Approximating Games and Economies by
Markets ∗
Myrna Holtz WOODERS†
Department of Economics, University of Toronto,
Toronto, Canada, M5S 1A1
e-mail mwooders@epas.utoronto.ca
iefa6@cc.uab.es
April 1994, June 1999
Abstract
It is shown that an arbitrary game with effective small groups is approximately a market game. Small groups are effective if all or almost all
gains to collective activities can be achieved by groups bounded in absolute size. Market games are games derived from markets where all agents
have continuous, concave payoff functions. Bounds on the closeness of the
approximation are stated in terms of the extent of substitution — the closeness of substitutes for each player - and other parameters characterizing
the games. In this respect our results resemble those of Starr (1969) and
Anderson (1978), for example, on approximate equilibrium and cores in
exchange economies. In contrast with the market-game equivalence results
of Shubik and Wooders (1982) and Wooders (1993,1994) no topology on
the space of player types is required — the results are for arbitrary games.
∗
Department of Economics, University of Toronto Working Paper No. 9415 Revised. This
paper is a revision of University of Toronto Department of Economics Working Paper Number
9305, March 1993.
†
The author is indebted to Vincent Crawford, Aleksander Kovalenkov, Lakshmi Raut, and
Jian Kang Zhang for helpful comments. Support from the Social Sciences and Humanities
Research Council of Canada is gratefully acknowledged.
We also show that independent of the number of types of commodities in
an economy, if small group effectiveness holds then the economy can be approximated by one where the number of types of commodities is no larger
than the number of approximate types of players. The following applications are demonstrated: (a) approximate monotonicity of equal-treatment
-core payoffs of large games; (b) cyclic monotonicity of core payoffs of
games satisfying a strict small group effectiveness condition; (c) nonemptiness of the set of approximate core payoffs treating similar players equally;
and (d) asymptotic equal treatment of players who are substitutes for each
other.1
Contents
1 Introduction . . . . . . . . . . . . . . . .
2 Games and markets . . . . . . . . . . . .
2.1 Games . . . . . . . . . . . . . . . .
2.1.1 Balanced cover games . . .
2.2 Markets . . . . . . . . . . . . . . .
2.3 Games induced by markets . . . . .
3 Approximating games by markets . . . .
3.1 Some remarks . . . . . . . . . . .
4 Equal treatment and monotonicity of the
4.1 Equal treatment . . . . . . . . . . .
4.2 Monotonicity . . . . . . . . . . . .
4.3 Asymptotic equal treatment . . . .
5 Proofs . . . . . . . . . . . . . . . . . . .
6 Conclusions . . . . . . . . . . . . . . . .
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approximate cores
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1. Introduction
The notion of a competitive economy presupposes many participants and many
substitutes for all commodities and services. It also assumes that there are no
significant gains in per capita benefits from increasing the size of the economy
— returns to scale of activities must be nearly exhausted. For such economies,
Adam Smith (1776) argued that the self-interested actions of individual economic
actors would lead to a desirable state of affairs, as if an “invisible hand” had
2
dictated and coordinated the actions of the many self-interested participants so
as to achieve an optimal outcome for society. In theories of competitive economies
it is also assumed that the number of participants is sufficiently large so that no
individual participant (or no small number of participants) can influence broad
economic outcomes. Thus, an individual participant might reasonably assume
that aggregate outcomes were independent of his own actions.
The notion of perfect competition has been well studied in the context of
models of exchange economies with an exogenously determined set of commodities.
But the question of which economic structures most encourage the emergence
of a set of commodities for which there is a competitive outcome is relatively
unexplored. The major response to this question is the “Hypothesis of Universality
of Markets” of Arrow. The Universality Hypothesis states that for some definition
of a sufficiently large set of commodities a price-taking equilibrium will be Paretooptimal. The Universality Hypothesis, however, does not distinguish between
various economic structures, for example, economies with pure public goods and
economies with local public goods.
Another approach towards the characterization of competitive economic structures was initiated by Shapley and Shubik (1969). Shapley and Shubik (1969)
proved the equivalence between totally balanced games and market games. A
totally balanced game is a game with the property that every subgame has a
nonempty core. A market game is a game derived from an economy where all
participants have continuous, concave utility functions. Their approach suggests
that in terms of existence of a price-taking equilibrium for some set of commodities, for any economy whose derived game is totally balanced, there is a set of
commodities such that, relative to those commodities, a price-taking equilibrium
exists. There are two difficulties with this approach. The first is that economies
may well not generate totally balanced games. The second is that their framework and results do not give any indication of whether or not individual economic
participants become negligible as the economy grows large. For example, Shapley
and Shubik (1969) do not address the questions of convergence or monotonicity
of cores.
In this paper we consider multi-participant economic structures satisfying the
condition of small group effectiveness. The economic structures are represented
as games in coalitional form. A game has effective small groups if all or almost
all gains to group formation, either for improvement upon outcomes or for the
3
realization of feasible outcomes, can be achieved by relatively small groups.2 In
contrast to previous research (for example, Shubik and Wooders 1982), in this paper the property of small group effectiveness is imposed directly on games; there is
no topological structure on the player set. We show that any game with effective
small groups is approximately represented by a market: there is at least one set
of commodities such that, relative to these commodities, an approximate competitive equilibrium exists. Moreover, the larger the number of close substitutes for
each player and the closer these players are to exact substitutes, the better the
approximation.
More precisely, we obtain a bound on the distance of a game from a market
game. The markets generating the market games can be chosen to be socially
homogeneous and to satisfy payoff-constant returns — all participants have the
same 1-homogeneous payoff function. The bound on the distance depends on the
parameters describing the closeness of substitutes and the size of groups required
to achieve almost all gains to group formation. Related results using a framework
with a topological structure on the total player set are contained in Wooders
(1993,1994). Since we give explicit bounds on the closeness of the approximation,
the results of this paper resemble those of Anderson (1978) and Manelli (1991),
for example, on convergence of cores of exchange economies, as opposed to those
of Hildenbrand (1974).
Recall that Arrow (1970) notes that when the set of commodities is expanded
so that all factors affecting costs and benefits are priced, markets may be too
“thin” for the price-taking hypothesis of perfect competition to hold. As a byproduct of our approach, however, we show that the number of commodities
required for a market approximating a large game is no more than the number of
approximate types of players in the game. Thus, no matter how many commodities are in the description of the economy there is an approximating market with
no more commodities than there are approximate player types.
We provide several applications of our framework and results. First, we recall
that for games with strictly effective small groups all payoffs in the core have the
equal-treatment property (Wooders (1983, Theorem 3)). We show that for an
arbitrary game where every player has sufficiently many close substitutes there
are approximate core payoffs that treat players of the same “approximate type”
2
Wooders (1992b, Propositions 3.1 and 3.8) shows this equivalence. When the conditions of
small group effectiveness and substitution are imposed directly on large games the equivalence
is maintained.
4
equally. We also show that core payoffs of large games with a finite number of
player types satisfy cyclic monotonicity. This implies monotonicity of core payoffs
with respect to changes in the population of players of any type; an increase in
the numbers of players of any type will not lead to an increase in core payoffs
to players of that types. Moreover, we show that equal-treatment cores of games
with sufficiently many players of each type and -effective B−bounded groups are
approximately monotonic.
Finally, we note that the next step in this research, showing closeness of cores
and approximate cores to competitive payoffs of representing markets, is beyond
the scope of this paper. In the pregame framework, however, it is shown in Wooders (1992b,c) that, under the condition of small group effectiveness, approximate
cores of large games and economies are close to competitive payoffs of representing
markets.3 Ongoing research considers closeness of approximate cores to competitive payoffs of representing markets without the topological structure imposed
by a pregame. In particular, it is shown that certain approximate cores are close
to competitive payoffs of socially homogeneous markets and the approximation
becomes sharp as the games become large (relative to the size of effective groups).
2. Games and markets
We first introduce the game theoretic framework and the notion of a market, as
defined in Shapley and Shubik (1969).
2.1. Games
Let (N, v) be a pair consisting of a finite set N , called the player set, and a function v, called the characteristic function, from subsets of N to the non-negative
real numbers with v(φ) = 0. The pair (N, v) is a game (with side payments).
Nonempty subsets of N are called groups.4 The game (N, v) is superadditive if for
3
This result extends prior results of Wooders (1979) and Wooders and Zame (1987). Since
the results require that there are many participants of each type, the condition of per capita
boundedness — finiteness of the sup of average payoffs — suffices.
4
We use the term “groups” instead of “coalitions” as we interpret the model and results
as pertaining to socio-economic structures rather than to cooperative behavior, which may be
suggested by the word ”coalition”.
5
all groups S ⊂ N and for all partitions {S k } of S it holds that
v(S) ≥
[
v(S k ).
k
Let (N, v) be a game. Let δ ≥ 0 be a positive real number. Two players i and
j are δ-substitutes if for every group S with i ∈
/ S and j ∈
/ S, it holds that
|v(S ∪ {i}) − v(S ∪ {j})| ≤ δ.
When δ = 0, players who are δ-substitutes are called exact substitutes.
Let δ ≥ 0 be a non-negative real number and let (N, v) be a game. A partition
{N [t] : t = 1, ..., T } of N into subsets with the property that all members of each
subset N[t] are δ-substitutes for each other is called a δ-substitute partition of N.5
When δ = 0 the set N[t] is called the type t and the members of N[t] are called
players of the same type. In general, the set N [t] is interpreted as an approximate
type, and players in the set are described as approximate substitutes (or, precisely,
δ-substitutes). A profile relative to {N [t]} is a vector f ∈ Z+T . A profile describes
a group of players in terms of the numbers of players of each type or approximate
type (when δ > 0) in the group. The norm of a profile f ∈ Z+T is denoted by f
S
and defined by f = t ft ; the norm of a profile is simply the number of players
in a group described by f. Let S be a subset of N . Define the profile of S relative
to {N [t]} as the vector s ∈ Z+T satisfying, for each t = 1, ..., T,
st = |S ∩ N [t]|,
where |S ∩ N [t]| denotes the cardinal number of the set.
Groups of players that are similar in terms of the numbers of players of each
approximate type in the group have similar payoffs. Let S0 = {i1 , i2 } and let
S1 = {j1 , j2 } where i1 and j1 are δ-substitutes and i2 and j2 are δ-substitutes.
Observe that
|v(S0 ) − v(S1 )|
≤ |v(S0 ) − v({i1 , j2 })| + |v({i1 , j2 }) − v(S1 )|
≤ 2δ.
5
Note that in general the partition {N [t]} of N into δ-substitutes is not uniquely determined.
6
In general, because all the players in each set N[t] are δ-substitutes for each other,
if there is a group S0 ⊂ N with profile s and another group S1 ⊂ N with the
same profile s, then
|v(S0 ) − v(S1 )| ≤ δ|S0 | = δ|S1 |.
(Note that we use | · | for both the cardinal number of a set and the absolute
value. This should not cause any confusion.) Let (N, v) be a game and let be a
non-negative real number. Let x be a vector in RN , called a payoff. Given S ⊂ N
S
we denote i∈S xi by x(S). A payoff x is feasible if
x(N) ≤ v(N).
Let ≥ 0 be a non-negative real number. A payoff x is in the -core of (N, v)
if it is feasible and if, for all S ⊂ N , it holds that
x(S) ≥ v(S) − |S|.
When = 0 the -core is called simply the core.
Let (N, v) be a game and let {N[t] : t = 1, ..., T } be a partition of N into
types. Let ≥ 0 be a non-negative real number. A feasible payoff x is in the
equal treatment −core if, for each t and all i, j ∈ N [t], it holds that xi = xj . For
each t define yt = xi for (any) i ∈ N[t]. The vector y ∈ RT represents a payoff in
the equal treatment −core.
Let C be a positive real number. A collection of games {(N, v)} has a per
capita payoff bound of C if, for all games (N, v) in the collection and all coalitions
S ⊂ N, it holds that
v(S)
≤ C.
|S|
2.1.1. Balanced cover games
Let (N, v) be a game, let S ⊂ N, and let β denote a collection of subsets of S.
The collection β is a balanced collection of subsets of S if there is a collection of
non-negative real numbers (ωS )S ∈β , called balancing weights,6 such that for each
6
Note that here we are allowing balancing weights of zero instead of requiring that all weights
be positive. This makes no substantive difference and is convenient in the current framework.
7
i ∈ N,
[
ωS = 1.
S : i∈S , S ∈β
Let ≥ 0 be a non-negative real number. A game (N, v) is -balanced if, for all
balanced collections β of subsets of N,
max
β
[
S∈β
ωS v(S) ≤ v(N ) + |N|.
Bondareva (1963) and Shapley (1967) have shown that a game has a nonempty
core if and only if it is balanced. Their result can be used to show that a game
has a nonempty -core if and only if it is -balanced (Wooders 1992b, Proposition
2.2.).
Let (N, v) be a game and let v b be the characteristic function defined for each
nonempty subset S of N by
v b (S) = max
β
[
ωS v(S ),
S ∈β
where the maximum is taken over all balanced collections β of S. Then (N, v b ) is
a balanced game, called the totally balanced cover of (N, v). The game (N, v) is
balanced if v b (N ) = v(N).
Let ≥ 0 be
a given positive real number. The game
b
(N, v) is −balanced if v (N) − v(N) < |S| .
2.2. Markets
M
A market is a pair (RM
+ , A) where R+ is the commodity space and A is an index
set of types of participants. A commodity bundle x is a vector x ∈ RM
+ . The set
i
i
i
A = {a : i = 1, ..., I} is a finite indexed collection of pairs, a = (ω , ui ), where
each ω i is a commodity bundle, called the endowment of participant i, and each
ui is a continuous, concave, and monotonic non-decreasing function from the set
of commodity bundles RM
+ to the reals, called the payoff function of participant i.
For convenience, we assume that payoff functions have been normalized so that
ui (ω i ) ≥ 0 for all i = 1, ..., I. The premarket satisfies payoff-constant returns if all
payoff functions are homogeneous of degree 1. The market is socially homogeneous
if there is a function u such that for all participants i, ui = u .
Socially homogeneous markets with payoff constant returns are extremely
tractable and have particularly pleasing properties. First, for any such market
8
an equilibrium exists, and equilibrium prices are given by the subgradients of the
indifference curve at the distribution of the total endowment. The equilibrium is
uniquely determined for almost all distributions of commodities since, from concavity of the utility function, it follows that the utility function is differentiable
almost everywhere on the unit simplex. It is easy to see that for such markets
an increase in the total endowment of one good leads to no increase (and possibly a decrease) in the equilibrium price of the good whose supply has increased,
suggesting the monotonicity results to follow. Since games with effective small
groups are approximated by socially homogeneous markets with payoff constant
returns, we can expect such games to behave approximately like such markets.
2.3. Games induced by markets
i
i
Let (RM
+ , A) be a market, where A = {a : i = 1, ..., I} and, for each i, a =
(ω i , ui ). The market induces a game (I, w) with player set I = {1, ..., I} and with
characteristic function w defined by
w(S) = sup
{xi }i∈S
where {xi }i∈S satisfies
[
i∈S
xi =
[
[
ui (xi ),
i∈S
ω i . The game (I, w) is called the game induced
i∈S
by the market. Any game which is induced by some market is called a market
game.
Theorem (Shapley and Shubik (1969)). A game (N, v) is a market game if and
only if it is totally balanced. Moreover, the market can be chosen to be socially
homogeneous and have the same number of commodities as players.7
Let (RM
+ , A) be a market, let T be some positive integer, and let {A[t] : t =
1, ..., T } be a partition of A into types, that is, for each t, for all i and i in A[t]
it holds that ai = ai . Thus, for each pair i and i in A[t], ui = ui and ω i = ω i ;
participants of the same type have the same payoff functions and endowments. We
call (RM
+ , A) a market with T types. By re-examining the Shapley-Shubik result,
it is easy to verify that a market with T types induces a game with T types
and conversely, a game with T (exact) types induces a market with T types (cf.,
Wooders (1994, Section 6)).
7
See also Shapley and Shubik (1975).
9
3. Approximating games by markets
Let ≥ 0 be a given positive real number, and let B be a given integer. Let (N, v)
be a game. The game (N, v) has -effective B-bounded groups8 if, for every group
S ⊂ N there is a partition {S k } of S into subgroups with |S k | ≤ B for each k and
v(S) −
[
k
v(S k ) ≤ |S|.
When = 0, −effective B-bounded groups are called strictly effective B −
bounded groups.
Let δ be a non-negative real number and let T be a positive integer. A game
(N, v) is a (δ, T )-type game if there is a T -member partition {N[t] : t = 1, ..., T }
of N into δ-substitutes — that is, a game (N, v) is a (δ, T )-type game if there are
T approximate types.
Let T and B be positive integers and let C be a positive real number. Let
Γ((δ, T ), C, ( , B)) be the class of all (δ, T )-type games that are superadditive and
have both a per capita bound of C and -effective B-bounded groups. Let K(T, B)
denote the number of distinct profiles f with norm less than or equal to B.
Theorem 1: Let δ ≥ 0 and ≥ 0 be non-negative real numbers and let B, C,
and T be positive integers. Let (N, v) be in Γ((δ, T ), C, ( , B)). Then there is
a socially homogeneous market with payoff-constant returns and T commodities
such that for all groups S in N it holds that
|w(S) − v(S)| ≤ K(T, B)BC + δ|S| + |S|,
where w denotes the characteristic function of the market game induced by the
market.9
Proofs of all our results are contained in Section 5.
An interesting aspect of Theorem 1 is that the games in Γ((δ, T ), C, ( , B))
may be derived from economies with arbitrarily large numbers of commodities,
8
This is the “small group effectiveness” of previous papers but without a topological structure
on the space of player types, and allowing to equal zero.
9
Note that since (N, w) is a market game, w equals wb , its totally balanced cover. Also, it is
possible that the bound K(T, B)BC may be improved; we have no non-trivial examples where
the bound is met.
10
but for every game in the class there is an approximating market with the number
of commodities in the approximating market equal to T, the number of approximate types of players. The games considered may themselves be derived from
economic structures where the contribution of a player to a coalition depends on
the attributes of that player, for example, his endowment, and the crowding effects (positive or negative) that he imposes on other members of the group. A
similar result obtains when the commodities in the representing market are the
attributes of players in the game.10
Our next Theorem relates games to their balanced covers. We are able to
obtain the same bound on the distance of a game from its balanced cover.
Theorem 2: Let δ ≥ 0 and ≥ 0 be non-negative real numbers and let B, C, and
T be positive integers. Let (N, v) be in Γ((δ, T ), C, ( , B)). Then for all groups S
it holds that
|vb (S) − v(S)| ≤ K(T, B)BC + δ|S| + |S|,
where v b is the totally balanced cover of v.
Since a game is −balanced if and only if it has a nonempty -core, the following
Corollary holds. The Corollary, a simple consequence of Theorem 2, subsumes
previous results in the literature on nonemptiness of approximate cores of games
with side payments, including those in Wooders (1992a), where players’ types
may come from a compact metric space of types and Wooders (1992c), where
commodities/attributes are interpreted as players.
Corollary 1: Let (N, v) be a game in Γ((δ, T ), C, ( , B)) and let
real number. If
∗
then the
∗
≥
∗
be a positive
K(T, B)B C
+δ+
|N|
-core of (N, v) is nonempty.
10
See Tauman (1988) for a survey of pricing of attributes/commodities in models of characteristic function games where the domain of the payoff function is a finite dimensional space.
Wooders (1993) discusses a model of a differentiated commodities economy allowing public goods
and collectively consumed and/or produced goods more generally where there may be many more
types of commodities than consumers and where the game derived from the economy satisfies
the conditions imposed in this paper.
11
Observe that the more substitution — the smaller δ and T -the better the
approximation. Also, K(T, B) is independent of the size of the game, so the
bound K(T, B) + δ|S| + |S| grows linearly with the size of the group S.
3.1. Some remarks
Remark 1: Dividing the inequality in the statement of Theorem 1 by |S| we obtain
v b (S)
v(S) K(T, B)B C
≤
−
+ δ+
|S|
|S| |S|
,
which is “small” if δ and are small and S is “large”; large games with effective
small groups are closely approximated, in per capita terms, by market games.
Remark 2. Using the pregame framework Wooders (1994, Theorem 4) shows that
when all players have sufficiently many close substitutes then per capita boundedness and small group effectiveness are equivalent. In the framework of this paper,
however, such a result cannot be obtained. The following examples illustrate that
per capita boundedness and small group effectiveness are distinct concepts.
Example 1: Let (N m , v) be a superadditive game with 2m players where every
pair of players can realize a payoff of 2m and v(N m ) = m2 . Clearly the
game (N m , v) has 2-player effective groups. But, since the per capita payoff
equals m, the per capita payoff becomes infinite as the games become large.
Example 2: Let (N m , v) be a superadditive game with m players where v(N m ) =
m and v(S) = 0 for all S = N. Then the games {(N, v)} all have one type,
many players of each type, and the same per capita bound of C. Yet the
games {(N, v)} do not have −effective B-bounded groups for any > 0
and B > 0.
Remark 3: Games of flow. Market games provide an economic description of
totally balanced games. Kalai and Zemel (1982) provide another description of a
totally balanced game as a “game of flow” and characterize such a game as the
minimum game of a finite number of additive games. Thus our results also show
an approximation of a game by games of flow and by minimum games of a finite
number of additive games.
12
Remark 4: NTU Games. Although in this paper we study only games with
side payments, it is clear that if we consider approximate cores where a small
“exceptional set” of players is ignored, then a similar result can be obtained for
games without side payments. (See Wooders (1983,1991) and references therein
for additional discussion of games without side payments.) Moreover, with some
assumption ensuring that payoffs can be “shared” among similar agents, closer
analogues can be obtained for games without side payments.
4. Equal treatment and monotonicity of the core and approximate cores
It is intuitive that in games with strictly effective small groups, matching games for
example, an increase in the number of substitutes for a player will not improve
the core payoff possibilities for that player. In this Section, we show that this
intuition holds for the core of an arbitrary game when subgroups of the total
player set can realize all gains to group formation and when there are sufficiently
many substitutes for each player. The result is obtained for the very strong
condition of cyclic monotonicity. Cyclic monotonicity can also be shown for the
core of the continuum limit game and an asymptotic cyclic monotonicity holds
for large finite games.11 Here we provide a result showing cyclic monotonicity
when all gains to group formation can be exhausted by subgroups of the total
player sets. Note that cyclic monotonicity implies monotonicity — a vector of
changes in the population proportions and a corresponding vector of changes in
core payoffs point in opposite directions. In addition we show that arbitrary games
satisfy an approximate monotonicity condition. These results are stated for core
payoffs which treat substitutes and close substitutes equally. Thus, we provide
some results on the equal treatment property of the core when strict subgroups
of the player set can realize all gains to collective activities and also results on
asymptotic equal treatment.
11
Montonicity in the pregame framework was suggested in Wooders (1978) for pregames with
strictly bounded effective group sizes and was shown in such a framework in Scotchmer and
Wooders (1988). The current result extends Wooders (1992b, Propositions 4.3 and 4.4). Extensions can also be demonstrated with a compact metric space of attributes.
13
4.1. Equal treatment
Let Γ(T, C, B) be the collection of all superadditive games with T (exact) types,
group sizes bounded by B, and a per capita bound of C. The following Theorem
shows that when there are more than B players of each type then the core has the
equal-treatment property — each payoff in the core treats all players of the same
type equally.
Theorem: (Wooders 1983, Theorem 3). Let (N, v) be in Γ(T, C, B) and let
{N [t]} denote a partition of N into types. Assume that for each t, |N [t]| > B.
Let x be a payoff in the core of (N, v). Then x has the equal-treatment property.
That is, if players i and j are of the same type, then xi = xj .
Our next result shows that as to be expected, in games with −effective
B−bounded groups there are approximate core payoffs with the property that
similar players are treated approximately equally.12
Theorem 3. Let δ ≥ 0 and ≥ 0 be non-negative real numbers and let B,
C, and T be positive integers. Let (N, v) be in Γ((δ, T ), C, ( , B)). Assume that
for each t, |N[t]| > B. Let ∗ be a positive real number satisfying ∗ ≥ K(T,B)BC
|N|
+δ + . Then there is a payoff x = (x1 , ...x|N | ) in the ∗ −core with the property
that for each t and all i and j in N[t],
xi = xj .
4.2. Monotonicity
In view of the equal-treatment property of the core of games with strictly effective
proper subgroups, for any game (N, v) in Γ(T, C, B) we can represent a payoff in
the core of (N, v) by a vector x ∈ RT+ , and we will do so in the following.
Theorem 4: Let (N, v) be in Γ(T, C, B) and let (S1 , v), ..., (SK , v) be subgames
of (N, v). Let {N[t]} denote a partition of N into types and for each k let f k
denote the profile of Sk relative to {N[t]}. Assume that ftk > B for each k and
12
In fact, proofs of nonemptiness of approximate cores with a finite number of types or with
a compact metric space of player types as in Wooders (1983,1992a) have relied on showing that
an equal treatment payoff in the core of the balanced cover of an approximating game with a
finite number of types is close to an equal-treatment payoff in the -core of the game.
14
each t. Assume also that for each k there is a vector xk ∈ RT+ representing a
payoff in the equal-treatment core of (Sk , v). Then
x1 · (f 1 − f 2 ) + x2 · (f 2 − f 3 ) + ... + xK · (f K − f 1 ) ≤ 0,
that is, the equal-treatment core correspondence satisfies cyclic monotonicity.
Let Γ(T, C, ( , B)) be the class of all superadditive games with per capita
bound of C, -effective B-bounded groups, and T exact types.
Theorem 5: Let (N, v) be in Γ(T, C, ( , B)) and let (S1 , v), (S2 , v) be subgames
of (N, v). Let {N [t]} denote a partition of N into types and for each k, let f 1
denote the profile of S1 and let f 2 denote the profile of S2 , both relative to {N[t]}.
Assume that ftk > B for each k and each t. Assume that the −core of each
subgame is nonempty and let xk ∈ RT+ represent a payoff in the equal-treatment
−core of (Sk , v), k = 1, 2. Then
(x1 − x2 ) · (f 1 − f 2 ) ≤ 2 f 1 + f 2 ,
that is, the −core correspondence is approximately monotonic.
4.3. Asymptotic equal treatment
Our results in this subsection establish that in large games with a finite number of
types the −core has approximately the equal treatment property. It is convenient
to present the result using the pregame framework, so first we formally define a
pregame.
Let Ω be a compact metric space, interpreted as a space of player types. For
purposes of this subsection a profile on Ω (sometimes called simply a profile) is a
function f from Ω to Z+ with finite support, that is,
σ(f) = {ω ∈ Ω : f (ω) = 0} is a finite set,
S
where σ(f ) is the support of f. The norm of a profile f is given by ω∈σ(f ) f (ω).
Let Ψ be a function from the set of profiles on Ω to R+ with Ψ(0) = 0,
where 0 denotes the profile which is identically zero. The pair (Ω, Ψ) is called
a pregame with characteristic function Ψ. The value Ψ(f ) is the total payoff a
group of players f can achieve by collective activities of the group membership.
The pregame is superadditive if
15
Ψ(f ) = max
[
Ψ(g),
g∈P
where P is a partition of f and the maximum is taken over all partitions of f.
A pregame is not a game since there is no specified total player set. When a
total player set is specified a pregame induces a game with that player set. A game
determined by a pregame (Ω, Ψ), called simply a game or a game in characteristic
form, is a pair [n, Ψ] where n is a profile, interpreted as a description of the total
player set in the game, and the characteristic function Ψ is restricted to subprofiles
of n. Note that for every positive integer r the pair [rn, Ψ] is also a game. Let
{ω1 , ..., ωT } denote the support of n and let nt = n(ωt ). We denote the total player
set by N = {(t, q) : t = 1, ..., T and q = 1, ..., n(ωt )}. A payoff for the game is
a vector x = (xtq : q = 1, ..., n(ωt ), t = 1, ..., T ) in RN . A payoff x is feasible if
nt
T S
S
t=1 q=1
xtq ≤ Ψ(n).
A pregame (Ω, Ψ) satisfies small group effectiveness if it is superadditive and
if, given any real number > 0, there is an integer η0 ( ) such that for each profile
f , for some partition {f k } of f :
f k ≤ η0 ( ) for each subprofile f k in the partition, and
Ψ(f ) −
[
k
Ψ(f k ) ≤
f .
Thus, for every profile f , almost all (within per capita) of the gains to collective activities can be realized by aggregating collective activities within groups of
participants bounded in absolute size.
A pregame (C, Ψ) satisfies per capita boundedness if there is a constant C such
that for all profiles f it holds that
Ψ(f )
≤ C.
f
The following Theorem states that given an initial profile n, for all sufficiently
large replications rn and sufficiently small non-negative real numbers , any payoff
x in the −core of [rn, Ψ] has the property that most players of the same type
16
are treated approximately equally. Since the statement of the Theorem imposes
that condition that there are many players of each type appearing in the game,
per capita boundedness suffices to obtain the result.13
Theorem 6. Let (Ω, Ψ) be a pregame satisfying per capita boundedness and let
n be a profile on Ω. Then given any real numbers γ > 0 and λ > 0 there is a
positive real number ∗ and an integer r∗ such that for each ∈ [0, ∗ ] and for any
r ≥ r∗ , if x ∈ RN is in the -core of [rn, Ψ], then for each t in the support of n,
|{(t, q) : |xtq − zt | > γ}| < λr ,
where zt =
1
rnt
rn
St
xtq , the average payoff received by players of type t.
q=1
Our next and final result follows easily from the preceding Theorem.
Theorem 7.
Let (Ω, Ψ) be a pregame satisfying small groups effectiveness.
Assume that Ω is a finite set. Then given any real numbers γ > 0 and any λ > 0
there is a positive real number ∗ and an integer ρ(γ, λ, ∗ ) such that for each
∈ [0, ∗ ] and for every game n with n ≥ ρ(γ, λ, ∗ ), if x is in the -core of the
game [n, Ψ] then
|{(t, q) : |xtq − zt | > γ}| < λ n ,
where zt =
1
nt
nt
S
xtq , the average payoff received by players of type t.
q=1
5. Proofs
We prove Theorems 1 and 2 by first establishing three Lemmas. The Lemmas proceed from the special case of bounded effective group sizes and exact substitutes
to the case of the Theorem.
Let Γ(T, C, B) be the collection of all superadditive games with side payments,
T (exact) types, group sizes bounded by B, and a per capita payoff bound of C.
Recall that K(T, B) denotes the number of profiles (vectors in Z+T ) with norm
less than or equal to B.
13
This result first appeared in Wooders (1980) for a more special class of games. In its current
form it has appeared in Wooders (1979, 1992b).
17
Lemma 1: Let T, C, and B be given positive integers. For every game (N, v) in
Γ(T, C, B) and every group S ⊂ N it holds that
|v b (S) − v(S)| ≤ K(T, B)BC.
Proof of Lemma 1: Let (N, v) be a game in Γ(T, C, B) and let {N[t]} denote
a partition of N into exact types. We first obtain a bound on v b (S) − v(S). Let
K(T,B)
{M k }k=1 denote the collection of all profiles mk relative to the partition {N[t]}
of N where mk ≤ B for each mk in the collection. Define a characteristic
function v̄ mapping profiles into RT+ , where
v̄(m) = v(M ) for any group M with profile m.
Observe that since mk ≤ B and C is a per capita bound, it holds that
v̄(mk ) ≤ C B for each k. Since (N, v) satisfies boundedness of effective group
sizes with bound B there is a balanced collection β of subprofiles of s where each
m ∈ β is in {mk } and, for some collection of balancing weights, (ωk ),
v b (S) =
[
ωk v̄(mk ).
k
S
From balancedness, it holds that k ωk mk = s.
Since there is a finite number of distinct profiles in the set {mk }, we can
write each ωk as an integer plus a fraction, say ωk = k + qk , where qk ∈ [0, 1).
S
Intuitively, we think of the players k qk mk as “leftovers” —it may not be possible
to fit these players (or any subprofile of them) into groups that can achieve core
payoffs (for the core of the balanced cover game) for their memberships.
Since the game (N, v) satisfies boundedness of effective group sizes, there
S
are non-negative integers rk satisfying k rk mkt = |S ∩ N [t]| for each t and
S
k
k rk v̄(m ) = v(S).
Now consider the difference v b (S) − v(S). Since v(M ) ≥ 0 for all subsets M ,
it follows that:
v b (S) − v(S) =
=
[
k
[
k
ωk v̄(mk ) −
ωk v̄(mk ) −
18
[
k
[
rk v̄(mk )
k
rk v̄(mk )
≤ min
β
[
k
(ωk −
k
k )v(m )
=
[
qk v̄(mk )
k
≤ K(T, B)BC.
Let Γ(T, C, ( , B)) denote the collection of all superadditive games with T exact
types, per capita bound of C, and -effective B-bounded groups. The next Lemma
shows that any game (N, v) with −effective B-bounded groups is approximated
by a balanced game with strictly effective B−bounded groups.
Lemma 2. Let (N, v) be a game in Γ(T, C, ( , B)). Then there is a game (N, w)
in Γ(T, C, B) such that for every group S ⊂ N it holds that:
|wb (S) − v(S)| ≤ K(T, B)BC + |S|.
Proof of Lemma 2: Suppose the statement of the Lemma is false. Then there is
a non-negative real number 0 , a game (N, v) in Γ(T, C, ( 0 B)), a partition {N[t]}
of N into types, and a group S ⊂ N such that
|wb (S) − v(S)| > K(T, B)BC +
0 |S|
for every game (N, w) in Γ(T, C, B).
Let (N, w) be the game formed from (N, v) by allowing only groups bounded
in size by B to be effective, that is, for each group S ⊂ N define
w(S) = max
k
{S }
[
v(S k )
k
where the maximum is taken over all partitions {S k } of S with |S k | ≤ B for each
k. Then the game {(N, w)} satisfies the conditions of Lemma 1. Therefore
|wb (S) − w(S)| ≤ K(T, B)BC,
where wb denotes the totally balanced cover of w.
From the fact that the game (N, v) has 0 -effective B-bounded groups, it follows that for each group S
19
0 ≤ v(S) − w(S) ≤
0 |S|
.
We now obtain the following estimate:
|wb (S) − v(S)|
≤ |wb (S) − w(S)| + |w(S) − v(S)|
≤ K(T, B)BC +
0 |S|,
which is a contradiction.
Let Γ((δ, T ), C, B) be the class of all superadditive (δ, T )−type games that
have a per capita payoff bound C and strictly effective B-bounded group sizes.
The following Lemma shows that (δ, T )−type games are approximated by games
with T exact types.
Lemma 3: Let δ ≥ 0 be a positive real number and let B be a positive integer.
Let (N, v) be in Γ((δ, T ), C, B) and let S ⊂ N . Then there is a totally balanced
game (N, wb ) in Γ(T, C, B) such that
|w b (S) − v(S)| ≤ K(T, B)BC + δ|S|.
Proof of Lemma 3. Let (N, v) be a game in Γ((δ, T ), C, B) and let {N[t] : t =
1, ..., T } be a partition of N into δ-substitutes. Suppose that for some partition
{N [t]}, for every totally balanced game (N, wb ) in Γ(T, C, B) there is a group
S ⊂ N such that
|wb (S) − v(S)| > K(T, B)BC + δ|S|
We define a new characteristic function on the domain {h ∈ Z+T : ht ≤ |N [t]|
for each t}, where there are T types of players (and all players of the same type
20
are exact substitutes). Specifically, for each profile h in Z+T with ht ≤ |N[t]| for
each t, define the value of the characteristic function w at h by
w(h) = max v(M ),
M
where the maximum is taken over all groups M in N with |M ∩ N [t]| = ht for
each type t. (We could also have chosen to take the minimum over all such groups
or, for example, a weighted average of the minimum and maximum; what we do
need is to assign to w(h) some sort of “representative” payoff.) Let w denote the
characteristic function defined, for each group M ⊂ N , by w (M ) = w(m) where
m is the profile of M relative to the partition {N [t]} of N . Let wc denote the
S
superadditive cover of w , that is, for each group S, wc (S) := max{S k } k w (S k )
where the maximum is taken over all partitions {S k } of S. We leave to the reader
the verification that (N, w c ) satisfies the conditions of Lemma 1. Therefore, for
all groups S ⊂ N with |S ∩ N[t]| > B,
|wb (S) − w c (S)| ≤ K(T, B)BC,
where wb denotes the totally balanced cover of w c . From the construction of w
and the fact that players of the same approximate types are δ-substitutes, for all
profiles s and for all groups S with profile s it holds that
|v(S) − w c (S)| = |v(S) − v(S ∗ )| < δ|S|,
where S ∗ is any group with the same profile, relative to {N[t]} as S and satisfying
v(S ∗ ) = w(s).
It now follows that
v(S) − w b (S) ≤
|v(S) − wc (S)| + wc (S) − wb (S) ≤ K(T, B)BC + δ |S| .
Proof of Theorem 1: Suppose the statement of the Theorem is false. Then
there are positive real numbers 0 > 0 and δ0 > 0, a positive integer B, a game
1
Result (d) initially appeared in Wooders (1979) and is related to the equal treatment result
of Wooders (1983, Theorem 3).
21
(N, v) in Γ((δ0 , T ), C, ( 0 , B)), and a partition {N [t]} of N into δ0 -substitutes,
such that for every totally balanced game (N, w b ) in Γ(T, C, ( 0 , B)), for some
group S ⊂ N ,
|wb (S) − v(S)| > K(T, B)BC + δ0 |S| +
0 |S|.
We will now again construct a game with the property that effective coalitions
are bounded in absolute size. Let (N, w) be the game formed by allowing only
groups bounded in size by B to be effective, that is,
w(S) = max
k
{S }
[
v(S k ),
k
where the maximum is taken over all partitions {S k } of S with |S k | ≤ B for each
S k in the partition. Then the game (N, w) satisfies the conditions of Lemma 3
and for each ν sufficiently large
|wb (S) − w(S)| ≤ K(T, B)BC + δ0 |S|.
From the definition of w it follows that
|v(S) − w(S)| <
0 |S|.
We now obtain the estimate
|wb (S) − v(S)|
≤ |wb (S) − w(S)| + |w(S) − v(S)|
≤ K(T, B)BC + δ0 |S| +
0 |S|.
Proof of Theorem 2: Let (N, v) satisfy the conditions of the Theorem. Define
the game (N, w) by
[
w(S) = max
v(Sk )
k
where the maximum is taken over all partitions {Sk } of S with |Sk | ≤ B. Let
(N, wb ) denote the balanced cover of (N, w). It holds that
w(S) ≤ v(S) ≤ v b (S) ≤ w(S) + |S| .
22
From Lemma 3 it holds that
b
w (S) − w(S) ≤ K(T, B)BC + δ |S| .
This two relationships yield the conclusion of the Theorem.
Proof of Corollary 1. Let ≥ 0 be a given non-negative real number, let (N, v)
be a game in Γ((δ, T ), C, ( , B)) and let ∗ be a positive real number satisfying
∗
≥
K(T, B)BC
+ δ+ .
|N |
From Theorem 2 the game (N, v) is ∗ -balanced and therefore is has a nonempty
∗
−core (Wooders 1992b, Proposition 2.2).
Proof of Theorem 3: Let (N, v) and ∗ satisfy the conditions of the Theorem.
Let {N[t]} be a partition of N into δ-substitutes. Define the game (N, wb ) as in
Lemma 3. Let y be in the equal treatment core of the game (N, wb ) (so if i, j are
(exact) substitutes for each other in the game (N, wb ) then yi = yj ). Observe
that y cannot be improved upon in the game (N, v) since y(S) ≥ w b (S) and from
the construction of wb it holds that wb (S) ≥ v(S).
For each player i define xi = yt − ∗ . Since y cannot be improved upon in the
game (N, v) it follows that x cannot be ∗ -improved in the game (N, v). From
Theorem 1 it follows that x(N ) ≤ v(N ). Therefore x is in the ∗ -core. Since y
treats all players in each set N[t] equally, x does also.
Proof of Theorem 4: Let (N, v), (Sk , v), f k , and xk , k = 1, ..., K satisfy the
conditions of the Theorem. For convenience, for any subprofile f define v(f ) =
v(S) where S is any subset of N with profile f . Since (N, v) and (Sk , v), k =
1, ..., K are in Γ(T, C, B) and ftk > B for each k it holds that,
xk · f k = v(f k ) and
xk · f k∗ ≥ v(f k∗ ) for all k ∗ .
To show this, first we observe that (Sk∗, v) is a balanced game. From the assumption that (Sk∗ , v) is in Γ(T, C, B) there exists a balanced collection {S j }
S
of subgroups of Sk∗ with weights ωj for S j such that v(Sk∗ ) = j ωj v(S j ) and
|S j | ≤ B for each j. Let sj denote the profile of S j . Since xk · g ≥ v(G) for all
23
subsets G of N with profiles g with g ≤ B it holds that xk · sj ≥ v(S j ) for each
S
S
S
j. Therefore xk · f k∗ = xk · j ωj sj = j ωj (xk · sj ) ≥ j ωj v(S j ) = v(Sk∗ ). From
these relationships it follows that
v(f 1 ) − x1 · f 1 ≥ v(f 2 ) − x1 · f 2
...
v(f k ) − xk · f k ≥ v(f k+1 ) − xk · f k+1
...
v(f K ) − xK · f K ≥ v(f 1 ) − xK · f 1 .
Adding these inequalities and simplifying, we obtain
−
[
k
k
k
x ·f ≥−
K−1
[
k=1
xk · f k+1 − xK · f 1 , and
x1 · (f 1 − f 2 ) + ... + xk · (f k − f k+1 ) + ... + xK · (f K − f 1 ) ≤ 0.
Proof of Theorem 5: As in the statement of the Theorem, let (N, v) be in
Γ(T, C, ( , B)) and let (S1 , v), (S2 , v) be subgames of (N, v) satisfying the conditions of the Theorem. Let f 1 denote the profile of S1 , let f 2 denote the profile of
S2 , and let x1 and x2 represent payoffs in the equal-treatment cores of the games
(so x1 , x2 are in RT ). Since ftk > B for each k and each t it follows that
and
x1 · f 2 ≥ v(f 2 ) − f 2 ≥ x2 · f 2 − 2 f 2 x2 · f 1 ≥ v(f 1 ) − f 1 ≥ x1 · f 1 − 2 f 1 .
Adding these two inequalities we obtain
x1 · f 2 + x2 · f 1 ≥ x2 · f 2 − 2 f 2 + x1 · f 1 − 2 f 1 24
or, equivalently,
(x1 − x2 ) · (f 1 − f 2 ) ≤ 2 f 1 + f 2 ,
the desired conclusion.
Proof of Theorem 6: Given real numbers λ and γ greater than zero, select
r∗ and r0 ≤ r∗ so that:
(a)
∗r
0
r∗
(b)
∗
<
λ
2 n
∗
,
;
> 0 and
nt = 0; and
∗
< mint { 4λγn , 2γnnt } where the minimum is over all t with
(c) for all r ≥ r0 , Ψr (rn)
−
n
∗
Ψb (r0 n) r0 n
∗
≤
∗
.
b
Since λ > 0 and γ > 0, and Ψr (rn)
− Ψr0(rn0 n) → 0 as r → ∞ and r0 → ∞,
n
such a selection is possible.
Select r ≥ r∗ and let x be in the ∗ -core of [rn, Ψ]. For each t, define zt as in
the statement of the Theorem. Since the -core is convex, z represents a payoff
in the equal-treatment ∗ -core of the game [rn, Ψ]. It follows then that for all
profiles s ≤ rn, z · s ≥ Ψ(s) − ∗ s and z · rn ≤ Ψ∗ (rn).
Let Nr = {(t, q) : t = 1, ..., T, q = 1, ..., nt }. It is convenient to establish the
convention that for each coalition S ⊂ Nr , St denotes the subset of players in S
of type t, i.e., St = {(t̂, q) : (t̂, q) ∈ S and t̂ = t} for each t = 1, . . . , T . We define
the profile of a coalition S by s ∈ Z+T with tth component given by |St | for each t.
Select a subset P of Nr so that the profile of P is r0 n and P contains the
“worst-off” players of each type; i.e., if (t, q) ∈
/ P then xtq ≥ xtq for all q with
(t, q ) ∈ P . Suppose that, for some type t∗ ,
|P ∩ {(t∗ , q) ∈ Nr : xt
∗q
< zt∗ − γ}| = r0 nt∗ ;
i.e. all players of type t∗ in P receive less than the average payoff for players of
that type minus γ. We then have
Ψb (r0 n) −
∗
r0 n ≤ x(P ) < r0 (z · n) − γr0 nt∗ ≤ r0
Ψ∗ (rn)
− γr0 nt∗ .
r
The first inequality follows from the fact that x in the -core of Nr . The second
∗
follows from the facts that zt∗ ≥ xt q − γ for each q with (t∗ , q) in Pt and x(Pt ) ≤
25
r0 zt nt for each t. The final inequality is from the feasibility of z; z · rn ≤ Ψ∗ (rn).
It now is apparent that the following relationships hold:
Ψb (r0 n) −
∗
r0 n < r0
Ψ∗ (rn)
− γr0 nt∗ .
r
Subtracting Ψb (r0 n) from both sides of the expression, adding γr0 nt∗ to both
sides, and dividing by r0 n we obtain
γnt∗
−
n
γnt∗
n
Ψ∗ (rn)
rn
From (b) above,
−
implies that ∗ <
−
∗
t = 1, . . . , T it holds that
∗
∗
<
>
Ψb (r0 n)
r0 n
∗
Ψ∗ (rn) Ψb (r0 n)
−
.
rn
r0 n
which, along with the preceding expression,
, a contradiction to (c). Therefore, for each
|P ∩ {(t∗ , q) ∈ Nr : xt
∗q
< zt∗ − γ}| < r0 nt∗ ;
of the worst off players of type t∗ , fewer than r0 nt∗ can be treated worse than the
average payoff for that type minus γ. This means that {(t, q) : xtq −z t < −γ} ⊂ P .
From the facts that:
r0
Ψ∗ (rn)
− 2 ∗ r0 n ≤ Ψb (r0 n) −
r
≤ x(P ) (since x is in the
∗
r0 n (from (c)),
∗
-core),
≤ r0 · n (from the definition of P ),
≤ r0
Ψ∗ (rn)
r
(from feasibility of x),
it follows that
0 ≤ r0 z · n − x(P ) ≤ 2 ∗ r0 n .
Informally, the above expression says that, for each t, on average players of type
t in P are receiving payoffs within 2 ∗ of zt .
We now turn to those players who are receiving payoffs significantly more
(more than γ) than the average for their types and put an upper bound on the
number of such players. Define the set of “best off” players (the “rich”) R by
R = {(t, q) ∈ Nr : xtq > zt + γ} .
26
Define the set of “middle class” players M by
S
Observe that, since
M = Nr /(R ∪ P ) .
(t,q)∈Nr
γ|R| ≤
(xtq − zt ) = 0, it follows that
[
(xtq − zt ) =
(t,q)∈R
[
(t,q)∈P ∪M
(zt − xtq ) .
From the preceding paragraph and the above expression,
γ|R| <
∗
r0 n +
[
(t,q)∈M
Obviously, the larger the value of
|R| to be. We claim that
S
(t,q)∈M
S
(t,q)∈M
(zt − xtq ) .
(zt − xtq ), the larger it is possible for
(zt − xtq ) ≤ 2 ∗ |M |. This follows from the fact
that the players in P are the worst off, and they are, on average, each within
2 ∗ of the average payoff for their types. Since those players in M are better off,
they must receive on average no less than the average for their types minus 2 ∗ .
S
Therefore,
(zt − xtq ) ≤ 2 ∗ |M|. It now follows that
(t,q)∈M
γ|R| ≤ 2 ∗ r0 n + 2 ∗ |M | .
From |M | + |R| = r n − r0 n , |M | ≤ r n − r0 n , and
γ|R| ≤ 2 ∗ r0 n + 2 ∗ (r n − r0 n ) ≤ 2 ∗ r n ,
∗
it follows that r|R|
≤2γ .
n
Counting the number of players who may be treated significantly differently
than the average we see that:
∗
|R|
λ
|P |
r0 2 ∗
+
≤
+
<
from (a) and (b) above.
rn
rn
r
γ
n
The conclusion of the Theorem is immediate from the observation that if x is
in the -core of rn for r ≥ r∗ and 0 ≤ ≤ ∗ , then x is in the ∗ -core of rn.
Proof of Theorem 7. Suppose the statement of the Theorem is false. Then
there is a pregame (Ω, Ψ) satisfying the condition of small group effectiveness and
27
real numbers γ > 0 and λ > 0 such that: for every integer K and some
non negative real number K with K < K1 , there is a game [nK , Ψ] with nK > K
and a payoff xK in the K -core of the game with the property that
t
where zK
=
1
nt
nt
S
q=1
tq
t {(t, q) : xK − zK
> γ} ≥ λ nK ,
xtq
K , the average payoff received by players of type t. Since Ω is
finite, without loss of generality we can assume that for each ωt ∈ Ω the sequence
{ n1K nK (ωt )} converges. Let nt = limK→∞ n1K nK (ωt ). By relabelling points in
Ω we can assume that for some T ≤ T it holds that nt > 0 for t = 1, ..., T and
nt = 0 for t = T + 1, ..., T.
Let h be a profile and let {rk } be a sequence of positive integers such that for
each t,
(a) h(ωt ) = 0 for t = T + 1, ..., T ;
(b) rK h(ωt ) ≤ nK (ωt ) for each t = 1, ..., T ;
(c) |z t − zt | < γ/2 where z t :=
(d)
T
S
1
h(ωt )
h(ω
St )
q=1
xtq
K ; and
(nK (ωt ) − rK h(ωt ))/ nK < λ.
t=+1
Define a sequence of games [rK h, ΛK ] by ΛK (rK h) = Ψ(nK ) and, for all profiles
s with s ≤ rK h, s = rK h, ΛK (s) = Ψ(s). It holds that the sequence of games
{[rK h, ΛK ]} satisfies per capita boundedness since in the pregame framework small
group effectiveness implies per capita boundedness (Wooders 1994, Theorem 4).
We can now apply the preceding Theorem to conclude that for all K sufficiently
large it holds that
tq
{(t, q) : xK − z tK > γ/2} < λ rK h .
tq
tq
tq
t t t t Now xtq
K − zK > γ implies that xK − z K > γ/2 since xK − zK ≤ xK − z K +
tq
t
t t |zK
− z tK | ≤ xtq
−
z
+
γ/2.
Therefore
{(t,
q)
:
x
−
z
K
K > γ} ⊂ {(t, q) :
K
K
tq
xK − z tK > γ} and
tq
t {(t, q) : xK − zK
> γ} < λ rK h < λ nK .
This is a contradiction.
28
6. Conclusions
In this paper we do not discuss any additional applications of our results. In
the context of pregames, the theory of large games with effective small groups
(with and without side payments) has been applied to economic models, including
exchange economies and ones with local public goods. Such applications of the
theory of large games with effective small groups are discussed in a survey of
literature in Wooders (1992b). The game theoretic framework has been especially
useful in the study of economies with collective goods and with production, as
in, for example, Bennett and Wooders (1979), Shubik and Wooders (1983) and
Barham and Wooders (1994) and papers referenced therein.
The discussion above and the model and results of this paper have been for
games with side payments. From results in Wooders (1983) it is apparent that the
nonemptiness of approximate cores can be extended to non-side-payments games
with effective small groups and substitution, without any topology on the space
of player types; this is very easy if we allow a small set of “left-over” players.2 The
market-game property is more problematic. Analogues of the result that totally
balanced games are market games, due to Shapley and Shubik (1969), have been
obtained for totally balanced games without side payments; see Billera (1970),
Billera and Bixby (1974), Hart (1982), and Mas-Colell (1975). By application
of these results for NTU games, analogues to the results of this paper can be
obtained if we allow infinite numbers of commodities in the markets or if we allow
more complicated consumption sets and two times the number of player types as
the number of commodities. With additional assumptions results closer to those
of this paper can be obtained.
References
[1] Anderson, R. M. (1978) “An Elementary Core Equivalence Theorem”, Econometrica 46, 1438-1487.
[2] Arrow, K.J. (1970) “Political and Economic Evaluation of Social Effects and
Externalities”, in The Analysis of Public Output, J. Margolis, ed. Columbia
University Press, New York and London.
2
Notes in manuscript.
29
[3] Barham, V. and M.H. Wooders (1994) “First and Second Welfare Theorems
for Economies with Collective Goods,” University of Toronto Working Paper.
[4] Bennett, E. and Wooders, M.H. (1979) ‘Income Distribution and Firm Formation’, Journal of Comparative Economics 3, 304-317.
[5] Billera, L.J. (1970) “Some Theorems on the Core of an n-Person Game without Side Payments”, SIAM Journal of Applied Mathematics, 18, 567-579.
[6] Billera, L.J. and R.E. Bixby (1974) “Market Representations of n-Person
Games”, Bulletin of the American Mathematical Society 80, 522-526.
[7] Bondareva, O.N. (1963) “Some Applications of Linear Programming Methods
to the Theory of Cooperative Games” (in Russian), Problemy Kibernetiki 10,
119-39.
[8] Hart, S. (1982) “The Number of Commodities Required to Represent a Market Game”, Journal of Economic Theory 27, 163-170.
[9] Hildenbrand, W. (1974) Core and Equilibria of a Large Economy Princeton,
NJ. Princeton University Press.
[10] Kalai, E. and E. Zemel (1982) “Totally Balanced Games and Games of Flow”,
Mathematics of Operations Research 7, 476-478.
[11] Manelli, A.M. (1991) “Core Convergence without Monotone Preferences and
Free Disposal”, Journal of Economic Theory 400-415.
[12] Mas-Colell, A. (1975) “A Further Result on the Representation of Games by
Markets”, Journal of Economic Theory 10, 117-122.
[13] Scotchmer, S. and Wooders, M.H. (1988) ‘Monotonicity in Games that Exhaust Gains to Scale’, IMSSS Technical Report No. 525, Stanford University.
[14] Shapley, L.S. (1967) “On Balanced Sets and Cores”, Navel Research Logistics
Quarterly 14, 463-460.
[15] Shapley, L.S. and M. Shubik (1969), “On Market Games”, Journal of Economic Theory 1, 9-25.
30
[16] Shapley, L.S. and M. Shubik (1975), “Competitive Outcomes in the Cores of
Market Games”, International Journal of Game Theory 4, 229-237.
[17] Shubik, M. and M.H. Wooders (1982) “Near markets and market games,”
Cowles Foundation Discussion Paper No. 657. Published as “Clubs, near
markets and market games, in Topics in Mathematical Economics and
Game Theory; Essays in Honor of Robert J. Aumann, eds. M. H. Wooders
American Mathematical Society Fields Communication Volume, on-line at
http://www.warwick.ac.uk/fac/soc/Economics/wooders/game-theory.html.
[18] Shubik, M. and Wooders, M.H. (1983) ‘Approximate Cores of Replica Games
and Economies: Part I. Replica Games, Externalities, and Approximate
Cores’, Mathematical Social Sciences 6, 27-48.
[19] Smith, A. (1776) An Inquiry into the Nature and Causes of the Wealth of
Nations. First Edition, London.
[20] Starr, R. M., “Quasi-equilibria in Markets with Nonconvex Preferences,”
Econometrica 37, 25-38.
[21] Tauman, Y. (1987) “The Aumann-Shapley Prices: A Survey”, in The Shapley Value: Essays in Honor of Lloyd S. Shapley, ed. A. Roth, Cambridge
University, Cambridge, MA.
[22] Wooders, M.H. (1978) “A Characterization of Approximate Equilibria and
Cores in a Class of Coalition Economies”, (Stony Brook Department of Economics Working Paper No. 184, Revised 1979).
[23] Wooders, M.H. (1979) ‘Asymptotic Cores and Asymptotic Balancedness of
Large Replica Games’ (Stony Brook Working Paper No. 215, Revised July
1980).
[24] Wooders, M.H. (1980) ‘The Tiebout Hypothesis: Near Optimality in Local
Public Good Economies’, Econometrica 48, 1467-1486.
[25] Wooders, M.H. (1983) “The Epsilon Core of a Large Replica Game ”, Journal
of Mathematical Economics 11, 277-300.
[26] Wooders, M.H. (1992a) “Inessentiality of Large Groups and the Approximate
Core Property; An Equivalence Theorem”, Economic Theory 2, 129-147.
31
[27] Wooders, M.H. (1992b) “Large Games and Economies with Effective Small
Groups”, Sonderforschingsbereich 303 University of Bonn Discussion Paper
No. 215, Revised September 1992 (to appear in Game Theoretic Approaches
to General Equilibrium Theory, eds. J.-F. Mertens and S. Sorin).
[28] Wooders, M.H. (1992c) “The Attribute Core, Core Convergence and Small
Group Effectiveness; The Effects of Property Rights Assignments of the
Core”, University of Toronto Department of Economics Working Paper No.
9304, forthcoming in Essays in Honor of Martin Shubik, eds. P. Dubey and
J. Geanakoplos.
[29] Wooders, M.H. (1993) “On Aumann’s Markets with a Continuum of Traders;
The Continuum, Small Group Effectiveness, and Social Homogeneity” University of Toronto Working Paper 9401.
[30] Wooders, M.H. (1994) “Equivalence of Games and Markets” Econometrica
62,1141-1160.
[31] Wooders, M.H. and W.R. Zame (1984) “Approximate Cores of Large
Games”, Econometrica 52, 1327-1350.
[32] Wooders, M.H. and W.R. Zame (1987) “Large Games; Fair and Stable Outcomes”, Journal of Economic Theory 42, 59-93.
32