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Working Paper Series
Coalition-Proof Allocations in Adverse
Selection Economies
WP 94-09
Jeffrey M. Lacker
Federal Reserve Bank of Richmond
John A. Weinberg
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
COALITION-PROOF ALLOCATIONS
IN ADVERSE SELECTION ECONOMIES
Jeffrey M. Lacker and John A. Weinberg*
Research Department, Federal Reserve Bank of Richmond
P.O. Box 27622, Richmond, VA 23261, 804-697-8205
September 20, 1995
ABSTRACT: We reexamine the canonical adverse selection insurance economy
first studied by Rothschild and Stiglitz (1976). We define blocking in a way
that takes private information into account, and define a coalition-proof
correspondence as a mapping from coalitions to allocations with the property
that allocations are in the correspondence if and only if they are not blocked
by any other allocations in the correspondence for any subcoalition. We prove
that the Miyazaki allocation--the Pareto-optimal allocation (possibly crosssubsidized) most preferred by low-risk agents--is coalition-proof. JEL No.
D82. Field designation: Information and Learning.
*
Helpful comments were received from the Editor and two anonymous
referees. The authors are solely responsible for the contents of the paper
and the views expressed do not necessarily reflect those of the Federal
Reserve Bank of Richmond or the Federal Reserve System.
Adverse selection economies have been particularly troublesome for
economists.
As Rothschild and Stiglitz (1976) showed, equilibrium, the way
one might naturally define it, often does not exist.
While modifications of
their definition of equilibrium have been proposed that solve the existence
problem, these take the form of specifying particular institutional
arrangements through which agents interact, with different specifications
yielding different predictions (Wilson 1979, Hellwig 1987).
It would be
desirable to have a predictive notion that is not heavily dependent on a
particular extensive form.
An alternative to specifying an explicit game form is to examine
solutions that are more cooperative in nature.
Miyazaki (1977) represents an
early example of this approach, applied to an adverse-selection labor market
environment.
The Miyazaki solution is built on the observation that in some
cases Pareto-efficiency requires cross-subsidization between types.
In the
insurance setting, the Miyazaki allocation is the Pareto-optimal allocation
most favored by the low-risk agents.
Since low-risk agents pay the subsidy,
Miyazaki argues that this equilibrium is supported by their threat to abandon
the arrangement and self-insure.
Several authors have presented core-like notions of equilibrium in
adverse selection environments that deliver the Miyazaki allocation (Boyd and
Prescott 1986, Boyd, Prescott and Smith 1988, and Marimon 1988).
The
extension of the core to adverse selection environments, however, is not
straightforward.
The most direct approach is to impose incentive
compatibility on agents' decisions about joining blocking coalitions, but this
results in a core that is empty in exactly those cases in which the
Rothschild-Stiglitz equilibrium fails to exist.
The authors cited above
address this problem by further restricting the notion of a successful
blocking coalition.
They require that a blocking coalition survive when the
2
agents left behind are allowed to react to the deviation.
In an adverse selection credit market environment with a continuum of
types (Lacker and Weinberg 1993) we explored a different approach to finding a
cooperative equilibrium, utilizing the notion of coalition-proofness
introduced by Bernheim, Peleg, and Whinston (1987, hereafter BPW).
Rather
than considering reactions by the complement of the deviating coalition,
coalition-proofness considers reactions by the deviating coalition itself.
A
"credible" blocking coalition must itself be immune from further deviations by
"credible" subcoalitions.
BPW introduced coalition-proofness as a refinement
of Nash equilibrium in finite normal-form games, and defined it recursively by
considering all possible sequences of subcoalitions of agents.
Greenberg
(1989) showed that coalition-proofness could be defined equivalently as a
stable set (see also Greenberg 1991, Kahn and Mookherjee 1992).
In a recent paper, Kahn and Mookherjee (1995, hereafter KM) present a
coalition-proof equilibrium for the standard adverse-selection insurance
environment.
Following BPW, they impose coalition-proofness as a refinement
of Nash equilibrium in an incomplete-information game of contract proposal and
acceptance.
Their approach might be viewed as a blend of cooperative and
noncooperative approaches to equilibrium in adverse selection economies.
The
unique result under their approach is the Rothschild-Stiglitz allocation--the
Pareto-optimum of the type-wise break-even, separating allocations.
The purpose of this note is to examine the implications of coalitionproofness in isolation for the adverse-selection insurance environment.
Given
our definition of blocking, we define coalition-proof allocations using
Greenberg's stable-set approach, as in Lacker and Weinberg (1993) and Lacker
(1994).
We show that the Miyazaki allocation is coalition-proof;
specifically, the correspondence consisting of the Miyazaki allocation for any
3
arbitrary coalition constitutes a stable set.
Therefore, it appears that it
is not coalition-proofness per se that rules out cross-subsidization in the KM
analysis.
In closely related work, Asheim and Nilssen (1994) present an extensive
form contracting game with renegotiation in which the unique perfect Bayesian
equilibrium is the Miyazaki allocation.
Their extensive form thus implements
the solution suggested by the cooperative literature.
Their result also
underscores the point made by Wilson (1979) and Hellwig (1987); the
noncooperative equilibrium in an adverse selection economy depends critically
on the extensive form.
The next two sections describe the environment and feasible allocations.
In Section 3 we define the coalition-proof correspondence.
define and characterize the Miyazaki allocations.
In Section 4 we
In Section 5 we prove that
they are the coalition-proof correspondence.
1.
The Economy
We study the simple insurance environment described in Rothschild and Stiglitz
(1976).
The economy is populated by a continuum of risk averse agents who
each receive a random endowment of a single consumption good.
All agents have
identical utility functions U(c), where U (c) > 0 and U (c) < 0 for all c
0.
Each agent receives a random endowment e drawn from the two-element set
{eg,eb}, where eg > eb > 0.
There are two types of agents; type H (high risk)
agents, and type L (low risk) agents.
Type H(L) agents receive the bad
endowment with probability pH(pL) and the good endowment with probability 1 pH (1 - pL).
We assume that 1 > pH > pL > 0.
type H(L) agents,
H
,
L
> 0, and let
= (
H
,
Let
H
(
L
) be the measures of
L
).
Although all agents know the distribution of types,
, each
4
individual's type is private information.
There is no technology available
for verifying or evaluating agents' types.
interacting with other agents.
Agents know their own types before
The endowment realization of each agent is
publicly observed, as are any contracts into which an agent may enter.
2.
Allocations
An allocation for this economy must describe the consumption of each agent in
every possible state of the world, but since there are a continuum of agents,
there is no uncertainty in the aggregate endowment.
We will consider only
allocations in which an agent's consumption depends on their own endowment.
We also restrict attention to allocations which treat all agents of a given
type identically, since agents are only different in a meaningful way through
their types.
Formally, an allocation consists of four nonnegative scalars: ciT ,
Let cL =
T=H,L, i=g,b, the consumption of a type T agent with endowment ei.
( cgL , cbL ) and cH = ( cgH , cbH ) be the allocations of type L and type H agents
respectively, and let c = (cL,cH).
We will want to define allocations and feasibility for arbitrary subsets
of the population.
For any arbitrary set of agents let
measures of type L and type H agents respectively, and let
L
and
= (
H
be the
L
,
H
).
We
will define feasibility for an allocation-coalition pair to encompass resource
and incentive feasibility.
An allocation-coalition pair (c, ) is resource
feasible if
H
[p HcbH
(1 p H)cgH]
L
[p LcbL
(1 p L)cgL]
(1)
H
[p Heb
(1 p H)eg]
L
[p Leb
(1 p L)eg]
5
This definition of resource feasibility makes use of the law of large numbers,
which here implies that the fraction of agents realizing any particular
outcome is equal to the probability of that outcome.
An allocation is
incentive feasible if
p Hu(cbH)
(1 p H)u(cgH)
p Hu(cbL)
(1 p H)u(cgL),
(2)
(1 p L)u(cgL)
p Lu(cbH)
(1 p L)u(cgH).
(3)
and
p Lu(cbL)
Define the set of feasible allocations C( ) for coalition
as the set of all
allocations satisfying (1) through (3).1
3.
The Coalition-Proof Allocations
Our definition of coalition-proof allocations for this economy differs from
the notion of "Coalition-Proof Nash Equilibrium" (CPNE) proposed by BPW and KM
in that we do not explicitly define a non-cooperative game to which the full
definition of the CPNE can be applied.
Instead, we simply seek allocations
which are "coalition proof" in the same sense that strategy profiles are
1
It would be possible to include in our economy another category of
agents capable of serving as insurance firms, as in KM. Such risk neutral
agents would be willing to provide insurance to coalitions of risk averse
agents only if their portfolio of insurance contracts earns nonnegative
expected profits. In our economy nonnegative expected profits for such a firm
is equivalent to the resource feasibility constraint (1). In addition, the
firm would face the same incentive feasibility conditions, (2) and (3), which
we impose on coalitions. If coalitions of risk averse agents can self-insure,
then the presence of insurance firms does not expand the set of attainable
allocations. If, on the other hand, coalitions of agents cannot self-insure,
the presence of insurance firms raises the possibility of allocations in which
such firms earn strictly positive profits.
6
coalition proof in the CPNE.
Coalition-proofness is based on the notion of a
blocking coalition, and so closely parallels the definition of the core of an
economy.
The notion of a blocking coalition must be adjusted, however, to
take account of the ex ante private information feature of our environment.
First, define the set of possible subcoalitions to a coalition
ˆ
( )
2
0 ˆL
, and 0 ˆ H
L
We will denote the set of all possible subcoalitions as
as
H
=
( ).
We will
denote the set of allocations that are feasible for some coalition as C, the
union of C( ) for
.
We adopt the following notion of a blocking coalition.
Definition 1:
ˆ
An allocation c C( ) for coalition
is blocked by a coalition
( ) and allocation c if:
ˆ
(a)
c C( ˆ) ;
ˆ
(b)
cb
for T=H,L, if ˆ T > 0 then p Tu(ˆT)
(c)
the inequality in (2) is strict for at least one T for which ˆ T > 0 ;
(d)
for T=H,L, if ˆ T <
T
(1 p T)u(c T)
ˆg
p Tu(cbT)
(1 p T)u(cgT) ;
(some type T agents are left behind) then
p Tu(cbT)
(1 p T)u(cgT)
p Tu(ˆH)
cb
(1 p T)u(c H) if ˆ H > 0 , and
ˆg
p Tu(cbT)
(1 p T)u(cgT)
p Tu(ˆL)
cb
(1 p T)u(c L)
ˆg
if ˆ L > 0 .
The first condition in the above definition states that the blocking
allocation is resource and incentive feasible for that blocking coalition.
Conditions (b) and (c) state the usual requirement that members of a blocking
coalition are at least as well off and some members are strictly better off.
The fourth condition is added to respect the constraints of private
information.
It states that if a deviating coalition intends to leave behind
7
any agents of type T, then those left behind cannot strictly prefer to join
the coalition (including by claiming to be another type).2
Some definitions of blocking in similar private information environments
(notably Boyd, Prescott and Smith 1987, and Marimon 1988), require the
deviating coalition to anticipate what will become of the complement coalition
after the deviation.
Our condition (d) requires only that those in the
complement coalition compare the status quo allocation to the proposed
deviation.
However, our "coalition-proof" requirement plays the role of
considering what might happen after a deviation occurs.
KM require blocking
allocations to be proper subsets of the original coalition, thus ruling out a
priori blocking by the coalition of the whole.
by the coalition of the whole.
Our approach allows deviations
KM also require strict inequality in (b), so
that all members of a deviating coalition must be made strictly better off.
Strict inequality in (b) would also imply that deviating coalitions must take
either all or none of any given type.
Our approach allows deviating
coalitions that strictly dominate only for a subset of their members.
The core of this economy--defined as the set of allocations that are
unblocked according to the above definition--may be empty.
The core is empty
in the cases in which the Rothschild-Stiglitz equilibrium fails to exist or is
Pareto dominated.
The Rothschild-Stiglitz equilibrium only exists when the
ratio of high- to low-risk agents exceeds a threshold below which there exists
a feasible pooling allocation that gives the low-risk agents the same expected
utility as in the Rothschild-Stiglitz equilibrium.
When the ratio lies in a
range just above this threshold, equilibrium exists but is Pareto dominated.
Coalition-proofness places more stringent conditions on blocking
2
This definition of blocking was adopted in Lacker and Weinberg (1993).
8
coalitions.
We require allocations to be immune only to deviations by
coalitions which are themselves immune to further deviations by similarly
immune sub-coalitions.
Definition 2:
A coalition-proof correspondence (CPC) is a mapping
:
C,
with the following properties:
(i)
(ii)
( ) C( ) for all
c
; and
( ) if and only if there does not exist a ˆ
( ) and c
ˆ
( ˆ) which
block c.
This definition states that any allocation-coalition pair which is not
coalition proof can be blocked by a subcoalition with an allocation which is
coalition proof.
On the other hand, if an allocation is coalition proof, then
any proposed deviation is deterred by a credible threat of further deviation
from the deviating coalition.
The requirement of coalition-proofness is
weaker than the requirement that an allocation be unblocked.
As a result, any
core allocation is also coalition-proof.
Our definition follows Greenberg's (1989) "von Neumann and Morgenstern
abstract stable set," defined as a partition of a stable system.
A stable
system is comprised of a set D, which in our case consists of allocationcoalition pairs (c, ), and a dominance relation, which in our case is
blocking.
In Greenberg's application a typical element of D is a coalition
together with a strategy profile.
He shows that, given a notion of blocking
for strategy profiles, the definition of a CPNE in terms of the coalitionproof correspondence is fully equivalent to the recursive definition given in
Bernheim, Peleg and Whinston (1987), allowing extension of the coalition-proof
9
concept to environments with an infinite number of agents.3
In our definition of the coalition-proof correspondence the conditions
required for a subcoalition to block a coalition are symmetric across
coalition-subcoalition pairs in the sense that they depend only on the
preferences, endowments and allocations within the initial coalition.
In
particular, blocking does not depend on whether the coalition being blocked is
the coalition of the whole or is itself a deviating coalition.
As a result,
the coalition-proof correspondence for a given coalition yields the coalitionproof allocation for an economy consisting of just that coalition.
This
incorporates the notion that deviations are private in the sense that only
those within a deviation can react to it (KM 1995, p. 118).
4.
The Miyazaki Allocation
Miyazaki (1977), in a labor market version of this economy, proposed that
equilibrium allocations in be defined as the solution to the following
programming problem.
The Miyazaki Problem:
p Lu(cbL)
MAX
s.t.
(1 p L)u(cgL)
(1), (2), (3), and
p Hu(cbH)
3
(1 p H)u(cgH)
UH
(4)
See also Greenberg (1991) and KM (1992). KM (1992) show that with an
infinite set of agents and infinite strategy sets the stable set and the CPNE
may fail to coincide and a stable set may fail to exist.
10
where
UH
p Hu(˜H)
cb
MAX
H
c b,˜H
˜ cg
p Hc H
˜b
s. t.
(1 p H)u(c H)
˜g
(1 p H)˜H
cg
p Heb
(1 p H)eg
The Miyazaki Problem maximizes the expected utility of the low risk agents
subject to resource and incentive feasibility.
The additional constraint,
(4), states that the high risk agents receive no less expected utility than
they would receive were they to band together and self insure, UH.
to achieve UH the high risk agents receive full insure at fair odds.
Note that
Define
c ( ) as the solution to the Miyazaki Problem, and call this the Miyazaki
allocation for the coalition
.4
Define full-insurance consumption at fair odds as c T for type T agents,
¯
¯
so that c T
p Teb
(1 p T)eg .
The Rothschild-Stiglitz allocation, c 0 , is the
type-wise break-even, separating allocation that provides full insurance at
H0
H0
¯
fair odds for the high risk agents ( cb = cg = c H ).
Low risk agents receive
fair odds but only partial insurance because of the incentive constraint (2)
that high risk agents do not prefer c L0 over c H0 .
L0
L0
Thus cb and cg are
defined jointly by equation (2) and
L0
p Lcb
4
(1
L0
p L)cg
p Leb
(1
p L)eg.
It is straightforward to generalize the Miyazaki allocation for n types.
For type 1, the highest risk, define U1 exactly the way UH is defined, and
define U2 exactly the way UL is defined. Assuming Uk has been defined for k <
j, Uj is the maximum expected utility of type j consumers subject to resource
and incentive feasibility and the constraints that all type k < j consumers
each receive at least Uk in expected utility.
11
Rothschild and Stiglitz (1976) identify this allocation as the Nash
equilibrium.
If the ratio of high-risk to low-risk agents is too small, then
a feasible pooling allocation exists that gives the low-risk agents as much
expected utility as c L0 , and their equilibrium does not exist.
define VS(cT) = p Su(cbT)
(1 p S)u(cgT) and ES[cT] = p ScbT
Finally,
(1 p S)cgT for S,T=L,H.
We collect together here some of the important properties of the
Miyazaki allocations, stated without proof (see Crocker and Snow (1985)).
(Properties of c ( ) )
Proposition 1:
cL (
1.
L
,0)
c L for all
¯
L
(0,
L
) , and c H (0,
H
)
c H for all
¯
H
(0,
H
)
(full insurance at fair odds for single-type coalitions).
2.
For all
H
s. t.
,
L
> 0:
(a) c ( ) is unique and continuous in
H
(b) cb ( )
H
cg ( )
;
c H (full insurance at fair odds or better for high¯
risk agents);
L
L
(c) cb ( ) < cg ( ) (incomplete insurance for low-risk agents);
(d) c ( ) satisfies (1) and (2) with equality (the resource constraint and
the high-risk incentive constraint bind);
(e) c ( ) satisfies (3) as a strict inequality (the low-risk incentive
constraint does not bind);
(f) there exists a
0
then c ( )
0
c
0
(0, ) s.t. if
H
/
L
0
c 0 , and if
then c ( )
3.
As
/
L
L
L
/
<
(type-wise break-even, separating allocation if and
only if the ratio of high- to low-risk agents is larger than
H
H
0, c ( ) c
¯
L
H
and c ( ) c
¯
L
).
0
(full insurance for all at low-risk
fair odds as the high-risk agents disappear).
VL(cL*( )) and VH(cH*( )) are strictly decreasing in
4.
H
/
L
for
H
/
L
<
.
0
Note that the Miyazaki allocation is scale-invariant, in the sense that
it depends only on the ratio of high-risk to low-risk populations. 5
5
A CPC is not required to be scale-invariant however.
When this
12
ratio is greater than
, the Miyazaki allocation is identical to the type-
0
wise break-even, separating allocation, the original Rothschild-Stiglitz
equilibrium.
When there are relatively few high-risk agents (
<
), low-
0
risk agents are willing to subsidize high-risk agents ( c H ( ) > c H ) in order
¯
to loosen incentive constraints.
In the limit, as the ratio of high- to low-
risk agents goes to zero the low-risk Miyazaki allocation approaches full
insurance at fair odds.
Contrast this with equilibrium notions that predict
the type-wise break-even separating allocation, which is invariant to the
number of high-risk agents.
In such equilibria, introducing a tiny number of
high-risk agents in an economy with just low-risk agents results in a discrete
jump in allocations.6
5.
The Miyazaki Allocation is Coalition-Proof
Our candidate CPC is just
*
( )
{c*( )}, for each coalition a set
consisting of the unique Miyazaki allocation.
To prove it we must select an
arbitrary coalition and an arbitrary feasible allocation and show that (1) if
it is not the Miyazaki allocation for that coalition then it can be blocked by
a Miyazaki allocation for some coalition, and (2) if it is the Miyazaki
allocation then it cannot be blocked by a Miyazaki allocation for any
subcoalition.
We briefly sketch the argument before formally stating the
result and the proof.
Two cases are handled quite easily.
First, if the expected utility of
6
If the economy includes insurance firms, then one could define a family
of Miyazaki-type allocations for a given coalition, indexed by the expected
profits of the insurance firms. Each allocation in the family satisfies
Proposition 1, with the full-insurance allocations defined net of expected
¯
p Leb (1 p L)eg
x , where expected firm profits per
firm profits (e.g. c L
insured agent are x). In such a setting, this entire family of Miyazaki-like
allocations would be coalition-proof.
13
the high-risk agents is lower in the arbitrary allocation than in the Miyazaki
allocation, then the coalition of the whole blocks the arbitrary allocation;
the Miyazaki allocation yields higher expected utility for both types of
agents.
Second, if the expected utility of the high-risk agents is greater in
the arbitrary allocation than they would receive with low-risk full insurance,
then the low-risk agents are doing quite poorly and a coalition of just lowrisk agents blocks the arbitrary allocation.
The difficult case lies between these two.
The trick is to note that
the expected utility of the high-risk agents in the Miyazaki allocation is
strictly decreasing in
.
A blocking coalition can be formed with all of the
low-risk agents and some of the high-risk agents such that the Miyazaki
allocation for that coalition gives the high-risk agents exactly the same
expected utility as they receive in the arbitrary allocation.
The high-risk
agents are indifferent between the status quo and the blocking allocation,
while the low-risk agents are strictly better off (unless the arbitrary
allocation is the Miyazaki allocation for the original coalition).
A Miyazaki allocation cannot be blocked by a Miyazaki allocation for any
deviating coalition.
If the deviating coalition has a lower ratio of high- to
low-risk agents, then either the allocations are identical (when
>
both allocations) or both types are strictly better off (when
0
deviating coalition).
<
0
in
in the
In the latter case, the deviating coalition would
attract all of the low-risk agents and all, not some, of the high-risk agents,
and thus would fail to reduce the ratio of high- to low-risk agents.
Similarly, any deviating coalition with a higher ratio of high- to low-risk
agents would make both types strictly worse off if the allocation was at all
different.
We can now state our central result.
14
Proposition 2:
Proof:
H
is a CPC.
Choose an arbitrary coalition
L
to
*
matters, we will use
coalition.7
and
, with
=
H
/
L
.
Since only the ratio
interchangeably to refer to a given
Choose an arbitrary c C( ).
c* then
We will prove that if c
there exists a subcoalition that blocks c with the Miyazaki allocation for
that subcoalition, and if c = c* then no such subcoalition exists.
L
= 0 or
H
= 0 the proof is trivial, so assume from now on that
First, suppose that VH(cH) < VH(cH*( )).
that VL(cL*( ))
satisfied.
If either
L
,
H
> 0.
The Miyazaki Problem implies
VL(cL), so conditions (b) and (c) of Definition 1 are
If we form a blocking coalition consisting of the entire coalition
, no agents are left behind, so we need not check blocking condition (d).
Therefore the entire coalition with allocation c*( ) blocks c.
Next suppose that VH(cH)
VH( c L ).
¯
c L > EH[e],
¯
This implies EH[cH]
which implies EL[cL] < EL[e], which implies VL(cL) < VL( c L ).
¯
Consider a
blocking coalition consisting of just the low-risk agents,
' = (
L
,0).
The
Miyazaki allocation for that coalition gives low-risk agents the full-
¯
insurance allocation c L .
¯
By construction c L is feasible for
condition (a) is satisfied.
satisfied.
', so blocking
¯
Since VL( c L ) > VL(cL), conditions (b) and (c) are
Since only high-risk agents are left behind, condition (d)
VH( c L ), which was assumed.
¯
requires VH(cH)
Therefore coalition
' = (
L
,0)
with Miyazaki allocation c L blocks c.
¯
Now suppose VH(cH*( ))
VH(cH) and
7
'
.
VH(cH) < VH( c L ).
¯
We know such a
Choose
' s.t. VH(cH*( ')) =
' exists from Proposition 1, properties
Note that our definition of coalition-proof correspondences allows for
mappings that are not scale-invariant.
15
(2a), (3) and (4).
VL(cL*( ))
The definition of the Miyazaki allocation implies that
VL(cL), and Property 4 implies that VL(cL*( '))
Therefore VL(cL*( '))
condition (b).
VL(cL*( )).
VL(cL) and VH(cH*( ')) = VH(cH), satisfying blocking
Since only high-risk agents are left behind, blocking
condition (d) just requires VH(cH)
VH(cL*( ')).
But this can be deduced from
the incentive compatibility of c*( '), which implies VH(cH*( '))
and the fact that VH(cH*( ')) = VH(cH) by construction.
VH(cL*( ')),
Obviously c*( ') is
feasible, so Definition 1 is satisfied except for condition (c).
Coalition
'
with Miyazaki allocation c*( ') blocks allocation c, therefore, if and only if
VL(cL*( ')) > VL(cL).
Suppose cH
cH*( ').
Since VH(cH)
VH(cH*( )) and u is strictly concave
we know that EH[cH] > EH[cH*( )], and thus EL[cL] < EL[cL*( )].
the unique solution to the Miyazaki Problem and cL
> VL(cL).
Therefore, VL(cL*( '))
cL*( ), we have VL(cL*( ))
VL(cL*( )) > VL(cL) and (c*( '), ') blocks c.
Now suppose cH = cH*( ') but cL
coalition
Since cL*( ) is
cL*( ').
Since c is feasible for
' and cL*( ') is the unique solution to the Miyazaki Problem, we
have directly that VL(cL*( ')) > VL(cL) and thus (c*( '), ') blocks c.
If cH = cH*( ') and cL = cL*( '), then c is the Miyazaki allocation for
, and (c*( '), ') fails to block c.
' =
We need to prove that no
subcoalition blocks c with the Miyazaki allocation for that subcoalition.
, then any coalition with
0
will thus fail to block.
0
,
has VT(cT*(
Any coalition with
<
If
)) = VT(cT*( )), T=H,L, and
0
has VT(cT*(
)) >
VT(cT*( )), T=H,L, attracts both types, and thus fails to obtain the proportion
.
Suppose
<
; then any coalition with
0
and attracts no low-risk agents.
>
has VL(cL*(
)) < VL(cL*( ))
The Miyazaki allocation for such a coalition
¯
of all high-risk agents gives them VH( c H )
VH(cH*( )).
Therefore, no
16
>
coalition with
with
<
has VL(cL*(
is capable of blocking c* in this case.
)) > VL(cL*( )) (since
<
Any coalition
) and will attract all of
0
the low-risk agents, but because VH(cH*( )) is strictly decreasing for
<
,
0
all of the high-risk agents are also attracted, so the allocation fails to
attract the correct proportions of high- and low-risk agents.
Therefore,
there is no coalition whose Miyazaki allocation blocks c.
One difference between our setup and KM is the absence here of a
separate class of agents acting as risk neutral insurance firms.
By itself,
the presence of firms does not affect our results since risk averse agents can
always split off from a firm making positive profits.
Under the additional
assumption that risk averse agents are incapable of joining together to selfinsure, such a deviation is not allowed and allocations with positive firm
profits can be sustained.
In this case our CPC must be expanded; for any
given collection of risk averse agents, the set consists of the family of
Miyazaki-type allocations indexed by the nonnegative level of firm profits.
(See note 6.)
Positive profits can be sustained because a set of agents
insured by a single firm cannot, by assumption, self-insure in any
subcoalition that does not include the firm.
However, the requirement that
insurance be obtained through firms does not change the basic characteristics
of coalition-proof allocations.
Miyazaki-like allocations are still
coalition-proof.
The other key differences between our results and those of KM stem from
our different definitions of coalition-proofness.
KM require that deviating
coalitions be proper subsets, whereas we allow deviations by an entire
coalition.
KM motivate strict set inclusion by the need to guarantee the
existence of a CPC (a "stable partition" in their terminology) in the presence
17
of infinite strategy spaces.
We have constructed a CPC using only weak set
inclusion, and thus strict set inclusion does not seem essential for existence
here.
Imposing strict set inclusion in our framework, the Miyazaki allocation
would not be coalition proof, and thus strict set inclusion does not seem to
be innocuous.
Finally, we have proven our results only for the case of two types.
The
existence problem identified by KM could well be more serious with n types.
This problem necessitates care in the details of how coalition-proofness is
defined.
Our results imply that in the two type case predictions can be
sensitive to seemingly technical details.
This suggests that other approaches
would be worth examining in the n type case as well.
research.
We leave this for future
18
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