On the Non-Robustness∗ of Nash
Implementation
Takashi Kunimoto†
Incomplete and Preliminary
First Version: March 2006
This Version: May 2006
Abstract. I consider the implementation problem under complete information and employ
Nash equilibrium as a solution concept. A socially desirable rule is given as a correspondence
from the set of states to the set of outcomes. This social choice rule is said to be implementable in Nash equilibrium if there exists a mechanism such that all (pure strategy) Nash
equilibrium outcomes are socially desirable. Maskin (1999) clarified the conditions on the
social choice rules under which he was able to construct a very general Nash implementing
mechanism when there are at least three players. This paper identifies the minimal amount
of incomplete information which prevents the Maskinian mechanism from being robustly implemented in Nash equilibrium. More precisely, with mild constrains on the social choice
rules, one can construct a canonical perturbation of the complete information structure under which a sequence of Bayesian Nash equilibria of the Maskinian mechanism supports a
non-Nash equilibrium outcome in the limit. Therefore, there is a minimal sense in which the
Maskinian mechanism is not robust to incomplete information. This non-robustness result
is also extended to the environment with two players.
JEL classification: C72, D78, D82, D83
Keywords: Approximate common knowledge; Implementation; Robustness; Nash
equilibrium
∗
I am thankful to Hassan Benchekroun, Dip Majumdar, and Roberto Serrano for the discussion.
I am grateful to financial support from McGill University through the startup grant and the James
McGill Professor fund.
†
Department of Economics, McGill University, 855 Sherbrooke Street West, Montreal, Quebec,
H3A2T7, CANADA, takashi.kunimoto@mcgill.ca
URL: http://people.mcgill.ca/takashi.kunimoto/
1
Introduction
Consider an environment with a set of players and a set of feasible outcomes, together with a set of possible states, each of which generates the players’ preferences
over these outcomes. In such an environment, implementation theory has provided a
number of characterizations of which social choice rules can be achieved by checking
if one can design a mechanism (or game form) such that all the equilibrium outcomes of the induced game end up being socially optimal. Thus, the implementing
mechanism must have the property that all equilibria of the mechanism lead to the
desired outcomes. This is sometimes called full implementation. As the answer for
this characterization relies crucially on the informational assumption and on solution concept, this paper only considers environments with complete information and
employs Nash equilibrium as a solution concept. Focusing on this special class of
setups, Maskin (1999) showed that if a social choice rule is monotonic and satisfies
no-veto-power (which I define later), it can be implemented in Nash equilibrium
when there are at least three players. 3 His proof is based on constructing a general
mechanism all of whose Nash equilibrium outcomes coincide with the social choice
rule.
Although Maskin’s theorem is general enough to encompass a wide range of
applications, it has had a limited impact on the more applied mechanism design
literature, such as auction theory and contract theory. This literature rather focuses
on the existence of some equilibrium of some mechanism that leads to the desired
outcomes. For this less demanding question, the revelation principle lends us the
following strong methodology: if the social choice rule can arise as an equilibrium
in some game form, then it will arise in a truth-telling equilibrium of the direct
mechanism (where each player truthfully reports his information and the designer
chooses an outcome assuming they are truthful). One reason for the unpopularity
of the Maskinian mechanism is that his result relies on a complicated indirect, or
“augmented,” mechanism (henceforth I abbreviate the Maskinian mechanism to the
M-mechanism) in which players report more than their information, despite the fact
that the multiplicity of equilibria is relevant in designing optimal mechanisms. In
fact, the M-mechanism uses the so-called “integer game”: The players are asked to
announce an integer, and if certain strategy profiles occur, the player who announced
the highest integer “wins.” Although the integer game is a powerful device to knock
down undesired outcomes, it has a very “unrealistic” nature making it impractical
to many applied researchers. Obviously, one possible route to tackle this problem is
to ask when one can design a “reasonable” mechanism without relying on the integer
game-like devices. This paper does not take this direction, however. Rather, I take
the M-mechanism seriously and I ask when the M-mechanism is no longer “robust.”
3
Maskin’s original paper had been circulated as a MIT working paper since 1977. However, his
proof was not as correct as his theorem exactly said.
1
There are at least three perspectives from which the Maskinian game form is not
robust: (1) the non-robustness to renegotiation; (2) the non-robustness to collusion;
and (3) the non-robustness to informational assumption. This paper pays special
attention to (3) the robustness to informational assumption. The initial assumption
this paper makes is complete information, which entails the common knowledge of
payoffs. Since this common knowledge requirement seems to be very stringent, I take
it as at best a simplifying assumption. Then, I will clarify if there is any assumption
weaker than common knowledge that is sufficient to sustain, or approximate, the
Nash implementability given by the M-mechanism. The answer turns out to depend
upon how I weaken the common knowledge assumption. I will follow and extend the
approach proposed by Monderer and Samet (1989).
The main result of this paper shows that, with mild constrains on the social
choice rules, one can construct a canonical perturbation of the complete information
structure under which a sequence of Bayesian Nash equilibria of the M-mechanism
supports a non-Nash equilibrium outcome in the limit. Furthermore, in terms of
topology, I quantify the minimal perturbation of the complete information needed
for the main result. Namely, if I perturb the complete information in such a way that,
with probability close to 1, there is approximate common knowledge in the sense of
Monderer and Samet (1989) – which I carefully define later – about which payoff
state is being realized, I obtain the non-robustness of the M-mechanism. This nonrobustness result gives us a precise sense in which the M-mechanism is not robust
if we believe that the perturbation of the complete information is an excessively
weak relaxation of the common knowledge assumption. If this is the case, the Mmechanism is not attractive for the more applied mechanism design literature not
necessarily because the integer game is unrealistic but because it is dubious to assume
that complete information is exactly satisfied in a given environment.
Following “Wilson’s doctrine,” Bergemann and Morris [B-M] (2005) have also
addressed some of the similar issues of robust implementation. 4 B-M fix a payoff
type space and extend it to a richer set of type spaces à la Harsanyi-Mertens-Zamir
as a relaxation of the common knowledge assumption. Then, B-M consider all richer
type spaces consistently extended from the original payoff type space so that they
define “robust implementation” as interim (Bayesian) implementation on all type
spaces. In particular, B-M show that robust implementation is possible if and only
if it is possible to implement the social choice rule using an iterative deletion of
strictly dominated strategies. If I restrict my attention to complete information,
B-M’s result says that their robust implementation is equivalent to implementation
in (correlated) rationalizability.
4
Wilson’s doctrine is summarized as the following comment by himself (1987): “I foresee the
progress of game theory as depending on successive reductions in the base of common knowledge
required to conduct useful analyses of practical problems. Only by repeated weakening of common
knowledge assumptions will the theory approximate reality.”
2
While this paper and B-M (2005) share the same robustness criterion as a relaxation of common knowledge, there are two features which make this paper distinct
from B-M (2005). First, using the equivalence class of type spaces induced by the
topology, I require that implementation be robust only to this equivalence class of
type spaces, and not to all type spaces. This makes the robustness of this paper
substantially weaker than B-M’s. Second, in order to make tight connections with
the results in the implementation literature, I require that the class of permissible
type spaces be restricted to be consistent with the use of Nash equilibrium under
complete information. This requirement further makes the robustness of this paper
weaker than B-M’s. I elaborate on this a little bit more. Suppose, for example, that
there are just two players i and j, the choice of strategy by i will depend on what i
believes j’s choice will be, which in turn will depend on what i believes j believes i’s
choice will be and so on. In games with complete information, this infinite regress
in beliefs can be generally “cut through” by the imposition of Nash equilibrium.
If I believe that the requirement of “being cut through” is too restrictive, the use
of Nash equilibrium is not appropriate under complete information and hence not
appropriate for implementation as well. If Nash implementation is not appropriate
under complete information, there is no point to ask when Nash implementation is
robust because it is not robust from the beginning. In this paper, I “assume” that
Nash implementation is appropriate under complete information. Then, I ask when
Nash implementation is robust or not.
The rest of the paper is organized as follows. In section 2, I formalize the setup
and definitions. In section 3, I introduce the M-mechanism (the Maskinian mechanism) and summarize some existing results in the literature. In section 4, I show
the main result. In section 5, I extend the main result to the environments with two
players. Section 6 concludes.
2
2.1
The Setup
The Environment
There is a finite set N = {1, . . . , n} of players. Let f : Θ → A be a social choice
correspondence, where Θ denotes the set of payoff states, and A denotes the set of
outcomes, which is fixed across the payoff states. I assume that the number of payoff
states is at least two, i.e., |Θ| ≥ 2. Associated with each state θ is a preference profile
θ , which is a list (θ1 , . . . , θn ) where θi is player i’s state θ preference relation over
A. I read a θ a as “a is at least as good as a in state θ.” I read a θ a as “a is
strictly preferred to a in state θ.”
Players do not observe the state directly but are informed of the state via signals.
Player i’s signal set is Si and I assume |Si | = |Θ| for each i ∈ N . A signal profile is
an element s = (s1 , . . . , sn ) ∈ S = ×i∈N Si . Let μ be a prior probability over Θ × S.
I designate sθ to be the signal profile in which each player’s signal sθi corresponds to
3
the state θ. 5 Complete information refers to the environment in which μ(θ, s) = 0
whenever s = sθ . This specification of the signal space is without loss of generality
as long as I only consider environments with complete information.
Given a mechanism Γ = (M, g), a designer is interested in the set of Nash
equilibrium (NE) outcomes under complete information. Here M ≡ ×i∈N Mi , Mi
is player i’s message space and g : M → A is the outcome function. The designer,
however, entertains the possibility that players face uncertainty about payoffs. Then,
he has to take into account the set of “nearby” incomplete information structures in
which the original complete information structure is subsumed. The designer then
employs Bayesian Nash equilibrium (BNE) as a solution concept for those nearby
incomplete information games. He wants to ascertain that his prediction about
players’ strategic behavior changes continuously with respect to some topology.
2.2
Definitions and Notations
Mathematically, the coarser the topology chosen, the larger the set of continuous correspondences with respect to it and therefore, the harder the achievement of robust
implementation with respect to it. By the same token, the finer the topology chosen,
the less it demands that the BNE correspondence be continuous with respect to it.
In order to talk about the topology, I have to explicitly expound the topological
structure of the domain of the BNE correspondence into which the players’ belief
structure is embedded. Let (Θ × S, μ) be a complete information structure. I shall
construct the set of states of the world, called Ω, that is consistently extended from
a given complete information environment. I want to keep track of the complete
information environment and let it be embedded in Ω so that we are able to coherently discuss the epistemic states of all players when the environment is subject to
incomplete information. 6
Definition 1 A space Θ × S is (algebraically) immersed in Ω if there exists a
one-to-one correspondence h : Θ × S → Ω. h is said to be an immersion of Θ × S
into Ω.
Definition 2 Assume that Θ × S is immersed in Ω. A probability space (Ω, Σ, P ∗ )
is a consistent extension from the complete information structure
(Θ × S, μ) if
∗
Θ×S
, μ , where P ∗ ≡ μ◦h−1 .
(Ω, Σ, P ) is equivalent to the probability space Θ × S, 2
Let (Ω, Σ, P ∗ ) be a probability space consistently extended from the complete
information structure (Θ × S, μ). I fix a measurable space (Ω, Σ) throughout the
argument while I change the probability distributions. In particular, I am interested
k
∗
in a net of probability distributions {P k }∞
k=1 for which P → P as k → ∞ in “a
certain” property preserving way. The very issue is that I have to be specific about
5
The notation sθ will be used extensively throughout the paper.
Players’ epistemic state dictates what they know or believe about the game and about each
other’s actions, knowledge, and beliefs.
6
4
with respect to which topology the two probability distributions are deemed to be
close. Let P be the space of all probability distributions over Ω. Let F be the
space of all partitions of Ω, the elements of which are in Σ. Let F ∗ be the subset
of F such that for any Π ∈ F ∗ , ω ∈ Π(ω) for each ω ∈ Ω. An element of the
form Π = (Πi )i∈N ∈ (F ∗ )n is called a partition structure. Let Π∗ : Ω → 2Ω be the
finest possibility correspondence that is coarser than Πi for each i ∈ N . Following
the result of Aumann (1976), I say that an event E is common knowledge at ω if
Π∗ (ω) ⊂ E. I fix a partition structure Π = (Πi )i∈N ∈ (F ∗ )n for the moment. Player
i’s strategy in the game Γ(P ) is a function σi : Ω → Mi which is Πi -measurable. Let
σ be a strategy profile in a game Γ(P ).
I take for granted Nash equilibrium as a reasonable solution in a game with
complete information. Therefore, players are assumed to choose their strategy independently of other players’ choice provided there is common knowledge about what
game being played. This implies that the amount of incomplete information I allow
for the robustness analysis should not contradict the use of Nash equilibrium under complete information. The formalization of this is summarized by the following
measurability condition on possible strategy profiles.
Definition 3 Let a partition structure Π ∈ F n and the associated game Γ(P ) be
given. A strategy profile σ is consistent with the complete information structure
if, for any ω, ω ∈ Ω, whenever there exists a profile (θ, sθ ) ∈ Θ × S such that
h(θ, sθ ) = ω̃ for any ω̃ ∈ Π∗ (ω) ∪ Π∗ (ω ), then we have σi (ω) = σi (ω ) for each
i ∈ N . When σ is consistent with the complete information structure, we simply say
that it is a consistent strategy profile.
By focusing only on consistent strategy profiles, I exclude the extent of correlation
of equilibrium strategies which invalidates the original equilibrium analysis under
complete information. Otherwise, there is a fundamental inconsistency: On the
one hand, I fix a type space as coarse as the complete information. On the other
hand, I allow for the strategy to vary arbitrarily more finely than the coarse type
space that I fixed in the beginning. If I anticipated the corresponding strategy will
vary arbitrarily more finely than the coarse type space, then I should question the
coarseness of the original type space rather than the use of Nash equilibrium. It must
be too coarse for the equilibrium analysis to be valid from the beginning. Thus, my
stance is consistent with the use of Nash equilibrium. 7 In other words, carrying out
Nash equilibrium analysis of the original complete information game by embedding
Θ in the larger type space incorporates the indirect restriction that two distinct types
of a player who share the same payoff state θ and hierarchy belief on θ choose the
same action.
7
Moreover, I restrict my attention only to pure strategy Nash equilibrium. This is consistent with
almost all papers in the implementation literature. However, this restriction is less problematic here
because this paper focuses on the impossibility result.
5
In this paper, I take ordinal preferences over the set of (pure) outcomes as a
primitive. In order to analyze players’ behavior under uncertainty, I shall extend
the ordinal preferences into preferences over “acts.” An act is a mapping α : Ω →
A. A belief is a probability distribution β on Ω. I read “α βi α ” as “player
i prefers α to α under his belief β.” I assume that a set of axioms is imposed
on preference relations over acts enough to obtain the subjective expected utility
representation. 8 Furthermore, I assume that any representation for preferences over
acts is permissible as long as it is consistent with the ordinal preferences over pure
outcomes. The act αΓσ induced by σ under Γ is defined by αΓσ (ω) = g(σ(ω)) for any
ω ∈ Ω. With these notations, I shall define Nash equilibrium NE and Bayesian Nash
equilibrium.
Definition 4 Let a partition structure Π = ×i∈N Πi ∈ (F ∗ )n and the associated
game Γ(P ) be given. A consistent strategy profile σ is a Bayesian Nash equilibrium (BNE) of Γ(P ) if σi is Πi -measurable for each i ∈ N , and for each i ∈ N , state
P (·|Πi (ω))
ω with P (Πi (ω)) > 0, and strategy σi which is Πi -measurable, we have αΓσ i
αΓ
.
σi ,σ−i
The above definition suffices for σ to be a Nash equilibrium of the game Γ(P )
if P is a complete information prior. Define the set of acts A ≡ AΩ . I define
ψΓBN E : P → A as the Bayesian Nash equilibrium correspondence associated with
the mechanism Γ. Here, A is endowed with product topology. I shall introduce a
topology which enables us to determine how close any two probability distributions
are. To define such topologies, I need some definitions and notations.
Monderer and Samet (1989) introduced the concept of “common p-belief ” as an
approximation to common knowledge, which is common 1-belief. Let Biq (E) ≡ {ω ∈
Ω|P (E|Πi (ω)) ≥ q} denote the set of states in which player i assigns probability at
least q to the event E. I call this player i’s q-belief operator. In particular, when
q = 1, I call Bi1 player i’s 1-belief operator corresponding to player i’s knowledge
operator. 9 An event E is said to be q-evident if E ⊂ Biq (E) for all i ∈ N . This
means that whenever E is true, everyone believes with probability at least q that E is
true. An event E is said
to be common q-belief at ω if there exists a q-evident event
F such that ω ∈ F ⊂ i∈N Biq (E). I will loosely say that an event E is approximate
common knowledge at ω if E is common q-belief at ω, for q close to 1.
Consider the notion of the closeness of probability distributions. Define d0 by
the rule
d0 (P, P ) = sup |P (E) − P (E)|.
E⊂Ω
8
More specifically, I assume that β satisfies completeness, transitivity, continuity, and the independence axiom.
9
Here we define “knowledge” as belief with probability 1.
6
Let P ∗ be the complete information prior. I will require extra conditions on conditional probabilities. Define G (η) to be the set of all states in which there is a
common (1 − η)-belief about what game is being played as follows:
G (η) = ω ∈ Ω ∃θ ∈ Θ such that Γ(θ) is common (1 − η)-belief at ω .
Let
d1 (P ) = inf{η | P (G (η)) = 1},
and
d∗ (P, P ) = max{d0 (P, P ), d1 (P ), d1 (P )}.
Note that d1 (P ∗ ) = 0 by definition. By construction, d∗ is non-negative and sym
metric, and d∗ (P, P ) = 0 if and only if P = P and both p and P are complete
information priors. d∗ generates a topology in the following sense: a net {P k | k ∈ K},
where K is a directed index set with partial order , converges to P ∗ if and only if
for any ε > 0, there is a k̄ ∈ K such that k k̄ implies d∗ (P k , P ∗ ) < ε. The reader
is referred to Kunimoto (2006) for the details on the topology induced by d∗ .
Definition 5 Let Γ be a mechanism. ψΓBN E is upper hemi-continuous at a complete information prior P ∗ with respect to the topology induced by d∗ if, ψΓBN E (P k ) →
ψΓN E (P ∗ ) as k → ∞ whenever d∗ (P k , P ∗ ) → 0 as k → ∞. Here ψΓN E (P ∗ ) denotes
the set of NE outcomes of the game Γ(P ∗ ).
The next result is given as a corollary of Theorem 1 in Kunimoto (2006).
Theorem 1 (Kunimoto (2006)) Suppose that Θ is finite. Then, the Bayesian
Nash equilibrium correspondence associated with any mechanism is upper hemi-continuous
at any complete information prior with respect to the topology induced by d∗ .
I want to apply the above result to the notion of robust Nash implementation. Let
N E(Γ(θ)) ⊂ M be the entire set of (pure strategy) Nash equilibria of the complete
information game Γ(θ). First, I shall define the notion of Nash implementation.
Definition 6 A social choice correspondence f is implementable in Nash equilibrium if there exists a mechanism Γ = (M, g) such that for any θ ∈ Θ, (1) for any
a ∈ f (θ), there is a Nash equilibrium mθ of the game Γ(θ) such that g(mθ ) = a and
(2) g(N E(Γ(θ)) ⊂ f (θ).
Condition (1) of Nash implementation requires that there be always a Nash
equilibrium outcome which coincides with any socially desirable outcome in each
state. Condition (2) of Nash implementation requires that every Nash equilibrium
outcome coincide with some socially desirable outcome in each state. Having defined
Nash implementation, I introduce the concept of robust Nash implementation as
follows.
7
Definition 7 A social choice correspondence f is robustly implementable relative
to d∗ (d∗∗ ) if (1) it is implementable in Nash equilibrium and (2) the Bayesian Nash
equilibrium correspondence associated with some Nash implementing mechanism is
upper hemi-continuous at any complete information prior with respect to the topology
induced by d∗ (d∗∗ ).
In particular, Theorem 1 implies that all Nash implementing mechanisms are
robust relative to d∗ . To strengthen this robustness result on Nash implementation,
I introduce a slightly coarser topology than that induced by d∗ . Let
d˜1 (P ) = inf η P (G (η)) ≥ 1 − η .
Define d∗∗ (P, P ) as follows:
d∗∗ (P, P ) = max d0 (P, P ), d˜1 (P ), d˜1 (P ) .
The reader is referred to Kunimoto (2006) for the details on the topology induced
by d∗∗ .
Definition 8 Let Γ be a mechanism. ψΓBN E is upper hemi-continuous at a complete information prior P ∗ with respect to the topology induced by d∗∗ if, ψΓBN E (P k ) →
ψΓN E (P ∗ ) as k → ∞ whenever d∗∗ (P k , P ∗ ) → 0 as k → ∞. Here ψΓN E (P ∗ ) denotes
the set of NE outcomes of the game Γ(P ∗ ).
3
The Maskinian Mechanism (the M-mechanism)
In his “classic” paper, Maskin (1999) showed that monotonicity is a necessary condition for Nash implementation.
Definition 9 A social choice correspondence f is monotonic if for every pair of
states θ and θ and any a ∈ A with a ∈ f (θ) such that for each player i and b ∈ A,
b θi a ⇒ b θi a,
then we have a ∈ f (θ ).
Maskin (1999) also provided sufficient conditions for Nash implementation when
there are at least three players. Before stating his result, I need one more definition.
Definition 10 A social choice correspondence f satisfies no veto power if, for all
θ ∈ Θ and all a ∈ A, whenever there exists i ∈ N such that, for all j = i and all
b ∈ A, if a θi b, then f (θ) = a.
No veto power says that if an outcome is at the top of n − 1 players’ preference
orderings, then the last player cannot prevent this outcome from being the social
optimum.
8
Theorem 2 (Maskin (1999)) Let the number of players be at least three, i.e.,
n ≥ 3. Then, any monotonic social choice correspondence f satisfying no veto
power is implementable in Nash equilibrium.
The proof of Theorem 2 is based on the construction of a canonical mechanism
which I call the Maskinian mechanism. Define the Maskinian mechanism Γ = (M, g),
in which each message mi ∈ Mi allowed to player i consists of an alternative, a payoff
state and a nonnegative integer. Thus, a typical message sent by player i is denoted
mi = (ai , θ i , z i ),
where ai ∈ A, θ i ∈ Θ and z i ∈ {0, 1, 2, . . . }. The outcome function g of the Mmechanism Γ is defined with the following rules, where m = (m1 , . . . , mn ):
1. If all players announce the same message, mi = (ai , θ i , z i ) = (a, θ, 0) for all
i ∈ N , and a ∈ f (θ), then g(m) = a.
2. If all players but one announce the same message, that is, mj = (a, θ, 0) for all
j = i with a ∈ f (θ) and player i announces mi = (ai , θ i , z i ) = (a, θ, 0), then
we can have three cases:
⎧
if a θi ai
⎨ ai
a
if ai θi a
g(m) =
⎩
a(i, θ) if a ∼θi ai
where a(i, θ) is the worst outcome for player i in state θ.
3. In all other cases, an integer game is played. That is,
∗
g(m) = ai ,
where i∗ is the lowest index among those who announce the highest integer,
∗
i.e., z i = maxj z j .
This paper makes a slight modification on Rule 2 of the original game form in
Maskin (1999). Rule 2 of the original canonical mechanism is as follows:
i
a if a θi ai
g(m) =
a if ai θi a
It is important to note that regardless of this modification, Maskin’s sufficiency result
(Theorem 2) continues to be valid.
4
An Impossibility Theorem
In this section, I pursue an impossibility theorem on robust Nash implementation.
To do so, I impose two assumptions on the social choice correspondences.
9
Assumption 1 A social choice correspondence f satisfies the following two conditions:
1. There are two states θ, θ such that there are a ∈ f (θ) and a ∈ f (θ ) for which
a θi a and a θi a for some player i ∈ N .
2. a θi a(i, θ) for each a ∈ f (θ), i ∈ N , and θ ∈ Θ.
Assumption 1-1 says that the social choice correspondence f describes the objective of a society in which there is some minimal degree of congruence of interests
across some states for some player. Namely, some player i prefers a to a across θ
and θ . Assumption 1-2 says that the social choice correspondence f never gives the
worst outcome for any player in any state. Most importantly, Assumption 1-2 with
Rule 2 of the M-mechanism ensures that the truth telling Nash equilibrium mθ is a
strict Nash equilibrium of the game Γ(θ) for any θ ∈ Θ. With the terminology provided by Kunimoto (2006), Monderer and Samet (1989) showed that any strict Nash
equilibrium is lower hemi-continuous with respect to the topology induced by d∗∗ . In
this sense, any truth telling equilibrium of the M-mechanism is robust to incomplete
information. I also impose some topological structure on the set of outcomes A.
Assumption 2 The set of outcomes A is compact.
Since the objective of this paper is to establish an impossibility result, the compactness of A is a fairly innocuous assumption. The next theorem is this paper’s
main result.
Theorem 3 Let the number of players be at least three, i.e., n ≥ 3. Let a social
choice correspondence f satisfy Assumption 1. Let Γ = (M, g) be the M-mechanism
which implements f in Nash equilibrium under complete information. Assume further that the set of outcomes A satisfies Assumption 2. Then, for any complete
information prior P ∗ , there is a net of probability distributions {P k }∞
k=1 with the
∗∗
k
∗
property that d (P , P ) → 0 as k → ∞ for which the Bayesian Nash equilibrium
correspondence associated with Γ exhibits a discontinuity at P ∗ .
Before going into the proof, I will sketch how to prove the theorem. First, I focus
on two states θ and θ given by Assumption 1-1. Second, I construct a perturbation of
the complete information only around θ and θ while the other states remain common
knowledge. Third, in a Bayesian game induced by the perturbation in the previous
step, I propose a strategy profile σ ∗ and show that σ ∗ constitutes a Bayesian Nash
equilibrium (Proposition 1). Finally, I show that σ ∗ supports a non-Nash equilibrium
outcome of the game Γ(θ) in the limit.
Proof of Theorem 3: Let θ and θ be two payoff states satisfying Assumption
1-1. Suppose that mθ is the truth telling Nash equilibrium of the game Γ(θ) where
mθi = (a, θ, 0) with a ∈ f (θ) for each i ∈ N and mθ is the truth telling Nash
10
equilibrium of the game Γ(θ ) where mθi = (a , θ , 0) with a ∈ f (θ ) for each i ∈ N .
Let us consider a canonical perturbation of the complete information structure. This
canonical perturbation is constructed so as to preserve the complete information
assumption in any state other than θ and θ . Therefore, without loss of generality,
we may continue the rest of the argument as if there were only two payoff states θ
and θ . Define as follows:
p = μ(θ, sθ |{θ, θ })
1 − p = μ(θ , sθ |{θ, θ })
We define Ω = {(0, 0), (1, 0), (1, 1), (2, 1), (2, 2)}. Let us define h as a mapping from
Θ × S to Ω.
h−1 (0, 0) = (θ , sθi , sθ−i ),
h−1 (1, 0) = (θ, sθi , sθ−i ),
h−1 (1, 1) = h−1 (2, 1) = h−1 (2, 2) = (θ, sθi , sθ−i ).
Therefore, h is an immersion from Θ×S into Ω. We consider the following probability
distribution P ε ∈ Ω:
• P ε (0, 0) = 1 − p;
• P ε (1, 0) = pε;
• P ε (1, 1) = pε(1 − ε);
• P ε (2, 1) = pε(1 − ε)2 ;
• P ε (2, 2) = p(1 − ε)3 .
Players’ partition structure Π is given as follows:
• Πi (0, 0) = {(0, 0)}, Πi (1, 0) = Πi (1, 1) = {(1, 0), (1, 1)}, Πi (2, 1) = Πi (2, 2) =
{(2, 1), (2, 2)};
• Πj (0, 0) = Πj (1, 0) = {(0, 0), (1, 0)},
Πj (1, 1) = Πj (2, 1) = Πj (2, 2) = {(1, 1), (2, 1), (2, 2)} for all j = i.
Fix this partition structure Π throughout the rest of the argument. The matrix
below displays the type space for the nearby incomplete information games we have
explained. The row is i’s signal and the column is everybody else’s signal.
i’s signal
0
1
2
j’s signal (∀j =
i)
0
1
2
1−p
0
0
pε
pε(1 − ε)
0
0
pε(1 − ε)2 p(1 − ε)3
11
In the appendix, I show that this perturbation of the complete information structure is characterized by a convergent net of probability distributions with respect to
the topology induced by d∗∗ . Namely, d∗∗ (P ε , P ∗ ) → 0 as ε → 0. 10 For strategy
profile σ to be consistent with complete information, any strategy profile σ : Ω → M
in the Bayesian game Γ(P ε ) must satisfy the property that σj (2, 1) = σj (2, 2) for any
j = i. The next proposition explicitly constructs a Bayesian Nash equilibrium σ ∗ of
the game Γ(P ε ) for any ε > 0 sufficiently small. Let mθi denote player i’s message
for which mθi = (a, θ, 0) for all i ∈ N , where a ∈ f (θ).
Proposition 1 There exists a sufficiently small ε̄ > 0 such that for any ε ∈ (0, ε̄]
there exists a Bayesian Nash equilibrium σ ∗ of the game Γ(P ε ) with the following
properties:
• σi∗ (0, 0) = mθi
• σi∗ (1, 0) = σi∗ (1, 1) = m∗i
• σi∗ (2, 1) = σi∗ (2, 2) = mθi
• σj∗ (0, 0) = σj∗ (1, 0) = mθj for any j = i
• σj∗ (1, 1) = σj∗ (2, 1) = σj∗ (2, 2) = mθj for any j = i
Proof of Proposition 1: Until the end of the proof, we keep assuming m∗i =
mθi . It will turn out that this assumption can be modified without changing all the
conclusions. First, we claim that for any j = i, σj∗ (0, 0) = σj∗ (1, 0) = mθj is a best
∗ (0, 0) = σ ∗ (1, 0) = mθ if ε is sufficiently small.
response to σ−j
−j
−j
Fix any player j = i. Playing mθj conditional upon {(0, 0), (1, 0)} gives the
following lottery:
• g(mθ ) with probability (1 − p)/(1 − p + pε) at (0, 0) in Γ(θ )
• g(mθ ) with probability pε/(1 − p + pε) at (1, 0) in Γ(θ)
Consider any other message mj . Playing mj conditional upon {(0, 0), (1, 0)} gives
the following lottery:
• g(mj , mθ−j ) with probability (1 − p)/(1 − p + pε) at (0, 0) in Γ(θ )
• g(mj , mθ−j ) with probability pε/(1 − p + pε) at (1, 0) in Γ(θ)
10
This perturbation is first used in an example in Kunimoto (2005) for achieving a different
objective.
12
Because mθ is the truth telling Nash equilibrium of the game Γ(θ ) and if we
apply Rule 2 of the M-mechanism Γ with Assumption 1-2, we have
g(mθ ) θj g(mj , mθ−j ) ∀mj = mθj .
By continuity of preferences over acts, we show that mθj is a best response to the
strategies of other players specified above for ε sufficiently small.
Second we claim that σj∗ (1, 1) = σj∗ (2, 1) = σj∗ (2, 2) = mθj for any j = i is a best
response to σi∗ (1, 1) = mθi and σi∗ (2, 1) = σi∗ (2, 2) = mθi and σk∗ (1, 1) = σk∗ (2, 1) =
/ {i, j}. Moreover, by the partition structure Π, we have
σk∗ (2, 2) = mθk for any k ∈
σj∗ (1, 1) = σj∗ (2, 1) = σj∗ (2, 2) for any j = i. Note that mθ constitutes the truth
telling Nash equilibrium of the game Γ(θ).
Fix any player j = i. Playing mθj conditional upon {(1, 1), (2, 1), (2, 2)} gives the
following lottery:
• g(mθi , mθ−i ) with probability ε at (1, 1) in Γ(θ)
• g(mθ ) with probability ε(1 − ε) at (2, 1) in Γ(θ)
• g(mθ ) with probability (1 − ε)2 at (2, 2) in Γ(θ)
Consider any other message mj . Playing mj conditional upon {(1, 1), (2, 1), (2, 2)}
gives the following lottery:
• g(mθi , mj , mθ−{i,j} ) with probability ε at (1, 1) in Γ(θ)
• g(mj , mθ−j ) with probability ε(1 − ε) at (2, 1) in Γ(θ)
• g(mj , mθ−j ) with probability (1 − ε)2 at (2, 2) in Γ(θ)
Because mθ is the truth telling Nash equilibrium of the game Γ(θ) and if we
apply Rule 2 of the Maskinian game form Γ with Assumption 1-2, we have
g(mθ ) θj g(mj , mθ−j ) ∀mj = mθj .
By continuity of preferences over acts, we show that mθj is a best response to the
strategies of other players specified above for ε sufficiently small.
Third, we claim that σi∗ (1, 0) = σi∗ (1, 1) = mθi “can” (not must) be a best
∗ (0, 0) = σ ∗ (1, 0) = mθ and σ ∗ (1, 1) = σ ∗ (2, 1) = σ ∗ (2, 2) = mθ .
response to σ−i
−i
−i
−i
−i
−i
j
At states {(1, 0), (1, 1)}, player i knows that the game Γ(θ) is played. However, player
i does not know if all other players know that the game Γ(θ) is played. Conditional
13
on {(1, 0), (1, 1)}, player i believes with probability z that all other players do not
know which game is being played and with probability 1 − z that all other players
know the game Γ(θ) is played. We can calculate z as follows:
z=
ε
1
=
> 1/2
ε + ε(1 − ε)
2−ε
as long as ε > 0
Playing mθi conditional upon {(1, 0), (1, 1)} gives the following lottery:
• g(mθ ) with probability z > 1/2 at (1, 0)
• g(mθi , mθ−i ) with probability 1 − z < 1/2 at (1, 1)
Playing mθi conditional upon {(1, 0), (1, 1)} gives the following lottery:
• g(mθi , mθ−i ) with probability z > 1/2 at (1, 0)
• g(mθ ) with probability 1 − z < 1/2 at (1, 1)
Applying Rule 2 of the M-mechanism Γ with Assumption 1-1, we can rewrite the
above expressions:
Playing mθi conditional upon {(1, 0), (1, 1)} gives the following lottery:
• g(mθ ) with probability z > 1/2 at (1, 0)
• g(mθi , mθ−i ) with probability 1 − z < 1/2 at (1, 1)
Playing mθi conditional upon {(1, 0), (1, 1)} gives the following lottery:
• g(mθ ) with probability z > 1/2 at (1, 0)
• g(mθ ) with probability 1 − z < 1/2 at (1, 1)
Since mθ is the truth telling Nash equilibrium of the game Γ(θ) and due to Rule
2 of the M-mechanism with Assumption 1-2, we have g(mθ ) θi g(mθi , mθ−i ). We
assign the following utility value to each outcome as follows:
ui (g(mθ ); θ) = ui (g(mθi , mθ−i ); θ) = 0,
ui (g(mθ ); θ) = 3, and
ui (mθi , mθ−i ; θ) = −1.
Playing mθi conditional upon {(1, 0), (1, 1)} gives the following expected utility:
0 × z + 0 × (1 − z) = 0
14
Playing mθi conditional upon {(1, 0), (1, 1)} gives the following expected utility:
3 × z − 1 × (1 − z) = 4z − 1 > 1 ∵ z > 1/2
Therefore, mθi is a “better” response to the belief specified above than mθi . In
other words, mθi is “not” a best response. Note that the assignment of utilities here
is not important. That is, even if we perturb the utilities slightly, the same argument
goes through. If mθi is indeed a best response, this shows σi∗ (1, 0) = σi∗ (1, 1) = mθi
can be a candidate for part of the Bayesian Nash equilibrium. Suppose, to the
contrary, that mθi is not a best response. Assume that ui (·; θ) : A → R is continuous.
Since Rule 2 of the M-mechanism is only relevant here, the choice of messages is
equivalent to the choice of outcomes. Since A is compact by Assumption 2, by
Weierstrass theorem, there must exist m∗i = mθi such that
m∗i ∈ arg max z × ui g(m̃i , mθ−i ); θ + (1 − z) × ui g(m̃i , mθ−i ); θ
m̃i ∈Mi
In this case, all we have to do is to replace mθi with m∗i and check that player j’s
best responses are unchanged at all states. We claim that for any j =
i, σj∗ (0, 0) =
σj∗ (1, 0) = mθj is a best response to σi∗ (0, 0) = mθi , σi∗ (1, 0) = σi∗ (1, 1) = m∗i and
/ {i, j} if ε is sufficiently small.
σk∗ (0, 0) = σk∗ (1, 0) = mθk for any k ∈
Fix any player j = i. Playing mθj conditional upon {(0, 0), (1, 0)} gives the
following lottery:
• g(mθ ) with probability (1 − p)/(1 − p + pε) at (0, 0) in Γ(θ )
• g(m∗i , mθ−i ) with probability pε/(1 − p + pε) at (1, 0) in Γ(θ)
Consider any other message mj . Playing mj conditional upon {(0, 0), (1, 0)} gives
the following lottery:
• g(mj , mθ−j ) with probability (1 − p)/(1 − p + pε) at (0, 0) in Γ(θ )
• g(m∗i , mj , mθ−{i,j} ) with probability pε/(1 − p + pε) at (1, 0) in Γ(θ)
Because mθ is the truth telling Nash equilibrium of the game Γ(θ ) and if we
apply Rule 2 of the M-mechanism Γ with Assumption 1-2, we have
g(mθ ) θj g(mj , mθ−j ) ∀mj = mθj .
By continuity of preferences over acts, we show that mθj is a best response to the
belief specified above for ε sufficiently small.
15
We claim that σj∗ (1, 1) = σj∗ (2, 1) = σj∗ (2, 2) = mθj for any j = i is a best response
to σi∗ (1, 1) = m∗i and σi∗ (2, 1) = σi∗ (2, 2) = mθi and σk∗ (1, 1) = σk∗ (2, 1) = σk∗ (2, 2) =
/ {i, j}.
mθk for any k ∈
Fix any player j = i. Playing mθj conditional upon {(1, 1), (2, 1), (2, 2)} gives the
following lottery:
• g(m∗i , mθ−i ) with probability ε at (1, 1) in Γ(θ)
• g(mθ ) with probability ε(1 − ε) at (2, 1) in Γ(θ)
• g(mθ ) with probability (1 − ε)2 at (2, 2) in Γ(θ)
Consider any other message mj . Playing mj conditional upon {(1, 1), (2, 1), (2, 2)}
gives the following lottery:
• g(m∗i , mj , mθ−{i,j} ) with probability ε at (1, 1) in Γ(θ)
• g(mj , mθ−j ) with probability ε(1 − ε) at (2, 1) in Γ(θ)
• g(mj , mθ−j ) with probability (1 − ε)2 at (2, 2) in Γ(θ)
Because mθ is the truth telling Nash equilibrium of the game Γ(θ) and if we
apply Rule 2 of the Maskinian game form Γ with Assumption 1-2, we have
g(mθ ) θj g(mj , mθ−j ) ∀mj = mθj .
By continuity of preferences over acts, we show that mθj is a best response to the
strategies of other players specified above for ε sufficiently small.
∗ (0, 0) =
Finally, we check that at state (0, 0), mθi is a best response to σ−i
∗ (1, 0) = mθ . Since mθ is the truth telling Nash equilibrium of the game Γ(θ )
σ−i
−i
and if we apply Rule 2 of the Maskinian game form with Assumption 1-2, we have
g(mθ ) θi g(mi , mθ−i ) ∀mi = mθi .
This completes the proof of Proposition 1. To complete the proof of the theorem, it remains to check that a non-Nash
equilibrium outcome is supported by the Bayesian Nash equilibrium σ ∗ . In fact, we
have the following:
g(σ ∗ (1, 1)) = g(m∗i , mθ−i ) = g(N E(Γ(θ))) = g(mθ ),
where h(θ, sθ ) = (1, 1). Because if we apply Rule 2 of the M-mechanism with Condition 2 of Assumption 1, we have
g(mθ ) θi g(m∗i , mθ−i ).
16
This implies the failure of the upper hemi-continuity of the Bayesian Nash equilibrium correspondence at any complete information prior. This completes the proof
of Theorem 3. When I combine Theorem 2 with Maskin’s sufficiency result on Nash implementation, called Theorem 2 in this paper, I have the following corollary.
Corollary 1 Let the number of players be at least three, i.e., n ≥ 3. Let the set
of outcomes A satisfy Assumption 2. Suppose that f is a monotonic social choice
correspondence satisfying no veto power and Assumption 1. Then, f is not robustly
implementable relative to d∗∗ by the M-mechanism.
The implication of the above corollary is as follows: If I am concerned with
achieving robust implementation relative to d∗∗ , I have to devise another canonical
mechanism which must be different from the Maskinian game from.
5
Two-Person Nash Implementation
In this section, I shall extend the non-robustness of Nash implementation to the case
of two players. This is an important extension because the two-person problem has
a bearing on a wide variety of bilateral contracting and bargaining problems. First,
note that no-veto-power is not a necessary condition for Nash implementation even
when there are more than two players. Furthermore, no-veto-power is unacceptably
strong in the environment with two players. Moore and Repullo (1990) fully characterized necessary and sufficient conditions for Nash implementation when there are at
least three players. Before turning my attention to two-person Nash implementation,
I briefly review the result of Moore and Repullo.
For any i ∈ N, θ ∈ Θ, and C ⊂ A, let Mi (C, θ) denote the set of maximal
elements in C for player i in state θ. Mi (C, θ) may, of course, be empty. For a given
mechanism Γ, I can select some message profile m(a, θ) of the game Γ(θ) whose
outcome is a. Given m(a, θ), let Ci (a, θ) denote the set of outcomes that player i
can generate by varying his own message, keeping the other players’ message choices,
m−i (a, θ), fixed:
Ci (a, θ) ≡ {c ∈ A| c = g (mi , m−i (a, θ)) for some mi ∈ Mi }
Definition 11 A social choice correspondence f satisfies Condition μ if there is a
set A∗ ⊂ A, and for each i ∈ N, θ ∈ Θ, and a ∈ f (θ), there is a set Ci (a, θ) ⊂ A∗ ,
with a ∈ Mi (Ci (a, θ), θ) such that for all θ ∗ ∈ Θ, the following three conditions are
satisfied:
1. (Monotonicity): If a ∈ Mi (Ci (a, θ), θ ∗ ) for each i ∈ N , then a ∈ f (θ ∗ ).
2. (Limited No Veto Power): If c ∈ Mi (Ci (a, θ), θ ∗ ) for some i ∈ N and c ∈
Mj (A∗ , θ ∗ ) for all j = i, then c ∈ f (θ ∗ ).
17
3. (Unanimity): If d ∈ Mi (A∗ , θ ∗ ) for each i ∈ N , then d ∈ f (θ ∗ ).
Theorem 4 (Moore and Repullo (1990)) Suppose there are three or more players, i.e., n ≥ 3. Then, a social choice correspondence is implementable in Nash
equilibrium if and only if it satisfies Condition μ.
Condition μ is not sufficient for two-person Nash implementation. Here I discuss
another condition which, together with Condition μ, is sufficient for two-person Nash
implementation. A social choice correspondence f is said to satisfy the nonempty
lower intersection property if, for all θ, θ ∈ Θ with θ = θ , and for all a ∈ f (θ) and
a ∈ f (θ ), there exists outcome c ∈ A such that a θ1 c and a θ2 c.
Theorem 5 (Dutta and Sen (1991)) Suppose there are two players, i.e., n =
2. Then, a social choice correspondence is implementable in Nash equilibrium if it
satisfies both Condition μ and the nonempty lower intersection property.
Dutta and Sen’s proof is done by constructing a mechanism. Define the extended
M-mechanism Γ̃ = (M̃ , g̃), in which each message m̃i ∈ M̃i contains one extra set of
messages, {F, N F }, as well as the previous three ingredients in the M-mechanism.
Here the set {F, N F } comprises the two elements, “flag” and “no flag.” Thus, a
typical message sent by player i is denoted
m̃i = (ai , θ i , r i , z i ),
where ai ∈ A, θ i ∈ Θ, r i ∈ {F, N F }, and z i ∈ {0, 1, 2, . . . , }. The outcome function
g̃ of the extended M-mechanism Γ is defined with the following rules, where m̃ =
(m̃1 , m̃2 ):
1. If m̃1 = (a, θ, N F, z 1 ), m̃2 = (a, θ, N F, z 2 ), and a ∈ f (θ), then g̃(m̃) = a.
2. If m̃1 = (a1 , θ 1 , N F, z 1 ) and m̃2 = (a2 , θ 2 , N F, z 2 ), then g̃(m̃) = x, where x is
2
1
chosen such that a2 θ1 x and a1 θ2 x.
3. If m̃i = (ai , θ i , F, z i ) and m̃j = (aj , θ j , N F, z j ), then we have three cases:
⎧
j
⎪
if aj θi ai
ai
⎨
j
g(m̃) =
aj
if ai θi aj
⎪
⎩ a(i, θ j ) if ai ∼θj aj
i
where a(i, θ j ) is the worst outcome for player i in state θ j .
4. In all other cases, an integer game is played. That is,
∗
g̃(m̃) = ai ,
where i∗ is the lowest index among those who announce the highest integer,
∗
∗
i.e., z i = max{z 1 , z 2 }. More specifically, z i = z 1 if there is a tie, i.e., z 1 = z 2 .
18
Rule 1, 3, and 4 of the extended M-mechanism are the same as Rule 1, 2, and 3 of
the M-mechanism, respectively. Rule 2 of the extended M-mechanism is well defined
as well as the social choice correspondence satisfies the nonempty lower intersection
property. Combining Theorem 3 with some minor modification and Theorem 5
(Dutta and Sen (1991)), I have the following result which is parallel to Corollary 1
in the previous section.
Corollary 2 Let the number of players be two, i.e., n = 2. Let the set of outcomes
A satisfy Assumption 2. Suppose that a choice choice correspondence f satisfies
Condition μ, the nonempty lower intersection property, and Assumption 1. Then, f
is not robustly implementable relative to d∗∗ by the extended M-mechanism.
6
Concluding Remarks
This paper shows that the M-mechanism is not a robust Nash implementing mechanism relative to d∗∗ . This is the minimal sense in which the M-mechanism is not
robust to incomplete information. As I already discussed in the introduction, Bergemann and Morris [B-M] (2005) also addressed the issues of robust implementation.
In sum, B-M’s robustness reduces to using much more robust solution concepts, such
as ex post equilibrium and rationalizability. I argue that their robustness requirement is too stringent to robustify the existing implementation results. It is true that
if I require that the mechanism be robust to all type spaces, the resulting robust
mechanism must be indeed robust in a practical sense. However, Jehiel, Moldovanu,
Meyer-ter-Vehn, and Zame (2005) recently showed that in environments where signals are multi-dimensional and the valuations are interdependent, generically only
constant social choice functions are implementable in ex post equilibrium. 11 No
matter how attractive B-M’s robustness is, if all we can robustly implement are
constant social choice functions, we should consider possibilities that there might
be some reasonably robust mechanisms that are not exactly robust in B-M’s sense.
This is the direction in which I would like to make further progress with the future
works.
7
Appendix
In this appendix, I will show that the canonical perturbation of the complete information structure is characterized by the topology induced by d∗∗ .
Lemma 1 There is approximate common knowledge at (0, 0) and (2, 2) about which
game being played.
11
Bikhchandani (2005) showed that the existence of ex post implementable mechanisms in models
with multi-dimensional signals and interdependent values is possible when there is no consumption
externality. Thus, the conclusion and implication of Jehiel, Moldovanu, Meyer-ter-Vehn, and Zame
(2005) should be weakened accordingly.
19
Proof: Since player i knows that the game Γ(θ ) is played at (0, 0), we denote
Bi1 ({(0, 0)}) = {(0, 0)}, i.e., {(0, 0)} is the state in which player i assigns probability
1 to event {(0, 0)}. Furthermore, we must have Biq ({(0, 0)}) = {(0, 0)} for any q < 1.
Since all other players do not know which game being played at (0, 0), we want to
know any such j’s (j = i) q-belief operator Bjq ({(0, 0)}). Set
q(ε) =
1−p
.
1 − p + pε
Note that q(ε) → 1 as ε → 0. Consider j’s 1-belief operator. Then we have
Bj1 ({(0, 0)}) = ∅. But we have for j’s q-belief operator that Bjq ({(0, 0)}) = {(0, 0), (1, 0)}
q(ε)
for q = q(ε). Note that {(0, 0)}
because {(0, 0)} ⊂ Bk ({(0, 0)}) for
is q(ε)-evident
q
q
k ∈ N . Since Bi ({(0, 0)}) ∩ j=i Bj ({(0, 0)}) = {(0, 0)} for q = q(ε), we claim that
when ε is sufficiently small, it is a common q(ε)-belif at (0, 0) that the game Γ(θ ) is
played.
Consider the event {(2, 2)}. Since all other players but i are able to distinguish
between {(2, 2)} and others, we have Bj1 ({(2, 2)}) = {(2, 2)} for any j = i. Furthermore, we have that Bjq ({(2, 2)}) = {(2, 2)} for any q < 1. Next consider player i. We
have i’s 1-belief operator as Bi1 ({(2, 2)}) = ∅. Next consider i’s q-belief operator. We
have that Biq ({(2, 2)}) = {(2, 1), (2, 2)} for q = 1 − ε. Since {(2, 2)} ⊂ Bkq ({(2, 2)})
for each k ∈ N
q = 1 − ε, the event {(2, 2)} is (1 − ε)-evident. Note that
and for
q
q
Bi ({(2, 2)})∩ j=i Bj ({(2, 2)}) = {(2, 2)} for q = 1 − ε. Hence, when ε is sufficiently
small, it is a common (1 − ε)-belief at state {(2, 2)} that the game Γ(θ) is played. Corollary 3 d∗∗ (P ε , P ∗ ) → 0 as ε → 0. That is, there exists a sequence {q k }∞
k=1
converging to 1 such that with probability at least q k , there is a common q k -belief
about which game being played.
Proof: Define {ε(k)}∞
k=1 as a seququence such that ε(k) → 0 as k → ∞. Let
qI (ε) = 1 − ε.
Then, there is a common qI (ε)-belief at (2, 2) that the game Γ(θ) being played. Let
qJ (ε) =
1−p
.
1 − p + pε
Then, there is common qJ (ε)-belief at (0, 0) that the game Γ(θ ) being played. Define
q ∗ (ε) as follows:
q ∗ (ε) = min{1 − p + p(1 − ε)3 , qI (ε), qJ (ε)}.
We set q k ≡ q ∗ (ε(k)) for each k. Then, with probability at least q k , there is a common
ε(k) }∞ .
q k -belief about which game being played for each k. Define {P k }∞
k=1 ≡ {P
k=1
k
∞
Therefore, we can confirm that {P }k=1 is a convergent net of probability distributions converging to the complete information prior P ∗ according to the topology
induced by d∗∗ . 20
References
[1] R.J. Aumann: “Agreeing to Disagree,” Annals of Statistics, 6, (1976), 12361239.
[2] D. Bergemann and S. Morris: “Robust Implementation: The Role of Large
Type Spaces,” mimeo, (June 2005).
[3] S. Bikhchandani: “The Limits of Ex-Post Implementation Revisited,” mimeo,
UCLA
[4] B. Dutta and A. Sen: “A necessary and Sufficient Condition for Two-Person
Nash Implementation,” Review of Economic Studies, 58, (1991), 121-128.
[5] P. Jehiel, M. Meyer-ter-Vehn, B. Moldovanu, and W. Zame: “The Limits of
Ex-Post Implementation,” (2005), forthcoming in Econometrica
[6] T. Kunimoto: “Robust Implementation under Approximante Common Knowledge,” mimeo, (November 2005).
[7] T. Kunimoto: “The Robustness of Equilibrium Analysis: The Case of Undominated Nash Equilibrium,” mimeo, (January 2006).
[8] E. Maskin: “Nash Equilibrium and Welfare Optimality,” Review of Economic
Studies, 66, (1999), 23-38.
[9] D. Monderer and D. Samet: “Approximating Common Knowledge with Common Beliefs,” Games and Economic Behavior, 1, (1989), 170-90.
[10] J. Moore and R. Repullo: “Nash Implementation: A Full Characterization,”
Econometrica, 58, (1990), 1083-1099.
[11] A. Rubinstein: “The Electronic Mail Game: Strategic Behavior under ‘Almost
Common Knowledge’,” American Economic Review, 79, (1989), 385-391.
21