Implementation of first best contracts under
ambiguity: partition model∗
Zhiwei Liu†
Nicholas C. Yannelis‡
October 15, 2015
Abstract
In a partition model, we show that each maximin individually rational
and ex-ante maximin efficient allocation of a single good economy is implementable. When there are more than one good, we characterize three
conditions. If none of the three conditions is satisfied, then a maximin
individually rational and ex-ante maximin efficient allocation may not be
implementable. But as long as one of the three conditions is satisfied, each
maximin individually rational and ex-ante maximin efficient allocation is
implementable. Our work complements de Castro, et. al. [6].
Keywords: maximin preferences, maximin efficient allocations, maximin equilibrium, implementation.
1
Introduction
We explore the implementation of maximin individually rational and ex-ante maximin efficient allocations in an ambiguous asymmetric information economy. This
paper differs from de Castro, et. al. [6] in that we consider a more general set up,
which includes de Castro, et. al. [6] as a special case.
∗
On different occasions, we have benefited from comments, discussions, suggestions and questions from Camelia Bejan, Jean-Marc Bonnisseau, Richard Hislop, Atsushi Kajii, John Ledyard,
Stephen Morris, Walter Trockel and Chi Leung Wong.
†
International School of Economics and Management, Capital University of Economics and
Business, Beijing, China. [email protected]
‡
Department of Economics, the University of Iowa, Iowa City, USA. [email protected]
1
In particular, we adopt a partition model. Our economy consists of a finite
set of states of nature, a finite set of agents, each of whom is characterized by an
information partition, a multi-prior set, a random initial endowment and an ex
post utility function. Furthermore, the agents have maximin preferences.
In an ambiguous asymmetric information economy, any efficient allocation is
incentive compatible with respect to the maximin preferences (de Castro-Yannelis
[4]). In other words, maximin preferences solve the conflict between incentive compatibility and efficiency. Recall that in the standard expected utility (Bayesian)
framework, an efficient allocation may not be incentive compatible as it was shown
by Holmström-Myerson [12]. Furthermore, de Castro, et. al. [6] show that if an
ambiguous asymmetric information economy can be represented by the standard
type model of the implementation literature, then each maximin individually rational and ex-ante maximin efficient allocation is implementable.
But, many economies are not representable by the standard type model. Our
partition model includes these economies. We illustrate with an example that in
such an economy, the agents may strictly prefer a maximin individually rational
and ex-ante maximin efficient allocation to their random endowment. Consequently, the problem of implementation in these economies is relevant and interesting.
The main result of the paper is that each maximin individually rational and
ex-ante maximin efficient allocation of a single good economy is implementable.
When there are more than one good, we characterize three sufficient conditions. As
long as one of the three conditions is satisfied, each maximin individually rational
and ex-ante maximin efficient allocation is implementable. We show by examples
that if none of the three conditions holds, then a maximin individually rational
and ex-ante maximin efficient allocation may not be implementable. However, the
three conditions are not necessary for implementation.
The paper is organized as follows. Section 2 and 3 define an ambiguous asymmetric information economy, and introduce the maximin individually rational and
ex-ante maximin efficient notions. In Section 4, we introduce the direct revelation mechanism, and the maximin equilibrium. Section 5 and 6 present the main
results of the paper. Finally, we conclude in Section 7.
2
Ambiguous asymmetric information economy
Let Ω denote a finite set of states of nature, ω ∈ Ω a state of nature, R`+ the ` good
commodity space, and I the set of N agents, i.e., I = {1, · · · , N }. An ambiguous
2
asymmetric information economy E is a set E = {Ω; (Fi , Pi , ei , ui ) : i ∈ I}.
Fi is a partition of Ω. Let EiFi ∈ Fi denote an event, and ω ∈ EiFi a state in
the event. Then, in the interim, if the state ω occurs, agent i only knows that
the event EiFi has occurred. We impose the standard assumption, that when a
state occurs, and all agents truthfully report their information, they will know the
realized state1 . That is,
T
F
F
Assumption 1. For each ω, j∈I Ej j (ω) = {ω}, where Ej j (ω) denotes the
element in Fj that contains the state ω.
Since the events are observable in the interim, it is natural to assume that at
ex ante each agent is able to form a probability assessment over his partition. For
each i, let µi : σ (Fi ) → [0, 1] be a probability measure defined on the algebra
generated by agent i’s partition.
Assumption 2. For each i and for each event EiFi ∈ Fi , µi EiFi > 0.
Each µi is a well defined probability. But it is not defined on every state of
nature. Indeed, if EiFi = {ω, ω 0 } with ω 6= ω 0 , then the probability of the event
EiFi is well defined, but not the probability of the event {ω} or the event {ω 0 }.
Let ∆i be the set of all probability measures over 2Ω that agree with µi . Formally,
∆i = probability measure πi : 2Ω → [0, 1] | πi (A) = µi (A) , ∀A ∈ σ (Fi ) .
Let Pi , a nonempty, closed and convex subset of ∆i , be agent i’s multi-prior set.
Agent i’s random initial endowment is ei : Ω → R`+ . Each agent receives his
endowment in the interim. We require ei to be Fi -measurable, meaning that ei (·)
is constant on each element of Fi . Formally,
Assumption 3. Let Fi be agent i’s partition and fix any ωk ∈ Ω. We have
ei (ω) = ei (ωk ) for any ω ∈ EiFi (ωk ).
This ensures that at each state ω, the event EiFi {ω} incorporates the information revealed by the endowment. Clearly, if each ei is state independent, then it
is automatically Fi -measurable. Assuming ei to be Fi -measurable is more general
than being constant.
Finally, ui : R`+ × Ω → R is agent i’s ex post utility function, taking the form
of ui (ci ; ω), where ci denotes agent i’s consumption. The ex post utility function
ui is strictly monotone in consumption2 . Also, we assume that each agent knows
1
This assumption is without loss of generality, since if there exist two different states ω and
ω 0 , such that no agent is able to distinguish them, then the two states may as well be treated as
one state.
2
For each fixed ω, we have ui (ci ; ω) < ui (ci + ; ω), whenever is a none zero vector in Rl+ .
3
his utility function in the interim, consequently, each agent’s utility function needs
to be Fi -measurable. Formally,
Assumption 4. For each i and for each fixed ci ∈ R`+ , ui (ci ; ·) is Fi -measurable.
That is, given any ci ∈ R`+ , and any two states ω, ω̂ ∈ Ω, with ω 6= ω̂, we have
ui (ci ; ω) = ui (ci ; ω̂), whenever ω ∈ EiFi (ω̂).
The Fi -measurability of the ex post utility functions is often assumed in games
with incomplete information. Indeed, one may regard Fi as agent i’s type space,
and EiFi ∈ Fi as a possible type of agent i. Then clearly, assuming the function
ui (ci ; ·) to be Fi -measurable, is the same as assuming ui to depend on agent i’s
type.
Now, let xi : Ω → R`+ denote agent i’s allocation (or in short, i-allocation).
Denote by Li the set of all possible allocations of agent i, and by x = (x1 , · · · , xN )
an allocation of the above economy E. An allocation x is said to be feasible, if for
P
P
each ω ∈ Ω, i∈I xi (ω) = i∈I ei (ω).
We postulate that the agents have maximin preferences (see Gilboa and Schmeidler [9]).
Definition 1. Take any two allocations of agent i, fi and hi , from the set Li .
P
Agent i prefers fi to hi under the maximin preferences (written as fi M
hi )
i
min
πi ∈Pi
X
ui (fi (ω) ; ω) πi (ω) ≥ min
πi ∈Pi
ω∈Ω
X
ui (hi (ω) ; ω) πi (ω) .
(1)
ω∈Ω
The general multi-prior model includes both the Bayesian and the Wald-type
maximin preferences of de Castro and Yannelis [4]3 as special cases. Indeed, if
Pi is a singleton set for each agent, then the multi-prior preferences become the
Bayesian preferences. If Pi = ∆i for each agent, then the multi-prior preferences
become the maximin preferences in de Castro and Yannelis [4], where the following
formulation is equivalent to (1),
!
X
F
Ei i ∈Fi
!
Fi
min ui (fi (ω) ; ω) µi Ei
Fi
≥
ω∈Ei
X
F
Ei i ∈Fi
min ui (hi (ω) ; ω) µi EiFi .
P
Furthermore, agent i strictly prefers fi to hi , fi M
i
P
P
not the reverse, i.e., fi M
hi but hi M
fi .
i
i
3
See also de Castro, et. al. [8].
4
Fi
ω∈Ei
(2)
hi , if he prefers fi to hi but
The interest of the preferences, used in de Castro and Yannelis [4], comes
from that under these preferences any efficient allocation is incentive compatible.
Furthermore, only these preferences have this property in general4 . Indeed, using
the other extreme case, i.e., the Bayesian preferences, an efficient allocation may
not be incentive compatible as it was shown by Holmström-Myerson [12].
3
Maximin individually rational and ex-ante maximin efficient allocations
The standard notions of individual rationality and efficiency, when applied to
agents with maximin preferences, can be stated as follows.
Definition 2. A feasible allocation x = (xi )i∈I is said to be (maximin) individually
P
rational, if for each i ∈ I, xi M
ei .
i
Definition 3. A feasible allocation x = (xi )i∈I is said to be ex-ante maximin
efficient, if there does not exist another feasible allocation y = (yi )i∈I , such that
P
P
yi M
xi for all i, and yi M
xi for at least one i.
i
i
Remark 1. It should be noted that the concepts maximin core allocations, maximin value allocations and maximin Walrasian expectations equilibrium allocations
defined in [4][1][11] are all individually rational and ex-ante maximin efficient.
De Castro, Liu and Yannelis [6] show that in an ` goods economy presentable by
a standard type model, each maximin individually rational and ex-ante maximin
efficient allocation can be reached by means of noncooperation. The standard
type model is less general than the partition model of this paper. Recall, in a
standard type model, each agent i has a type set Ti , and the set of states of nature
is T = T1 × · · · × TN . This construction insures that there is always an agreed
state, regardless of the agents’ reported types. Now, let Ti = Fi and T = Ω. A
partition model can be represented by the standard type model if and only if the
economy satisfies a stronger assumption than Assumption 1:
4
In addition to the fact that maximin preferences solve the conflict between efficiency and
incentive compatibility, it has been shown in [1] [4] [7] [8] [11] that the adoption of the maximin preferences provides new insights and superior outcomes than the Bayesian preferences.
Furthermore, the maximin preferences solve the Ellsberg Paradox (see for example [5]).
5
F
Assumption 5. For any i ∈ I and Ej j ∈ Fj ,
T
F
j∈I
Ej j = {ω} for some ω ∈ Ω.
However, Assumption 5 is not satisfied by many economies such as the example
below. In this economy, agents are strictly better off if they trade, and therefore
the problem of implementation is relevant and interesting.
Example 1. There are two agents, one good, and three possible states of nature
√
Ω = {a, b, c}. The ex post utility function of each agent i is ui (ci ; ω) = ci . The
agents’ random initial endowments, information partitions and multi-prior sets
are:
(e1 (a) , e1 (b) , e1 (c)) = (5, 5, 1); F1 = {{a, b} , {c}}
(e2 (a) , e2 (b) , e2 (c)) = (5, 1, 5); F2 = {{a, c} , {b}}
2
and π1 ({c}) =
P1 = probability measure π1 : 2Ω → [0, 1] | π1 ({a, b}) =
3
2
P2 = probability measure π2 : 2Ω → [0, 1] | π2 ({a, c}) =
and π2 ({b}) =
3
1
3
1
3
.
.
A maximin individually rational and ex-ante maximin efficient allocation is
x=
!
x1 (a) x1 (b) x1 (c)
=
x2 (a) x2 (b) x2 (c)
!
5 4.8 1.2
.
5 1.2 4.8
Note, both agents strictly prefer the allocation x to the initial endowment e under
the maximin preferences. Indeed, we have for each i,
n√ √ o 1 √
n√ √ o 1 √
2
2
min
5, 5 +
1 = 1.824 < min
5, 4.8 +
1.2 = 1.826.
3
3
3
3
Could one provide a noncooperative foundation for the maximin efficient and
maximin individually rational notions in a general partition model? We address
this question in the next section. Our work complements de Castro, Liu and
Yannelis [6].
4
The direct revelation mechanism and the maximin equilibrium
A direct revelation mechanism, associated with an allocation and its underlying
ambiguous asymmetric information economy, is a noncooperative game, in which
agents (players) decide what to report after a state of nature is realized.
6
In the interim, a state of nature ω is realized. Each player i privately observes
the event EiFi (ω) and receives the initial endowment ei (ω). Then, each player i
reports EiFi ∈ Fi , but the report EiFi may not be truthful.
Definition 4. Suppose the realized state (the true state) is ω. Then, a report of
player i, EiFi ∈ Fi , is a lie, if it differs from the event EiFi (ω).
Definition 5. A strategy of player i is a function si : Fi → Fi .
Let Si denote player i’s strategy set. Denote by S = ×i∈I Si the strategy set,
and let s ∈ S denote a strategy profile. With a slightly abused notation, we
use s (ω) to denote the players’ reports, when they adopt the strategy profile s,
ΠN
and the realized state is ω. That is, s (ω) = s1 E1Π1 (ω) , · · · , sN EN
(ω) .
Clearly, for any ω ∈ Ω, s (ω) ∈ ×i∈I Fi .
The players’ reports are announced simultaneously. Based on the reports,
redistribution takes place. Figure 1 shows the time line.
A planned redistribution (net transfer) is the adjustments needed to go from
the initial endowment e to a planned allocation x.
Definition 6. Let x be the planned allocation. The planned redistribution is given
by x − e.
The actual redistribution, on the other hand, depends on the planned redistribution, and the players’ reports. From Assumption 1, we know for any collection
T
F
FN
of reports E1F1 , · · · , EN
, the j∈I Ej j is either singleton or empty.
T
F
FN
Definition 7. We say the reports E1F1 , · · · , EN
are compatible, if j∈I Ej j =
{ω̃}, for some ω̃ ∈ Ω. Furthermore, we refer the state ω̃ as the implied state (the
agreed state).
7
The implementation literature often assumes that the set of feasible alternatives is independent of the state of nature. It follows that if the realized state
is ω, and the players’ agreed state is ω̃, then the players end up with the social
choice x (ω̃). The implicit assumption is that the social choice x (ω̃) is feasible at
the state ω. In our context, the players receive initial endowment first, and then
redistribute the endowments based on their reports. The relevant feasibility condition is that each player is rich enough to participate in the revelation mechanism.
That is, for each i, ω and ω̃, we have ei (ω) + xi (ω̃) − ei (ω̃) ∈ R`+ .
FN
When the reports E1F1 , · · · , EN
are compatible, the players will end up with
e (ω) + x (ω̃) − e (ω̃), where ω̃ is the agreed state, and x (ω̃) − e (ω̃) is the planned
redistribution specified for the state ω̃. Clearly, if all the players tell the truth, then
ω̃ = ω and the players get what they planned to get, e (ω) + x (ω) − e (ω) = x (ω).
But, since some player may successfully lie, ω̃ may not be the true state. As a
consequence, e (ω) + x (ω̃) − e (ω̃) may differ from x (ω), i.e., the players may not
end up with the planned allocation.
When the reports are not compatible at the realized state ω, the lies are detected. There are many ways to resolve the players’ payoffs. In a single good
economy, the mechanism designer (MD) could appropriate minω∈Ω {xi − ei } from
each agent i. Note the MD does not know the realized state, but he knows the
planned redistribution x − e, so he knows minω∈Ω {xi − ei } for each agent i. In
an ` goods economy, the mechanism designer could randomly pick a state ω̃ and
enforce the net transfer x (ω̃) − e (ω̃). Furthermore, no trade is also an option,
that is, each agent keeps his initial endowments.
FN
Let Di x − e, E1F1 , · · · , EN
denote the actual redistribution of player i. It
depends on the planned redistribution x − e and the players’ reports. Its exact
form will be defined in Sections 5 and 6.
Definition 8. Let gi : F1 × · · · × FN × Ω → R`+ be the outcome function of player
i. It takes the form of
gi
FN
FN
E1F1 , · · · , EN
, ω = ei (ω) + Di x − e, E1F1 , · · · , EN
.
FN
where ei (ω) + Di x − e, E1F1 , · · · , EN
i ends up consuming.
(3)
is the bundle of the goods, that player
Finally, each player i has a final payoff function. It tells us the final payoff that
player i ends up, given a list of reports and a realized state of nature. Formally,
8
Definition 9. Denote by vi : F1 × · · · × FN × Ω → R, the final payoff function of
player i. It takes the form of
vi
FN
FN
E1F1 , · · · , EN
; ω = ui ei (ω) + Di x − e, E1F1 , · · · , EN
;ω .
A direct revelation mechanism associated with a planned allocation x and its
underlying ambiguous asymmetric information economy E = Ω; (Fi , Pi , ei , ui )i∈I
is a set Γ = I, S, x − e, {gi }i∈I , {vi }i∈I .
In view of the players’ Wald type preferences, we adopt the maximin equilibrium of de Castro, et. al[6]. It says that every player adopts a criterion a la Ward
[13]. That is, each player maximizes the payoff that takes into account the worst
actions of all the other players against him and also the worst state that can occur.
Definition 10. In a direct revelation mechanism Γ = I, S, x − e, {gi }i∈I , {vi }i∈I ,
a strategy profile s∗ = (s∗1 , · · · , s∗N ) constitutes a maximin equilibrium (ME), if for
each player i, his strategy s∗i maximizes his interim payoff lower bound, that is,
the function s∗i : Fi → Fi satisfies, for each EiFi ∈ Fi ,
F
min
Fi
E−i−i ∈F−i ; ω 0 ∈Ei
F
vi s∗i EiFi , E−i−i ; ω 0 ≥
F
min
Fi
E−i−i ∈F−i ; ω 0 ∈Ei
F
vi ÊiFi , E−i−i ; ω 0 , (4)
F
for all ÊiFi ∈ Fi ; where E−i−i denotes the reports from all the other players, so
F
E−i−i ∈ F−i = ×j6=i Fj .
The maximin equilibrium has the flavor of the robust control of Hansen and
Sargent [10] in which the decision maker maximizes his payoff taking into account
the worst possible model5 . In contrast to the restricted maximin equilibrium
notion of Dasgupta, Hammond and Maskin [3] and the consistent planning equilibrium of Bose and Renou [2], the maximin equilibrium notion does not need
each player to correctly guess his opponents’ strategies to reach an equilibrium.
Furthermore, the maximin equilibrium is unique, whenever truth telling is optimal for each player. This is not necessarily the case with the restricted maximin
equilibrium notion or the consistent planning equilibrium notion.
Now, we say an allocation x is implementable, if x can be realized through a
maximin equilibrium of the direct revelation mechanism Γ = I, S, x − e, {gi }i∈I ,
5
The robust control approach presumes that decision makers are able to specify the set of
possible models, but are either unable or unwilling to form a prior over the forms of model
misspecification.
9
{vi }i∈I .
Let ME (Γ) denote the set of maximin equilibria of the mechanism Γ.
Definition 11. Let x be an allocation of an ambiguous asymmetric information economy E, and ME (Γ) the set of maximin equilibria of the mechanism
Γ = I, S, x − e, {gi }i∈I , {vi }i∈I . We say the allocation x is implementable as
a maximin equilibrium of the mechanism Γ if,
∃s∗ ∈ ME (Γ) , such that gi (s∗ (ω) , ω) = xi (ω) ,
for each ω ∈ Ω and for each i ∈ I.
Definition 12. We say a strategy profile s is truth telling, if according to it, every
player reports the truth whenever it is his turn to report. That is, for each player i
and for each state ω, if he observes the event EiFi (ω) from nature, then his action
is to report the true event, i.e., si EiFi (ω) = EiFi (ω). We denote such a strategy
profile by sT .
Remark 2. Clearly, if the truth telling strategy profile sT constitutes a maximin equilibrium of the mechanism Γ, i.e., sT ∈ ME (Γ), then the allocation x is
implementable as a maximin equilibrium of the mechanism Γ.
Indeed, under the truth telling strategy profile sT , the list of reports associated
FN
(ω) . That is, the players always
with each state ω is sT (ω) = E1F1 (ω) , · · · , EN
tell the truth. As a consequence, we have
gi sT (ω) , ω
= gi
FN
E1F1 (ω) , · · · , EN
(ω) , ω
FN
= ei (ω) + Di x − e, E1F1 (ω) , · · · , EN
(ω) , ω
= ei (ω) + xi (ω) − ei (ω) = xi (ω) ,
for each ω ∈ Ω and for each i ∈ I – the requirement of Definition 11.
Furthermore, when sT ∈ ME (Γ), we say Γ has a truth telling maximin equilibrium.
5
5.1
Implementation in a single good economy
Implementation
We show that each individually rational and ex-ante maximin efficient allocation x
in a single good economy is implementable through its corresponding mechanism
Γ = I, S, x − e, {gi }i∈I , {vi }i∈I .
10
FN
Let x−e denote a planned redistribution, and E1F1 , · · · , EN
a list of reports.
Definition 13. The actual redistribution of player i is given by
FN
Di x − e, E1F1 , · · · , EN
=
F



xi (ω̃) − ei (ω̃)
if ∩j∈I Ej j = {ω̃}


minω0 ∈Ω {xi (ω 0 ) − ei (ω 0 )}
if ∩j∈I Ej j = ∅
F
That is, when reports are not compatible, the mechanism designer (MD) appropriates minω∈Ω {xi − ei } from each agent i.
Theorem 1. Denote by x a maximin individually rational and ex-ante maximin
efficient allocation of a single good economy, and ME (Γ) the set of maximin
equilibria of the direct revelation mechanism Γ = I, S, x − e, {gi }i∈I , {vi }i∈I .
Then, there exists a truth telling maximin equilibrium sT , which is the unique
maximin equilibrium of the mechanism Γ ( i.e., sT = ME (Γ) ), for which we
have gi sT (ω) , ω = xi (ω), for each ω ∈ Ω and for each i ∈ I, i.e., the allocation
x is implementable as a maximin equilibrium of its corresponding mechanism Γ.
Since maximin core allocations, maximin value allocations and maximin Walrasian expectations equilibrium allocations are individually rational and ex-ante
efficient, it follows from the main theorem that they are implementable as a maximin equilibrium.
5.2
Heuristic proof
Now we provide a heuristic proof by means of an example. A complete proof will
be given in the next section.
Example 2. Recall Example 1. There are two agents, one commodity, and three
possible states of nature Ω = {a, b, c}. The ex post utility function of each agent i
√
is ui (ci ; ω) = ci . The agents’ random initial endowments, information partitions
and type sets and multi-prior sets are:
(e1 (a) , e1 (b) , e1 (c)) = (5, 5, 1); F1 = {{a, b} , {c}}
(e2 (a) , e2 (b) , e2 (c)) = (5, 1, 5); F2 = {{a, c} , {b}}
2
P1 = probability measure π1 : 2Ω → [0, 1] | π1 ({a, b}) =
and π1 ({c}) =
3
2
P2 = probability measure π2 : 2Ω → [0, 1] | π2 ({a, c}) =
and π2 ({b}) =
3
11
1
3
1
3
.
.
A maximin individually rational and ex-ante maximin efficient is
x=
!
x1 (a) x1 (b) x1 (c)
=
x2 (a) x2 (b) x2 (c)
!
5 4.8 1.2
,
5 1.2 4.8
Let the planned allocation be x. Then, the planned redistribution x − e is
(x1 (a) − e1 (a) , x1 (b) − e1 (b) , x1 (c) − e1 (c)) = (0, −0.2, 0.2)
and
(x2 (a) − e2 (a) , x2 (b) − e2 (b) , x2 (c) − e2 (c)) = (0, 0.2, −0.2) .
The game tree is presented in Figure 2, in which for simplicity, we let A1 =
{a, b}, c1 = {c}, A2 = {a, c}, and b2 = {b}. We will show that the truth telling
strategy profile constitutes the only maximin equilibrium of the game, and the
immediate consequence is that the allocation x is implemented. Formally, we will
show that the strategy profile s = (s1 (A1 ) = A1 , s1 (c1 ) = c1 ; s2 (A2 ) = A2 , s2 (b2 )
= b2 ), constitutes the only maximin equilibrium of the game.
0
We look at player 1 first, she has two information sets I1 and I1 . (Figure 3)
If she is at I1 , then she must have seen the event A1 from nature. She can
either tell the truth A1 or the lie c1 . Player 1 cannot distinguish the two decision
nodes within the set I1 , so her action is common at the two nodes. Figure 3a shows
12
that, being truthful (reports A1 ), she may end up with, from the left to the right,
5, 4.8, 5 or 4.8 units of the good. That is, at the information set I1 , if player 1 tells
the truth, then she may go down one of the four paths ‘aA1 A2 ’, ‘aA1 b2 ’, ‘bA1 A2 ’
and ‘bA1 b2 ’, for which she ends up with g1 ((A1 , A2 ) , a) = e1 (a) + x1 (a) − e1 (a) =
5+0 = 5, g1 ((A1 , b2 ) , a) = e1 (a)+x1 (b)−e1 (b) = 5−0.2 = 4.8, g1 ((A1 , A2 ) , b) =
e1 (b) + x1 (a) − e1 (a) = 5 + 0 = 5, and g1 ((A1 , b2 ) , b) = e1 (b) + x1 (b) − e1 (b) =
5 − 0.2 = 4.8 units of the good respectively. Similarly, by lying (reports c1 ), she
may end up with, from the left to the right, 5.2, 4.8, 5.2 or 4.8 units of the good6 .
Clearly, when player 1 observes the event A1 , telling the truth (reports A1 ) gives
her a lower bound payoff of
min {v1 (A1 , A2 ; a) , v1 (A1 , b2 ; a) , v1 (A1 , A2 ; b) , v1 (A1 , b2 ; b)}
n√ √
√ √ o √
5, 4.8, 5, 4.8 = 4.8;
= min
lying (reports c1 ) gives her a lower bound payoff of
min {v1 (c1 , A2 ; a) , v1 (c1 , b2 ; a) , v1 (c1 , A2 ; b) , v1 (c1 , b2 ; b)}
n√
√
√
√ o √
= min
5.2, 4.8, 5.2, 4.8 = 4.8.
So when she observes the event A1 , she has no incentive to lie, i.e., s1 (A1 ) = A1
constitutes part of a maximin equilibrium of the game.
0
If player 1 is at I1 , then she must have seen the event c1 from nature. Fig6
Note, both path ‘ac1 b2 ’ and path ‘ac1 b2 ’ lead to incompatible reports. Therefore, the actual
redistribution of player 1 is minω0 ∈Ω {x1 (ω 0 ) − e1 (ω 0 )} = −0.2.
13
ure 3b shows that telling the truth (reports c1 ) gives her a lower bound pay√ √
√
off of min {v1 (c1 , A2 ; c) , v1 (c1 , b2 ; c)} = min
1.2, 0.8 = 0.8; lying (reports A1 ) gives her a lower bound payoff of min {v1 (A1 , A2 ; c) , v1 (A1 , b2 ; c)} =
√ √ √
1, 0.8 = 0.8. So when she observes the event c1 , she should report the
min
event c1 , i.e., s1 (c1 ) = c1 constitutes part of a maximin equilibrium of the game.
0
Now, turn to player 2. He has two information sets also, I2 and I2 . (Figure 4)
If player 2 is at I2 , then he must have seen the event A2 from nature. Figure
4a shows that, being truthful (reports A2 ), he may end up with, from the left to
the right, 5, 4.8, 5 or 4.8 units of the good; and by lying (reports b2 ), he may end
up with, from the left to the right, 5.2, 4.8, 5.2 or 4.8 units of the good. Clearly,
telling the truth (reports A2 ) gives him a lower bound payoff of
min {v2 (A1 , A2 ; a) , v2 (c1 , A2 ; a) , v2 (A1 , A2 ; c) , v2 (c1 , A2 ; c)}
n√ √
√ √ o √
= min
5, 4.8, 5, 4.8 = 4.8;
lying (reports b2 ) gives her a lower bound payoff of
min {v2 (A1 , b2 ; a) , v2 (c1 , b2 ; a) , v2 (A1 , b2 ; c) , v2 (c1 , b2 ; c)}
n√
√
√
√ o √
= min
5.2, 4.8, 5.2, 4.8 = 4.8.
14
So when he observes the event A2 , he has no incentive to lie, i.e., s2 (A2 ) = A2
constitutes part of a maximin equilibrium of the game.
0
If player 2 is at I2 , then he must have seen the event b2 from nature. Figure 4b shows that telling the truth (reports b2 ) gives him a lower bound pay√ √
√
1.2, 0.8 = 0.8; lying (reoff of min {v2 (A1 , b2 ; b) , v2 (c1 , b2 ; b)} = min
ports A2 ) gives him a lower bound payoff of min {v2 (A1 , A2 ; b) , v2 (c1 , A2 ; b)} =
√ √ √
min
1, 0.8 = 0.8. So when he observes the event b2 , he should report the
event b2 , i.e., s2 (b2 ) = b2 constitutes part of a maximin equilibrium of the game.
Now, put together, the strategy profile s = (s1 (A1 ) = A1 , s1 (c1 ) = c1 ; s2 (A2 ) =
A2 , s2 (b2 ) = b2 ) is a maximin equilibrium of the game. It is, in fact, the only
maximin equilibrium of the game7 . The equilibrium report paths are s (a) =
(s1 (A1 ) , s2 (A2 )) = (A1 , A2 ), s (b) = (s1 (A1 ) , s2 (b2 )) = (A1 , b2 ) and s (c) =
(s1 (c1 ) , s2 (A2 )) = (c1 , A2 ), as marked in Figure 2. It can be easily checked
that the individually rational and ex-ante maximin efficient allocation x is implemented, since we have g1 (s (a) , a) = g1 ((A1 , A2 ) , a) = 5 + 5 − 5 = 5 = x1 (a),
and similarly, we have g2 (s (a) , a) = 5 = x2 (a), g1 (s (b) , b) = 4.8 = x1 (b),
g2 (s (b) , b) = 1.2 = x2 (b), g1 (s (c) , c) = 1.2 = x1 (c), g2 (s (c) , c) = 4.8 = x2 (c).
These outcomes are illustrated in Figure 2, as pairs following the equilibrium
paths.
5.3
Proof of theorem 1
To ease the explanation, we introduce some notations. We use F−i
= ×j6=i Fj to deFi−1
F−i
,
note the action set of all the players except player i, and E−i = E1F1 , · · · , Ei−1
Fi+1
FN
∈ F−i reports of all the players expect player i.
Ei+1
, · · · , EN
F
F
Furthermore, we write ω ∈ E−i−i or E−i−i (ω), if the state ω belongs to each
T F
Fi+1
Fi−1
FN
, · · · , EN
; and we use EiFi E−i−i =
, Ei+1
element in the list E1F1 , · · · , Ei−1
T
Fj
j∈I Ej to denote the information revealed by the reports of all the players.
Let x be an individually rational and ex-ante maximin efficient allocation.
Suppose that the mechanism Γ does not have a truth telling maximin equilibrium.
Then, there must exist a player i, an event EiFi , and a lie ẼiFi ∈ Fi (clearly,
ẼiFi 6= EiFi ), such that when the player i observes the event EiFi , he can insure a
7
We assume that a player lies, only if he can benefit from doing so.
15
better lower bound payoff by lying, i.e.,
min
F
E −i ∈F−i ;
−i
F
ω 0 ∈Ei i
n o
F
vi EiFi , E−i−i ; ω 0
<
min
F
E −i ∈F−i ;
−i
F
ω 0 ∈Ei i
n o
F
vi ẼiFi , E−i−i ; ω 0 .
(5)
We will show, in Step 1 and 2, that (5) cannot hold, and therefore every game Γ
has a truth telling maximin equilibrium.
To ease the explanation, denote the left hand side of (5) by
∗
vi EiFi , E−i
; ω∗ =
F
min
Fi
E−i−i ∈F−i ; ω 0 ∈Ei
n o
F
vi EiFi , E−i−i ; ω 0 ,
∗
where E−i
∈ F−i and ω ∗ ∈ EiFi solve the minimization problem above.
∗
Step 1 We will show that if EiFi ∩ E−i
= {ω̃} for some ω̃, and (5) holds, then
x fails to be an ex-ante maximin efficient allocation.
Clearly, ω̃ ∈ EiFi , and we have
∗
; ω ∗ = ui (ei (ω ∗ ) + xi (ω̃) − ei (ω̃) ; ω ∗ ) = ui (xi (ω̃) ; ω ∗ ) = ui (xi (ω̃) ; ω̃) ,
vi EiFi , E−i
as the initial endowment ei and the utility function ui are Fi -measurable.
∗
; ω ∗ = ui (xi (ω̃) ; ω̃) implies8
Notice, vi EiFi , E−i
n o
F
∗
vi EiFi , E−i
; ω ∗ = min vi EiFi , E−i−i (ω 0 ) ; ω 0 .
Fi
(6)
ω 0 ∈Ei
Also, (5) implies9
n o
F
∗
vi EiFi , E−i
; ω ∗ < min vi ẼiFi , E−i−i (ω 0 ) ; ω 0 .
Fi
(7)
ω 0 ∈Ei
8
Note ω̃ ∈ EiFi , and
∗
vi EiFi , E−i
; ω∗
=
min
F−i
E
∈F−i ;
−i
F
ω 0 ∈Ei i
n o
n o
F
F
vi EiFi , E−i−i ; ω 0
≤ min vi EiFi , E−i−i (ω 0 ) ; ω 0
Fi
ω 0 ∈Ei
F
≤ vi EiFi , E−i−i (ω̃) ; ω̃ = ui (xi (ω̃) ; ω̃) ,
imply that we must have equality throughout.
9
Since by the definition of a minimum, we have that
n o
n o
F
F
vi ẼiFi , E−i−i ; ω 0
≤ min vi ẼiFi , E−i−i (ω 0 ) ; ω 0 .
min
F−i
E
∈F−i ;
−i
F
ω 0 ∈Ei i
Fi
ω 0 ∈Ei
16
Now, (6) and (7) together imply
min
Fi
ω 0 ∈Ei
n o
n o
F
F
vi EiFi , E−i−i (ω 0 ) ; ω 0
< min vi ẼiFi , E−i−i (ω 0 ) ; ω 0 .
Fi
(8)
ω 0 ∈Ei
Finally, Lemma 1 (see below) shows that if (8) holds, then x fails to be an
ex-ante maximin efficient allocation, a contradiction.
∗
Step 2 We will show that if EiFi ∩ E−i
= ∅, then (5) cannot hold. Now we have
∗
vi EiFi , E−i
; ω ∗ = ui (ei (ω ∗ ) + xi (ω̂) − ei (ω̂) ; ω ∗ ), where ω̂ minimizes xi − ei . If
ω̂ ∈ EiFi , we are back to Step 1. Here, we discuss the cases of ω̂ ∈
/ EiFi . We show
by Lemma 2 that in these cases, we have
min
F
E −i ∈F−i ;
−i
F
ω 0 ∈Ei i
n o
F
vi ẼiFi , E−i−i ; ω 0
= ui (ei (ω ∗ ) + xi (ω̂) − ei (ω̂) ; ω ∗ ) ,
for every ẼiFi 6= EiFi . That is, (5) does not hold.
Therefore, we conclude that the mechanism Γ has a truth telling maximin
equilibrium, i.e., sT ∈ ME (Γ).
We now show that the truth telling maximin equilibrium is the only maximin
equilibrium of the mechanism Γ, i.e., sT = ME (Γ). So suppose otherwise,
that is, suppose both sT and s∗ are maximin equilibria of the mechanism Γ, and
sT 6= s∗ .
The truth telling strategy profile sT is different from the strategy profile s∗ ,
implies that there must exist a player i and an event EiFi , such that
sTi EiFi = EiFi 6= ẼiFi = s∗i EiFi .
(9)
But s∗i EiFi = ẼiFi 6= EiFi holds, only if lying makes player i strictly better
off upon observing the event EiFi , i.e.,
min
F
E −i ∈F−i ;
−i
F
ω 0 ∈Ei i
n o
F
vi EiFi , E−i−i ; ω 0
<
min
F
E −i ∈F−i ;
−i
F
ω 0 ∈Ei i
n o
F
vi ẼiFi , E−i−i ; ω 0 ,
a contradiction to the fact that the truth telling strategy profile constitutes a
maximin equilibrium of the mechanism.
Clearly, the individually rational and ex-ante maximin efficient allocation x is
implemented. Indeed, under the truth telling strategy profile sT , the list of reports
17
associated to each state ω is
FN
sT (ω) = E1F1 (ω) , · · · , EN
(ω) .
That is, the players always tell the truth. As a consequence, we have
gi sT (ω) , ω
= gi
FN
E1F1 (ω) , · · · , EN
(ω) , ω
FN
= ei (ω) + Di x − e, E1F1 (ω) , · · · , EN
(ω) , ω
= ei (ω) + xi (ω) − ei (ω) = xi (ω) ,
for each ω ∈ Ω and for each i ∈ I – the requirement of Definition 11.
Lemma 1. Given a direct revelation mechanism Γ = I, S, x − e, {gi }i∈I , {vi }i∈I ,
if the allocation x is a maximin individually rational and ex-ante maximin efficient
allocation, then there does not exist a player i, an event EiFi , and a lie ẼiFi ∈ Fi
(clearly, ẼiFi 6= EiFi ), such that10
min
Fi
ω 0 ∈Ei
o
n o
n F
F
< min vi ẼiFi , E−i−i (ω 0 ) ; ω 0 .
vi EiFi , E−i−i (ω 0 ) ; ω 0
Fi
(10)
ω 0 ∈Ei
That is, for all i, given that all other players tell the truth, it is optimal for player
i to tell the truth11 .
Proof. Suppose that there exist a player i, an event EiFi , and a lie ẼiFi 6= EiFi ,
such that (10) holds. We will show that the feasible allocation x fails to be ex-ante
efficient under the maximin preferences – an idea similar to the one in theorem
4.1 of de Castro-Yannelis [4].
Note, for each ω 0 ∈ EiFi , we have
F−i
Fi
0
0
vi Ei , E−i (ω ) ; ω = ui (xi (ω 0 ) ; ω 0 ) ,
and therefore, the left hand side of (10) can be rewritten as
min
Fi
ω 0 ∈Ei
n o
F
vi EiFi , E−i−i (ω 0 ) ; ω 0
= min {ui (xi (ω 0 ) ; ω 0 )} .
Fi
ω 0 ∈Ei
10
In words, (10) says that if all the other players are truthful, then player i can insure a higher
lower bound payoff by lying under the event EiFi .
11
In other words, truth telling for all i turns out to be a fixed point
18
Define an i-allocation of player i, zi (·), such that for each ω 0 ∈ EiFi ,
F
vi ẼiFi , E−i−i (ω 0 ) ; ω 0 = ui (zi (ω 0 ) ; ω 0 ) ,
and therefore, the right hand side of (10) can be rewritten as
n o
F−i
Fi
0
0
min vi Ẽi , E−i (ω ) ; ω
= min {ui (zi (ω 0 ) ; ω 0 )} .
Fi
Fi
ω 0 ∈Ei
ω 0 ∈Ei
It follows from (10) that
min {ui (xi (ω 0 ) ; ω 0 )} < min {ui (zi (ω 0 ) ; ω 0 )} ,
(11)
Fi
Fi
ω 0 ∈Ei
ω 0 ∈Ei
which then implies that,
for each ω 0 ∈ arg min
Fi
ω 00 ∈Ei
n 00 00 o
,
ui xi ω ; ω
we have ui (xi (ω 0 ) ; ω 0 ) < ui (zi (ω 0 ) ; ω 0 ) .
(12)
00 00 For (12) to hold, it must be the case that for each ω 0 ∈ arg minω00 ∈E Fi ui xi ω ; ω
,
i
there exists a state ω̃, such that
F
1. ẼiFi ∩ E−i−i (ω 0 ) = {ω̃},
2. zi (ω 0 ) = ei (ω 0 ) + xi (ω̃) − ei (ω̃) 6= xi (ω 0 ).
00 00 Let {ω 0 , ω̃} denote a set, containing a state ω 0 ∈ arg minω00 ∈E Fi ui xi ω ; ω
i
and its corresponding12 ω̃. It follows by 1 above that, for each
ω 0 ∈ arg min
Fi
ω 00 ∈Ei
n 00 00 o
,
ui xi ω ; ω
F
the set {ω 0 , ω̃} is a subset of E−i−i (ω 0 ).
Now, we are ready to define an allocation y that Pareto improves x under the
maximin preferences. Define for each j ∈ I, the j-allocation yj (·) by
(
yj (ω 0 ) =
zj (ω 0 ) = ej (ω 0 ) + xj (ω̃) − ej (ω̃)
00 00 if ω 0 ∈ arg minω00 ∈E Fi ui xi ω ; ω
xj (ω 0 )
otherwise.
12
i
To avoid n confusion,
is worthwhile to re-emphasize that
00 it00 o
arg minω00 ∈E Fi ui xi ω ; ω
may be matched with a different ω̃.
i
19
different
ω0
∈
Notice that the allocation y is feasible.
00 00 , we have
Indeed, for a state ω 0 ∈
/ arg minω00 ∈E Fi ui xi ω ; ω
i
X
yj (ω 0 ) =
j∈I
X
xj (ω 0 ) =
j∈I
X
ej (ω 0 ) ;
j∈I
00 00 and for a state ω 0 ∈ arg minω00 ∈E Fi ui xi ω ; ω
, we have
i
X
yj (ω 0 ) =
j∈I
X
zj (ω 0 ) =
j∈I
X
ej (ω 0 ) +
j∈I
X
xj (ω̃) −
j∈I
X
ej (ω̃) =
j∈I
X
ej (ω 0 )
j∈I
(recall that x is a feasible allocation at the state ω̃).
From (12) and the definition of yi , we have
min {ui (yi (ω 0 ) ; ω 0 )} > min {ui (xi (ω 0 ) ; ω 0 )}
Fi
Fi
ω 0 ∈Ei
(13)
ω 0 ∈Ei
under the event EiFi ; and for any other event ÊiFi ∈ Fi , we have
min {ui (yi (ω 0 ) ; ω 0 )} = min {ui (xi (ω 0 ) ; ω 0 )} .
Fi
Fi
ω 0 ∈Êi
ω 0 ∈Êi
Therefore, combined with the assumption on µi (·) (Assumption 2), we conclude
that, for the player i,
X Ei ∈Fi
X 0
0
min
ui (xi (ω ) ; ω ) µi (Ei ) .
min
ui (yi (ω ) ; ω ) µi (Ei ) >
0
0
0
0
ω ∈Ei
Ei ∈Fi
ω ∈Ei
(14)
Here we abuse the notations in (14) slightly, in particular, Ei denotes an arbitrary
event in Fi . That is, player i strictly prefers the i-allocation yi to the i-allocation
xi under the maximin preferences. Now, it remains to show that for any other
player k 6= i, we have yk is preferred to xk under the maximin preferences.
Fix an arbitrary player k 6= i, and an arbitrary event that player k may observe,
Fk
Ek ∈ Fk . Notice, if the event EkFk contains a state
ω 0 ∈ arg min
Fi
ω 00 ∈Ei
n 00 00 o
ui xi ω ; ω
,
then it contains the set {ω 0 , ω̃}. So, by the Fk -measurability of ek , we have
zk (ω 0 ) n= ek (ω 0 ) + xk (ω̃)o− ek (ω̃) = nxk (ω̃). Now, for othe event EkFk , define
Xk = xk (ω 0 ) : ω 0 ∈ EkFk and Yk = yk (ω 0 ) : ω 0 ∈ EkFk . We have Yk ⊂ Xk .
20
00 00 , then
Indeed, if ω 0 ∈ EkFk and ω 0 ∈ arg minω00 ∈E Fi ui xi ω ; ω
i
yk (ω 0 ) = zk (ω 0 ) = xk (ω̃) ∈ Xk ;
00 00 , then
and if ω 0 ∈ EkFk and ω 0 ∈
/ arg minω00 ∈E Fi ui xi ω ; ω
i
yk (ω 0 ) = xk (ω 0 ) ∈ Xk .
Therefore, with Assumption 4, we have that
min {uk (yk (ω 0 ) ; ω 0 )} ≥ min {uk (xk (ω 0 ) ; ω 0 )} .
F
ω 0 ∈Ek k
F
ω 0 ∈Ek k
Since the event EkFk ∈ Fk is arbitrary, we conclude that
!
min uk (yk (ω 0 ) ; ω 0 ) µk EkFk ≥
X
F
F
Ek k ∈Fk
ω 0 ∈Ek k
!
min uk (xk (ω 0 ) ; ω 0 ) µk EkFk .
X
F
F
Ek k ∈Fk
ω 0 ∈Ek k
Also, since player k 6= i is arbitrary, we have for every player k 6= i, yk is preferred
to xk under the maximin preferences.
Thus, the feasible allocation y Pareto improves the allocation x under the
maximin preferences, i.e., x fails to be an ex-ante maximin efficient allocation.
This contradiction completes the proof of Lemma 1.
Lemma 2. Let
vi
∗
; ω∗
EiFi , E−i
=
F
min
Fi
E−i−i ∈F−i ; ω 0 ∈Ei
n o
F−i
Fi
0
vi Ei , E−i ; ω
.
∗
If EiFi ∩ E−i
= ∅, then
min
F
E −i ∈F−i ;
−i
F
ω 0 ∈Ei i
n o
F−i
Fi
0
vi Ei , E−i ; ω
=
min
F
E −i ∈F−i ;
−i
F
ω 0 ∈Ei i
n o
F−i
Fi
0
vi Ẽi , E−i ; ω
,
(15)
for each ẼiFi ∈ Fi .
Proof. By construction, a player anticipates the worst net transfer in the case
21
of incompatible reports. Suppose that there exists ẼiFi 6= EiFi such that ẼiFi is
F
compatible with all E−i−i ∈ F−i . We will show that arg minω∈Ω {xi − ei } ∩ ẼiFi is
non-empty. Then, since any state in ẼiFi can be an agreed state, player i can end
up with same net transfer as when the reports are not compatible. That is, (15)
holds.
Assume otherwise, i.e., arg minω∈Ω {xi − ei } ∩ ẼiFi = ∅. Now define an allocation y. For each ω 0 ∈ arg minω∈Ω {xi − ei }, let y (ω 0 ) = e (ω 0 ) + x (ω̃) − e (ω̃),
F
where {ω̃} = E−i−i (ω 0 ) ∩ ẼiFi . Note, ω̃ is well defined, since by construction ẼiFi
F
is compatible with all E−i−i ∈ F−i . Also note, for each ω 0 ∈ arg minω∈Ω {xi − ei },
we have ui (ei (ω 00 ) + xi (ω 0 ) − ei (ω 0 ) ; ω 00 ) < ui (ei (ω 00 ) + xi (ω̃) − ei (ω̃) ; ω 00 ) for
each ω 00 ∈ Ω by construction. Now, for each ω 0 ∈
/ arg minω∈Ω {xi − ei }, let
0
0
y (ω ) = x (ω ).
Clearly, the allocation y is feasible. Also, y Pareto improves x. Indeed, at each
0
ω ∈ arg minω∈Ω {xi − ei }, we have
min
F
ω 00 ∈Ei i (ω 0 )
ui (yi (ω 00 ) ; ω 0 ) >
min
F
ω 00 ∈Ei i (ω 0 )
ui (xi (ω 00 ) ; ω 0 ) .
At each ω 0 with EiFi (ω 0 ) ∩ arg minω∈Ω {xi − ei } = ∅, we have
min
F
ω 00 ∈Ei i (ω 0 )
ui (yi (ω 00 ) ; ω 0 ) =
min
F
ω 00 ∈Ei i (ω 0 )
ui (xi (ω 00 ) ; ω 0 ) .
Combined with Assumption 2, yi gives player i a strictly higher ex ante maximin
payoff, i.e., player i strictly prefers yi to xi at ex ante under the maximin preferences. Let j be an arbitrary player different from i. By construction, for each
F
ω 0 ∈ arg minω∈Ω {xi − ei }, its corresponding ω̃ is in the set E−i−i (ω 0 ). That is,
player j cannot distinguish ω 0 and its corresponding ω̃. By the same argument as
in Lemma 1, we have at each ω 0 ∈ arg minω∈Ω {xi − ei }, we have
min
F
ω 00 ∈Ej j (ω 0 )
uj (yj (ω 00 ) ; ω 0 ) ≥
min
F
ω 00 ∈Ej j (ω 0 )
uj (xj (ω 00 ) ; ω 0 ) .
F
At each ω 0 with Ej j (ω 0 ) ∩ arg minω∈Ω {xi − ei } = ∅, we have
min
F
ω 00 ∈Ej j (ω 0 )
uj (yj (ω 00 ) ; ω 0 ) =
min
F
ω 00 ∈Ej j (ω 0 )
uj (xj (ω 00 ) ; ω 0 ) .
That is, player j prefers yj to xj at ex ante under the maximin preferences.
Therefore, the allocation x is not ex ante maximin efficient. This contradiction
22
completes the proof of Lemma 2.
6
Implementation in an economy with more than
one good
In an economy with more than one good, the actual redistribution (Definition 13)
of section 5 is not well defined. We need a new payout rule. Let x − e denote a
FN
planned redistribution and E1F1 , · · · , EN
a list of reports.
Definition 14. The actual redistribution of player i is given by
FN
Di x − e, E1F1 , · · · , EN
=
F



xi (ω̃) − ei (ω̃) if ∩j∈I Ej j = {ω̃}


xi (ω̂) − ei (ω̂) for some ω̂, if ∩j∈I Ej j = ∅.
F
Now, whenever the players’ reports are not compatible, the mechanism designer
(MD) decides the net transfer. Being maximin, the players anticipate the worst.
That is, for each i and ω, player i anticipates the payoff
min
ui (ei (ω) + xi (ω 0 ) − ei (ω 0 ) ; ω)
0
ω ∈Ω
in the case of incompatible reports.
Theorem 2. Denote by x a maximin individually rational and ex-ante maximin
efficient allocation in an economy with more than one good. Its corresponding
direct revelation mechanism Γ = I, S, x − e, {gi }i∈I , {vi }i∈I has a truth telling
maximin equilibrium sT , which is the unique maximin equilibrium of the mechanism Γ, if one of the following hold:
1. the economy satisfies Assumption 5.
2. whenever incompatible reports can occur when a player tells the truth, then
incompatible reports can occur when he tells a lie. Formally, let ω be the
F
realized state of nature. If there exists E−i−i ∈ F−i , such that EiFi (ω) ∩
F
F
E−i−i = ∅, then for every ÊiFi 6= EiFi (ω), there exists Ê−i−i ∈ F−i , such that
F
ÊiFi ∩ Ê−i−i = ∅.
23
3. for each i the set Mi is not empty, where
Mi = {ω m : ui (ei (ω) + xi (ω m ) − ei (ω m ) , ω) ≤
ui (ei (ω) + xi (ω̃) − ei (ω̃) , ω) for all ω, ω̃ ∈ Ω} .
Proof. Condition 1 rules out the case in Step 2 of the proof of Theorem 1. Therefore, the result follows from Step 1 of the proof of Theorem 1 and Lemma 1. See
also de Castro, Liu and Yannelis [6].
If an economy satisfies condition 2, the result follows from the proof of Theorem
∗
1 and Lemma 1. The case in Step 2 (i.e., EiFi ∩ E−i
= ∅) of the proof of Theorem
1 can occur, but condition 2 guarantees that in this case no lie can give the player
a higher maximin payoff.
Finally, when an economy satisfies condition 3, we can replace the state ω̂
in Step 2 of the proof of Theorem 1 with a ω m ∈ Mi , and replace the notion
arg minω∈Ω {xi − ei } in Lemma 2 with the set Mi . Now, the result follows from
the proof of Theorem 1, Lemma 1 and Lemma 2.
Example 3 shows that if all three conditions of Theorem 2 fail, then a maximin individually rational and ex-ante maximin efficient allocation may not be
implementable in Γ.
Example 3. There are two agents, I = {1, 2}, two goods, and six
states of nature Ω = {a, b, c, d, e, f }. The ex post utility function
p
p
of agent 1 is u1 (x1,1 (ω) , x1,2 (ω) ; ω) = x1,1 (ω) + x1,2 (ω), ω ∈
p
p
{a, b, c, d, e}, and u1 (x1,1 (f ) , x1,2 (f ) ; f ) = 0.5 x1,1 (f ) + 2 x1,2 (f ),
where the second index refers to the good. The ex post utility funcp
p
tion of agent 2 is u2 (x2,1 (ω) , x2,2 (ω) ; ω) = x2,1 (ω) + x2,2 (ω), ω ∈
p
p
{a, c, e, f }, and u2 (x2,1 (ω) , x2,2 (ω) ; ω) = 1.01 x2,1 (ω) + x2,2 (ω),
ω ∈ {b, d}. The agents’ random initial endowments, information partitions and multi-prior sets are: e1 (ω) = (10, 10) for all ω; e2 (ω) =
(10, 10) for ω ∈ {a, b, c, d}; and e2 (ω) = (11, 9) for ω ∈ {e, f }.
F1 = {{a, b} , {c, d, e} , {f }} ; F2 = {{a, c} , {b, d} , {e, f }}.
1
Ω
P1 = probability measure π1 : 2 → [0, 1] | π1 ({a, b}) = π1 ({c, d, e}) = π1 ({f }) =
.
3
24
1
Ω
P2 = probability measure π2 : 2 → [0, 1] | π2 ({a, c}) = π2 ({b, d}) = π2 ({e, f }) =
.
3
A maximin individually rational and ex-ante maximin efficient allocation is
!
x1 (a) x1 (b) x1 (c) x1 (d) x1 (e) x1 (f )
x =
x2 (a) x2 (b) x2 (c) x2 (d) x2 (e) x2 (f )
=
!
(10, 10) (9.950311, 10.049813) (10, 10) (9.99, 10) (10.5, 9.5) (3.469, 14.439)
.
(10, 10) (10.049689, 9.950187) (10, 10) (10.01, 10) (10.5, 9.5) (17.531, 4.561)
This example does not satisfy any of the three conditions of Theorem 2. It turns out that the mechanism Γ has a unique maximin
equilibrium, in which player 1 (agent 1) lies when he is in the event
{a, b}13 , and player 2 (agent 2) always reports the true event. That
is, s1 ({a, b}) = s1 ({c, d, e}) = {c, d, e}, s1 ({f }) = {f }, s2 ({a, c}) =
{a, c}, s2 ({b, d}) = {b, d}, s2 ({e, f }) = {e, f } constitute the unique
maximin equilibrium. For each i and ω, let yi (ω) = gi (x − e, s (ω) , ω).
That is, the allocation y is realized through the unique maximin equilibrium:
!
y1 (a) y1 (b) y1 (c) y1 (d) y1 (e) y1 (f )
y =
y2 (a) y2 (b) y2 (c) y2 (d) y2 (e) y2 (f )
=
!
(10, 10) (9.99, 10) (10, 10) (9.99, 10) (10.5, 9.5) (3.469, 14.439)
.
(10, 10) (10.01, 10) (10, 10) (10.01, 10) (10.5, 9.5) (17.531, 4.561)
13
When player 1 is in the event {a, b}, the worst net transfer for him is x1 (f ) − e1 (f ). He
may get the worst transfer if the players’ reports are not compatible, or if the agreed state is
f . The lie {c, d, e} allows him to avoid the worst net transfer, and gives him the highest lower
bound payoff. Indeed, reporting {a, b} gives him a payoff of
min {v1 ({a, b} , {a, c} ; a) , v1 ({a, b} , {b, d} ; a) , v1 ({a, b} , {e, f } ; a) ,
v1 ({a, b} , {a, c} ; b) , v1 ({a, b} , {b, d} ; b) , v1 ({a, b} , {e, f } ; b)}
= min {6.32455, 6.32455, 5.66239, 6.32455, 6.32455, 5.66239} = 5.66239;
reporting {c, d, e} gives him a payoff of
min {6.32455, 6.32297, 6.32258, 6.32455, 6.32297, 6.32258} = 6.32258;
and reporting {f } gives him a payoff of
min {5.66239, 5.66239, 5.66239, 5.66239, 5.66239, 5.66239} = 5.66239.
Clearly, the payoff of reporting {c, d, e} is the highest.
25
Clearly, the allocation y differs from x. That is, the allocation x is not
implemented.
However, the three conditions of Theorem 2 are not necessary conditions. The
following example does not satisfy any of the three conditions, and its maximin
individually rational and ex ante maximin efficient allocation is implementable.
Example 4. There are two agents, I = {1, 2}, two goods, and three
states of nature Ω = {a, b, c}. The ex post utility function of agent 1
p
p
is u1 (x1,1 (ω) , x1,2 (ω) ; ω) = x1,1 (ω) + 2 x1,2 (ω), for all ω, where
the second index refers to the good. The ex post utility function of
p
p
agent 2 is u2 (x2,1 (ω) , x2,2 (ω) ; ω) = 2 x2,1 (ω) + x2,2 (ω), for all ω.
The agents’ random initial endowments, information partitions and
multi-prior sets are:
(e1 (a) , e1 (b) , e1 (c)) = ((8, 8) , (10, 2) , (10, 2)); F1 = {{a} , {b, c}}
(e2 (a) , e2 (b) , e2 (c)) = ((4, 8) , (4, 8) , (15, 6)); F2 = {{a, b} , {c}}
1
Ω
P1 = probability measure π1 : 2 → [0, 1] | π1 ({a}) = π1 ({b, c}) =
.
2
1
Ω
.
P2 = probability measure π2 : 2 → [0, 1] | π2 ({a, b}) = π2 ({c}) =
2
A maximin individually rational and ex-ante maximin efficient allocation is
!
!
x1 (a) x1 (b) x1 (c)
(2.4, 12.8) (2.8, 8) (5, 6.4)
x=
=
.
x2 (a) x2 (b) x2 (c)
(9.6, 3.2) (11.2, 2) (20, 1.6)
This example does not satisfy any of the three conditions of Theorem 2.
It turns out that the mechanism Γ has a unique maximin equilibrium,
in which the players (agent 1 and 2) always report the true event. That
is, s1 ({a}) = {a} , s1 ({b, c}) = {b, c}, s2 ({a, b}) = {a, b}, s2 ({c}) =
{c} constitute the unique maximin equilibrium. Consequently, the
allocation x is realized through the unique maximin equilibrium.
26
7
Concluding remarks
We characterize conditions under which each maximin individually rational and
ex-ante maximin efficient allocation is implementable by means of noncooperative
behavior under ambiguity. That is, given any arbitrary individually rational and
ex-ante maximin efficient allocation, its corresponding direct revelation mechanism
yields the allocation as its unique maximin equilibrium outcome.
It follows that each maximin core allocation, maximin value allocation and
maximin Walrasian equilibrium allocation is implementable in a single good economy. These allocations are implementable in a multi-goods economy, provided
that at least one of three sufficient conditions in Theorem 2 is satisfied.
Our implementation results depend on the payoff rules of the direct revelation
mechanism. In particular, the players’ net transfers (redistribution) when their reports are not compatible. In this paper, the players associate incompatible reports
with the worst planned net transfer, i.e., we impose a ‘punishment’ whenever the
reports are not compatible. It is an open question, if altering the payoff rules can
make each maximin individually rational and ex-ante maximin efficient allocation
implementable in a multi-good economy. Changing our ‘punishment’ to ‘no trade’
14
does not help. In particular, the allocation of Example 3 is not implementable
under the ‘no trade’ rule. Furthermore, it follows easily from our proofs that if
the mechanism designer knows the realized state of nature ω in the interim, and
carry out the correct planned net transfer x (ω) − e (ω) whenever the reports are
not compatible, then each maximin individually rational and ex-ante maximin
efficient allocation is implementable. However, one needs to extend the game of
this paper to capture the learning of the mechanism designer about the realized
state of nature.
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14
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27
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