that g(mh ; m h ) = g(m). Since h 2 H, (m0h ; m h ) R;H
(mh ; m h ). Hence, m 2
=
h
N E( ; (Ti )i2N ; %R;H ).
Case 2. There is i 2 N such that for each j 6= i, mj = (a; 0; k j ) such that a 2
= F (R) and
mi = (b; 0; k i ) such that b 6= a.
By Rule 1, g(m) = a 2
= F (R). Under Assumption 2, since jHj 2 there exists a partially
honest agent h 2 Hnfig. Without loss of generality, let i = 1 and h = 2. Let m02 = (a02 ; 1; k 02 )
be such that a02 2 F (R) and ( j6=2 k j + k 02 )(mod 3) + 1 = 3. By the de nition of the truthtelling correspondence, m2 2
= T2 (R; F ) and m02 2 T2 (R; F ). By Rule 2, g(m02 ; m 2 ) = a3 = a
0
(m2 ; m 2 ).
so that g(m2 ; m 2 ) = g(m). Since agent 2 is partially honest, (m02 ; m 2 ) R;H
2
R;H
Hence, m 2
= N E( ; (Ti )i2N ; % ).
Case 3. There is i 2 N such that for each j 6= i, mj = (a; 0; k j ) such that a 2
= F (R) and
mi = (b; 1; k i ) such that b 6= a.
By the same argument as Case 2 of Lemma 6, m 2
= N E( ; (Ti )i2N ; %R;H ).
Case 4. In all other cases, Rule 2 is applied.
Suppose g(m) 2
= F (R). Since F satis es unanimity, there is i 2 N and b 2 A such that
0
bPi g(m). Let mi = (b; ; k 0i ) 6= mi be such that ( j6=i k j + k 0i )(mod n) + 1 = i. By the same
argument as Case 3 of Lemma 4, m 2
= N E( ; (Ti )i2N ; %R;H ).
9
Remark 2. While in Dutta and Sen [6]'s mechanism and Kimya [8]'s mechanism, each
agent reports a preference pro le, an outcome, and a positive integer, in our mechanism each
agent reports an outcome and a positive integer from 1 to n. We succeed in a signi cant
reduction in the size of the message space required for partially honest implementation in
Nash equilibria in the sense of Saijo [20].
Remark 3. When n = 3, our outcome mechanism such that Mi = A N for each i 2 N
does not work. Actually, when n = 3 and jF (R)j = 1, a message pro le m 2 M such that
g(m) 2
= F (R) may be a Nash equilibrium with partially honest agents in ( ; (Ti )i2N ; %R;H ).
When there is i 2 N such that for each j 6= i, mj = (a; k j ) such that a 6= F (R) and
mi = (b; k i ) such that b = F (R), there exists a partially honest agent h 2 Hnfig. Without
loss of generality, let i = 1 and h = 2. By Rule 1, g(m) = a. Let m02 = (b; k 02 ). By Rule 1,
(m02 ; m 2 ). On the
g(m02 ; m 2 ) = b. If aP2 b and aR2 c for each c 2 Anfa; bg, (m2 ; m 2 ) R;H
2
other hand, agent 1 cannot change the outcome. Hence, m 2 N E( ; (Ti )i2N ; %R;H ).
In order to solve this problem, we need to ask agents to report an objective ag in addition
to an outcome and a positive integer from 1 to n for three agents case.
5
Concluding Remarks
In this paper, we succeed in strategy space reduction. We introduce an outcome mechanism
in which each agent only reports an outcome and a positive integer from 1 to n. We show
that if there are at least three agents and there exists at least one partially honest agent, then
an outcome mechanism can partially honest implement any SCC satisfying no veto power in
Nash equilibria. Also, we show that if there are at least three agents and there are at least
two partially honest agents, then an outcome mechanism can partially honest implement any
SCC satisfying unanimity in Nash equilibria.
The next step is to investigate if outcome mechanisms work in practice. Corchon et
al. [1] conduct an experimental study on implementation in Nash equilibria by a simpler
mechanism than Maskin[17]'s mechanism, using three subjects in non-repeated groups, as well
as three outcomes, three states of nature, and three integer choices. Outcome mechanisms
are so simple and intuitive. Therefore, it may be not di cult that subjects understand the
mechanism.
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