MONOTONICITY IN GAMES THAT
EXHAUST GAINS TO SCALE
Suzanne Scotchmer
and
Myrna Holtz Wooders
Working Papers in Economics E-89-23
The Hoover Institution
Stanford University
August 1989
This work was supported partially by the Social Sciences and Humanities Research Council of
Canada, National Science Foundation Grant 86 10021, National Science Foundation Grant SES
87-06967 at the Institute for Mathematical Studies in the Social Sciences at Stanford University.
Authors thank Wulf Albers, Robert Anderson, and Andreu Mas-Colell for helpful discussions.
The views expressed in this paper are solely those of the authors and do not necessarily reflect the
views of the staff, officers, or Board of Overseers of the Hoover Institution.
1
Hoover Institution
Working Paper No. E-89-23
August 1989
MONOTONICITY IN GAMES THAT
EXHAUST GAINS TO SCALE
Abstract
We compare core payoffs in two cooperative games in which coalitions contained in both player
sets have the same feasible payoffs in both games. The players are characterized by their “types,"
and players of the same type are substitutes for each other. If the (finite) set of players in each game
is large enough to exhaust gains to scale in a sense we define, the abindance of one type of player
and the core utility achieved by that type cannot change in the same direction. If the core utility is
interpreted as the price of the player, the interpretation is that price decreases with supply. We
present a geometric exposition and extend our results to games derived from exchange economics.
Suzanne Scotchmer
National Fellow
Hoover Institution
(415) 723-9175
and
University of California, Berkeley
Myrna Holtz Wooders
University of Toronto
2
1.
Introduction
A basic idea in economics is that a decrease in the supply of a commodity causes an increase in
its price.1 A related conjecture related to game theory is that a decrease in the number of players of
one type will cause an increase in their core payoff ("price"). In this paper, we give a condition
under which this conjecture is true for games with a finite number of players
We compare the payoffs in two games that differ only in the populations of players, but have the
same "types" of players. The characteristic functions for the games are determined by a "worth"
function that specifies the total payoff achievable by any coalition of players. Thus, the characteristic
functions for the games we compare have the same values for coalitions with the same numbers of
players of each type. We show that, in two games with the same worth function and different
numbers of players of type-t, no core utility vector for the game with more type-t players gives more
utility to type-t than any core utility vector for the game with fewer type-t players.
Few comparative static results have emerged for games in which core payoffs can only be
achieved in the grand coalition.2 Our result follows because we assume that all gains to scale can be
exhausted by coalitions strictly contained in the grand coalition of each game. This is a natural
assumption in games derived from club economies3 and coalition production economies, and may
also apply to games derived from finite private-goods exchange economies, as we discuss in Section
5.
1
We know of only one expression of this in a general equilibrium exchange context: When all goods are gross
substitutes, a decrease in the supply of a commodity causes its equilibrium price to rise.
2
Recently, Ichiishi (1987a,b) has presented some comparative results on core payoffs when the set of players is
fixed but the feasible payoff to each coalition changes in a systematic way. The present paper holds the feasible
payoffs to members of each coalition fixed, but changes the set of players.
3
The premise of club theory is that there are gains to scale because players can pool their private
goods to produce a shared facility and crowding costs eventually dominate these gains.
3
Some comparative results have emerged for assignment games, where there are two "sides" to the
market.4 Kelso and Crawford (1982) consider payoffs that favor one side of the market, and show
that if the number of players on the other side of the market increases, payoffs to players on the
favored side increase. For the case that core payoffs can be achieved only in coalitions consisting of
one player from each side of the market (as in marriage models), Mo (1986) and Roth and
Sotomayor (1987) present comparative static results that apply to all core payoffs in the two
assignment games compared. Roughly speaking, an increase in the number of players on one side of
the market decreases the payoff to some players on that side, and increases the payoff to some
players on the other side. Our comparison below also applies to any pair of core payoffs for the two
games under consideration. We comment further in the Conclusion on how the games we consider
differ from assignment games.
When a coalition can be described by a vector of real numbers ( i.e., when players are not
necessarily integral), rather than by only a vector of integers, it is particularly easy to give conditions
for which games exhaust gains to scale. We therefore present this case in Section 2.
Section 3 extends the result to the case that a coalition is represented by a vector of integers.
Whether or not coalitions must have integral players , the core may be empty. Section 4
discusses when the core is nonempty and shows that, when gains to scale are exhaustible, an
approximate core--more specifically, the strong epsilon core--is nonempty for large populations of
players. Therefore, we extend our comparative static result to approximate cores. Section 5
extends our result to finite exchange economies, and Section 6 discusses conjectures related to the
comparative statics we present here.
4
Since the two sides are often thought of as buyers and sellers, assignment games are frequently
referred to as markets. The study of markets as cooperative games was initiated by Shapley
(1955). In (1961), he showed that players on one side of the market are “substitutes” and ones on
the opposite sides, “complements”, but these are defined in terms of the characteristic function rather
than by the solution concept.
4
2 . Games with Exhaustible Gains to Scale and Divisible Players
We shall denote the population or player set by N = (N 1,..,Nt,..,NT ), where Nt > 0 is a mass of
players of type t. A coalition is a vector n = (n1..,nt,..nT ) in RT + . A nonnegative function V [ • ],
called the worth function, is defined on the domain R T+ . A game5 is a pair [N,V], where the worth
function V[ • ] i s restricted to coa1itions n<N. A partition of N is a finite collection of coalitions {nk
}, where nk = (n1 k ,...,nT k ), such that, ∑K nK = N. A payoff for the game [N,V] is a vector u in
k
k
k
R T+ such that, for some partition {n } , u•N = ∑kV[n ] . A partition (n } achieves a payoff u if
u•nk = V[nk ] for each k. A payoff u can be improved upon by a coalition n<N if u•n<V[n] . A
payoff u is in the core if, for all coalitions n < N, u•n >V[n]; i e. , u cannot be improved upon by any
coalition n<N. A core partition is a partition of N that achieves the core utilities u. A core coalition
is a member of a core partition.
We have defined only payoffs that treat players of the same type identically. This involves no
substantive loss, since core payoffs for populations that exhaust gains to scale will have this
property. 6
In order to define what it means for a game to exhaust gains to scale, we need some additional
notation. The composition of a coaIition n is a point s in the T-dimensional simplex, S, that
represents the relative numbers of players of each type in a coalition; i.e., s=n/n where n=∑tnt .
Figure 1 shows the simplex for two types of players, a and b, so that s=(sa sb). In Figure 1, all
coalitions with the same composition s, are represented by a ray from the origin. We define v(s) =
supr V[rs]/r, the highest per-capita payoff available to any coalition with composition s. The curve
v(s) is drawn in Figure 1 above the simplex. If there is a maximizer for V[rs]/r, it is denoted by r(s),
and is possibly not unique.
5
We observe that each game is a coalition-structure game with characteristic function V[ •]. Coalition-structure
games are not necessarily superadditive, and core payoffs are typically achieved by cooperation within
coalitions in a partition of the set of players, rather than in the coalition of the whole. See Auman and Dreze
(1974) and Proposition 1 below.
6
Cf., Scotchmer and Wooders (1987)
5
A bounding hyperplane to the worth function V is a linear function u•n, u > 0,7 for which inf n > 0
u•n-V[n] = 0.
The discussion that follows relies on the geometry of Figures 1 and 3.8 With the worth function V
fixed, we consider games [N,V] large enough so that, for each composition s, at least one "efficient"
coalition sr(s) is feasible in the population N. In Figure 1, such populations N lie to the "south" of the
line sr(s).9 Proposition 1 shows that for every such game, a core utility vector defines a bounding
hyperplane u•n that lies above V[n] on the entire domain R T+ (not only on the domain n<N) and
above v( s ) .Furthermore, one can see from the geometry that every bounding hyperplane defines a
core utility vector for some game [N,V].10 A core partition that achieves utilities u contains only
coalitions at which the bounding hyperplane touches the worth function, and therefore only coalitions
sr(s) can appear in a core partition of any game that exhausts gains to scale. Coalitions with
compositions in a "dip" of v(s) can appear in
no core partition, because the worth function touches no bounding hyperplane there. The set SC in
Figure 1 represents all compositions that could appear in a core partition of some game that
exhausts gains to scale.
We now define these ideas formally. In order to define SC it is useful first to define the smallest
concave function larger than v(s), denoted by vo (s) and represented by the dotted line in Figure 1.
In Section 5, where we comment on exchange economies, we show that if V[n] is superadditive,
then v(s) is concave and vo (s)=v(s) for all s.
1) vo (s) = minu > 0 {u•s  u•s' > v(s'), all s' in S}
(2) Sc = { s in S  v(s) = vo (s) }
We refer to this linear function as a bounding hyperplane because the hyperplane {(n,v) u •n- ν =0 } bounds
the sets {(n,ν) ν < V[n] , n > 0 } and {(s,ν) s in S, ν < v (s) }.
8
Other discussion of this geometry is in Scotchmer (1986)
9
We might perhaps say the “deep south” is implied by our definition below of when the game [N,V] exhausts
gains to scale.
10
For market games with a finite number of players, the competitive payoffs, and hence the core, have been
related to the gradient of V and, with a continuum of players, to the gradient of V[N]/N (c.f. Aumann and
Shapley (1974), Aubin (1981), and Shapley and Shubik (1975)). The gradient is the coefficient vector on our
bounding hyperplane. Our analysis relies on a crucial additional property: That when gains to scale are
exhausted, the bounding hyperplane lies above V on the domain RT+ , rather than only on a specific player set.
7
6
The definitions of bounding hyperplane, vo (s) and SC refer only to the worth function V and do
not depend on the population of players, which we shall allow to vary.
We say that V has exhaustible gains to scale if v(s) and vo (s) exist for each s in the sirnplex, and
if, for each bounding hyperplane u, there is an s and r(s) that achieves u, i.e., u•sr(s)=V[sr(s)]. That
v(s) exists means that per-capita payoffs for coalitions with composition s are bounded. That vo (s)
exists means that a bounding hyperplane u•s exists at every s (i.e., the bounding hyperplane has finite
coefficients u). That each bounding hyperplane is achievable at some composition s means that the
coefficients u of the bounding hyperplane are core utilities for some game. For each s that is a
vertex of the simplex, representing compositions with just one type, there is always a bounding
hyperplane that contains the vertex, and therefore, if V has exhaustible gains to scale, the set SC
contains vertices. We say the game [N,V] exhausts gains to scale if V has exhaustible gains to scale
and if, for each s in SC there is r(s) such that sr (s) < N.
If V is homogeneous of degree one and a coa1ition with each composition s is feasible in the
game, then the game automatically exhausts gains to scale. Thus, games derived from convex
exchange economies with an atomless measure space of players automatically exhaust gains to scale.
This is the case described by Aumann and Shapley (1974).
Our main interest is in finite games, which are discussed further in Sections 3 through 5. When nt
is not necessarily integral, it can be considered as a continuous approximation to a finite, integral
number of players of type-t.11 We do not provide an interpretation of V and v when N is interpreted
as an infinite player set.12 We include the continuous case mainly because it provides intuition for the
finite case. The definition of exhaustion and the comparative statics argument are more transparent
in the continuous case.
11
To formalize this approximation, one might consider a non-integral n as allowing part-time membership or
allowing players to participate in a coalition with less than full “intensity”.
12
If N is interpreted as an infinite player set of finite measure, and we view V as defined on proportions of N, then
exhaustion of gains to scale by finite coalitions would mean that the supremum of V[rs]/r is “achieved” at r=0.
But r=0 for any finite coalition, so the condition r=0 is not rich enough to distinguish the finite coalition that
maximizes per capita payoff from any other finite coalition.
7
Proposition 1: If a game [N,V] exhausts gains to scale, a payoff u and a partition {nk } of N are a
core payoff and a core partition if and only if (3) and (4) hold.13
(3) u•n > V[n]
for all n in RT +
(4) u•nk = V[nk ] for nk in {nk } a partition of the player set N.
Proof of Proposition 1:14 [If] Since u•n > V[n] for all n in RT + , no feasible coalition n can
produce more payoff for any of its members, and therefore no coalition can improve upon u. Since
(4) holds, there is a partition of N that achieves u. Thus, u is a core payoff.
[Only If] If {nk } is a core partition and u are core payoffs, then no feasible coalition can improve
upon u. This means that u•n > V[n] for all feasible n (n<N), and, since the game exhausts gains to
scale, u•r(s)s > V[r(s)s] = r(s)v(s), for some r(s) and all s in Sc and therefore u•s > v(s) for all s in Sc.
We only need to show that u•s > v(s) for all s in S. It then follows that for all n in R T+ u•n = u•rs
>rv(s) > V[rs] = V[n] , where s is the composition of n.
If u=0, then V[n]=0 for all n<N. But if V[n]>0 for an n that is not contained in N, then for some
s, v(s)>0, and the game [N,V] does not exhaust gains to scale. Hence, if u=0, V[n]=0 for all n.
For the case that some component ut of u is positive, we argue by contradiction.
Suppose there exists s' not in Sc for which u•s'<v(s'). Then there exists a scalar λ>1 for which infS
λu•s-v(s)=0, and λu•s-v(s)>0 for all s in S. Since λu defines a bounding hyperplane, and the game
exhausts gains to scale, there is a composition s for which λu•s=v(s), and therefore this s is in Sc.
But this is a contradiction, since λu•s (s)>0 for all s in Sc. Q.E.D.
Proposition 1 enables us to easily prove the next proposition.
13
When n is restricted be n < N, this is simply a restatement of the usual definition of the core, c.f. Auman and
Dreze (1974) for superadditive cover games and Albers (1974). That condition (3) holds for all vectors n in RT+ ,
rather than only for n < N, follows from our assumption that [N, V] exhausts gains to scale.
14
We use the fact that payoff can be transferred among individuals. Thus, any coalition that can increase payoff
of one member can redistribue to increase payoff of all members.
8
Proposition 2: Consider two games [N*,V] and [N' V] that exhaust gains to scale and have
nonempty cores, {u*) and {u' } , respectively. Then for any utility vectors u* and u' from these sets,
(u'-u*)•(N'-N*) < 0. That is, if the number of type-t players is smaller in one game (and the numbers
of players of other types are the same), then the per-capita payoff achieved by type-t in the core
cannot be smaller and may be larger.
Proof of Proposition 2: Let (n* k} and {n' k } be core partitions that achieve core payoffs u* and
u'. Then, since (3) holds for all nonnegative n, it follows from (3) and (4) that
u*•n*k =
V(n* k) < u'•n* k . Summing over k, we have (u*-u')•N* < 0. Reversing the roles of 'star' and
'prime', we also have (u'-u*)•N' < 0. Adding these together, (u'-u*)•(N'-N*) < 0. If N'=N* in every
component except Nt, then the number of type-t and the per-capita payoff to players of type-t
cannot move in the same direction. Q.E.D.
Proposition 2 is illustrated by Figure 2, which shows payoffs available to type-t in the core payoff
vectors {u*} and {u'). If these sets overlapped other than at a point, there would be payoff vectors
from the collections {u*) and {u'} for which (u*-u')•(N*-N') < 0 would be violated.
Intuitively, the notion that gains to scale are exhausted means that increases in population size
cannot make everybody better off. We state this as a Proposition:
Proposition 3: Provided the games [N*,V] and [N' V] exhaust gains to scale, it cannot be that u*
> u' or u' > u* with strict inequality in some component, where u* is a core utility vector for [N*,V]
and u' is a core utility vector for [N',V].
Proof of Proposition 3: This follows directly from our observations in the previous proof that (u*u')•N* < 0 and (u'-u*)•N' < 0. Q.E.D.
Figure 3 illustrates that when the abundance of type-a increases, the per-capita payoff achieved
by players of type-a cannot increase. Each tangent line above v(s) is the cross-section of a
9
bounding hyperplane u•n and its slope is ua-ub.15 When s'a < s*a , the slope u* a - u* b is less than the
slope u'a - u' b . Since u* cannot be less than u' in all components, u* a < u 'a Type-a's per-capita
payoff cannot increase when the abundance of type-a increases.
The inverse relationship between abundance and payoff achieved in the core, as states in
Proposition 2, can only be guaranteed if gains to scale are exhausted. To see this, consider a
homogeneous population for which V[r]/r increases with r, the population size. Then the equaltreatment core payoff is V[r]/r, and increasing the abundance of players will increase their equaltreatment payoff. This would occur, for example, if the gains to scale arise from sharing a public
good, and those gains to scale are never offset by crowding costs. This example also illustrates that
when gains to scale are not exhausted, a change in the population may yield higher core payoffs for
all players.
The famous glove example of Shapley (1955) and Shapley and Shubik (1969) illustrates
Proposition 2. Suppose there are two types of players, called left-hand-gloves and right-handgloves, and a coalition of one each has worth one. All other coalitions have worth zero. If there are
fewer right-hand-gloves than left-hand-gloves, the core gives payoff one to each right-hand-glove,
but when more right-hand-gloves are added, their core payoff stays the same until there are equal
numbers of right- and lefthand-gloves, and then it falls to zero.
3 . Games with Exhaustible Gains to Scale and integral Players
With divisible players, any coalition rs in R T+ is possible. When coalitions n must be vectors of
integers, rs must be integral. There is a dense set of points in the simplex for which rs is integral for
some r; namely the rational points. In this section, we retain all the previous structure, except that
we restrict attention to integral n (or rs) and rational s.
15
The tangent line is the function f(s a ,s b) - f(s a ,1-s a ) = u a s a + u b (1-s a ), so its slope, as s a increases, is u a -u b.
10
With integral players, a subtlety arises in identifying conditions on the worth function V for which a
game [N,V] with finite population exhausts gains to scale.
For any population N, there exist
rational points s in the simplex for which the only integral vectors rs are greater than N.
For example, as the percentage of type-t becomes small, the smallest coalition that has one integral
type-t player becomes large. If the set SC contained all the rational points in the simplex, no finite
game [N,V] would exhaust gains to scale. We can accommodate this problem by modifying the
definition of exhaustion, as in Section 5, or by finding restrictions on V which ensure that SC is a
finite subset of the simplex. In this section we do the latter.
Consider V for which there is an upper bound, k, on the per capita utility achievable in "large"
coalitions of any composition. That is, there exists r such that, for each rational s, if r > r, V[rs]/r <
k. Assume also that, for each type of player, an appropriately large, but finite, coalition containing
only that type of player can achieve at least as much per capita payoff as k. (The composition of a
coalition with a single type of player is a vertex of the simplex.) It follows from these two restrictions
(Lemma 1) that SC is a finite subset of the rational points in S, that vo (s)>k is piecewise linear, and
that a large enough finite game of any rational composition exhausts gains to scale.
The assumptions on v do not mean, of course, that core payoffs will be achieved by coalitions
containing only single types of players. Coalitions containing many types of players may do strictly
better than the minimum per capita payoff, k, guaranteed by coalitions containing only single types of
players.
An example of a game that satisfies the restrictions on V would be a game derived from a club
economy in which, for any composition of coalition, there are gains in per capita payoff as the
coalition is enlarged, because members can pool private resources to produce public goods, but
eventually the gains are dominated by crowding costs, so that per capita payoff falls. Another
example would be a coalition production economy in which different types of players have
complementary skills, and for each composition there is an optimal size firm.
11
Lemma 1: Suppose there exists k>0 and r>0 such that (i) if s is a vertex in the simplex, v(s)>k>O,
and (ii) if r>r, V[rs]/r < k for each rational s. Then (a) V has exhaustible gains to scale and (b) there
exists a finite population vector η for which all games [N,V] with N>η exhaust gains to scale.
Proof of Lemma 1: We will argue that only coalitions with compositions contained in a fixed, finite
set of compositions can achieve the utilities that define bounding hyperplanes. Then we use a linear
programming argument.
By (i), min u>0 { u•s  u•s' > v( s' ) for all s' in S } > k > 0, and thus vo(s) > k > 0 for all s.
Define S* to be the finite collection S* = {s in S  rs is integral for some r, 0 < r < r }. By (ii), if s is
not in S*, v(s) < k < vo (s), and therefore SC is a subset of S*.
Since (1) becomes a linear program, with the constraints u•s > (s) restricted to the finite set S* and
v(s) is bounded below by zero, vo (s) exists at each s, with finite coefficients u. Furthermore, every
bounding hyperplane u has the property that u•s=v(s) for some s these two restrictions (Lemma 1)
that SC is a finite subset of the rational points in S, that vo (s)>k is piecewise linear, and that a large
enough finite game of any rational composition exhausts gains to scale.
The assumptions on v do not mean, of course, that core payoffs will be achieved by coalitions
containing only single types of players. Coalitions containing many types of players may do strictly
better than the minimum per capita payoff, k, guaranteed by coalitions containing only single types of
players.
An example of a game that satisfies the restrictions on V would be a game derived from a club
economy in which, for any composition of coalition, there are gains in per capita payoff as the
coalition is enlarged, because members can pool private resources to produce public goods, but
eventually the gains are dominated by crowding costs, so that per capita payoff falls. Another
example would be a coalition production economy in which different types of players have
complementary skills, and for each composition there is an optimal size firm.
12
Lemma 1: Suppose there exists k>0 and r>0 such that (i) if s is a vertex in the simplex, v(s)>k>O,
and (ii) if r>r, V[rs]/r < k for each rational s. Then (a) V has exhaustible gains to scale and (b) there
exists a finite population vector η for which all games [N,V] with N>η exhaust gains to scale.
Proof of Lemma 1: We will argue that only coalitions with compositions contained in a fixed, finite
set of compositions can achieve the utilities that define bounding hyperplanes. Then we use a linear
programming argument.
By (i), min u>0 { u•s  u•s' > v( s' ) for all s' in S } > k > 0, and thus vo(s) > k > 0 for all s. Define
S* to be the finite collection S* = {s in S  rs is integral for some r, 0 < r < r }. By (ii), if s is not in
S*, v(s) < k < vo (s), and therefore SC is a subset of S*.
Since (1) becomes a linear program, with the constraints u•s > (s) restricted to the finite set S*
and v(s) is bounded below by zero, vo (s) exists at each s, with finite coefficients u. Furthermore,
every bounding hyperplane u has the property that u•s=v(s) for some s in this finite collection, since
otherwise u•s>v(s) for all s in the finite collection, and the coefficients u could be reduced, thus
reducing u•s for all s, and thus reducing vo (s). This concludes part (a).
But then, since SC is contained in S*, part (b) follows immediately. Any game [N,V], with
N>[r,...,r] exhausts gains to scale. Q.E.D.
Proposition 2 applies to games with integra1 p1ayers , and this Lemma gives a transparent
condition under which we can compare the payoffs in two games with integral players and finite
populations N* and N'.
4. Approximate Cores and Integral Players
A condition under which the core of a game [N,V] is nonempty for every N is that players are
continuously divisible and the characteristic function V[n] is homogeneous of degree one; that is,
every r maximizes V[rs]/r. This has been shown for exchange economies with transferable utility (c.
f. , Aumann and Shapley (1974) and also Aubin (1981)) and for other economic models (c.f.,
13
Shaked (1982) for a continuum private-goods exchange economy where there are externalities
among players that depend only on the composition of the group).
The core may be empty for two reasons, relating to (i) integral coalitions n and (ii)
nonhomogeneity of V. Figure 3 shows why the core may be empty when coalitions are integral. If
the composition of the population is so, core coalitions must have compositions s' and s".
Nonemptiness of the core implies that there are real numbers k' and k" for which k' s' +
k" s" = No where No is the population, and the elements of k's' and k"s" are integral. This may
not be possible. This problem will not arise if V is concave as well as homogeneous of degree one.
On the other hand, the core may be empty even with continuously divisible players, when V is
not homogeneous of degree one. This is easiest to see with a homogeneous population (i.e., in a
symmetric game). If ro uniquely maximizes V[r]/r, and if the population is larger than r o , the core is
empty unless the population size is an integer multiple of r o . Otherwise, some players will be in a
coalition that does not achieve the core payoff V[r o ] /r o and can bribe other players to join them in
forming a new coalition of size r o.
When either of these two problems is present, so that the core is empty, it may be appropriate to
consider approximate cores or epsilon cores. Since approximate cores may be nonempty when the
exact core is empty, we extend our comparative static result to approximate cores. The remainder
of this section restricts attention to integral players, and the results also apply to the easier case of
divisible players.
Two concepts of approximate cores, the weak ε-core16 and the strong ε-core, were introduced
by Shapley and Shubik (1966), who gave conditions under which each of these approximate cores
16
A payoff is in the weak ε-core if it cannot be improved upon by c for each member of any
coalition. Nonemptiness of weak ε-cores of large (TU and NTU) games was shown by
Wooders (1983). (See also Kaneko and Wooders (1982), Shubik and Wooders (1983) and
Wooders and Zame (1984).)
14
is nonempty, for private-goods economies with transferable utility (and nonconvexities). Again using
the condition that "large" coalitions cannot achieve maximum per-capita payoffs, we show here that
in large (but finite) games that exhaust gains to scale, the strong ε-core is nonempty.
The strong ε-core can be defined as follows. Given ε > 0, a payoff u* in R T is in the strong- εcore if, for all coalitions n < N, u* •n + ε > V[n]. (Recall that, since u* is a payoff, there is a partition
{n k } that achieves u*; i.e., u* •nk = V[nk ] for each k.) We prove the strong ε-core is nonempty,
using the same condition as in Lemma 1, that for each composition, large coalitions achieve strictly
less per-capita payoff than the maximum, v(s).17 We need the following lemma.
Lemma 2: Suppose there exists k > 0 and r > 0 such that (i) if s is a vertex of the simplex, v(s) > k
> 0, and (ii) if r > r, V[rs]/r < k, for each rational s. Then for any (rational) composition s, there
exists ro such that each game [l ro s,V], for any positive integer l, has the same nonempty core.
Proof of Lemma 2: For each s in the simplex, we can describe vo(s) as the solution t∅ to (1), where
s' is restricted to S*, defined in the proof of Lemma 1, that is, to a finite number of points.18 Hence
we have a finite number of constraints, and a linear programming problem. Let u be any solution to
this problem. Letting i index the vectors in S*, the dual to this linear programming problem19 is
maximize λ > 0 ∑ i λi v (si ) subject to the constraints s > ∑ i λi si . Let λ* denote a solution.
We
The ε in the strong ε -core can be interpreted as a cost of forming a coalition, since ε is
attached to a coalition and not to a player. For superadditive cover games, a cost of ε per player,
as in the weak ε -core, might better describe the cost of forming coalitions.
Each "large"
coalition in the superadditive cover game would achieve its core payoff by subdividing into smaller
coalitions, the number of which is approximately proportional to the size of the coalition. Since the
assumptions (i) and (ii) in Lemma 2 and Proposition 4 exclude super-additive games, the strong εcore is the approprite notion to capture the cost of coalition formation.
17
18
Finiteness of S* and SC is crucial to the linear programming argument and also to Lemma 2. If SC contained a
continuum of points, it might not be true that for each bounding hyperplane, there is a sequence of games for
which the coefficients u are core utilities. Suppose, for example, there are two types, a and b, and SC = {s s a <.5,
s a = 1}, v(s) = s a for s a < . 5, v(s) = 0 for s a < . (5). Then, for s such that .5 <s<1, it is not true that v o(s) = εi λi v(s i)
for a set {s i} in SC for which εi λi s i = s. There is no sequence of games for which the bounding hyperplane
defined by u=(.5, .5) describes core utilities. Continuity of V[n] and v(s) would eliminate examples such as this.
19
The characterization of the core based on the dual linear program is due to Bonzareva (1962, 1963) and,
independently, to Shapley (1967).
15
can assume that λ* i >0 for at most T components i and that λ* i is rational.20 Using the facts that
the optimal values of the objective functions coincide, that
λ*i >0 implies that u•si = v ( si ) , and that si is in S* and therefore in the simplex, we conclude that
vo(s) = ∑ i λ*i v (si ) = ∑ i λ*i u•si , and ∑ i λ*i = 121
The dual program picks out a set of points {si} in Sc, of which s is a convex combination with
weights {λ* i }, which have the property that coalitions with the chosen compositions in SC achieve
utilities u. (In Figure 3, the dual program picks out the coalitions s' and s" in Sc, and
vo(so) = λ' v (s') + λ" v (s").)
Let ro be an integer with the property that ro λi si is integral for each i. Then ro s is also integral.
The payoff u is in the core of [l ro s,V], for each integer l, since there is a partition of the player set
l ro s containing l coalitions ro λi si for each i , each of which achieves the payoff u. No coalition
could achieve payoff greater than u. Therefore, the core of [l ro s,V], contains u.
Now let u be in the core of [l ro s,V].Then u is a solution to the linear programming problem
above: u minimizes user u' • l ro s for any positive integer l. From the above argument, u is in the
core of [l ro s, V] for any positive integer l, since it is feasible and cannot be improved upon by any
coalition, by Proposition 1. Hence the cores of [l ro s,V] for all integers l , coincide. Q.E.D.
Proposition 4: Suppose there exists k>0 and r >0 such that (i) if s is a vertex of the simplex,
v(s)>k>0, and (ii) if r > r, V[rs]/r < k for each rational s. For each rational s and for every ε > 0
there is an r (ε) such that, if r > r (ε) and rs is integral, then (a) the game [rs,V] has a nonempty
The equation ∑ i λi s i = s can be written xλ=s, where a column of x is s i , the column vector λ is (λ1 ,... λσ), and σ is
the number of vector in S*. There is a solution to the linear program with no more that T positive λ* i ‘ s.
Eliminating the columns of X that correspond to λ* i = 0 and eliminating redundant rows if necessary) an optimal
λ* can be written x-1 s. Since all the components of X, s, and {s i} are rational (since feasible coalitions are
integral), all the components of λ* are rational.
i
i
21
This is because the {s i} and s are in the simplex. Therefore, ∑ i λi s t = st , and thus ∑ i λi [∑t s t ] = ∑ i
20
λi = ∑t st = 1.
16
strong ε-core, and (b) u*•n > V[n] -ε for all n (not just for n < rs), for any payoff u* in the strong
ε-core of [rs,V].
Proof of Proposition 4: Given a rational s, let uεRT + be in the cores of the sequence of games
[l ro s,V ] defined in Lemma 2.
We will refer to the size of the player set as r, rather than l ro
Choose any ε > 0. Choose an integer r1 large enough so that (i) u is in the core of [ r1 s,V], (ii) r1
> r, (iii) ε/ r1 < b, where b - v0 (s) - sup r>r V[rs]/r. The latter quantity is positive, since V[rs]/r < k
< v0 (s) for r > r.
Choose r (ε ) so that (iv) r (ε ) is an integer multiple of r1 and (v) r1 u•s/r (ε )
< ε / r1 . (And r (ε ) > r1 .)
To show that the strong ε-core is nonempty, we must show that there is a u* that is both
achievable and cannot be ε-improved by any feasible coalition in the game rs, for any r >r (ε). First
we will show that we can choose u* so that ut > u*t > ut - ε/r1 , and so that u* is achievable by
some partition of the player set rs. Since any group r1 s can achieve core utilities u, we can partition
the player set rs so that there are less than r1 =  r1 ,s players in a group, with composition s, which
cannot achieve u. Therefore, there is a feasible payoff u* for which ru*•s > (r-r1 )u•s and U* > ((rr1) /r) u. (Those players who are in a group which cannot achieve u can achieve at least zero.) For
such a u*, u* > u-(r1 /r)u > u - (r1 / r(ε )) u and u*•s > u - ε/r1 .
We now show that no coalition contained in rs can ε-improve on payoffs u* with the property that,
for each t, ut > u* t > ut - ε/r1 , i.e., V[rs' ] < u*•rs' + ε for all coalitions rs', where s' is a
composition in the simplex. For r < r1 and for any s' in the simplex, V[rs' ] < r u•s' < r u*• s'
+ r ε/r1 < r u*•s' + ε. For r > r1 , V[rs']/r < vo (s') - b < u• s' - ε / r1 < u*• s' < u*• s' + ε/r, or
V[rs' ] < r u*•s' + ε
This completes the proof of Part (a) and also proves Part (b), since the
argument applies to any coalition with r > r1 . Q.E.D.
The extension of Proposition 2 to approximate cores is then
17
Proposition 5: Given ε > 0, let [N*,V] and [N' V] be two games that exhaust gains to scale, for
which the strong ε-cores are nonempty. Then for any u* and u' in the strong ε-cores of these
games respectively, (u'-u*) • (N'-N*) < 2ε.22
Proof of Proposition 5: Let {n k *} and {nj '} be core partitions that achieve ε-core utilities u*
and u' for the games [N*, V] and [N' V] , respectively. Then u* •nj ' > V[nj ' ] - ε = u' • n j ' - ε.
which implies that (u*- n ') • > N' > - ε. Reversing the roles of "prime" and "star",
(u'-u*) • N* >-ε. Adding these together, (u'-u*) • (N'-N*) < 2ε. Q.E.D.
The interpretation of Proposition 5 is that, while sets of approximate core payoffs may overlap for
two games that differ only in the number of players of one type, the area of overlap is "small" if either
the player sets are "large" or if the difference N't - N*t is large.
5. Games Derived from Exchange Economies
Exchange economies fit into the framework discussed in this paper, with a slight modification of
our definition of exhaustion. Suppose there are private goods (x,y) where x is a scalar and y is a
vector. We suppose utility is transferable in the good x, and Ut (xt y t) = x t + h t (y t ) is the
utility function of an agent of type t. Then the total transferable utility available to a coalition n is
V[n] = sup{x t y t } ∑t n t Ut (x t y t ), where the total consumption of x and y, ∑t n t x t and ∑t n t y t
, does not exceed the endowment of coalition n.
Games derived from exchange economies are superadditive. By the following Proposition, with
superadditivity every rational composition s is in SC. For every game [N,V], there are rational
compositions s for which the minimum integral coalition rs is so large that it is not contained in N,
and therefore no finite game [N,V] with superadditive V exhausts gains to scale. We shall therefore
modify the definition slightly to accommodate this fact. First we establish an additional Proposition.
An analogous proposition could be demonstrated for the weak ε-core with fewer assumptions, but the bound
would depend on the sizes of the player sets. We showed nonemptimess of the strong ε-core in Proposition 4 in
order to motivate this stronger comparative static result.
22
18
Proposition 6: If V is superadditive, and if v(s) exists at each rational s, then v(•) (whose domain
is the rational points in the simplex) is concave and SC includes all of the rational points in the
simplex.23
Proof of Proposition 6: If a rational composition s is a convex combination of rational {s i } in S,
there are positive rational {λi } that sum to one for which ∑ i λ i si = s. Choose any ε > 0. For
each si, there exists rational r i for which r i si is integral and v(si ) > V[r i si} / r i > v(si ) - ε/kλ i ,
where k is the numbers of vectors si . Choose r so that λ i r/ri is integral for all i. Then rs = ∑ i (r λ i
/ r i ) (r i si ). By superadditivity, V [rs] > ∑ i (r λ i / r i ) [r i si ]. After dividing by r, this implies
v(s) > V[rs]/r >∑ i λ i v( si ) - ε. Since this is true for all ε > 0, and for every rational s
that is a convex combination of an arbitrary collection of rational {si }, v(s) is concave.
Since v is concave, v(s)= vo (s) for all rational s, and therefore SC contains all the rational points in
the simplex. Q.E.D.
We say that V is homogeneous if V[rs]/r = v(s) for all r such that rs is integral. If V is
homogeneous and superadditive, vo (s) =v(s) for all s, and v(s) is concave. The core of every game
[N,V] with homogeneous and superadditive V will be nonempty, and a core utility vector will satisfy
u•rs = V[rs] = rv(s). Homogeneity means that, as we move out along a ray with fixed composition
in Figures 1 and 3, all the points V[rs] for which rs is integral lie on a linear function. There are no
per capita gains or losses in doubling the number of players if we keep the composition fixed. An
exchange economy with T types of traders, each with quasiconcave preferences and transferable
utility, generates a worth function V that is homogeneous as well as superadditive.
A finite game with superadditive and homogeneous V might not exhaust gains to scale, and
therefore our comparative static result does not always apply. The reason for this is illuminated by
Figure 4.
23
We could show concavity of v defined on the entire simplex if we extended v by defining v(s) = lim n→∞ v (s n),
for any sequence of rationals {s n} converging to an irrational point in the simplex s. (This limit exists and is
unique)
19
In Figure 4, if there is only one player of each of types a and b, represented by composition
s=(1/2,1/2), then there are only three feasible points in the simplex, {(0,1),(1/2,1/2),(1,0)}. Core
utility vectors for this game are represented by bounding hyperplanes balanced on the peak of an
inverted V, with vertex at s=(1/2,1/2). Those hyperplanes sweep through the "cone" designated
"core utilities for the game [(1,1),V]," and correspond to core utilities in the "1ense" of the
Edgeworth box. Now increase the number of type-a players from one to two, and consider the
resulting game [(2,1),V]. We have added the feasible composition (2/3,1/3), and the game has core
utilities designated by a different "cone". The important thing to notice is that an arbitrary utility
vector (bounding hyperplane) for the first game may have greater or lesser slope than an arbitrary
utility vector of the second game, that has more type-a players. (The sets of core utility vectors
overlap.) For two arbitrarily chosen utility vectors from the cores of these games, we cannot say
whether type-a's utility is greater or smaller in one than in the other. Thus, Propositions 2 and 5 do
not apply to these two games.
Nevertheless, our comparative static result extends in a slightly modified form to games with
homogeneous and superadditive V. The modification is that there is a lower bound on the difference
in numbers of type-a players required in the two games being compared. The larger the games, the
smaller is the required difference in numbers of type-a players.
To see why sizes of the games matter, refer back to Figure 4. We will argue that the utility
achieved by type-a in any utility vector in the core of [(100,100),V] cannot be smaller than the utility
achieved by type-a in any utility vector in the core of [(200,100),V]. This comparison cannot be
made for the games [(1,1),V] and [(2,1),V], which have the same compositions. There are more
blocking opportunities in the larger games, so that the cores are smaller. In particular, a blocking
coalition with composition between (1/2,1/2) and (2/3,1/3) (e.g., (3/5,2/5)) is feasible in both of the
larger games, but not in the smaller games. The possibility
of blocking by a coalition with composition (3/5, 2/5) in the game [(100,100),V] means that the
core of [(100,100),V] does not include the entire cone labeled "core utilities for [(1,1),V]." The
20
feasibility of blocking by a coalition with composition (3/5,2/5) in each game prevents the "cones"
that represent the cores from overlapping.
Figure 5 shows bounding hyperplanes for limit economies with proportions (1/2,1/2) and
(2/3,1/3), respectively. As in Figure 3, the equal-treatment core utilities for types a and b can be
compared. One would suspect that the comparison could be made for sufficiently large games as
well . We now introduce a notion of approximate exhaustion to show that this is so.
We will let Sε (s) be an ε-ball in the simplex centered at s:
Sε (s) = {s' ε S   s' - s < ε }
where the double bars denote a metric. We say that the game [rs,V] ε-exhausts gains to scale if V
has exhaustible gains to scale and if there exists w < r for which ws is integral and
[ws1 /[ ws + (r-w) st ],... rs t /[ ws + (r-w) st ] , ... ws T /[  ws + (r-w)s t ]] is contained in
S ε (s), t = 1,...T. Thus, if we move from the center of the ε - ball in the direction of an "edge" of
the simplex, so that we keep the relative numbers of all types except type-t fixed, we will find a
composition within the ε -ball that is feasible in the game [rs,V]. If ε is small, the game [rs,V] must
be large in order to ensure this.
For our comparative static result to hold, there must be enough type-a players in the larger game
so that the composition of the larger game lies outside the ε-ball. In Figure 4, the composition s* =
(1/2,1/2) is at the center of the ε-ball, the composition of the new game with more type-a players is
s'=(2/3,1/3), and a feasible composition in an ε -ball would be (3/5, 2/5).
Proposition 7: Consider two games [N*,V] and [N',V], with compositions s* and s'
respectively. If (i) V is homogeneous and superadditive and [N*,V] ε-exhausts gains to scale, (ii)
u* is in the core of [N*,V] and u' is in the core of [N',V], (iii) N*t = N' t for t other than a and N*a
< N' a , and (iv) s' is not in Sε (s*), then u* a > u'a.
21
Proof: First, if s = λs*+ (1-λ)s' , o<λ<1, and v (s*)-u•s* > v(s) - u•s then, from concavity of
v, v (s*) -u•s* > v(s') - u•s'.
Since [N*,V] ε-exhausts gains to scale, and since we have chosen N' a large enough so that s'
is outside the ε-ball centered at s*, there is a convex, combination of s* and s' , which we shall call s
, such that a coalition with composition s is feasible in the game [N*,V] and s is in the ε-ball.
Therefore u*•s* = v (s*) and v (s) - u* • s < 0. It then follows from Lemma 3 that v(s') - u*•s' < 0.
Since s* is feasible in the larger game [N', V] it is also true that v (s*) - u'• s* < 0.
Hence, u* •N'
= u*•r' s' > v (s' ) r' = u' •r' s' - u' •N' and u' •N* = u' •r*s* > v v (s*)r* = u* •r* s* = u* •N*. By
adding the inequality u*•N' - u' •N' > 0 to the inequality u'•N* - u* •N* > 0, we see that (u*u')•(N*-N') = (u* a - u' a )• (N*a - N'a ) < 0. Q. E. D.
Games derived from exchange economies with nonconvex preferences will not necessarily have
homogeneous V, and, in addition, the cores of such games may be empty. Then Proposition 7 does
not apply. (But Proposition 6 does apply.) When the strong ε-core is nonempty, as it may be for a
large enough game derived from an exchange economy with nonconvex preferences (c.f. Shapley
and Shubik (1966)), we could derive an approximate version of our comparative static result, similar
to that given in Proposition 5.
6 . Conjectures and Conclusions
It is, of course, unrealistic to assume that the population duplicates individuals (has types) even if the
number of types is large. We have studied a game with types of players because it makes the
comparative statics straightforward. In the more realistic case that all players differ, one could
imagine that, although individuals themselves are not duplicated, their attributes, such as skills or
education, occur repeatedly in the population. One could then decompose core payoffs as payoffs
to attributes of players, as in the vast hedonic-price literature. We conjecture that in such a model,
provided gains to scale are exhausted in an appropriate sense, changing the population so that one
attribute is more heavily represented will cause the payoff to that attribute to fall.
22
The results of Mo (1986) and Roth and Sotomayor (1987) are in the spirit of our conjecture
above. In the assignment model, we can think of players on one side of the market as having one
common attribute (for example, sex), although they may differ in all other respects, and thus are not
necessarily perfect substitutes for each other. In a sense, gains to scale are exhausted (although not
exactly in the sense defined in this paper) , since only one-person and two-person coalitions, with
one person from each side of the market, are valuable. Introducing another player on one side of
the market causes the payoff to that player's attribute to fall.
Our model differs from assignment models in the sense in which gains to scale are exhausted.
Our comparative static results, Propositions 2 and 5, would extend to the assignment game if each
player had an identical twin and we increased a set of twins to a set of triplets, for example.
A second, related conjecture is that when gains to scale are exhausted in an appropriate sense,
adding a player to the player set causes a decrease in the core payoffs to sufficiently similar
players.24
24
To formalize the notion of similar players, we have in mind a metric space of attributes, a point of which
describes a player. Two players are similar if their distance apart in this metric space is small. The framework
used by Wooders and Zame (1984) to show nonemptiness of the weak ε-core is suitable for this, since each
player’s “type” is a point in a compact metric space called the space of attributes.
23
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