Notes on Exact and Approximate Bayesian Implementation

Takashi Kunimoto∗

This Version: December 2010Very Incomplete

1 Preliminaries

Let N = {1, . . . , n} denote the set of agents and Ti be the set of finite types of agent i.Denote T ≡ T1 × · · · × Tn, and T−i ≡ T1 × · · · × Ti−1 × Ti+1 × · · · × Tn.12 Each agent ihas a prior (probability measure) qi defined over T . These beliefs agree on states withzero probability: for all i, j ∈ N and for all t ∈ T , qi(t) = 0 if and only if qj(t) = 0. LetT ∗ = {t ∈ T |qi(t) > 0 ∀i ∈ N} be the set of states with positive probability.

Let A denote the finite set of pure outcomes, which are assumed to be independentof the information state.3 Let Δ(A) be the set of probability distributions over A.

Agent i’s state dependent von Neumann-Morgenstern utility function is denoted ui :Δ(A) × T → R.

We can now define an environment as E = (A, {ui, Ti, qi}i∈N ), which is implicitlyunderstood to be common knowledge among the agents.

A (possibly stochastic) social choice function (SCF) is a function f : T → Δ(A). LetF be the set of all possibly stochastic SCFs. Consider the following metric on SCFs: forany two SCFs f, h ∈ F ,

d(f, h) = sup{|f(a|t) − h(a|t)||a ∈ A, t ∈ T ∗}where f(a|t) denotes the probability that outcome a ∈ A is realized at state t ∈ T ∗ Forε ≥ 0, I shall say that two SCFs f and h are ε-approximate (f ≈ε h) if d(f, h) ≤ ε. Ishall say that two SCFs f and h are equivalent (f = h) if they are 0-approximate. Thismeans that the two SCFs “coincide” for every t ∈ T ∗. The interim expected utility ofagent i of type ti that pretends to be of type t′i corresponding to an SCF f is defined as:

Ui(f ; t′i|ti) ≡∑

t−i∈T−i

qi(t−i|ti)ui(f(t′i, t−i)); (ti, t−i)).

∗Dept. of Economics, McGill University and CIREQ, Montreal, Quebec, Canada and Dept. ofEconomics, Hitotsubashi University, Kunitachi, Tokyo, Japan; takashikunimoto9@gmail.com

1Similar notation will be used for products of other sets.2The finiteness of the type space is assumed for the sake of simplicity. See Serrano and Vohra (2010)

for how to relax this assumption.3This finiteness is also made for simplicity. See Serrano and Vohra (2010) for how to relax this

assumption.

1

Denote Ui(f |ti) = Ui(f ; ti|ti).A mechanism Γ = ((Mi)i∈N , g) describes a (nonempty, countable) message space

Mi for agent i and an outcome function g : M → Δ(A), where M = ×i∈NMi.4 Letσi : Ti → Δ(Mi) denote a (mixed) strategy for agent i and Σi his set of mixed strategies.Here Δ(Mi) denotes the set of probability measures over Mi. Let

Ui(g ◦ σ|ti) ≡∑

t−i∈T−i

qi(t−i|ti)ui(g(σ(t−i, ti)); (t−i, ti)).

Given a mechanism Γ = (M,g), let Γ(T ) be an incomplete information game associ-ated with a type space T .

Definition 1 (mixed Bayesian Nash Equilibrium (mBNE)) A strategy profile σ ∈Σ is a (mixed strategy) Bayesian Nash equilibrium (BNE) if for any agent i ∈ N ,any type ti ∈ Ti, and any pure strategy σ

′i ∈ Σi,

Ui(g ◦ σ|ti) ≥ Ui(g ◦ (σ′i, σ−i)|ti).

Given an incomplete information game Γ(T ), let mBNE(Γ(T )) ⊆ Σ be the entireset of mixed strategy Bayesian Nash equilibria of the game Γ(T ).

Definition 2 (mixed Bayesian Implementability) An SCF f is exactly Bayesianimplementable in mixed strategies if there exists a mechanism Γ = (M,g) with thefollowing two properties: (1) BNE(Γ(T )) �= ∅; and (2) for any σ ∈ mBNE(Γ(T )),f = g ◦ σ.

Definition 3 (mixed Virtual Bayesian Implementability) An SCF f : T → Δ(A)is virtually Bayesian implementable in mixed strategies if there exists ε > 0 suchthat for any ε ∈ (0, ε], there exists an SCF f ε with the following two properties: (1)f ≈ε f

ε (f and f ε are ε-approximate) and (2) f ε is exactly Bayesian implementable inmixed strategies.

2 Exact Implementation

Definition 4 (Incentive Compatibility (IC)) An SCF f : T → Δ(A) satisfies sat-isfy incentive compatibility (IC) if for every i ∈ N, ti, t

′i ∈ Ti,

Ui(f |ti) ≥ Ui(f ; t′i|ti)A mixed deception is a profile of functions, α = (αi)i∈N , where αi : Ti → Δ(Ti),

αi(ti) �= ti for some ti ∈ Ti for some i ∈ N . (Note that the identity function I : Θ → Θis not a deception.) Let A be the set of all mixed deceptions. For an SCF f ∈ F and a

4I can also handle the case of uncountable message space. In doing so, I must impose some suitablemeasurability condition on M so that the corresponding strategy spaces and interim preferences arewell-defined.

2

mixed deception α ∈ A, f◦α denotes the SCF such that for each t ∈ T , [f◦α](t) = f(α(t))where α(t) = Πi∈Nαi(ti) denotes the probability measure on T induced by α in state tand f(α(t)) imposes f(t′) on each of the realizations t′ of the measure α. For any SCFf and any mixed deception α, I use f ◦ α �= f to denote the case where f(t) �= f(α(t))for some t ∈ T .

For a type ti ∈ Ti, an SCF f ∈ F , and a deception α ∈ A, let fαi(ti)(t′) = f(t

′−i, αi(ti))

for all t′ ∈ T where f(t′−i, αi(ti)) is defined as above.

Definition 5 (mixed Bayesian Monotonicity (mBM)) An SCF f satisfies mixedBayesian monotonicity (mBM) if for any mixed deception α ∈ A, whenever f ◦α �=f , there exist i ∈ N, ti ∈ Ti, and an SCF y ∈ F such that

Ui(y ◦ α|ti) > Ui(f ◦ α|ti) and Ui(f |t′i) ≥ Ui(yαi(ti)|t′i) ∀t′i ∈ Ti. (∗)

Proposition 1 If an SCF f is exactly Bayesian implementable in mixed strategies, thenthere exists an SCF f which is equivalent to f and satisfies IC and mBM.

Proof of Proposition 1: Let Γ = (M,g) Bayesian implements the SCF f . DefineF = {f ∈ F|f = f}.

Since the necessity of incentive compatibility can be shown in the same way as Jackson(91) does, we shall show that mixed Bayesian monotonicity is necessary for exactlyimplementing an incentive compatible SCF in mixed strategies.

Take any f ∈ F that satisfies IC. Let σ be an equilibrium such that f = g◦σ. Supposethat there exists some (mixed) deception α such that f �= f ◦ α. By the requirementof Bayesian implementation, it must be that σ ◦ α is not a (mixed) equilibrium at somet ∈ T . Therefore, there exist i ∈ N and mi ∈Mi such that

Ui(g ◦ ((σi, σ−i) ◦ α)|ti) > Ui(g ◦ (σ ◦ α)|ti) =︸︷︷︸�f=g◦σ

Ui(f ◦ α|ti),

where σi(ti) ≡ mi for each ti ∈ Ti. Note that σi is a pure strategy. Let y = g ◦ (σi, σ−i) ∈F . Hence, we have

Ui(y ◦ α|ti) > Ui(f ◦ α|ti).

Since σi is constant, we have that yαi(ti) = y = g ◦ (σi, σ−i). Because σ is an equilibrium,it follows that Ui(f |t′i) ≥ Ui(yαi(ti)|t

′i) for each t

′i ∈ Ti. This shows that f satisfies mixed

BM. �

I begin by introducing some additional pieces of notation. For every SCF f ∈ Fand every mixed deception α ∈ A such that f �= f ◦ α, a test-agent is any i ∈ N forwhom condition (∗) in the definition of mixed BM holds. Denote by Di(f) the set ofmixed deceptions for which i is a test-agent at f . For each test-agent i and each mixeddeception α ∈ Di(f), fix an SCF yαi satisfying (∗) for agent i of type ti. Notice that

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condition (∗) concerns the SCF y only in those states in which agent i is of type αi(ti).There is, therefore, no loss of generality in assuming that yαi is of the form:

yαi (t−i, t′i) = yαi (t−i, ti) ∀t−i ∈ T−i, ∀t′i ∈ Ti.

Thus, yαi is constant over Ti.For each f ∈ F and each i ∈ N , let

Cfi = {f} ∪⋃

α∈Di(f)

{yαi } .

Of course, it follows that if agent i is not a test-agent for any mixed deception α atf ∈ F , I set Cfi = {f}.

In what follows, I shall make two regularity assumptions on environments. First, Iadapt the “no-total-indifference” (NTI) assumption to my environments.

Definition 6 (NTI) An environment E satisfies no-total-indifference (NTI) if forevery agent i ∈ N and every ti ∈ Ti, there exist a, a′ ∈ A such that∑

t−i∈T−i

qi(t−i|ti)ui(a; t−i, ti) �=∑

t−i∈T−i

qi(t−i|ti)ui(a′; t−i, ti).

In addition, I shall make the economic environment assumption made in Jackson(1991). Before defining an economic environment, let me define a splicing of two SCFsx, z ∈ F along a set S ⊆ T . The SCF x

/Sz is defined as follows: for any t ∈ T ,

[x/Sz](t) =

{x(t) if t ∈ Sz(t) otherwise

Definition 7 (Economic Environment (E)) An environment E satisfies E if for anySCF f ∈ F and any t ∈ T ∗, there exist two agents i, j ∈ N with i �= j and two SCFsx, y ∈ F such that

Ui(x/Sf |ti) > Ui(f |ti) and Uj(y

/Sf |tj) > Uj(f |tj)

for all S ⊆ T for which t ∈ S.

Theorem 1 Suppose that an environment E satisfies NTI and E. Then, an SCF f isBayesian implementable in mixed strategies if and only if there exists an SCF f whichis equivalent to f and satisfies IC and mBM.

Remark: It is worthwhile to mention that this result holds for any number of agents.This exhibits a stark contrast with the result of Jackson (1991).

Proof of Theorem 1: We have already shown the necessity part in Proposition 1.So, we focus here on the sufficiency part. Define F = {f ∈ F|f = f}. Fix any f ∈ F and

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assume that f satisfies IC and mBM. We will propose a mechanism and will show thatthe proposed mechanism exactly Bayesian implements the SCF f in mixed strategies.For each agent i ∈ N , define

Mi = Ti × {0, 1} × C fi × F × Z+,

where Z+ = {0, 1, 2, . . . } denotes the set of nonnegative integers. Agent i’s typicalmessage mi ∈Mi can be decomposed into (m1

i ,m2i ,m

3i ,m

4i ,m

5i ). Let M = M1×· · ·×Mn.

Let M = M1 ×M2 ×M3 ×M4 ×M5. Denote by m3i the part of the third component

of mi that is an element of C fi and m by a message profile. Fixing a constant δ ∈ (0, 1)and denoting by a the uniform lottery over A. The outcome function g : M → Δ(A) isdefined by the following rules:

• Rule (1): if m2i = 0 for each agent i ∈ N , then

g(m) = f(m1).

• Rule (2): if there exists an agent i ∈ N such that m2i = 1 and for any agent j �= i,

m2j = 0, then,

g(m) = (1 − δ)f(m1) +δ

n

[(n− 1)f(m1) +m3

i (m1)

].

• Rule (3): Otherwise, denoting by j the agent with the lowest index among thosewho announce the highest integer m5

j ,

g(m) = (1 − δ)f(m1) + δ

⎡⎣ m5

j

m5j + 1

m4j(m

1) +1

n(m5j + 1)

⎛⎝a+

∑i�=j

m3i (m

1)

⎞⎠

⎤⎦

where a denotes the uniform lottery over A.

The proof of the theorem consists of a series of lemmas.

Lemma 1 There exists an equilibrium σ ∈ BNE(Γ(T )) where ∀i ∈ N, ∀ti ∈ Ti, σi(ti) =(ti, 0, f , hi, 0).

Proof of Lemma 1: Let σ be such a strategy profile defined in the above statement.Note that this strategy profile induces Rule (1) and g ◦ σ = f . It is trivial to see thatthere is no unilateral deviation from σ that can trigger Rule (3) and therefore, m4

i andm5i has no effect on the outcome. The only way an agent i can change the outcome is

by changing his announcement of m1i or m2

i and m3i . Since f satisfies IC, reporting a

false type only in m1i is not a profitable deviation for any agent. By condition (∗) in

the definition of mixed BM and the very definition of C fi , it is not profitable to onlychange in m2

i and m3i . (Ui(f |t′i) ≥ Ui(yαi(ti)|t

′i) ∀t′i ∈ Ti) By both IC and condition (∗)

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in the definition of mixed BM, it is impossible to profit by changing m1i ,m

2i , and m3

i all

together because yαi ∈ C fi is constant over Ti. More specifically, we argue as follows.Consider agent i of type ti. One can define an arbitrary deviation σ

′i: for any ti ∈ Ti,

σ′i(ti) =

{(αi(ti), 1, yαi , σ

4i (ti), σ

5i (ti)) if ti = ti

σi(ti) otherwise.

Recall that m4i and m5

i has no effect on the outcome. Since f satisfies IC and mixed BM,we know that for any t

′i ∈ Ti,

Ui(f |t′i) ≥ Ui(fαi(ti)|t′i)︸ ︷︷ ︸

�f satisfies IC

and Ui(f |t′i) ≥ Ui(yαi |t′i)︸ ︷︷ ︸

Part of Condition (∗) of mixed BM

.

Thus, Ui(g ◦σ|ti) ≥ Ui(g ◦ (σ′i, σ−i)|ti) for any σ

′i. Since the choice of i and ti is arbitrary,

this shows that there are no profitable deviations from σ by changing m1i ,m

2i , and m3

i

all together. Thus, σ is indeed an equilibrium and it follows that f = g ◦ σ. �

Lemma 2 There exists no equilibrium σ ∈ mBNE(Γ(T )) that induces Rule (3) withpositive probability.

Proof of Lemma 2: We argue by contradiction. Let T ⊆ T ∗ be a nonempty set ofstates where σ induces Rule (2) with positive probability. Define

z = supt∈T

maxk∈N

sup{m5k ∈ Z+| m5

k is in the support of σ5k(tk)}.

Note that z is well-defined because the type space is finite. Let j be the lowest indexedagent who, without loss of generality, announces z in some state in T . Thus, there existsa state t ∈ T ∗ in which the integer game is played with positive probability, and is wonby agent j of type tj who announces the integer z. Define

T−j = {t−j ∈ T−j|(tj , t−j) ∈ T}.

By construction, agent j of type tj is the lowest indexed agent who announces z in eachstate in {tj} × T−j and by announcing zj = z, he wins the integer game in each statewithin {tj}×T−j . If agent j of type tj changes his announcement of the integer to z

′j > z,

he continues to be the winner of the integer game in all states in {tj} × T−j . Consideragent j’s following deviation strategy σ

′j : for any tj ∈ Tj,

σ′j(tj) =

{(σ1j (tj), σ

2j (tj), σ

3j (tj), h

′j , z

′j) if tj = tj

σj(tj) otherwise,

where z′j > z and the SCF h

′j ∈ F is chosen appropriately using NTI to yield a strictly

preferred lottery in each state in {tj} × T−j where agent j of type tj wins the integer

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game.5 Since σ induces Rule (3) with positive probability, this would increase type tj ’sexpected payoff. That is,

Uj(g ◦ (σ′j , σ−i)|tj) > Uj(g ◦ σ|tj).

This contradicts the hypothesis that σ is an equilibrium. �

Lemma 3 There exists no equilibrium σ ∈ mBNE(Γ(T )) that induces Rule (2) withpositive probability.

Proof of Lemma 3: We argue by contradiction. Suppose that there exists anonempty set of states T ⊆ T ∗ where m2

i = 1 and m2j = 0 for each j �= i. Let agent i of

type ti be such an agent. Suppose that this could happen with positive probability overT . In that case, the corresponding outcome is

g(m) = (1 − δ)f (m1) +δ

n

[(n− 1)f(m1) +m3

i (m1)

].

Then, since the environment satisfies E, there exists agent j �= i with type tj for whomthis equilibrium outcome is not top ranked. Then, any such agent j of type tj canannounce m2

j = 1. Fix such agent j of type tj. Let T−j = {t−j ∈ T−j |(tj , t−j) ∈ T}.Define

z = supt∈T

maxk∈N

sup{m5k| m5

k is in the support of σ5k(tk)}.

Note that z is well-defined because the type space is finite. We can assume, without lossof generality, that agent j is the lowest indexed agent who announces z in each state in{tj} × T−j . Consider the deviation σ

′j : for any tj ∈ Tj,

σ′j(tj) =

{(σ1j (tj), 1, σ

3j (tj), h

′j , z

′j) if tj = tj

σj(tj) otherwise,

where z′j > z and h

′j is an SCF. We will soon describe how we choose z

′j and h

′j . By

choosing z′j sufficiently high so that agent j continues to be the winner in the integer

game in all states in {tj} × T−j, g ◦ (σ′j , σ−j) induces the following lottery:

g(m) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(1 − δ)f(m1) + δ

[z′j

z′j+1

h′j(m

1) + 1n(z

′j+1)

(a+

∑i�=jm

3i (m

1))]

if m2i = 1

(1 − δ)f(m1) + δn

[(n− 1)f(m1) +m3

i (m1)

]if m2

i = 0

Since the environment satisfies E, we can choose an SCF h′j to approximate an optimal

outcome. Due to the linearity of expected utility, we obtain

Uj(g ◦ (σ′j , σ−j)|tj) > Uj(g ◦ σ|tj).

However, this contradicts the hypothesis that σ is an equilibrium. �5Here NTI plays an important role. Without NTI, one may not be able to find a strictly better SCF

h′j .

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Lemma 4 Let σ ∈ mBNE(Γ(T )) be an equilibrium that induces Rule (1) such that foreach i ∈ N and each ti ∈ Ti, σi(ti) = (αi(ti), 0, ·, ·, ·) where α ∈ A is the mixed deceptionunder σ. Then, f = f ◦ α.

Proof of Lemma 4: Suppose not, that is, there is such an equilibrium σ describedin the statement under Rule (1) but f �= f ◦ α. In this case, for any t ∈ T ∗, we have

g(σ(t)) = f(σ1(t)) = f(α(t)).

Since f satisfies mixed BM, there exist agent i ∈ N , type ti ∈ Ti, and an SCF y satisfyingcondition (∗):

Ui(y ◦ α|ti) > Ui(f ◦ α|ti) and Ui(f |t′i) ≥ Ui(yαi(ti)|t′i) ∀t′i ∈ Ti. (∗)

Consider the following deviation σ′i: for any ti ∈ Ti,

σ′i(ti) =

{(αi(ti), 1, y, f , 0) if tj = tj

σi(ti) otherwise.

Let T−i = {t−i ∈ T−i|(ti, t−i) ∈ T ∗}. g ◦ (σ′i, σ−i) induce the following lottery at any

state t ∈ {ti} × T−i:

g((σ′i, σ−i)(t)) = (1 − δ)f (α(t)) +

δ

n

((n− 1)f (α(t)) + y(α(t))

).

Due to the linearity of expected utility, we obtain that

Ui(g ◦ (σ′i, σ−i)|ti) > Ui(g ◦ σ|ti).

This contradicts the hypothesis that σ is an equilibrium. �Lemmas 1,2,3, and 4 together prove that the proposed mechanism exactly implements

f . This completes the proof of Theorem 1. �

3 Approximate Implementation

Definition 8 An SCF f satisfies mixed virtual monotonicity (mVM) if there existsan incentive compatible SCF x ∈ F such that for any mixed deception α ∈ A, wheneverf �= f ◦ α, there exist i ∈ N, ti ∈ Ti, and an SCF y ∈ F for which

Ui(y ◦ α|ti) > Ui(x ◦ α|ti) and Ui(x|t′i) ≥ Ui(yαi(ti)|t′i) ∀t′i ∈ Ti. (∗∗)

Theorem 2 Suppose that an environment E satisfies NTI and E. Then, an SCF f isvirtually Bayesian implementable in mixed strategies if and only if it is equivalent to anSCF f that satisfies IC and mVM.

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Remark: Provided that an SCF f satisfies IC, it is easy to see that mixed BM impliesmixed VM. In other words, virtual Bayesian implementation is more permissive thanexact Bayesian implementation.

Proof of Theorem 2: (NECESSITY): Fix any SCF f ∈ F . By our hypothesis,for every ε > 0, there exists a nearby SCF f ε that is exactly Bayesian implementable inmixed strategies. By Theorem 1, we know that f ε must satisfy IC. Using the standardcontinuity argument and the continuity of expected utility, for every ε > 0 sufficientlysmall, f satisfies IC if and only if f ε satisfies IC.

Consider an arbitrary mixed deception α ∈ A such that f �= f ◦ α. Since, by ourhypothesis, f ε is exactly Bayesian implementable in mixed strategies, f ε satisfies mBM.If we choose ε > 0 small enough, we have that f ε �= f ε ◦ α. Then, the fact that f ε

satisfies mixed BM implies that there exist i ∈ N, ti ∈ Ti, and an SCF y ∈ F such that

Ui(y ◦ α|ti) > Ui(f ε ◦ α|ti) and Ui(f ε|t′i) ≥ Ui(yαi(ti)|t′i) ∀t′i ∈ Ti.

We have already shown that f ε satisfies IC. Letting x = f ε, we show that f satisfiesmVM.

(SUFFICIENCY): Let F = {f ∈ F|f = f}. Fix any f ∈ F . Suppose that the SCFf satisfies IC and mVM. This proof exploits the properties assumed and Theorem 1 toobtain a direct proof. Thus, it remains to show that for every ε > 0, there exists an SCFf ε such that f ≈ε f

ε and f ε satisfies IC and mBM.By mixed VM, there exists an incentive compatible SCF x and an SCF y exhibiting

the appropriate preference reversal as in (∗∗) for every mixed deception α satisfyingf �= f ◦α. Let f ε = (1− ε)f + εx and y′ = (1− ε)f + εy. Since both f and x satisfy IC,by construction, f ε also satisfies IC.

We claim that for every ε > 0, we claim that f ε satisfies mixed BM. Due to thelinearity of expected utility, we have

Ui(y′ ◦ α|ti) − Ui(f ε ◦ α|ti) = ε [Ui(y ◦ α|ti) − Ui(x ◦ α|ti)] .It follows from (∗∗) that

Ui(y′ ◦ α|ti) > Ui(f ε|ti).From (∗∗), we also know that

Ui(x|t′i) ≥ Ui(yαi(ti)|t′i) ∀t′i ∈ Ti.

By adding (1− ε)Ui(f |t′i) to ε weighted both hand sides of the above inequality, we have

(1 − ε)Ui(f |t′i) + εUi(x|t′i) ≥ (1 − ε)Ui(f |t′i) + εUi(yαi(ti)|t′i) ∀t′i ∈ Ti.

Moreover, since f satisfies IC,

Ui(f |t′i) ≥ Ui(fαi(ti)|t′i) ∀t′i ∈ Ti.

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Plugging this back into the previous inequality, we obtain

(1 − ε)Ui(f |t′i) + εUi(x|t′i) ≥ (1 − ε)Ui(fαi(ti)|t′i) + εUi(yαi(ti)|t

′i) ∀t′i ∈ Ti.

By the linearity of expected utility and taking into account the definitions of f ε and y′,we can rewrite the inequality as

Ui(f ε|t′i) ≥ Ui(y′αi(ti)|t′i) ∀t′i ∈ Ti.

To sum up, we have the following: for any mixed deception α ∈ A, whenever f ε �= f ε ◦α,there exist i ∈ N, ti ∈ Ti, and an SCF y′ ∈ F such that

Ui(y′ ◦ α|ti) > Ui(f ε|ti) and Ui(f ε|t′i) ≥ Ui(y′αi(ti)|t′i) ∀t′i ∈ Ti.

This implies that f ε satisfies mixed BM. The rest of the proof will be completed bychecking that f ε ≈ε f and by Theorem 1, f ε is exactly Bayesian implementable in mixedstrategies. �

4 Pure V.S. Mixed Strategies

In this section we will show that mixed Bayesian monotonicity is equivalent to the usualBayesian monotonicity, provided that the given SCF satisfies IC. This implies that aslong as we consider an environment satisfying E and NTI and stochastic SCFs, there areno extra constraints in terms of implementability to move from pure strategies to mixedstrategies

Proposition 2 Let f be an SCF that satisfies IC. Then, f satisfies mixed Bayesianmonotonicity if and only if it satisfies Bayesian monotonicity.

Proof of Proposition 2: By definition, it is clear that mixed BM implies BMbecause it imposes the requirement of the appropriate preference reversal over all mixeddeceptions, which include pure deceptions as particular cases.

We now show that BM implies mixed BM. Consider a non-pure deception α ∈ Asuch that f �= f ◦ α. Let β0 ∈ A be a pure deception in the support of α satisfying thatf �= f ◦ β0. Since β0 is a pure deception and f satisfies BM, there exist i ∈ N, ti ∈ Ti,and an SCF y ∈ F such that

Ui(y ◦ β0|ti) > Ui(f ◦ β0|ti) and Ui(f |t′i) ≥ Ui(yβ0i (ti)|t

′i) ∀t′i ∈ Ti.

Since α can be expressed as a function of β0 and the other pure deceptions {βk}Kk=1, letα(t) =

∑Kk=0 pkβ

k(t) for each t ∈ T ∗ such that pk > 0 for each k = 0, 1, . . . ,K, and∑Kk=0 pk = 1. Define an SCF y ◦ α ∈ F as follows:

y ◦ α =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

y ◦ β0 w.p. p0

f ◦ β1 w.p. p1

f ◦ β2 w.p. p2...

...f ◦ βK w.p. pK

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Due to the linearity of expected utility, we have

Ui(y ◦ α|ti) = p0Ui(y ◦ β0|ti) +K∑k=1

pkUi(f ◦ βk|ti).

f ◦ α can be expressed as follows:

f ◦ α =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

f ◦ β0 w.p. p0

f ◦ β1 w.p. p1...

...f ◦ βK w.p. pK

Once again, due to the linearity of expected utility, we have

Ui(f ◦ α|ti) = p0Ui(f ◦ β0|ti) +K∑k=1

pkUi(f ◦ βk|ti).

Thus, we have

Ui(y ◦ α|ti) > Ui(f ◦ α|ti).Note that αi can be expressed as follows: for any ti ∈ Ti,

αi(ti) =K∑k=0

πki βki (ti)

where πki ≥ 0 for each k = 0, 1, . . . ,K and∑K

k=0 πki = 1. Due to the way we define y ◦α,

we can express yαi(ti) as follows:

yαi(ti) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

yβ0i (ti)

w.p. π0i

fβ1i (ti)

w.p. π1i

......

fβKi (ti)

w.p. πKi

where πki ≥ 0 for each k = 0, 1, . . . ,K and∑K

k=0 πki = 1. Thus, we have

Ui(yαi(ti)|t′i) = π0

iUi(yβ0i (ti)|t

′i) +

K∑k=1

πki Ui(fβki (ti)

|t′i).

Since f satisfies IC, we have

Ui(f |t′i) ≥ Ui(fβki (ti)

|t′i) ∀k = 1, . . . ,K.

This enables us to conclude that

Ui(f |t′i) ≥ Ui(yαi(ti)|t′i) ∀t′i ∈ Ti.

Thus, f also satisfies mixed BM. �

11

5 Complete Information

Here we aim to take care of complete information environments as a special class ofincomplete information environments.

Definition 9 An environment E satisfies the complete information assumption ifthere exists a set T0 with the following two properties:

1. for each agent i ∈ N , there exists a bijection φi : Ti → T0; and

2. for each i ∈ N and each t ∈ T ,

qi(t−i|ti) ={

1 if φj(tj) = φi(ti) for each j �= i0 otherwise.

Lemma 5 Suppose that an environment E satisfies the complete information assump-tion. Let f be an SCF that satisfies IC. Then, f satisfies mixed Bayesian monotonicityif and only if it satisfies Maskin monotonicity.

Remark: This result shows that we do not need more than IC and Maskin monotonicitywhen we want to take care of mixed strategies. This is consistent with Maskin’s (1999)Theorem 3.

Proof of Lemma 5: Suppose that there exist two states t, t′ ∈ T such that φi(ti) =t0 ∈ T0 and φi(t

′i) = t

′0 for each i ∈ N and t0 �= t

′0. Assume further that f(t) �= f(t′).

Maskin monotonicity requires the existence of an agent i and an alternative y satisfyingthat

ui(f(t′); t′) ≥ ui(y; t′) and ui(y; t) > ui(f(t′); t).

Let α be a pure deception satisfying α(t) = t′ and y be an SCF such that y(t) = y(t′) = yand y(t′′) = f(t′′) for any t′′ ∈ T\{t, t′}. By this construction and because f satisfies IC,we have

Ui(y ◦ α|ti) > Ui(f ◦ α|ti) and Ui(f |t′i) ≥ Ui(yαi(ti)|t′i) ∀t′i ∈ Ti.

Hence, Bayesian monotonicity also obtains. By the previous proposition, we show theequivalence between mixed BM and BM. This concludes that the equivalence of mixedBM and Maskin monotonicity under complete information. �

Lemma 6 Suppose that an environment E satisfies the complete information and thereare at least three agents. For any SCF f ∈ F , there exists an SCF f ∈ F which isequivalent to f and satisfies IC.

Corollary 1 Suppose that an environment E satisfies E and the complete informationassumption. Assume further that there are at least three agents, i.e., n ≥ 3. Then, if anSCF f satisfies Makin monotonicity, it is Nash implementable in mixed strategies.

12

6 How Restrictive is mixed Virtual Monotonicity?

6.1 Type Diversity

Recall that A = {a1, . . . , aK}. Henceforth, I will find it convenient to identify a lotteryx ∈ Δ(A) as a point in the (K − 1) dimensional unit simplex ΔK−1 = {(x1, . . . , xK) ∈RK+ | ∑K

k=1 xk = 1}. Define V ki (ti) to be the interim expected utility of agent i of type

ti for the constant SCF that assigns ak in each state in T , i.e.,

V ki (ti) =

∑t−i∈T−i

qi(t−i|ti)ui(ak; ti, t−i).

Let Vi(ti) = (V 1i (ti), . . . , V K

i (ti)).Next, I define the condition of type diversity in an environment, which is proposed

in Serrano and Vohra (2005).

Definition 10 (TD) An environment E satisfies type diversity (TD) if there do notexist i ∈ N, ti, t

′i ∈ Ti with ti �= t′i, β ∈ R++ and γ ∈ R such that

Vi(ti) = βVi(t′i) + γe,

where e is the unit vector in ΔK−1.6

The reader is referred to Serrano and Vohra (2005) to find an appraisal of the connec-tions of type diversity with the conditions of interim value distinguished types (Palfreyand Srivastava (1993, definition 6.3)), incentive consistency (Duggan (1997)), and withthe algorithm behind measurability due to Abreu and Matsushima (1992c).

In environments satisfying TD and NTI, Serrano and Vohra (2005) show the followingcritical lemma.

Lemma 7 (Serrano and Vohra (2005)) Suppose an environment E satisfies TD andNTI. Then there exist constant SCFs

((i(ti))ti∈Ti

)i∈N such that for every i ∈ N and

ti, t′i ∈ Ti with ti �= t′i,

Vi(i(ti)|ti) > Vi(i(t′i)|ti).

Remark: All that is needed for this lemma is the assumption that the individual prefer-ences over lotteries are monotone in the sense that any shift of probability weight from aless preferred to a more preferred pure alternative yields a lottery which is preferred. The

6 If A is a separable metric space, let A∗ = {a1, a2, . . . } be a countable dense subset of A. Now, wecan define

Vi(ti) = (V ki (ti))

∞k=1 ∈ �∞

We also define e as the countable unit base in A with ‖e‖ = 1. With these qualifications, first-order TDis also well defined for separable metric spaces.

13

axiom that preferences are monotone is, of course, much weaker than the independenceaxiom, and is implied by the von Neumann-Morgenstern utility representation.

Proof : Consider the constant SCF x, which prescribes in each state the lottery x,assigning equal probability to each alternative in A , i.e., x(t) = (1/K, . . . , 1/K) for allt ∈ T . We will use induction on the number of types of agent i.

First, we show that for i ∈ N , and for two types ti, t′i ∈ Ti with ti �= t′i, there existconstant SCFs x and x′, close to x, such that

Vi(x|ti) > Vi(x′|ti) and Vi(x′|t′i) > Vi(x|t′i). (1)

The interim indifference curve of agent i of type ti through x is described by ahyperplane, H, in R

K−1+ :

H =

{(x1, . . . , xK−1) ∈ R

K−1+

∣∣∣∣ K−1∑k=1

pk(ti)xk = u

},

where pk(ti) = (V ki (ti) − V K

i (ti)) for k = 1, . . . ,K − 1.Let p(ti) = (p1(ti), . . . , pK−1(ti)) ∈ R

K−1. Consider the interim indifference hyper-plane through x of agent i of type t′i where ti �= t′i:

H ′ =

{(x1, . . . , xK−1) ∈ R

K−1+

∣∣∣∣ K−1∑k=1

pk(t′i)xk = u′},

Given NTI, we must have p(ti) �= 0 and p(t′i) �= 0. We claim that p(ti) �= cp(t′i) for anyc > 0. Suppose not; that is, there is c > 0 such that p(ti) = cp(t′i). This implies thatVi(ti) = cVi(t′i) + γe, which contradicts TD. Thus, either p(ti) = cp(t′i) where c < 0 orthere does not exist c �= 0 such that p(ti) = cp(t′i). In the former case, it is easy to see(using NTI) that any point which lies above H must be below H ′ and, choosing twopoints (one above H and one below it) close to x, one finds constant SCFs which satisfy(1). In the latter case, it is clear that we can choose two constant SCFs which satisfy(1).

Now, according to the induction hypothesis, suppose that for the first |Ti| − 1 typesof agent i, i.e., for all ti ∈ Ti \ {t0i }, we have been able to find |Ti| − 1 constant SCFsnear x, say x(ti), such that for every ti ∈ Ti \ {t0i }, Vi(x(ti)|ti) > Vi(x(t′i)|ti) for everyt′i ∈ Ti \ {t0i , ti}. Consider type t0i . Choose the constant SCF among the collection(x(ti))ti∈Ti\{t0i } that is ranked highest by type t0i (without loss of generality, there isonly one). Call it x(ti). By arguments similar to the ones in the previous paragraph,because of NTI and TD, one can find a constant SCF near x(ti), call it x(t0i ), such thattypes ti and t0i satisfy (1). To construct a set of new constant SCFs i[ti] that wouldseparate every type, consider compounding lotteries (x[ti])ti∈Ti\{t0i } and x[ti] (to obtain(i[ti])ti∈Ti\{t0i }) and x[ti] and x[t0i ] (to obtain i[t0i ]). Since all inequalities concerning(x[ti])ti∈Ti\{t0i } are strict, the weights can be chosen so that the collection of constant

14

SCFs (i(ti))ti∈Ti satisfy all the inequalities in the statement of the lemma, so the proofis complete.7 �

Proposition 3 Suppose that an environment E satisfies NTI and TD. Then, there existsan SCF x that satisfies IC and mixed Bayesian monotonicity.

Proof : By Lemma 7, we construct the following SCF x: for any t ∈ T ,

x(t) =1n

∑i∈N

i(ti).

By construction, it is easy to see that the SCF x satisfies IC. Thus, it only remains toshow that x also satisfies mixed BM. Fix any deception α ∈ A such that x �= x ◦α. Thisimplies that there exist i ∈ N and ti ∈ Ti for whom αi(ti) �= ti. Define an SCF y asfollows: for any t ∈ T ,

y(t) =1ni(ti) +

1n− 1

∑j �=i

j(tj).

Note that y is constant over Ti by construction. Thus, we have

Ui(y ◦ α|ti) − Ui(x ◦ α|ti) =1n

[Ui(i(ti)|ti) − Ui(i(αi(ti))|ti)]> 0 (∵ Lemma 7).

We also have that for any t′i ∈ Ti,

Ui(x|t′i) − Ui(yαi(ti)|t′i) =

1n

[Ui(i(t

′i)|t

′i) − Ui(i(ti)|t′i)

]≥ 0 (∵ Lemma 7).

In sum, for any α ∈ A, whenever x �= x◦α, there exist i ∈ N, ti ∈ Ti, and an SCF y ∈ Fsuch that

Ui(y ◦ α|ti) > Ui(x ◦ α|ti) and Ui(x|t′i) ≥ Ui(yαi(ti)|t′i) ∀t′i ∈ Ti.

Therefore, x satisfies mixed BM. �

Corollary 2 Suppose that an environment E satisfies E, NTI and TD. Then, an SCFf is virtually Bayesian implementable in mixed strategies if and only if there exists anSCF f which is equivalent to f and satisfies IC.

7If A is a separable metric space, the modification we must make to the previous argument is the waywe define the lottery x(t):

x(t) = (xk(t))∞k=1

where xk(t) = (1 − δ)δk−1, and 0 < δ < 1.

15

Proof : Thanks to Theorem 2, all we have to show is that every SCF satisfies mixedvirtual monotonicity. The previous proposition shows that one can construct an SCF xsatisfies IC and mixed BM. Thus, any SCF satisfies mixed virtual monotonicity. �

Proposition 4 Assume that |A| ≥ 3. Let Ω be the entire parameter space. Then, thereexists an open and dense subset Ω∗ ⊆ Ω for which TD holds. E satisfies TD generically.

Remark: This result seems to crucially depend on the finiteness of the type space.

Proof : Recall that A is a finite set consisting of K alternatives, and recall ourdefinition of the interim utility Vi(ti) = (V k

i (ti))k=1,... ,K . Let K ≥ 3 (if K = 2, aviolation of TD happens when ordinal preferences are the same across types, a propertythat is certainly preserved for small perturbations).

Let Vi : Ti → RK be an agent i’s interim utilities over all constant SCFs. For each

type ti, assume there exist two alternatives ak, ak′ ∈ A such that V ki (ti) < V k

′i (ti), and

choosing one such pair of alternatives with extreme values, normalize expected utilitiesso that V k

i (ti) = 0, V k′

i (ti) = 1, and Vi(ti) ∈ [0, 1]K .Let |Ti| denote the cardinality of the set of types for agent i. Call S =

∑i∈N |Ti|.

With this notation, one can associate a normalized environment E with a point on Ω,the unit cube in R

(K−2)S with vertices at the points (0, . . . , 0) and (1, . . . , 1). Endow Ωwith the uniform metric, and define open balls using this metric relative to Ω. Since theproperty of TD is defined by a finite number of inequalities, one can easily see that theset of points in Ω satisfying it is an open and dense subset of Ω. That is,

• for each environment in Ω that satisfies TD, there exists an open ball around itcontaining only the environments in which the property is maintained, and

• for each environment in Ω that violates TD and for each open ball around it, therealways exists an environment satisfying TD in that ball.

Suppose therefore that the planner does not know which types will be chosen bynature, i.e., which point in Ω will be chosen, and suppose she can specify an ex-anteprobability measure over such nature choices. The assumption that she can confineherself to Ω uses the innocuous normalization of expected utilities and assumes furtherthat she knows that she will be dealing only with “finite worlds,” a finite number of typesfor each agent (perhaps due to complexity issues, in specifying payoffs and probabilities,agents stop after a finite number of decimals). �

6.2 Abreu-Matsushima (AM) Measurability

Denote by Ψi a partition of the set of types Ti, where ψi is a generic element of Ψi andΠi(ti) is the element of Ψi that includes type ti. Let Ψ = ×i∈NΨi and ψ = ×i∈Nψi.

Definition 11 An SCF f is measurable with respect to Ψ if, for every i ∈ N andevery ti, t′i ∈ Ti, whenever Πi(ti) = Πi(t′i),

f(ti, t−i) = f(t′i, t−i) ∀t−i ∈ T−i.

16

Measurability of f with respect to Ψ implies that for any player i, f does not distin-guish between any pair of types in the same cell of the partition Ψi.

Definition 12 A strategy σi for player i is measurable with respect to Ψi if for everyti, t

′i ∈ Ti,

Πi(ti) = Πi(t′i) =⇒ σi(ti) = σi(t

′i).

A strategy profile σ is measurable with respect to Ψ if, for every i ∈ N , σi is mea-surable with respect to Ψi.

For every i ∈ N, ti, t′i ∈ Ti, and (n − 1) tuple of partitions Ψ−i, I say that ti is

equivalent to t′i with respect to Ψ−i if, for every f and every f which are measurablewith respect to Ti × Ψ−i,

Vi(f |ti) ≥ Vi(f |ti) ⇐⇒ Vi(f |t′i) ≥ Vi(f |t′i).

Let ρi(ti,Ψ−i) be the set of all elements of Ti that are equivalent to ti with respectto Ψ−i, and let

Ri(Ψ−i) = {ρi(ti,Ψ−i) ⊂ Ti| ti ∈ Ti} .

Note that Ri(Ψ−i) forms an equivalence class on Ti, that is, constitutes a partitionof Ti. We define an infinite sequence of n-tuples of partitions, {Ψh}∞h=0, where Ψh =×i∈NΨh

i in the following way. For every i ∈ N ,

Ψ0i = {Ti},

and recursively, for every i ∈ N and every h ≥ 1,

Ψhi = Ri(Ψh−1

−i ).

Note that for every h ≥ 0, Ψh+1i is the same as, or finer than, Ψh

i . Define Ψ∗ as follows:

Ψ∗ ≡∞⋂h=0

Ψh.

Since Ti is finite for each agent i ∈ N , Lemma ?? guarantees that there exists apositive integer L such that Ψh = ΨL for any h ≥ L. Therefore, we can write Ψ∗ = ΨL.For each i ∈ N and ti ∈ Ti, define Π∗

i : Ti → Ψ∗i as the partition over Ti, which is

computed by the AM measurability algorithm.

Definition 13 An SCF f satisfies AM measurability if it is measurable with respectto Ψ∗.

17

Note how the partitions Ψ0, Ψ1, ..., and hence, the final partition Ψ∗ used in A-M measurability are really nothing but a property of the environment. The aim isto “treat equally” those types that are “indistinguishable” according to their interimpreferences. Thus, we start considering constant SCFs, i.e., SCFs that are measurablewith respect to the coarsest possible partition, and we separate types who have differentinterim preferences over this class of SCFs. This gives us a new partition of the set oftypes for each agent (iteration 1). Next, we consider SCFs measurable with respect tothese new partitions, and ask the same question: are there types that, having the samepreferences over constant SCFs, now can be separated because they exhibit differentinterim preferences over the enlarged class of SCFs considered? If the answer is No, theprocess ends and we have found Ψ∗. If it is Yes, we proceed to make the induced finerpartition of each set of types (iteration 2), and so on. The process ends after a finitenumber of steps with the identification of Ψ∗, which provides the maximum possibledegree of type separation or distinguishability in terms of interim preferences. A-Mmeasurability simply asks that the SCF not distinguish between different types that are“indistinguishable” according to Ψ∗.

When I considered type diversity, I have defined a vector Vi(ti) of player i’s valuationsof each alternative ak. When the algorithm that determines Ψ∗ does not stop in the firststep, we need to consider a more complicated “version” of ak, that we define below.Define

F = {h | h(t) is a degenerate lottery for all t ∈ T} .Recall that Ti is finite for every i ∈ N , and that A is finite. Then, F becomes a finitefunctional space. Define also

F (Ψ) = {h ∈ F | h is measurable with respect to Ψ} .Let |F (Ti ×Ψ−i)| = K.8 Define V k

i (ti; Ψ−i) to be the interim expected utility of agent iof type ti for each SCF fk ∈ F (Ti × Ψ−i), i.e.,

V ki (ti; Ψ−i) =

∑t−i∈T−i

qi(t−i|ti)ui(fk(ti, t−i); ti, t−i)).

Let Vi(ti; Ψ−i) = (V 1i (ti; Ψ−i), . . . , V K

i (ti; Ψ−i)).The next lemma follows simply from the definitions of F (Ψ) and of equivalent types.

Its proof is omitted:

Lemma 8 ti is equivalent to t′i with respect to Ψ−i if and only if there exist β > 0 andγ ∈ R such that

Vi(ti; Ψ−i) = βVi(t′i; Ψ−i) + γe,

where e is the unit vector in ΔK−1.8This is a slight abuse of notation, since K was defined in previous sections as the finite number of

alternatives in the set A. In part, we choose to use the same symbol here to enhance the parallels acrossthe arguments in the different sections. Also, it should not cause any confusion.

18

One can make the following useful observation:

Lemma 9 (TD and NTI ⇒ AM measurability, Serrano and Vohra (2005)) Supposean environment E satisfies NTI and TD. Then, every SCF satisfies AM measurability.

That is, if the environment satisfies NTI and TD, the algorithm that separates typesin the definition of measurability arrives at the finest partition in the first round.

Recall the recursive construction behind AM measurability, and, in particular, thepartitions Ψh

i for i ∈ N and h = 0, 1, . . . . For each i ∈ N, ti ∈ Ti, and h ≥ 0, let Πhi (ti)

be the element of Ψhi that includes ti.

The next lemma provides SCFs that will help us separate first-order types, as allowedby the h-th iteration in the measurability construction. It is a generalization of Lemma??.

Lemma 10 Suppose an environment E satisfies NTI. Then, for every i ∈ N and everyh = 1, 2, . . . , L, there exists a collection of SCFs {xhi [ψhi ]}ψh

i ∈Ψhi, which are measurable

with respect to Ψhi × Ψh−1

−i , and such that for every ti ∈ Ti and ψhi ∈ Ψhi \Πh

i (ti),

Vi

(xhi [Π

hi (ti)]

∣∣ti) > Vi

(xhi [ψ

hi ];ψ

hi

∣∣ti) ,Recall that

Vi(xhi [·]; ·|ti) ≡∑

t−i∈T−i

qi(t−i|ti)ui(xhi [·](·, t−i); ti, t−i)).

Proof of Lemma ??: Again we recall that A is finite. Fix iteration h in the AMmeasurability algorithm. Consider the SCF xh, which prescribes in each state the lotteryxh, assigning equal probability to each SCF in F (Ψh

i × Ψh−1−i ), the space of degenerate

SCFs measurable with respect to Ψhi × Ψh−1

−i . That is,

xh(t) =1Kh

f1(t) + . . .+1Kh

fKh(t)

for all t ∈ T . Here, |F (Ψhi × Ψh−1

−i )| = Kh. By construction, xh is measurable withrespect to Ψh

i × Ψh−1−i , and, abusing notation, we can write xh(t) = xh(Πh(t)).

We claim that for every i ∈ N , every ti, t′i ∈ Ti, with Πh

i (ti) �= Πhi (t

′i), there exist

SCFs xhi [Πhi (ti)] and xhi [Π

hi (t

′i)] that are measurable with respect to Ψh

i ×Ψh−1−i , close to

xh, such that

Vi(xhi [Πhi (ti)]|ti) > Vi(xhi [Π

hi (t

′i)]; t

′i|ti) and Vi(xhi [Π

hi (t

′i)]|t′i) > Vi(xhi [Π

hi (ti)]; ti|t′i). (2)

We can prove this claim by using the same argument as in Lemma ??. That is, considerthe (Kh − 1)-dimensional unit simplex, whose extreme points are the elements of thefunctional space F (Ψh

i × Ψh−1−i ). Note how the interim expected utility of each extreme

point is well defined for each type, and thus, one can consider the corresponding hyper-planes as the level curves of such interim utility. By construction of the h-th iteration

19

of measurability, types ti and t′i can be separated in their interim preferences over SCFsin F (Ψh

i × Ψh−1−i ) whenever Πh

i (ti) �= Πhi (t

′i). Then, using the argument in the proof of

Lemma ??, one can find two SCFs to separate the two types as written in (2). The restof the argument is based on an induction step on the number of elements of Ψh

i , exactlyas in the proof of Lemma ??. �

Proposition 5 Suppose that an environment E satisfies NTI. Then, there exists an SCFx that satisfies IC and AM measurability. In particular, the constructed SCF x satisfiesthe following property: for any h = 1, . . . , L, i ∈ N, ti ∈ Ti, and ψi ∈ Ψh

i \Πhi (ti),

Ui(x|ti) > Ui(x;ψi|ti).

Proof : Define an SCF x : T → Δ(A) by

x(t) =α

n

∑i∈N

L∑h=1

δh−1xhi [Πhi (ti)](t) ∀t ∈ T

where α is defined as

α ≡ 11 + δ + δ2 + · · · + δL−1

,

and xhi [Πhi (ti)] are as constructed in Lemma ?? for each h = 1, . . . , L; 0 < δ < 1.

Note how x(·) ≡ xL(·) satisfies AM measurability by construction. Recall that, thanksto AM measurability, we can abuse notation and write, for any ψ ∈ Ψ∗, x(ψ) = x(t)whenever ψ = Π∗(t).

For every SCF y, define

Gi(y) = maxt,t′∈T

∣∣ui(y(t′); t) − ui(y(ti, t′−i); t)∣∣.

Choose δ > 0 small enough so that for every i ∈ N and every h = 1, . . . , L, thereexists λ such that

δh mini∈N,ti∈Ti,ψh

i �=Πhi (ti)

Vi(xhi [Πhi (ti)]|ti) − Vi(xhi [ψ

hi ];ψ

hi |ti) > λ >

∑i′∈N

L∑k=h+1

δkGi(xki′). (∗)

Fix i ∈ N, ti ∈ Ti and ψi ∈ Ψ∗i \Π∗

i (ti) arbitrarily. Suppose that ψi /∈ Π1i (ti). Then,

by condition (∗), we know that

Ui(x|ti) > Ui(x;ψhi |ti).

By the induction hypothesis, we assume that for each h = 1, . . . , L − 1, ti ∈ Ti, andψi ⊆ Ψh

i \Πhi (ti),

Ui(x|ti) > Ui(x;ψ|ti).

20

Without loss of generality, we can assume that ψi ⊆ ΠL−1i (ti) but ψi ∈ ΨL

i \ΠLi (ti). Then,

we have

Ui(x|ti) − Ui(x;ψi|ti) =αδL−1

n

[Ui(xLi |ti) − Ui(xLi ;ψi|ti)

]> 0.

This shows that for any i ∈ N, ti ∈ Ti, and ψi ∈ Ψ∗i \Π∗

i (ti),

Ui(x|ti) > Ui(x;ψi|ti).

�

Proposition 6 (AM measurability ⇒ mixed VM) Suppose that an environment Esatisfies NTI. Then, if an SCF f satisfies AM measurability, it also satisfies mixed virtualmonotonicity.

Proof : By the previous lemma, we construct the following SCF x: for any t ∈ T ,

x(t) =α

n

∑i∈N

L∑h=1

δh−1xhi [Πhi (ti)](t).

Note that x satisfies IC and AM measurability. Fix any (possibly mixed) deceptionα ∈ A such that f �= f ◦ α. Since f satisfies AM measurability, we lose nothing to focuson the case that there exist i ∈ N and ti ∈ Ti for whom the support of αi(ti) contains atype ti /∈ Π∗

i (ti). Define an SCF y as follows: for any t ∈ T ,

y(t) =α

n

L∑h=1

δh−1xhi [Πhi (ti)](ti, t−i) +

α

n− 1

∑j �=i

L∑h=1

δh−1xhj [Πhj (tj)](ti, t−i).

Note that y is constant over Ti by construction. By Lemma ??, we can show

Ui(y ◦ α|ti) − Ui(x ◦ α|ti) =α

n

L∑h=1

δh−1[Ui

(xhi [Π

hi (ti)]

∣∣ti) − Ui

(xhi [Π

hi (αi(ti))]

∣∣ti)]> 0

By Lemma ??, we also have that for any t′i ∈ Ti,

Ui(x|t′i) − Ui(yαi(ti)|t′i) =

α

n

L∑h=1

δh−1[Ui

(xhi [Π

hi (t

′i)]

∣∣t′i) − Ui

(xhi [Π

hi (αi(ti))]

∣∣t′i)]≥ 0.

In sum, for any deception α ∈ A, whenever f �= f ◦ α, there exist i ∈ N, ti ∈ Ti, and anSCF y ∈ F such that

Ui(y ◦ α|ti) > Ui(x ◦ α|ti) and Ui(x|t′i) ≥ Ui(yαi(ti)|t′i) ∀t′i ∈ Ti.

Hence, if f satisfies AM measurability, it also satisfies mixed virtual monotonicity. �

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Corollary 3 Suppose that an environment E satisfies E and NTI. Let f be an SCFf satisfies AM measurability. Then, an SCF f is virtually Bayesian implementable inmixed strategies if and only if there exist an SCF f which is equivalent to f and satisfiesIC.

Proof : The necessity part is quite obvious given the previous proposition. So, wehere pay attention to the sufficiency part. Let F = {f ∈ F|f = f}. Fix any f ∈ F .This proof exploits Theorem 1 to obtain a direct proof. By the previous proposition, wehave shown that every SCF satisfies mixed virtual monotonicity. Suppose that the SCFf satisfies IC and mixed VM. Thus, it remains to show that for every ε > 0, there existsan SCF f ε such that f ≈ε f

ε and it satisfies IC and mixed BM.Let f ε = (1 − ε)f + εx and y′ = (1 − ε)f + εy. Since both f and x satisfy IC and

AM measurability, f ε also satisfies IC and AM Measurability. The rest of the proof iscompleted the same as Theorem 2 is shown. �

6.3 Implementation by Regular Mechanisms

The next definitions are borrowed from Abreu and Matsushima (1992c):For every i ∈ N and every partition Ψi, let Σi(Ψi) denote the set of mixed strategies

of player i that are measurable with respect to Ψi.

Definition 14 (pseudo Bayesian Equilibrium) The profile σ ∈ Σ1(Ψ1)×· · ·×Σn(Ψn)is a pseudo Bayesian Nash equilibrium (pseudo BNE) with respect to Ψ in Γ(T )if for all i ∈ N and all ψi ∈ Ψi, there exists some ti ∈ Ti with ti ∈ ψi such that

Ui(g ◦ σ|ti) ≥ Ui(g ◦ (σ′i, σ−i)|ti) ∀σ′i ∈ Σi

Definition 15 (Regular Mechanisms) A mechanism Γ is regular if for each Ψ thereexists a pseudo BNE with respect to Ψ in Γ(T ).

This condition amounts to the property that best response correspondence alwayshas nonempty values. In particular, finite mechanisms are regular. Mechanisms thatrely on the use of integer games – e.g., like the one constructed in Serrano and Vohra(2010) – are not regular.

The next result is a result in Abreu and Matsushima (1992c):

Proposition 7 If an SCF is virtually Bayesian implementable in mixed strategy by aregular mechanism, then it satisfies AM measurability.

Proof : Since f is virtually Bayesian implementable in mixed strategies, there existsf ε that is exactly Bayesian implementable in mixed strategies and d(f, f ε) < ε for ε > 0sufficiently small. Consider a “regular” mechanism Γ = (M,g) that exactly Bayesianimplements the SCF f ε in mixed strategies. Let σ ∈ ×i∈NΣi(Ψ∗

i ) be a pseudo BNE withrespect to Ψ∗. Note that σ is measurable with respect to Ψ∗. What we want to showhere is that σ is a BNE as well.

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If mi = σi(ti) is a best response for player i of type ti, then mi is also a best responsefor player i of any type t′i such that t′i ∈ ρi(ti,Ψ∗

−i). That is, this implies that for anyψi ∈ Ψ∗

i and any ti, t′i ∈ Ti with ti, t′i ∈ ψi, the best responses of player i of type ti and t′ito any σ−i that is measurable with respect to Ψ∗

−i are the same. Then, it follows that anypseudo BNE σ that is measurable with respect to Ψ∗ is in fact a BNE. Since f ε = g ◦ σby our hypothesis and f ε is exactly Bayesian implementable in mixed strategies, f ε ismeasurable with respect to Ψ∗ and therefore it must satisfy AM measurability. Finally,for a sufficiently small ε > 0, it follows that f satisfies AM measurability if and only iff ε satisfies AM measurability. �

7 Implementation in Iteratively Undominated Strategies

Given a mechanism Γ = (M,g), let Hi be a subset of Σi.9

Definition 16 (Strict Dominance) A strategy σi ∈ Hi is strictly dominated for agenti with respect to H = ×j∈NHj if there exist ti ∈ Ti and σ

′i ∈ Σi such that for every

σ−i ∈ ×j �=iHj,

Ui(g ◦ (σ′i, σ−i)|ti) > Ui(g ◦ (σi, σ−i)|ti).

For any subsets H,H ′ ⊆ Σ, where H ′ ⊆ H, we use the notation H → H ′ (read: His reduced to H ′) to signify that for any σ ∈ H\H ′, some σi is strictly dominated withrespect to H. Let λ0 denote the first element in an ordinal Λ; let ≥ be the linear orderon Λ; and let λ′ denote A successor to λ in Λ.10 Let {Kλ}λ∈Λ be a finite, countablyinfinite, or uncountably infinite family of subsets of the strategy space Σ satisfying thefollowing properties: (1) Kλ0 = Σ; (2) Kλ → Kλ′ where Kλ =

⋂λ′′<λKλ′′ for a limit

ordinal λ and any successor λ′; and (3) K∗ ≡ ⋂λ∈Λ Kλ → K only for K = K∗.

Definition 17 (Iterative Dominance) A strategy profile σ ∈ Σ is iteratively undom-inated if σ ∈ K∗.

Definition 18 (Exact Implementability in IUS) An SCF f is said to be exactlyimplementable in iteratively undominated strategies if there exists a mechanism Γ =(M,g) such that there exists a unique {σ} = K∗ for which g(σ(t)) = f(t) for all t ∈ T .

Remark: If σ is a unique iteratively undominated strategy profile, we gain nothing byallowing the agents to use “mixed strategies.” Given the definition of implementabilityabove, we automatically guarantee the existence of a unique iteratively undominated

9Our notation seems to assume that a message space M can be either finite or countable. However,we can also handle the case of uncountable M . In doing so, we must impose some suitable measurabilitycondition on M so that the corresponding strategy space Σi and interim preferences Ui(g ◦ σ|θi) arewell-defined. See Duggan (1997) and Serrano and Vohra (2010) for this treatment.

10An ordinal Λ is a well-ordered set in the order-isomorphic sense. In particular, the well-ordered setof natural numbers is called the first infinite ordinal. By saying that λ′ is a successor of λ, we mean thatλ′ > λ. A limit ordinal is an element in Λ which is not a successor.

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strategy profile, which is a unique Bayesian Nash equilibrium as well. This equilibriumis furthermore in pure strategies (see Chen, Long and Luo (2007)).

Definition 19 (Approximate Implementability in IUS) An SCF f is said to bevirtually or approximately implementable in iteratively undominated strategies if, thereexists ε > 0 such that for any ε ∈ (0, ε], there exists an SCF f ε for which d(f, f ε) < εand f ε is exactly implementable in iteratively undominated strategies.

Definition 20 (Quasi-Transferability) An environment E satisfies quasi-transferability(QT) if, there exists η > 0 such that for any η ∈ (0, η], there exists a collection of con-stant lotteries {ai(η, ψ)}i∈N,ψ∈Ψ∗ satisfying the following two conditions:

1. ui(x; t) − ui(ai(η, ψ); t) ≥ η ∀i ∈ N, ∀ψ ∈ Ψ∗, ∀t ∈ ψ;

2. ui(ai′(η, ψ); t) ≥ ui(x; t) ∀i, i′ ∈ N with i �= i′, ∀ψ ∈ Ψ∗, ∀t ∈ ψwhere x is a lottery that assigns equal probability to each alternative in A.

Remark: If an environment E satisfies QT, it also satisfies NTI. However, QT neitherimplies nor is implied by E.

Theorem 3 (Abreu and Matsushima (1992)) Suppose that an environment E sat-isfies TD and QT. Then, an SCF f is virtually implementable in iteratively undominatedstrategies if and only if there exists an SCF f which is equivalent to f and satisfies IC.

Corollary 4 (Abreu and Matsushima (1992)) Suppose that an environment E sat-isfies TD and QT. Then, an SCF f is virtually Bayesian implementable in mixed strate-gies if and only if there exists an SCF f which is equivalent to f and satisfies IC.

Theorem 4 (Abreu and Matsushima (1992)) Suppose that an environment E sat-isfies QT. Then, an SCF f is virtually implementable in iteratively undominated strate-gies if and only if there exists an SCF f which is equivalent to f and satisfies IC andAM measurability.

Corollary 5 (Abreu and Matsushima (1992)) Suppose that an environment E sat-isfies QT. Then, an SCF f is virtually Bayesian implementable in mixed strategies bya regular mechanism if and only if there exists an SCF f which is equivalent to f andsatisfies IC and AM measurability.

8 Bayesian Implementation with Full Transferability

Definition 21 An environment E satisfies full transferability (FT) if there exist aset X, {Yi}i∈N where Yi ⊆ R, and a collection of von Neumann-Morgenstern utilityfunctions {vi}i∈N where vi : X × Yi → R such that

1. A ≡ X × Y where Y = Y1 × · · · × Yn;

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2. for each i ∈ N, t ∈ T, a = (x, y1, . . . , yn) ∈ X × Y , ui(a; t) ≡ vi(x, yi; t); and

3. vi is strictly increasing in the second argument. That is, for any x ∈ X, t ∈ T ,and yi, y

′i ∈ Yi,

vi(x, yi; t) > vi(x, y′i; t) ⇔ yi > y

′i.

Remark: FT implies NTI and E. Moreover, FT also implies QT.

References

Abreu, D. and H. Matsushima (1992), “Virtual Implementation in Iteratively Undomi-nated Strategies: Incomplete Information,” Mimeo, Princeton University.

Artemov, G., T. Kunimoto, and R. Serrano (2010), “Robust Virtual Implementationwith Incomplete Information: Towards a Reinterpretation of the Wilson Doctrine,”(Revised) Economics Working Paper 2007-06, Brown University.

Jakcson, M. (1991), “Bayesian Implementation,” Econometrica, 59, 461-477.Maskin, E. (1999), “Nash Equilibrium and Welfare Optimality,” Reveiw of Economic

Studies, 66, 23-38.Matsushima, H. (1993), “Bayesian Monotonicity with Side Payments,” Journal of Eco-

nomic Theory, 59, 107-121.Serrano, R. (2004), “The Theory of Implementation of Social Choice Rules,” SIAM

Review 46, 377-414.Serrano, R. and R. Vohra (2005), “A Characterization of Virtual Bayesian Implementa-

tion,” Games and Economic Behavior 50, 312-331.Serrano, R. and R. Vohra (2010), “Multiplicity of Mixed Equilibria in Mechanisms:

A Unified Approach to Exact and Approximate Implementation,” forthcoming inJournal of Mathematical Economics

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