Nash Implementation and Tie-Breaking Rules∗
Mert Kimya
†
Koç University, Department of Economics, Rumeli Feneri Yolu, Sarıyer 34450, Istanbul,
Turkey
October 5, 2016
Abstract
I study Nash implementation when agents might use a tie-breaking rule to
choose among the messages they are materially indifferent between. If the planner
is endowed with the knowledge of the rule, this might expand or shrink the set of
implementable social choice correspondences depending on the particular rule used
by the agents. If the planner is not endowed with the knowledge of the rule, then
the problem of implementation is almost equivalent to double implementation in
Nash and strict Nash equilibrium.
JEL classification: C72; D01; D70; D78
Keywords: Nash Implementation; Behavioral Mechanism Design; Double Implementation
1
Introduction
Both through introspection and empirical evidence we know that individuals are not
solely interested in their welfare. They also have considerations such as caring for the
welfare of others, an inclination to conform or a tendency to tell the truth. In a nutshell,
this paper investigates the problem of Nash implementation when such considerations can
lexicographically affect the preference of individuals as tie-breakers in case of indifference.
To understand the idea better, suppose agent i has lexicographic preference for Pareto
efficiency. The Pareto concern of the individual will only kick in when he is materially
indifferent between sending two messages. Suppose agent i believes that the other agents
will send the message profile m−i . If he is indifferent between the outcome he receives
when he announces the message mi and the message m0i , but if the outcome corresponding
to (mi , m−i ) Pareto dominates the outcome corresponding to (m0i , m−i ) then the agent will
strictly prefer to send the message mi . This is what is meant by lexicographic preference
for ‘Pareto efficiency’ or any other concern the individual might have.
The following examples demonstrate the wide range of tie-breaking rules agents might
use.
∗
I am grateful to Geoffroy de Clippel, Roberto Serrano and Rajiv Vohra for helpful comments and
suggestions. I would also like to thank the advisory editor, whose comments have improved the paper
considerably.
†
E-mail Address: mert [email protected]
1
Example 1. (Partial Honesty (Dutta and Sen (2012), Matsushima (2008))) An
individual is said to be partially honest if he prefers to announce the true state of the
world whenever lying does not strictly improve his welfare.
Example 2. (Pareto) An individual with Pareto concerns breaks indifference in favor
of Pareto dominating outcomes.
Example 3. (Dissent) An individual is said to be a dissident if he breaks indifference
against the objectives of the planner.1
Throughout, it is assumed that there is complete information and the focus is on Nash
implementation. The paper is divided into two main sections. In one of those sections I
will assume that the designer is aware of the particular rule used by the agents to break
indifference. I will demonstrate that depending on the rule used this might expand or
shrink the set of implementable social choice correspondences (SCC).
In the second main section I look at the problem of implementation when the designer
is not aware of the tie-breaking rule used by the agents and wants to come up with a
mechanism that implements his objectives for every possible rule the agents might use.
It is shown that this leads to a restrictive result. Strict Maskin monotonicity (originally defined in Bergemann, Morris and Tercieux (2011)) is necessary for implementation.
This condition is a non-trivial strengthening of Maskin monotonicity. Furthermore, robust implementation in this sense is closely related to double implementation in Nash and
strict Nash equilibrium. A characterization of double implementation is also obtained as
a byproduct of this section.
I will start with the literature review in Section 1.1. In Section 2, I will introduce
the model. In Section 3, I will assume that the designer is aware of the particular tiebreaking rule used by the agents. In Section 4, I will consider the case where the designer
is not aware of the tie-breaking rule, hence wants to come up with a mechanism that
implements the SCC no matter what tie-breaking rule the agents might be using.
1.1
Relationship to the Literature
The paper is most closely related to the literature on implementation when agents have
lexicographic preferences for honesty (partial honesty). Dutta and Sen (2012) has provided an almost complete characterization of Nash implementation under this assumption. They show that if the planner knows that at least one agent is partially honest then
any SCC satisfying no veto power (NVP) can be Nash implemented.
Matsushima (2008a) shows that if there are at least three agents and if the planner can
impose fines, then any social choice function can be implemented through iterative elimination of dominated strategies if the agents are partially honest. Whereas Matsushima
(2008b) studies the question in a Bayesian environment. Kartik et al. (2014) demonstrate
that when agents have small preferences for honesty, under a condition called ‘separable
punishment’ any social choice function can be implemented through two rounds of iterative elimination of strictly dominated strategies. It is also worth noting that unlike most
of the literature, their results only require two players. Ortner (2015) shows that under
partial honesty, if there are at least 5 individuals then any social choice function is implementable in fault tolerant equilibrium (closely related to Eliaz (2002)) and stochastically
1
Formal definition of dissent is provided in Section 2.
2
stable equilibrium (see Kandori et al. (1993), Young (1993)).2
Kartik and Tercieux (2012a) study the problem of Nash implementation when the
agents can voluntarily provide evidence. Their model includes certain tie-breaking rules,
such as partial honesty, as a special case.
Aside from these, there is a growing literature on behavioral implementation. De
Clippel (2014) generalizes implementation theory to the case where the agents’ choice
need not be consistent with the maximization of a preference relation. In his work, the
choices of the agents depend only on the set of options available to them, hence it does
not contain my work as a special case. Eliaz (2002) studies Nash implementation when
agents might be faulty, in the sense that they may fail to act optimally. In Cabrales and
Serrano (2011) agents myopically adjust their actions in the direction of best responses.3
The paper is also related to the literature on double implementation, which seeks to
find mechanisms that implement a given SCC in two different solution concepts. Yamato
(1993) studies double implementation in Nash and undominated Nash equilibrium, Saijo
et al. (2007) in Nash and dominant strategy equilibrium and Schmeidler (1980) and Suh
(1997) in Nash and strong Nash equilibrium.4
Finally, I show that robust implementation is closely related to double implementation
in Nash and strict Nash equilibrium. To my knowledge there does not exist a characterization of double implementation in Nash and strict Nash equilibrium, hence the current
paper is also the first to provide this. Nash implementation has been characterized by
Maskin (1999) and strict Nash implementation has been characterized by Cabrales and
Serrano (2011).
2
Preliminaries
The environment is {N, Z, Θ}, where N is the finite set of agents, Z is the set of alternatives and Θ is the set of states. Each agent i ∈ N has a complete, transitive and
reflexive preference ordering Riθ over Z for all θ ∈ Θ. Siθ and Iiθ denote the asymmetric
and symmetric components of Riθ . I assume that there is complete information.
A social choice correspondence (SCC) is a mapping f that assigns a nonempty set of
alternatives to each state, i.e. f : Θ → 2Z \ ∅. A mechanism g = {M, π} =
Q{(Mi )i∈N , π}
consists of a message set Mi for each agent i and an outcome function π : i∈N Mi → Z.
Let G denote the set of all mechanisms. In each state θ a mechanism g induces a game.
Let N (g, Rθ ) denote the set of Nash equilibrium outcomes of the corresponding game
in state θ. A SCC f is implementable in Nash equilibrium if there exists a mechanism
g ∈ G such that for all θ ∈ Θ, f (θ) = N (g, Rθ ).
Definition 1. Tie-breaking Rule
T is a tie-breaking rule if for any g = (M, π) ∈ G and θ ∈ Θ, T (g, θ) is an asymmetric
and transitive relation on M × M . Let T denote the set of all tie-breaking rules.
I will denote the tie-breaking rule used by agent i as Ti and let T = (T1 , T2 , ..., Tn ) ∈
T denote the profile of tie-breaking rules.
We can think of the tie-breaking rule as a second strict and transitive preference
relation the agent consults only when he is indifferent between sending two different
n
2
For more on implementation under the assumption of partial honesty, see Korpela (2014) and Lombardi and Yoshihara (2011).
3
Also see Bierbauer and Netzer (2014), Glazer and Rubinstein (2012) and Korpela (2012)
4
For more on double implementation see Peleg (1996a, 1996b) and Corchon and Wilkie (1996).
3
messages. The rule may depend on the mechanism and the state of the world. It might
be incomplete. For example it might be empty, in which case we would be back to Nash
implementation.
The idea is that if an individual is using the tie-breaking rule Ti ∈ T , then given a
mechanism g and a state of the world θ if he is indifferent between sending the message mi
and m0i when the message profile announced by other agents is m∗−i then he will prefer to
send mi over m0i if ((mi , m∗−i ), (m0i , m∗−i )) ∈ Ti (g, θ). The definition of extended preference
below formalizes this notion.
Definition 2. Extended Preference (Ti ,θ )
Given a profile of tie-breaking rules T ∈ T n , a state θ ∈ Θ and a mechanism g =
(M, π), the extended preference of agent i, Ti ,θ , is defined as:
• If π(m)Siθ π(m0 ) then m Ti ,θ m0 .
• If π(m)Iiθ π(m0 ) and (m, m0 ) ∈ Ti (g, θ) then m Ti ,θ m0 .
• If π(m)Iiθ π(m0 ) and (m, m0 ) ∈
/ Ti (g, θ) then m ∼Ti ,θ m0 .5
T ,θ denotes the profile of extended preferences.
Consistent with the paragraph above, the only place in which the extended preference
would not agree with the underlying preference is when there is indifference in the underlying preference. Note that although the tie-breaking rule is assumed to be transitive,
the extended preference may fail to be transitive. For an example see the Pareto rule
and Footnote 7. Below, I explicitly write down the tie-breaking rules discussed in the
Introduction.
Partial Honesty: An individual is said to be partially honest if he prefers to announce the true state of the world whenever lying does not strictly improve his welfare.
Formally, a partially honest individual i uses the following tie-breaking rule. Ti (g, θ) =
{(m, m0 )|m1i = θ, m0i 1 6= θ}6 if g is such that Mi = Θ × Fi for some set Fi , otherwise
Ti (g, θ) = ∅. Note that partial honesty would only have a bite in direct mechanisms or
mechanisms where the message space is augmented with the state space.
Pareto: An individual with Pareto concerns breaks indifference in favor of Pareto
dominating outcomes, i.e. for any g and θ we have (m, m0 ) ∈ Ti (g, θ) iff π(m) Pθ D π(m0 ),
where for a, b ∈ Z, a Pθ D b iff aRiθ b for all i ∈ N and aSjθ b for some j ∈ N .7
Dissent: Individual i is said to be a dissident if he breaks indifference against the
objectives of the planner. In particular for any g and θ we have (m, m0 ) ∈ Ti (g, θ) iff
π(m) ∈
/ f (θ) and π(m0 ) ∈ f (θ).
3
With the Knowledge of the Tie-Breaking Rule
The following is the definition of Nash implementation when the designer knows the tiebreaking rules used by the agents. It is basically the definition of Nash implementation
with the extended preferences.
,θ
,θ
,θ
As usual, ∼T
and T
denote the symmetric and asymmetric components of T
i
i
i .
1
mi denotes the first component of agent i’s message.
7
The extended preference corresponding to this rule may fail to be transitive. For example let Z =
{a, b, c}, N = {1, 2, 3}, let the underlying preferences in state θ be aI1θ bI1θ c, cS2θ aS2θ b and aI3θ bS3θ c. Let
g = (M, π) be a mechanism that includes the messages m1 , m2 and m3 , where π(m1 ) = a, π(m2 ) = b
and π(m3 ) = c. Then if agent 1 is using the Pareto rule to break indifference, his extended preference
will be the following which is intransitive: m1 θ1 m2 ∼θ1 m3 ∼θ1 m1 .
5
6
4
Definition 3. T -Implementation
Given T ∈ T n , we say that a SCC f is T -Implementable in Nash equilibrium if there
exists a mechanism g = (M, π) such that for all θ ∈ Θ, f (θ) = N (g, T ,θ ).8
Some tie-breaking rules, such as Pareto and dissent, have the special structure that
they do not depend on the particular mechanism used by the designer. Specifically,
the agent has a fixed (state dependent) ordering over the alternatives that he consults
whenever he is indifferent. I call such rules ‘outcome-based rules’9 . Note that partial
honesty is not an outcome-based rule as it only has a bite when the mechanism includes
the state space and it does not depend on the alternatives.
A characterization for outcome-based rules can be obtained simply by considering
the extended preference, which is done in the Appendix.10 The results in the Appendix
together with the results in the literature imply that the use of tie-breaking rules might
make the implementation problem easier (partial honesty), non-comparable (Pareto)11 or
harder (dissent). Here, I will focus on ‘dissent’ for this salient role it plays.
A SCC is dissent-implementable if it is T -implementable when each agent is using
dissent as their tie-breaking rule. The following modification of Maskin monotonicity is
a necessary condition for dissent-implementation.
Definition 4. D-Monotonicity (D-Mon)
A SCC f satisfies D-Mon if for all θ, θ0 ∈ Θ whenever a ∈ f (θ) and a ∈
/ f (θ0 ), there
0
exists i ∈ N and b ∈ Z such that bSiθ a and either aSiθ b or aIiθ b and b ∈ f (θ).
It is immediate that D-Mon is a stronger condition than Maskin monotonicity. D-Mon
is necessary for dissent-implementation and with NVP it is sufficient. Furthermore, the
set of dissent-implementable SCCs is a strict subset of the set of Nash implementable
SCCs.
Lemma 1. .
• If f is dissent-implementable then f satisfies D-Mon. Furthermore if n ≥ 3 and if
f satisfies D-Mon and NVP then f is dissent-implementable.
• The set of SCCs that are dissent-implementable is a strict subset of the set of SCCs
that are Nash-implementable.
4
Without the Knowledge of the Rule
Given the existence of several intuitive tie-breaking rules, one might want to find the conditions under which a designer is able to implement a given SCC without the knowledge
of the tie-breaking rule used by the agents. Therefore, he has to design a mechanism that
implements the SCC for every possible rule agents might use.
For any θ ∈ Θ we have a ∈ N (g, T ,θ ) if there exists a message profile m∗ with π(m∗ ) = a such that
,θ
for all i ∈ N , m∗ T
(mi , m∗−i ) for all mi ∈ Mi
i
9
Formal definition is provided in the Appendix.
10
Although the extended preference may fail to be transitive, Hurwicz (1986) has shown that this has
no consequence for the characterization.
11
For instance when all the agents use the Pareto rule to break ties then the strong Pareto correspondence is Pareto-implementable, which is not Nash implementable. Whereas dictatorship is not
Pareto-implementable.
8
5
The definition below formalizes this. Note that according to the definition agents
might be using different tie-breaking rules or some agents might not be using any rule at
all.
Definition 5. Robust T-Implementation (RT-implementation)
A SCC f is RT-Implementable in Nash equilibrium if there exists a mechanism g such
that for all θ ∈ Θ and for all T ∈ T n , f (θ) = N (g, T ,θ ).12
It is immediate that RT-Implementation implies Nash implementation. Furthermore,
by Lemma 1 we also know that Nash Implementation does not imply RT-implementation.
Definition 6. Strict Maskin Monotonicity (strict monotonicity)
A SCC f satisfies strict monotonicity if for all θ, θ0 ∈ Θ whenever a ∈ f (θ) and
0
a∈
/ f (θ0 ), there exists i ∈ N and b ∈ Z such that aSiθ b and bSiθ a.
Strict monotonicity, defined in Bergemann, Morris and Tercieux (2011) for social
choice functions, is a nontrivial strengthening of Maskin monotonicity. For example
weak Pareto correspondence and dictatorship satisfy Maskin monotonicity but neither of
these satisfy strict monotonicity.13 Bergemann, Morris and Tercieux (2011) have shown
that strict monotonicity is necessary for implementation in rationalizable strategies, it
turns out that it is also necessary for RT-implementation.
Lemma 2. If f is RT-implementable then f satisfies strict monotonicity.
For sufficiency, we need the following additional condition.
Definition 7. No Worst Alternative (NWA)
A SCC f satisfies NWA whenever for every i ∈ N , θ ∈ Θ and a ∈ f (θ) there exists
an outcome zia,θ such that aSiθ zia,θ .
NWA requires that for each agent there is always a strictly worse outcome than
the outcome prescribed by the SCC. In many situations such as exchange economies,
NWA is naturally satisfied. Although it can also be a very restrictive assumption under
certain conditions, such as collective choice between two alternatives. NWA also appears
as a sufficient condition for implementation in best-response dynamics and strict Nash
equilibrium (Cabrales and Serrano (2011)), implementation in rationalizable strategies
(Bergemann, Morris and Tercieux (2011)) and LLQRE implementation (Tumennasan
(2013)).
Finally, strict monotonicity, NWA and NVP is sufficient for RT-implementation.
Proposition 1. If n ≥ 3 then f is RT-implementable if f satisfies strict monotonicity,
NVP and NWA.
12
One might be interested in a slightly weaker definition which requires the mechanism to be ruledependent, i.e. f is implementable in this sense if for each T ∈ T n there exists a mechanism g that
T -implements f . All of the results would still go through with this definition (sufficiency is immediate,
for necessity the proof of Lemma 2 also applies directly to this definition).
0
13
To see this take N = {1}, Z = {a, b}, Θ = {θ, θ0 }, aI1θ b and bS1θ a. In this example both the
dictatorship and the weak Pareto correspondence is f (θ) = {a, b} and f (θ0 ) = {b}. It violates strict
monotonicity.
6
Double Implementation in Nash and Strict Nash Equilibrium
One interesting property of RT-implementation is that it is closely related to double
implementation in Nash and strict Nash equilibrium and in this section I will show this.
Let SN (g, Rθ ) denote the set of strict Nash equilibrium outcomes of mechanism g in
state θ.
Definition 8. Double Implementation
A SCC f is doubly implementable in Nash equilibrium and strict Nash equilibrium if
there exists a mechanism g such that for all θ ∈ Θ, f (θ) = N (g, Rθ ) = SN (g, Rθ ).
The following is a strengthening of RT-implementation. It will be useful, as it is
equivalent to double implementation and closely related to RT-implementation. Just like
RT-implementation, it requires that for all θ ∈ Θ and for all T ∈ T n , f (θ) = N (g, T ,θ ).
But on top of this, it also requires that for each θ and a ∈ f (θ), there is a rule independent
equilibrium of the mechanism g in state θ that implements a. Let N̂ (g, T ,θ ) denote the
set of Nash equilibria of g with the extended preferences corresponding to T .
Definition 9. Strong Robust T-implementation(SRT-implementation)
A SCC f is SRT-implementable if there exists a mechanism g such that
• For all θ and a ∈ f (θ) there exists a message profile m∗ such that for all T ∈ T n
we have m∗ ∈ N̂ (g, T ,θ ) and π(m∗ ) = a.
• If a ∈ N (g, T ,θ ) for some T ∈ T n and θ ∈ Θ then a ∈ f (θ).
The definition directly implies RT-implementation. However, the reverse is not true.
An example showing this is available upon request from the author. The proposition below establishes the equivalence between double implementation and SRT-implementation.
Proposition 2. A SCC f is SRT-implementable iff f is doubly implementable in strict
Nash and Nash equilibrium.
Trivially, strict monotonicity is a necessary condition for SRT-implementation. Furthermore it is sufficient together with NWA and NVP.
Proposition 3. If n ≥ 3 and if a SCC f satisfies strict monotonicity, NWA and NVP
then it is SRT-implementable (hence, doubly implementable in Nash and strict Nash).
5
Concluding Remarks
Remark 1. (Stochastic Tie-Breaking Rules) The tie-breaking rule is defined as a deterministic rule, but it can also be defined as a probabilistic rule in which the agents use
a certain tie-breaking rule with a certain probability. As, under robust implementation
the mechanism implements the SCC for any rule, the results on robust implementation
would not change even if we include probabilistic tie-breaking rules in T .
Remark 2. (Stochastic Mechanisms) Benoit and Ok (2008) and Bochet (2007) have
shown that introduction of stochastic mechanisms may aid the problem of Nash implementation. It turns out that this is even more true for RT-implementation. Specifically
suppose that ∆(Z) represents the set of lotteries on Z and assume that preferences can
be represented in the expected utility form. Bergemann, Morris and Tercieux (2011) have
7
shown that in this environment, strict monotonicity reduces down to monotonicity if the
SCC satisfies NWA. But then it can easily be shown that Maskin monotonicity, NWA
and NVP is sufficient for RT-implementation and double implementation.
Remark 3. (Mixed Strategies) The mechanisms used for sufficiency proofs do not work
when mixed Nash equilibria are employed. As usual, under certain assumptions on preferences over lotteries (see Kartik and Tercieux (2012b)) mixed strategies can be adapted
with mechanisms similar to the ones used by Maskin (1999) and Maskin and Sjostrom
(2002).
Appendix A.
Outcome-based Rules
Definition 10. Outcome-based Rule
T ∈ T is an outcome based rule if there exists an asymmetric and transitive relation
θ
BT on Z for each θ such that for each g ∈ G and θ ∈ Θ we have (m, m0 ) ∈ T (g, θ) iff
π(m) BθT π(m0 ).
Since the preferences of the agents do not depend on the particular mechanism (messages) used, implementation under outcome-based rules reduces down to implementation
under a reflexive and complete (but possibly intransitive) preference relation ≥θi on the
outcomes which agrees with the extended preferences Ti ,θ .14 Hurwicz (1986) has shown
that Maskin’s theorem is still valid even when the preferences of the agents fail to be
transitive. Hence, all we need to do is to translate Maskin monotonicity and NVP from
the extended preferences to the underlying preferences.
The definition of T -Monotonicity below corresponds to Maskin monotonicity when
the agents are using an outcome-based rule. By the argument above, it is a necessary
condition for T -implementation for outcome-based rules.
Definition 11. T -Monotonicity (T -Mon)
Let T ∈ T n be a profile of outcome-based rules. A SCC f satisfies T -Mon if for all
θ, θ0 ∈ Θ whenever a ∈ f (θ) and a ∈
/ f (θ0 ), there exists i ∈ N and b ∈ Z such that
• aRiθ b and either aSiθ b or b 6 BθTi a
0
0
0
• And bRiθ a and either bSiθ a or b BθTi a
Note that T -Mon is neither stronger nor weaker than Maskin monotonicity. T -Mon
will be sufficient for implementation with the following modification of NVP. (This condition is weaker than NVP)
Definition 12. T -NVP
Let T ∈ T n be a profile of outcome-based rules. A SCC satisfies T -NVP if for all
θ ∈ Θ, a ∈ Z if there exists i ∈ N such that either aSjθ b or aIjθ b and b 6 BθTj a for all j 6= i
and for all b ∈ Z then a ∈ f (θ).
Proposition 4. Let T ∈ T n be a profile of outcome-based rules. If f is T -implementable
then f satisfies T -Mon. Furthermore, if n ≥ 3 and if f satisfies T -Mon and T -NVP
then f is T -implementable.
14
,θ
That means, for any θ ∈ Θ and g = (M, π) ∈ G we have m T
m0 iff π(m) ≥θi π(m0 )
i
8
θ
θ0
1
aIb
bSa
2
aSb
aSb
3
bSa
bSa
4
aSb
aIb
Table 1: f (θ) = a and f (θ0 ) = b
Proof. As mentioned above, the result follows from Hurwicz (1986) and Definitions 2 and
10.
Appendix B. Proofs
Proof of Lemma 1. The rule is an outcome based rule and (m, m0 ) ∈ T (g, θ) iff π(m) ∈
/
0
f (θ), but π(m ) ∈ f (θ). Hence the first part follows from Proposition 4.
For the second part, it is immediate that if f is dissent-implementable by a mechanism
g, g also Nash implements f . Finally, consider the example in Table 1. f satisfies Maskin
monotonicity and no veto power, hence it is Nash implementable, but f does not satisfy
D-Mon, therefore it is not dissent-implementable.
Proof of Lemma 2. Suppose f is RT-implementable and g implements f . Take any
θ, θ0 ∈ Θ, a ∈ Z such that a ∈ f (θ) and a ∈
/ f (θ0 ). As g implements f for every
possible rule, it also implements f when each agent uses the following rule: T (g, θ) =
{(m, m0 )|π(m) 6= a, π(m0 ) = a} and T (g, θ0 ) = {(m, m0 )|π(m) = a, π(m0 ) 6= a}. a is
an equilibrium outcome of g under the extended preferences in state θ and there exists
a deviation from the corresponding equilibrium in θ0 to an alternative b. But with the
extended preferences every agent strictly prefers to stay with a in θ0 instead of getting
an alternative indifferent to a, hence we should have that there exists b ∈ Z and i such
0
that bSiθ a. And with the extended preferences every agent strictly prefers to deviate to
an alternative indifferent to a in θ, hence we should have that aSiθ b.
Proof of Proposition 1. See Proposition 3 and note that SRT-implementation implies
RT-implementation.
Proof of Proposition 2. .
Step 1: f is SRT-implementable ⇒ f is doubly implementable
Suppose that f is SRT-implementable. Let g be a mechanism that implements f . I
will show that g doubly implements f in Nash and strict Nash equilibrium.
Step 1-a: Show that for all θ ∈ Θ we have f (θ) ⊆ SN (g, Rθ ) (hence, f (θ) ⊆ N (g, Rθ ))
Take any θ ∈ Θ and a ∈ Z such that a ∈ f (θ). Since g SRT-implements f , in state θ
there exists a message profile m∗ such that for all T ∈ T n we have m∗ ∈ N̂ (g, T ,θ ) and
π(m∗ ) = a. I will show that m∗ is a strict Nash equilibrium. Suppose not, then there
exists i and m0 ∈ M such that m0 = (m, m∗−i ) 6= m∗ for some m ∈ Mi and π(m0 )Riθ π(m∗ ).
Let Ti be such that Ti (g, θ) = {(m0 , m∗ )}, but then m0 Ti i ,θ m∗ and hence m∗ is not a
Nash equilibrium of g with the extended preferences corresponding to Ti . A contradiction.
Step 1-b:Show that for all θ ∈ Θ we have N (g, Rθ ) ⊆ f (θ) (hence, SN (g, Rθ ) ⊆ f (θ))
Let T = (T1 , T2 , .., Tn ) be such that for all i ∈ N we have Ti (g, θ) = ∅ for all g and θ.
Note that N (g, Rθ ) = N (g, T ,θ ) ⊆ f (θ) for all θ ∈ Θ.
Step 2: f is doubly implementable ⇒ f is SRT-implementable.
9
Suppose that f is doubly implementable in strict Nash and Nash equilibrium. Let g
be a mechanism that implements f . I will show that g SRT-implements f .
Step 2-a:Show that for all θ ∈ Θ and a ∈ f (θ) there exists a message profile m∗ such
that for all T ∈ T n we have m∗ ∈ N̂ (g, T ,θ ) and π(m∗ ) = a.
Take any θ ∈ Θ and a ∈ Z such that a ∈ f (θ). Since g doubly implements f , in state
θ there exists a strict Nash equilibrium m∗ of g such that π(m∗ ) = a. I will show that
m∗ ∈ N̂ (g, T ,θ ) for all T ∈ T n . As the extended preferences agree with the underlying
preferences on the asymmetric part of the underlying preference and since m∗ is a strict
Nash equilibrium, we have that there is no deviation from m∗ for any possible rule.
Step 2-b: Show that for all θ ∈ Θ and T ∈ T n we have N (g, T ,θ ) ⊆ f (θ).
For all θ ∈ Θ and T ∈ T n we have N (g, T ,θ ) ⊆ N (g, Rθ ) ⊆ f (θ). This is because an
equilibrium under a profile T is always a refinement of Nash equilibrium and the second
inequality is because g Nash implements f .
Proof of Proposition 3. Suppose f satisfies strict monotonicity, NWA and NVP. By
NWA, for each θ ∈ Θ, i ∈ N and a ∈ f (θ), there exists zia,θ ∈ Z such that aSiθ zia,θ . I will
use a slight variant of the canonical mechanism to prove the result. Let Mi = Θ × Z × N
for every i ∈ N . The outcome function is defined by the following rules:
Rule 1: If for all i ∈ N , mi = (θ, a, 0) where a ∈ f (θ) then π(m) = a.
Rule 2: If for all j 6= i, mj = (θ, a, 0) with a ∈ f (θ) and mi = (θ0 , b, n) 6= (θ, a, 0) then
0
π(m) = b if aSiθ b and bSiθ a, otherwise π(m) = zia,θ .
Rule 3: In all other cases the outcome is the one announced by the agent announcing
the highest integer. (Ties are broken in favor of the agent with the lowest index).
I will show that the mechanism doubly implements the SCC in Nash and strict Nash,
which implies that the mechanism SRT-implements the SCC, which implies that the
mechanism RT-implements the SCC.
Step 1: For all θ ∈ Θ, we have f (θ) ⊆ SN (g, Rθ ) (hence, f (θ) ⊆ N (g, Rθ ))
Take any θ ∈ Θ and a ∈ f (θ). Then mi = (θ, a, 0) for all i is a strict Nash equilibrium
in θ, since the agents can only deviate to an alternative strictly worse than a.
Step 2: For all θ ∈ Θ, we have N (g, Rθ ) ⊆ f (θ) (hence, SN (g, Rθ ) ⊆ f (θ) )
Suppose there is a Nash equilibrium in Rule 1 in which everybody announce (θ, a, 0)
and θ is not the true state of the world. Suppose the true state of the world is θ0 6= θ.
0
By strict monotonicity there exists i ∈ N and b ∈ Z such that aSiθ b and bSiθ a, but i can
induce b, hence there is a profitable deviation. A contradiction.
By NVP, if a is an equilibrium outcome in state θ under Rule 2 or Rule 3 then a ∈ f (θ).
References
Benoit, J. P., Ok, E. A. (2008). Nash implementation without no-veto power. Games
and Economic Behavior, 64(1), 51-67.
Bergemann, D., Morris, S., Tercieux, O. (2011). Rationalizable implementation. Journal
of Economic Theory, 146(3), 1253-1274.
Bochet, O. (2007). Nash implementation with lottery mechanisms. Social Choice and
Welfare, 28(1), 111-125.
10
Bierbrauer, F., Netzer, N. (2014). Mechanism design and intentions. University of Zurich
Department of Economics Working Paper, (66).
Cabrales, A., Serrano, R. (2011). Implementation in adaptive better-response dynamics:
Towards a general theory of bounded rationality in mechanisms. Games and Economic
Behavior, 73(2), 360-374.
Corchon, L., Wilkie, S. (1996). Double implementation of the ratio correspondence by
a market mechanism. Economic Design, 2(1), 325-337.
Dutta, B., Sen, A. (2012). Nash implementation with partially honest individuals.
Games and Economic Behavior, 74(1), 154-169.
de Clippel, G. (2014). Behavioral implementation. The American Economic Review,
104(10), 2975-3002.
Eliaz, K. (2002). Fault tolerant implementation. The Review of Economic Studies, 69(3),
589-610.
Glazer, J., Rubinstein, A. (2012). A model of persuasion with boundedly rational agents.
Journal of Political Economy, 120(6), 1057-1082.
Hurwicz, L. (1986). On the implementation of social choice rules in irrational societies.
In: Essays in Honor of Kenneth J. Arrow (Vol I), ed. by Heller et al., Cambridge
University Press.
Kandori, M., Mailath, G. J., Rob, R. (1993). Learning, mutation, and long run equilibria
in games. Econometrica: Journal of the Econometric Society, 29-56.
Kartik, N., Tercieux, O. (2012a). Implementation with evidence. Theoretical Economics,
7(2), 323-355.
Kartik, N., Tercieux, O. (2012b). A note on mixed-Nash implementation. Mimeo.
Kartik, N., Tercieux, O., Holden, R. (2014). Simple mechanisms and preferences for
honesty. Games and Economic Behavior, 83, 284-290.
Korpela, V. (2012). Implementation without rationality assumptions. Theory and decision, 72(2), 189-203.
Korpela, V. (2014). Bayesian implementation with partially honest individuals. Social
Choice and Welfare, 43(3), 647-658.
Lombardi, M., Yoshihara, N. (2011). Partially-honest Nash implementation. Social
Choice and Welfare, 32, 171-179.
Maskin, E. (1999). Nash equilibrium and welfare optimality. The Review of Economic
Studies, 66(1), 23-38.
Maskin, E., Sjostrom, T. (2002). Implementation theory. Handbook of social Choice
and Welfare, 1, 237-288.
Matsushima, H. (2008a). Behavioral aspects of implementation theory. Economics Letters, 100(1), 161-164.
11
Matsushima, H. (2008b). Role of honesty in full implementation. Journal of Economic
Theory, 139(1), 353-359.
Peleg, B. (1996a). A continuous double implementation of the constrained Walras equilibrium. Economic Design, 2(1), 89-97.
Peleg, B. (1996b). Double implementation of the Lindahl equilibrium by a continuous
mechanism. Economic Design, 2(1), 311-324.
Saijo, T., Sjostrom, T., Yamato, T. (2007). Secure implementation. Theoretical Economics, 2(3), 203-229.
Schmeidler, D. (1980). Walrasian analysis via strategic outcome functions. Econometrica: Journal of the Econometric Society, 1585-1593.
Suh, S. C. (1997). Double implementation in Nash and strong Nash equilibria. Social
Choice and Welfare, 14(3), 439-447.
Ortner, J. (2015). Direct implementation with minimally honest individuals. Games and
Economic Behavior, 90, 1-16.
Tumennasan, N. (2013). To err is human: Implementation in quantal response equilibria.
Games and Economic Behavior, 77(1), 138-152.
Yamato, T. (1993). Double implementation in Nash and undominated Nash equilibria.
Journal of Economic Theory, 59(2), 311-323.
Young, H. P. (1993). The evolution of conventions. Econometrica: Journal of the Econometric Society, 57-84.
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