In each state with positive probability there is

exactly one consumer who is fully informed; consumer i is the informed agent

in state ti.

The endowments are as follows:

!i(ti) = ( (1, 0) if ti = bi

(0.5, 0.5) if ti = ai

For x = (x1, x2) 2 IR2

+, the state-dependent utility functions are as follows.

u1(x, t) = ( 1.5(x1 + x2) ift = t3

x1 + x2 otherwise

u2(x, t) = ( 1.5(x1 + x2) ift = t1

x1 + x2 otherwise

u3(x, t) = ( 1.5(x1 + x2) ift = t2

x1 + x2 otherwise

Notice that both commodities are perfect substitutes. For an allocation x

let  (x) denote the sum of the two commodities allocated to each consumer,

i.e.,  i(x, t) = xi1(t)+xi2(t). It can be shown that x belongs to the incentive

compatible coarse core if and only if

 (x, t1) = (1+_1, 2−_1, 0),  (x, t2) = (0, 1+_2, 2−_2),  (x, t3) = (2−_3, 0, 1+_3)

where _1, _2, _3 2 [0, 1/3].

The incentive compatible, coarse core contains in particular the allocation

¯x(t), where ¯xi(t) = !i(t) for t /2 T_ and

¯x(t1) = ((0.5, 0.5), (2, 0), (0, 0))

¯x(t2) = ((0, 0), (0.5, 0.5), (2, 0))

¯x(t3) = ((2, 0), (0, 0), (0.5, 0.5))

This allocation is not in the incentive compatible, fine core.10 Consumers 1

and 3 have a fine objection over the event {t1} since  1(¯x, t1)+ 3(¯x, t1) = 1,

10In fact, it can be shown, that in this example the fine core (with or without incentive

constraints) is empty. The main difference between this example and Wilson’s (1978)

Example 2 is that in our example each agent’s endowment depends on his own type.

while their aggregate endowment of the two commodities is 2. If private

information can be shared, as is implicit in the notion of the fine core, then

clearly ¯x is not viable in state t1. But, in the present example more can be

said to justify a fine objection by agents 1 and 3. Suppose the state is t1,

which consumer 1 knows. Consumer 3 knows that the true state is either

t1 or t2. Consider an offer from consumer 1 to consumer 3 of the allocation

˜x(t), where

(˜x1(t), ˜x3(t)) = ( ((1.1, 0), (0.4, 0.5)) if t = t1

((!1(t), !3(t)) otherwise

In state t1, the corresponding net-trades are z1(t1) = (0.6,−0.5), z3(t1) =

(−0.6, 0.5). In state t1, the informed agent gives up 0.5 units of commodity

2 for 0.6 units of commodity 1. Note that t1 is the only state in which

her endowments permit her to make this trade. While 3 does not know

whether the true state is t1 or t2, she does know that the informed agent

would be better off with this allocation only if the true state is t1; if the

state is actually t2, the net-trade (0.6,−0.5) is infeasible for agent 1. The

informed agent’s claim, that the state is t1, is credible and should, therefore,

be accepted by agent 3. Acceptance of this allocation requires only that

agent 3 infer (correctly) from the allocation that the state is t1, not that 1’s

private information becomes explicitly available to agent 3.11 In this respect

this allocation offers a sensible objection to the status-quo. Agents should be

able to coordinate on an event that can be inferred simply by the fact that

all members of the coalition are willing to sign an allocation that is to their

benefit only on the given event. In the present example, this makes it hard

to justify the coarse core as the appropriate core notion.

We now give an example which shows that unlimited pooling of information,

which is implicit in the definition of the fine core, may not be very

appropriate under some circumstances.

Example 3.2

Consider a simpler version of Example 3.1 in which there is only one

commodity, and each consumer has an endowment of 1 unit in each state.

The information structure is the same as in the previous example.

The state-dependent utility functions are as follows.

u1(x, t) = ( 1.5x if t = t3

x otherwise

u2(x, t) = ( 1.5x if t = t1

x otherwise

u3(x, t) = ( 1.5x if t = t2

x otherwise

It is easy to see that the incentive compatible, coarse core contains ¯x(t),

where ¯xi(t) = 1 for t /2 T_, and

¯x(t1) = (1, 2, 0), ¯x(t2) = (0, 1, 2), ¯x(t3) = (2, 0, 1).

This allocation is not in the incentive compatible, fine core.12 Consumers

1 and 3 have a fine objection over the event {t1} with an allocation ˜x(t1)

such that ˜x1(t1) = 1+_ and ˜x3(t1) = 1 − _ for _ 2 (0, 1). In fact, every fine

objection must be of this form. But agent 3 cannot infer from this allocation

that consumer 1 is of type a1 because consumer 1 would prefer the net trade

_ in both states t1 and t2. Moreover, if the true state is t2, consumer 3 by

agreeing to the allocation ˜x, and accepting 1’s claim that she is of type a1,

would be worse off compared to the status-quo ¯x. In this sense, the fine

objection is not credible. The same argument holds for any fine objection to

an allocation that belongs to the coarse core. In this example, therefore, the

coarse core seems more reasonable than the fine core.

4 The Credible Core

The essential message from the previous examples is that the pooling of

private information between members of a coalition should be permitted if

and only if it can be justified as being credible. We now develop a notion of

objections which incorporates this consideration.

Suppose each i in coalition S claims, independently, not to be of any type

ˆti /2 Ei. This type, ˆti, cannot be ruled out by agent j 2 S, with her private

information, if

for some t 2 E, ˆti /2 Ei, q(t−i, ˆti) >0 (4.1)

For each i 2 S let Vi(E) _ Ti \Ei denote the set of all ˆti satisfying (4.1). Of

course, if the event E is not a common knowledge event, Vi(E) 6= ; for some

i 2 S.

Our credibility criterion imposes the restriction that none of the types in

Vi(E) should select (or pretend) to be some type in Ei.

Given an admissible event E for coalition S define for each i 2 S and

ˆti 2 Vi(E),

q(t−i | ˆti,E) =

q(t−i, ˆti)

Pt0

−i2E−i q(t0

−i, ˆti)

.

Note that this expression is well-defined given the definition of Vi(E).

For an event admissible for coalition S, we can now define for each i 2 S

and a type ˆti 2 Vi(E), the conditional expected utility (conditional on E),

of an allocation x as

Ui(x | ˆti,E) = X t0

−i2E−i

q(t0

−i | ˆti,E)ui(x(t0

−i,ˆti), (t0

−i, ˆti))

Similarly, define the conditional expected utility of x to ˆti 2 Vi(E) if ˆti

pretends to be of type si 2 Ti as

Ui(x, si | ˆti,E) _ X t0

−i2E−i

q(t0

−i | ˆti,E)ui(xi(t0

−i, si | ˆti), (t0

−i, ˆti))

Note that this is well-defined since ˆti 2 Vi(E).

Suppose x 2 AN, y 2 AS and E is an admissible event for coalition S.

An allocation y is said to satisfy self-selection with respect to x over E if

Ui(y, si | ˆti,E) _ Ui(x | ˆti,E) for all ˆti 2 Vi(E), si 2 Ei for all i 2 S (SS).

This constraint can be seen as an extension of (IC) to those types who are

not supposed to be part of the objecting coalition. Notice that, as in (D)

and (IC), the probabilities used in computing conditional expected utility,

are those corresponding to the event E over which the objection is supposed

to take place. By the argument used in proving proposition 2.1, it can be

shown that condition (SS) is equivalent to one in which this inequality is

required to hold for all si 2 Ti, not just all si 2 Ei.

Coalition S is said to have a credible objection to an incentive compatible

allocation x 2 AN if there exists y 2 AS and an admissible event E such that

(D), (IC) and (SS) are satisfied.

The credible core consists of all incentive compatible allocations to which

there does not exist a credible objection.

We use Example 3.1 again to illustrate the nature of condition (SS). Consider

again the allocation ¯x(t) which we pointed out was not in the incentive

compatible fine core. Let S = {1, 3}. Then, the event {t1} is admissible

for S since 1 can discern that t1 is the true state of the world. Consider

the allocation ˜x specified there. Since 1 is the only informed agent, we need

only check that (SS) is satisfied for 1. Note that V1(t1) = {b1}. As we have

pointed out earlier, the net trade involved in ˜x is not feasible for 1 when her

type is b1. Hence, ˜x would give her a utility of −1 if she is of type b1, but

claims to be of type a1. This shows that (SS) is satisfied, and so ¯x is not in

the credible core.

Remark 1. The credible core contains the incentive compatible, fine core,

and is contained in the incentive compatible, coarse core. The first inclusion

follows from the observation that an incentive compatible, fine objection is

not required to satisfy (SS). To see the second inclusion, notice that if E is

a common knowledge event for S then Vi(E) = ; for all i 2 S, and (SS)

is, therefore, vacuously satisfied. In example 3.1, the credible core coincides

with the incentive compatible, fine core, and in example 3.2 it coincides with

the incentive compatible, coarse core. In the next section we will present

an example in which these inclusions are strict, and all three cores are nonempty.

Remark 2. If incentive constraints, (IC), were to be dropped from the

conditions defining the credible core, it would become identical to the fine

core. This is so because a fine objection y over E by coalition S is then

equivalent to one in which agent i is assigned 0 in every state t such that

t−i 2 E−i and ti 2 Vi(E).13 Indeed, if types are verifiable as in Wilson (1978),

then the fine core becomes a more appealing concept - there is no reason why

members of a blocking coalition cannot share all their information since false

communication will be detected.

The basic logic underlying our notion of the credible core is related to

similar ideas used in other contexts. Most notably, it is similar to the concept

of credible updating used by Grossman and Perry in defining a perfect

sequential equilibrium. See also the discussion of the intuitive criterion in

Cho and Kreps (1987) and the discussion in Kahn and Mookherjee (1995)

regarding coalition proof Nash equilibrium under incomplete information.14

It is also related in spirit to the notion of durability studied by Holmstr¨om

and Myerson (1983).