Aggregate monotonicity of cooperative solutions is widely accepted as a desirable property, and examples where certain solution concepts (such as the nucleolus) violate this property are scarce and have no economic interpretation. We provide an example of a simple four-player game that points out at a class of economic contexts where aggregate monotonicity is not appealing.Cooperative games, aggregate monotonicity, axiomatic solution, core, Shapley value, nucleolus

Andriy Zapechelnyuk

Yair Tauman

English

Research Papers in Economics

On (non-) monotonicity of cooperative
solutions
Yair Taumany
Tel Aviv University and Stony Brook University
Andriy Zapechelnyukz
University of Bonn and KSE-KEI
October 22, 2009
Abstract
Aggregate monotonicity of cooperative solutions is widely accepted
as a desirable property, and examples where certain solution concepts
(such as the nucleolus) violate this property are scarce and have no
economic interpretation. We provide an example of a simple four-
player game that points out at a class of economic contexts where
aggregate monotonicity is not appealing.
Keywords: Cooperative games, aggregate monotonicity, axiomatic
solution, core, Shapley value, nucleolus
JEL classication: C71, C78
We thank two anonymous referees for helpful comments.
y Recanati School of Business, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel
z University of Bonn, Economic Theory II, Lenn estrasse 37, 53113 Bonn, Germany.
E-mail: zapechelnyuk@hcm.uni-bonn.de
11 Introduction
A solution of a cooperative game is said to be aggregate monotonic (Megiddo,
1974) if no player is worse o whenever the worth of the grand coalition
increases while the worth of every other coalition remains unchanged. Ag-
gregate monotonicity is a very mild requirement that is broadly considered
a desirable and natural property. Among well known solution concepts, the
Shapley value, the egalitarian (equal division) rule,1 the core (on the class
of games with nonempty core) and the per-capita nucleolus (Grotte, 1970;
Young et al., 1982) are aggregate monotonic (though the last two violate a
slightly stronger requirement of coalitional monotonicity2), while the nucle-
olus (Schmeidler, 1969) and the kernel (Davis and Maschler, 1965) are not
(see Megiddo, 1974; Hokari, 2000).3 Maschler (1992) comments a lack of
aggregate monotonicity of the nucleolus:
\This is certainly an undesirable feature, and it bothered some
people. One has a feeling that in any \fair" outcome all players
should benet if v(N) [the worth of the grand coalition] increases
and other coalitions stay put. For that reason, there was a sug-
gestion (Young et al., 1982) to use the per-capita nucleolus, which
yields a monotonic one-point outcome in the core for games with
a nonempty core. This is not going to be of much help, because
even the per-capita nucleolus does not satisfy a slightly stronger,
but not less intuitive coalitional monotonicity property. [...]
1 The egalitarian solution equally divides the worth of the grand coalition among the
players.
2 A solution is said to be coalitional monotonic (Young, 1985) if for any coalition no
member of that coalition is worse o whenever its worth increases while the worth of every
other coalition remains unchanged.
3 Hokari (2000) shows that on the class of convex games the nucleolus is not aggregate
monotonic. The same statement applies to the kernel, since on this class of convex games
these solution concepts coincide (Maschler et al., 1971).
2Surprisingly, Young (1985) proves that for the class of games
with nonempty core there does not exist a one-point coalitional
monotonic solution which always lies in the core. [...] There
is no escape from this fact: if you want a unique outcome in the
core, you must face some undesirable monotonicity consequences.
On the other hand, if you feel that monotonicity is essential, say,
because it \provides incentives" if imposed on a society (Young,
1985), then you should sometimes discard the core, and the nu-
cleolus is not a solution concept that you should recommend."
(Maschler, 1992, pp. 613{614)
Whether there is a trade-o between monotonicity and other desirable
properties of a solution or not depends on the context from which a coop-
erative game arises. Some contexts may narrow down the class of games
to a subclass where the nucleolus is monotonic. Other contexts may prove
the monotonicity requirement completely unreasonable, for instance, Moulin
and Thomson (1988) show that in an exchange economy the resource mono-
tonicity of a solution is incompatible with Pareto optimality and some weak
requirements of fair division. Aumann (2000) argues:
\Most axioms appearing in axiomatizations do seem reason-
able on the face of it, and many of them are in fact quite com-
pelling. The fact that a relatively small selection of such axioms
is often categoric (determines a unique solution concept), and
that dierent such selections yield dierent answers, implies that
all together, these reasonable sounding axioms are contradictory.
[...] Any given kind of a counterintuitive example can be elim-
inated by an appropriate choice of solution concept, but only
at the cost of another quirk turning up. Dierent solution con-
cepts can therefore be thought of as results of choosing not only
3which properties one likes, but also which examples one wishes
to avoid." (Aumann, 2000, p. 77)
Unfortunately, the examples of non-monotonicity of solution concepts in
the literature (Megiddo, 1974; Young, 1985; Hokari, 2000) can hardly be put
in any economic context. They serve as warnings rather than tools that help
one to understand whether the monotonicity property is crucial for one's
needs. In this note we present a simple example with a clear economic inter-
pretation where the aggregate monotonicity property of a solution concept
is not convincing. The purpose of this note is not just to provide another
example where the nucleolus is not aggregate monotonic, or to argue that
sometimes the nucleolus is more appealing than the Shapley value. Instead,
our main goal is to point out at a certain class of simple economic contexts
or settings where the monotonicity property of a solution concept is less
appealing than it may seem at rst glance.
2 The Example
A game in coalitional form (a cooperative game) is a pair (N;v), where func-
tion v associates with every coalition of players S  N its worth v(S), with
the convention v(?) = 0. Let V be a class of games in coalitional form. A
solution on class V is a mapping  that maps every game (N;v) 2 V to a
payo vector (N;v) for all players in N.
A solution  is said to be aggregate monotonic if whenever two games
(N;v0) and (N;v00) satisfy v00(N)  v0(N) and v00(S) = v0(S) for all S   N,
the solution assigns to every player in v00 at least as high payo as in v0, i.e.,
i(N;v00)  i(N;v0) for all i 2 N.
Let N = f0;1;2;3g be the set of players, where player 0 is an employer
who possesses a production technology and the other players are employees
who use this technology to produce output. The employer on his own can
4produce zero units of output, but if he hires k workers (k = 1;2;3), they can
produce f(k) units. We dene a cooperative game, (v;N), as follows: every
coalition that contains either the employer alone or the workers without the
employer has zero worth; every coalition that contains the employer and k
workers has worth f(k), i.e., v(S) = f(k) if S 3 0 and jSj = k+1, k = 1;2;3,
and otherwise v(S) = 0.
Let us compare two production functions, f0 and f00. The rst production
function is f0(1) = 1 and f0(2) = f0(3) = 2, that is, the total production is the
same whether there are two or three workers. This denes a game (N;v0) as
follows: v0(0;i) = 1 and v0(0;i;j) = v0(N) = 2 for all distinct i;j 2 f1;2;3g,
and v0(S) = 0 otherwise. The second production function is f00(k) = k,
that is, every worker is able to produce one unit independently of how many
workers are employed. Now the game is given by v00(S) = jSj   1 whenever
S 3 0 and otherwise v00(S) = 0.4
The employer and the workers can be considered as complementary inputs
(capital and labor) of a production technology. In the rst case, v0, there is
a shortage of capital, and the employer can press the wages down by playing
out the workers against each other. In contrast, in the second case, v00,
there is no shortage of capital, thus the employer and the workers are on
equal terms in negotiations. Since the bargaining position of the employer is
signicantly stronger in the rst situation, it seems plausible that he should
obtain a higher payo in v0. However, any solution concept that assigns to the
employer a higher payo in v0 than in v00 violates aggregate monotonicity, as
v0(N) < v00(N) and v0(S) = v00(S) for all S   N. The aggregate monotonicity
requirement in this context is less appealing.
4 These two games belong to the class of glove market games (Shapley, 1959; Apartsin
and Holzman, 2003). We thank an anonymous referee for pointing out this fact.
53 Analysis
We will now analyze how some solution concepts perform in the above ex-
ample. Let N = f0;1;2;3g, let 0  z  1, and suppose that vz is given
by vz(0;i) = 1, vz(0;i;j) = 2 for all distinct i;j 2 f1;2;3g, vz(N) = 2 + z,
0  z  1, and vz(S) = 0 for every other coalition S. Note that the func-
tions v0 and v00 in the above example coincide with vz for z = 0 and z = 1,
respectively.
The core5 of (N;vz) contains payo vectors that allocate to every worker
i = 1;2;3 a payo between 0 and z and to the employer the rest of the
surplus, i.e.,
C(N;vz) =
(
(x0;x1;x2;x3)
 
  
xi 2 [0;z]; i = 1;2;3;
x0 = 2 + z   x1   x2   x3:
 
  
)
Thus, with z = 0 the core is a singleton, C(N;v0) = f(2;0;0;0)g, the unique
payo vector in the core assigns zero to each worker and the entire surplus,
2, to the employer. For every z > 0, C(N;v0) is not a singleton and con-
tains payo vectors that assign positive payos to all workers. Notice that
the core does not violate aggregate monotonicity on the set of games under
consideration, since there is a selection in the core that is weakly increasing
in z.6 A simple example is the allocation (2 + z=4;z=4;z=4;z=4) that yields
a strict improvement to all players as z goes up.
The Shapley value is given by Sh0(N;vz) = 5
4 + z
4 and Shi(N;vz) = 1
4 + z
4,
i = 1;2;3, and the per-capita nucleolus yields PN 0(N;vz) = 2 + z
4 and
PN i(N;vz) = z
4, i = 1;2;3. These two solution concepts assign increasing
payos to all players as the worth of the grand coalition, vz(N) = 2 + z,
grows, thus obeying the aggregate monotonicity requirement. In contrast,
5 We omit the denition of the core, as well as other solution concepts, referring to the
classical literature, e.g., Maschler (1992).
6 A set-valued solution is said to be aggregate monotonic if it possesses a single-valued
selection that is aggregate monotonic.
6the nucleolus, which yields N0(N;vz) = 2   z
2 and Ni(N;vz) = z
2, i = 1;2;3,
is not aggregate monotonic, since the payo to the employer decreases as
vz(N) goes up. The kernel is not aggregate monotonic either: the described
set of games (N;vz), 0  z  1, belongs to the class of clan games where the
nucleolus is a unique kernel element (Potters et al., 1989; Arin and Feltkamp,
1997).
Recall that a higher value of z is less favorable for the employer: every
worker knows that by refusing to work she can make the others lose z, so
she can exert pressure on the employer in negotiations, and the pressure is
higher when z is larger. One can therefore expect from a solution to account
for this attribute of the problem by assigning smaller payos to the employer
for higher values of z, as the nucleolus and the kernel do. Thus, one who
is convinced that the discussed attribute of the solution is essential should
abolish monotonicity and avoid monotonic solutions, such as the Shapley
value and the per-capita nucleolus.
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9

Can everyone bene from growth? Two diculties.

Clan games.

Collected Papers.

Computation of and observations on the nucleolus and the central games.

Cost allocation in water resources development.

Monotonic solutions of cooperative games.

On the nonmonotonicity of the bargaining set, the kernel and the nucleolus of a game.

On (Non-) Monotonicity of Cooperative Solutions

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