Coalitional fairness under strategic behavior
Nicholas Ziros
Abstract
This paper studies the relationship between coalitional fairness and allocations
that result from Nash equilibrium strategies in economies where trade is conducted
via the rules of strategic market games. In particular, we show that coalitional-fair
allocations can be approximately decentralized as Nash equilibrium. We argue that
this exercise is essential because it implies that if all individuals have access to the
same trading institutions and rules, then strategic behavior is compatible with a fair
distribution of commodities.
JEL classi…cation: D43; D50; C72
Keywords: Strategic market games; Coalitional fairness; Nash equilibrium.
Department of Economics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
E-mail: [email protected]
1
1
Introduction
The properties of allocations that result from trade is the main analytical tool in general
equilibrium theory. Indeed, such allocations have been characterized as exhausting the
gains from trade (that is Walrasian allocations), e¢ cient (e.g., Pareto optimal allocations)
or coalitional stable (e.g., core allocations) depending on their properties. In late 1960’s
and early 1970’s a number of researchers have examined under what conditions an allocation of commodities can be characterized as fair. An early attempt to de…ne fair allocations
appeared in Foley (1967), where an allocation is characterized as equitable if each trader
prefers her commodity bundle to that of any other trader. Since then, ’fairness’has been a
subject of an extensive discussion in the literature, which resulted to a number of variants
of Foley’s concept, that usually go under di¤erent names and have brought into light various interpretations, implications and limitations. For instance, Varian (1974) de…ned as
envy-free allocations those in which no trader envies the bundle of another trader and as
fair allocations the ones that are both envy-free and e¢ cient. Schmeidler and Vind (1972)
introduced the concept of fair net trades, according to which each trader should prefer
her net trade to those of other traders. Similarly, Mas-Colell (1985) de…ned anonymous
allocations as the ones in which no trader would be better o¤ with the net trade of any
other trader.1
A complementary concept to the ones described above is coalitional-fair (c-fair from
now on) allocations, which have been introduced by Gabszewicz (1975). An allocation of
commodities is said to be c-fair if no group of traders envies the net trade of some disjoint
other group of traders. In other words, if an allocation is c-fair then no coalition of traders
can redistribute the net trade of any other coalition in a way that would make each of its
members better o¤. Hence c-fair allocations have an appealing stability property and are
particularly useful for a complete characterization of competitive market equilibria.2
The contribution of the present paper is to examine whether strategic behavior can
1
2
The same allocations are characterized as self-selective in Mas-Colell et al. (1995).
The interested reader could consult Yannelis (1985) or Zhou (1992) for such results.
2
produce c-fair allocations. To this end, we employ the methodology of strategic market
games, which provides a complete speci…cation of trading rules and transaction constraints
with respect to which a noncooperative equilibrium is reached. Indeed, these games, originating in Shubik (1973) and Shapley and Shubik (1977), feature an explicit description of
the formation of prices and of the distribution of commodities and give a clear insight into
the strategic possibilities and incentives of individuals to in‡uence market outcomes. In
that sense, this class of games can be thought of as modern formulation of the Cournotian
oligopoly tradition. They have also have helped in the articulation of a theory of competition as a number of papers have examined the relationship between Nash equilibrium
outcomes of strategic market games to Walrasian ones.3
We believe that establishing a relation between coalitional fairness and strategic behavior, as we do in this paper, is very important for two main reasons. First, fairness
besides from being a key issue of confrontation in the political economy debate,4 is also
the objective of many intended policies. Hence, our investigation has practical implications as it proves that such allocations can be achieved even in a non-Walrasian context,
where individuals are able to exercise market power or to manipulate prices. Second, in
the existing literature, the various concepts of fairness have been analyzed in frameworks
unconstrained by any kind of institutional mechanisms of trade. On the other hand, our
approach considers a framework where all agents have access to the same trading institutions and rules. This observation is closely related to the original idea of fairness, that no
traders are treated discriminatory by the market.
Speci…cally in this paper we use the market game framework in order to provide a
non-Walrasian version of c-fair allocations. Indeed, Proposition 1 shows that Nash equilibrium strategies can produce c-fair allocations, once the planning authority is allowed
to announce appropriate income redistributions, by determining budget constraints (i.e.,
tax credits/liabilities) imposed on individuals. In our setup there is no inbuilt hypothesis
3
See, for instance, Dubey and Shubik (1978), Postlewaite and Schmeidler (1978), Mas-Colell (1982) and
Koutsougeras (2009).
4
Where it is viewed as the theoretical foundation of some schools of thought.
3
of price taking. Moreover, the redistribution policies announced induce traders to change
their demands and supplies to a speci…c direction but do not deprive them from their
ability to in‡uence market outcomes. Consequently, traders are allowed, in principle, to
use their market power in order to change market prices if they …nd it pro…table to do so.
Then we proceed to further examine the relationship between c-fair allocations and Nash
equilibrium allocations without imposing any income redistributions. In Proposition 2 we
exhibit that any c-fair allocation can be obtained as a Nash equilibrium of the standard
strategic market game, de…ned on a ’nearby’ economy. The method to formulate ’proximity’between economies draws on Postlewaite and Schmeidler (1981) and Koutsougeras
and Ziros (2011), and an essential implication of our result is that the approximation to
the original economy becomes …ner as the number of individuals becomes larger.
The presentation in this paper proceeds by developing the model in Section 2. Section
3 features the results of the paper. Further discussion follows in Section 4.
2
The model
The economy consists of a …nite set L of commodities and a …nite set H of agents. Agents
are characterized by (i) their consumption sets, which are identi…ed with <L
+ , (ii) their
initial endowments eh 2 <L
+ and (iii) their preferences, which are represented by a utility
function uh : <L
+ ! <. Concerning preferences we assume the following:
Assumption: Preferences are convex, C 2 , di¤ erentiably strictly monotone 5 and indifference surfaces through the endowment do not intersect the axes.
Let P 2 denote the set of C 2 di¤erentiably strictly monotone preferences that satisfy
the boundary condition stated in the assumption above, and let the set of characteristics T
P2
<L
++ be compact. An economy is de…ned as a mapping E : H !
T and the set of feasible allocations for this economy is de…ned as a mapping F =
P
P
x 2 <LH
+ :
h2H xh =
h2H eh .
The de…nitions of Walrasian, fair and c-fair allocations for this economy are as follows.
5
i.e., if u represents
i
then for all x 2 <L
++ , @uh =@x > 0 for each i = 1; 2; :::; L:
4
De…nition 1 A pair (p; x) 2 <L
+
P
P
(i) h2H xh = h2H eh ;
(ii) xh 2 arg maxfuh (y) : py
<LH
+ is called a Walrasian equilibrium if and only if:
peh g:
De…nition 2 An allocation x 2 <LH
+ is fair if there exist no individuals h and k; and an
allocation y 2 <LH
+ , such that uh (yh ) > uh (xh ) and (yh
eh ) = (xk
ek ).
De…nition 3 An allocation x 2 <LH
+ is c-fair if there exist no coalitions of traders S1 ; S2 ;
with S1 \ S2 = ?, and an allocation y 2 <LH
+ , such that uh (yh ) > uh (xh ) 8h 2 S1 and
P
P
eh ) = h2S2 (xh eh ).
h2S1 (yh
Let us state here, as a fact for future reference, that the set of Walrasian allocations is
contained in the set of c-fair allocations. We proceed to describe trade according to the
rules of the strategic market game studied in Postlewaite and Schmeidler (1978) and in
Peck et al. (1992).
2.1
The market game
Trade in this economy is organized via a system of trading posts where individuals o¤er
commodities for sale and place bids for purchases of commodities. Bids are placed in terms
eih ;
<L
of a unit of account. The strategy set of each agent is Sh = f(bh ; qh ) 2 <L
+ : qh
+
P
P
i
i
i
i = 1; 2; :::; Lg. Given a strategy pro…le let B i =
h2H qh . Also
h2H bh and Q =
P
P
for each agent h let B i h = k6=h bik , Qi h = k6=h qki . The distribution of commodities
i = 1; 2; :::; L for each agent h 2 H is determined as follows:
8
< ei
h
xih (b; q) =
:
0
qhi +
bih i
Q
Bi
Bi i
q
Qi h
otherwise
if
PL
i=1
PL
i
i=1 bh
;
where the term 0/0 is equal to zero whenever it appears in the above expressions.
The interpretation of this allocation mechanism is that commodities are distributed
among non-bankrupt agents in proportion to their bids (o¤ers). Note also that when
B i Qi 6= 0 the fraction
i (b; q)
= B i =Qi can be interpreted as the market clearing ’price’.
5
In this framework, agents are viewed as solving the following problem:
maxfuh ((xih (b; q))L
i=1 ) : (bh ; qh ) 2 Sh g
An equilibrium is de…ned as a collection of bids and o¤ers f(bh ; qh ) 2 Sh : h 2 Hg
Q
that form a Nash equilibrium. Let N(E)
h2H Sh denote the set of Nash equilibrium
strategy pro…les of the strategic market game and N (E)
<LH
+ the set of consumption
allocations corresponding to the elements of N(E):
Notice that, due to the bankruptcy rule above, at an equilibrium with positive bids and
o¤ers individuals can be viewed as solving the following problem:
max fuh ((xih (b; q))L
i=1 ) :
(bh ;qh )2Sh
PL
i=1 (B
i i
qh =Qi )
PL
i
i=1 bh g
As shown in Peck et. al. (1992), equilibria where agents choose strategies lying in the
interior of their strategy sets always exist. In other words, there exist equilibria where the
…rst order conditions in the maximization problem of agents are operative.
2.2
An example
In order to outline the relationship between c-fairness and the (equilibrium) outcomes of
market games, let us consider the following example.
Example 1 Consider the exchange economy consisting of two commodities i = ;
and
three agents h = 1; 2; 3 with initial endowments eh = (eh ; eh ). The 3 agents have identical
utility functions uh (xh ; xh ) = xh xh whereas their initial endowments are e1 = (4:5; 74)
and e2 = e3 = (45; 12).
For this economy, a possible exchange of the two commodities can yield the following
(individually rational) allocation of commodities x1 = (13:5; 59), x2 = (41:5; 22) and
x2 = (41:5; 17). However, this allocation is not c-fair because trader 3 envies the net
trade of agent 2. It should be noted that, due to the di¤erent prices at which the exchange
of the two commodities takes place, this allocation cannot be supported by the market
6
game mechanism. In other words, an allocation of commodities that is not c-fair due to
unequal prices cannot be an (equilibrium or out of equilibrium) outcome of the market
game described above.
Let us consider now the following pro…le of bids and o¤ers for the three traders:
(b1 ; q1 ; b1 ; q1 ) = (16:5; 4:5; 99; 74), (b2 ; q2 ; b2 ; q2 ) = (39; 45; 24; 12), (b3 ; q3 ; b3 ; q3 ) = (39;
45; 24; 12). These strategies satisfy the book-keeping constraints of each trader, give rise
to prices p
= 1 and p
= 3=2 and to the following c-fair allocation of commodities
x1 = (16:5; 66), x2 = x3 = (39; 16). However, they do not satisfy the …rst order conditions
that characterize a Nash equilibrium of the market game.6
Finally, we show that the above c-fair allocation of commodities can be achieved by
a Nash equilibrium strategies. Let us consider the following pro…le of bids and o¤er
for the three traders (b1 ; q1 ; b1 ; q1 ) = (16; 4; 24; 24), (b2 ; q2 ; b2 ; q2 ) = (6; 12; 9; 2),
(b3 ; q3 ; b3 ; q3 ) = (6; 12; 9; 2). The above strategies satisfy the book-keeping constraints
of each trader, give rise to market clearing prices p = 1 and p = 3=2 and to the c-fair
allocation of commodities x1 = (16:5; 66), x2 = x3 = (39; 16). It is easy to check, that
they also satisfy the conditions that characterize a Nash equilibrium.
Hence, it is evident that a c-fair allocations can be decentralized as outcomes of the
market-game mechanism described above, and, moreover, they could also result from Nash
equilibrium strategies.
2.3
The market game with transfers
L
Let E be an economy and (p; x) 2
F be a price vector and a feasible allocation for
this economy. Consider a market game on this economy, which is de…ned exactly as in the
previous section except that the allocation rule is now as follows:
x
^ih (b; q; p; x) =
6
That is,
8
<
:
eih
qhi +
bih i
Q
Bi
0
@uh =@xh
(p )2 Q h B
=
(
)(
(p )2 B a h Q
@uh =@xh
h
bih i
Q
Bi
otherwise
if
PL
i=1 (
) for h = 1; 2; 3.
h
7
qhi )pi
p(xh
eh )
(1)
This modi…ed allocation rule di¤ers from the original one, only in that individuals are
required to satisfy a di¤erent budget constraint, which involves credits/liabilities imposed
on individuals, determined by the pre-speci…ed allocation x along with the vector p. By
virtue of the fact that the pre-speci…ed allocation x is feasible, those lump-sum transfers
cancel out on the aggregate, so the unit of account remains in zero net supply (money is
still ’inside’). Of course, if p(xh
eh ) = 0 for all h 2 H, then the allocation rule (1) is the
same as the original one. Moreover, for arbitrarily small transfers the budget restrictions
imposed on individuals are arbitrarily close to those imposed by the original allocation
rule.
De…nition 4 A market game with transfers on the economy E is the strategic market
game induced by the strategic outcome function de…ned in (1).
We denote by N(p;x) (E) the set of Nash equilibrium strategy pro…les of the corresponding
(p; x)-transfer market game and N(p;x) (E) the set of corresponding consumption allocations.
There are a few interesting points that should be stressed here. First, the planner does
not interfere with the functioning of markets. The allocation of commodities takes place
as before via decentralized trades, the planner only distributes the unit of account across
individuals. Second, the parameters (p; x) are set a priori and known to individuals at the
time they make their market decisions. In other words the planner sets the appropriate
policy parameters and does not interfere after individual actions in markets have been executed to further a¤ect the result of the economy. Hence, the interference of the planning
authority is in terms of ’policy’rather than ’regulation’. We argue that this is an appropriate rule in an economy where some authority is allowed to redistribute wealth. Third,
the transfers are in monetary rather than commodity units, because if it were possible for
the government to achieve the desired allocation through commodity transfers, then there
would be no need for markets.
Notice also that (1) is a legitimate strategic outcome function. In order to verify this
for any (b; q) 2
h2H Sh
let Z denote the set of individuals whose budget is satis…ed, i.e.
8
Z = fh 2 H :
PL
i=1 (
bih i
Q
Bi
qhi )pi
p(xh
P
i
h2H xh (b; q; p; x) =
=
eh )g. We have:
bih i
Q
qhi )
Bi
P
P
bih i
i
Q)
e
+
(
h2Z h
h2H
Bi
P
P
i
i
i
h2Z eh + Q
h2Z qh
P
i
h2H (eh +
P
i
h2Z eh
P
h2H
3
+
eih .
P
i
h2Z qh
P
i
h2Z qh
Results
We begin with the …rst result of the paper, which shows that any c-fair allocation can be
decentralized via a Nash equilibrium of a market game with transfers.
Proposition 1 Let E be an economy and x 2 <LH
+ be a c-fair allocation of this economy.
Then there is (b; q) 2 N(p;x) (E) such that x(b; q; p; x) = x and (b; q) = p:
Proof. Since x is a c-fair allocation there exists p 2
L
such that Duh (x) =
h p,
where
Duh (x) is the gradient of uh (x).
Now let us consider the (p; x)-transfer strategic market game. Let (b; q) 2
h2H Sh
be
the strategy pro…le de…ned as follows: bih = pi xih , qhi = eih , for each i = 1; 2; :::; L. In this
case:
i
Therefore
i (b; q)
P
P
i i
i
Bi
h2H p xh
h2H xh
i
P
(b; q) = i = P
=
p
= pi :
i
i
Q
e
e
h2H h
h2H h
= pi . Furthermore, for each h 2 H :
PL
i
i=1 (bh
Bi i
q ) = p(xh
Qi h
eh ):
Hence it follows that since no individual is bankrupt:
p i xi
Qi
xih (b; q; p; x) = eih qhi + bih i = eih eih + i h = xih ; 8h 2 H:
p
B
9
(2)
A Nash equilibrium for the (p; x)-transfer strategic market game is characterized by the
following necessary and su¢ cient conditions for each h 2 H :
PL
i=1 (
dxi
@u
(xh ) ih
i
@xh
dbh
d
(
dbih
PL
h(
The last equation yields:
bih i
Q
Bi
i=1 (
qhi )pi = p(xh
bih i
Q
Bi
B i h Qi
@uh
(x
)
h
(B i )2
@xih
Let us try the strategy pro…le (b; q)
qhi )pi
i
i
i B hQ
hp
(B i )2
h2H Sh
p(xh
eh )
(3)
eh ))) = 0 i = 1; 2; :::; L:
= 0 i = 1; 2; :::; L:
along with the Lagrange multipliers (
h )h2H .
Clearly by (2) the set of constraints (3) is satis…ed for each h 2 H: We also have for that:
B i h Qi
@uh
(x
)
h
(B i )2
@xih
i
hp
B i h Qi
@uh
= [ i (xih )
i
2
(B )
@xh
i
hp ]
B i h Qi
=0
(B i )2
where the last equality follows by virtue of the fact that Duh (x) =
h p:
Therefore, by the su¢ ciency of those conditions it follows that (b; q) 2 N(p;x) (E);
i (b; q)
= pi and x(b; q; p; x) = x as desired.
In the following result we associate c-fair allocations of a given economy with the Nash
equilibria of a ’nearby’economy.
Proposition 2 Let E be an economy and x
^ 2 <LH
a coalitional fair allocation of this
+
economy. Then there is an economy E 0 : H ! P 2
<L
^ 2 N (E 0 ). Moreover,
+ such that x
> 0 there is N such that if #H > N then (E,E 0 ) < :
for every
Proof. Since c-fair allocations are Pareto e¢ cient, by the second welfare theorem there
is p^ 2
L
that supports x
^, de…ned (up to a normalization) by the (common) marginal
rates of substitution of individuals at x
^. For each h 2 H let:
^h =
@uh
(^
xh )
@xih
.
p^i
We now de…ne:
i
h
p^i (^
xi
= Ph
eih )
^ih
k6=h x
10
0
<L
++ , de…ned as E (h) = (vh ; eh ), where for
2
and we construct a new economy E 0 : Psm
each h 2 H
vh (x) = uh (x) + ^ h
PL
i i
hx
i=1
We show that when the number of individuals is su¢ ciently large, the economy E 0
constructed in the above proof, is close in some sense to the original economy E: Clearly,
it su¢ ces to show that for a large number of individuals the perturbation of preferences
de…ned by
= (
i )L
h i=1 h2H
is arbitrarily small. Denote by 1L the L dimensional vector
with all coordinates equal to one and let r^ > 0 be a number such that, 8 y 2 A; ( 1L )h2H
y.
Clearly, x
^ 2 A so the following inequalities are true 8 h 2 H : xh
1L ; 1L
eh
r^1L .
Therefore,
p^i (^
xih
eih )
p^i (
h
eih )
p^i (
h
r)
r^
L.
where the last inequality follows from the fact that p^ 2
Moreover by a theorem in Mas-Colell [proposition 7.4.3, page 279] we have that since
p^ 2
that
L
supports x
^, which is a coalitional fair allocation, there is a constant M > 0 so
PL
^i (^
xih
i=1 p
eih )
M
. Therefore,
#H
p^i (^
xih
eih ) =
M P j j
p (eh
#H j6=i
M
+ (^
r
#H
M
+ r^
#H
Hence, we can conclude that p^i (^
xih
M
+ r^
#H
eih )
x
^jh )
)
P
p^j
j6=i
. Also
Therefore,
i
h
1
#H 1
M
r^
+
#H
11
1 :
P
^ih
k6=h x
(#H
1) .
Now let us consider the strategic market game de…nes on the economy E 0 : Let (^b; q^) 2
h2H Sh
be a strategy pro…le de…ned as follows: ^bih = p^i x
^ih , q^hi = eih , for each i = 1; 2; :::; L.
Note that:
i
P
P
ix
i
^i
p
^
^
^ih
B
h2H
h2H x
h
i
P
(^b; q^) =
= P
=
p
^
= p^i :
i
i
i
^
e
e
Q
h2H h
h2H h
Furthermore for each h 2 H :
^i
PL ^i
PL i i
P
P
B
^x
^h = L
^i eih = L
^hi
i=1 bh =
i=1 p
i=1 p
i=1 ^ i q
Q
Hence it follows that since no individual is bankrupt:
xih (^b; q^) = eih
q^hi + ^bih
^i
Q
= eih
i
^
B
eih +
p^i x
^ih
=x
^ih
p^i
(4)
Recall now that a Nash equilibrium is characterized by the following necessary and
su¢ cient conditions for each h 2 H :
@vh B i h Qi
@xi (B i )2
PL
Qi h
= 0 i = 1; 2; :::; L
Qi
i
i=1 bh =
Let us try the strategy pro…le (^b; q^)
h2H Sh
PL
i=1
Bi i
q
Qi h
along with the Lagrange multipliers ( ^ h )h2H .
Clearly by (4) the second set of constraints is satis…ed for each h 2 H. Furthermore by
the de…nition of functions vh ( ) we have:
12
^i Q
^i
B
@vh
h
(^
x
)
h
^ i )2
@xi
(B
^h
^i
Q
h
^i
Q
=
=
=
=
=
=
^i Q
^i
B
h
^ i )2
(B
^i
Q
h
^i
Q
#
"
^i Q
^i
^ i (B
^ i )2 B
Q
@uh
h
h
i
^
^
(^
xh ) + h h
h
^ i )2 B
^i
^ i )2
@xi
(Q
(B
h
"
#
P
i
^i Q
^i
e
B
@uh
h6
=
k
h
h
i 2
^ h i ^ h (^
P
(^
x
)
+
p
)
h
h
^ i )2
@xi
p^i h6=k x
^ih (B
!#
"
P
i
^i Q
^i
B
@uh
h6=k eh
h
i
i
^
P
(^
x
)
p
^
h
h
h
^ i )2
@xi
^ih
(B
h6=k x
"
!#
P
i
^i Q
^i
B
(^
xih eih )
@uh
h6=k eh
h
^
P
P
(^
x
)
h
h
^ i )2
@xi
^ih
^ih
(B
h6=k x
h6=k x
"
!#
P
i
^i Q
^i
x
^ih
B
@uh
h6=k eh
h
i
^
P
(^
x
)
p
^
h
h
^ i )2
@xi
^ih
(B
h6=k x
= [
@uh
(^
xh ) + ^ h
@xi
@uh
(^
xh )
@xi
i
h
^ h p^i ]
^
^i
^i Q
B
h
^ i )2
(B
= 0
Therefore by su¢ ciency of those conditions it follows that (^b; q^) 2 N(E 0 ),
i (^
b; q^)
= p^i and
x(^b; q^) = x
^ as desired.
4
Concluding remarks
We have attempted here to articulate some results that relate Nash equilibria of strategic
market games to c-fair allocations. The …rst intention in this paper was to study whether
an imperfectly competitive setup allows for fairness. To this end, we proved that any
c-fair allocation can be supported as a Nash equilibrium of a market game with transfers.
It is important to mention that the choice of the right appropriate redistribution policies
is crucial for the proper functioning of markets and eventually for achieving e¢ ciency.
Indeed, notice that the price vector which is at the heart of the redistribution policy
must be compatible with the targeted optimal commodity allocation. Otherwise, the
resulting strategic market game does not have a Nash equilibrium. Finally, in order to
provide a complete examination of the relationship between the two concepts, we have also
shown that c-fair allocations of an economy are Nash equilibrium allocations for a nearby
13
economy, with the proximity between the two economies becoming …ner as the number
of individuals becomes larger. This result complements very well several existing ones in
the literature and allows us to make inferences about the asymptotic limits of one concept
from the asymptotic behavior of the other.
References
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