Weak differential marginality and the Shapley value

HHL Working Paper
No. 155 July 2016
Weak Differential Marginality
and the Shapley Value
André Casajusa, Koji Yokoteb
a
Prof. Dr. André Casajus is a Research Professor at the Chair of Economics and Information
Systems, HHL Leipzig Graduate School of Management, Leipzig, Germany
Email: [email protected]
b
Koji Yokote is a Ph.D. student at the Graduate School of Economics, Waseda University,
Tokyo, Japan.
Abstract:
The principle of differential marginality for cooperative games states that the differential of two
players’ payoffs does not change when the differential of these players’ productivities does not
change. Together with two standard properties, efficiency and the null player property, differential marginality characterizes the Shapley value. For games that contain more than two players,
we show that this characterization can be improved by using a substantially weaker property than differential marginality. Weak differential marginality requires two players’ payoffs to
change in the same direction when these players’ productivities change by the same amount.
Weak di¤erential marginality and the Shapley value✩
André Casajusa, , Koji Yokoteb
a
b
HHL Leipzig Graduate School of Management, Jahnallee 59, 04109 Leipzig, Germany
Graduate School of Economics, Waseda University, 1-6-1, Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050,
Japan
Abstract
The principle of di¤erential marginality for cooperative games states that the di¤erential of
two players’payo¤s does not change when the di¤erential of these players’productivities does
not change. Together with two standard properties, e¢ ciency and the null player property,
di¤erential marginality characterizes the Shapley value. For games that contain more than
two players, we show that this characterization can be improved by using a substantially
weaker property than di¤erential marginality. Weak di¤erential marginality requires two
players’payo¤s to change in the same direction when these players’productivities change
by the same amount.
Keywords: TU game, Shapley value, di¤erential marginality, weak di¤erential marginality
2010 MSC: 91A12
JEL: C71, D60
1. Introduction
The Shapley value (Shapley, 1953) probably is the most eminent one-point solution
concept for cooperative games with transferable utility (TU games). Besides its original
axiomatic foundation by Shapley himself, alternative foundations of di¤erent types have been
suggested later on. Important direct axiomatic characterizations are due to Myerson (1980)
and Young (1985). Hart and Mas-Colell (1989) suggest an indirect characterization as the
marginal contributions of a potential (function).1 Roth (1977) shows that the Shapley value
can be understood as a von Neumann-Morgenstern utility. As a contribution to the Nash
program, which aims at building bridges between cooperative and non-cooperative game
theory, Pérez-Castrillo and Wettstein (2001) implement the Shapley value as the outcomes
✩
We are grateful to Yukihiko Funaki and Frank Huettner for valuable comments on this paper. Financial support by the Deutsche Forschungsgemeinschaft for André Casajus (grant CA 266/4-1) is gratefully
acknowledged. Financial support by the Japan Society for the Promotion of Science (JSPS) for Koji Yokote
is gratefully acknowledged.
corresponding author
Email addresses: [email protected] (André Casajus), [email protected] (Koji Yokote)
URL: www.casajus.de (André Casajus)
1
Calvo and Santos (1997) and Ortmann (1998) generalize the notion of a potential.
Preprint submitted to HHL Working Paper series
July 9, 2016
of the sub-game perfect equilibria of a combined bidding and proposing mechanism, which
is modeled by a non-cooperative extensive form game.2
Among the one-point solution concepts for TU games, the Shapley value can be viewed
as the measure of the players’own productivity in a game. This view is strongly supported
by Young’(1985) characterization via three properties: e¢ ciency, strong monotonicity, and
symmetry.3 E¢ ciency says that the worth generated by the grand coalition is distributed
among the players. Strong monotonicity requires a player’s payo¤ not to decrease whenever
her productivity, measured by her marginal contributions to coalitions of the other players,
weakly increases. Symmetry ensures that equally productive players obtain the same payo¤.
In order to allow for solidarity among players, Casajus and Huettner (2013) suggest a
di¤erential version of strong monotonicity called di¤erential monotonicity. This property requires two players’payo¤ di¤erential not to decrease whenever their productivity di¤erential
doesn’t decrease. Di¤erential monotonicity is a strengthened version of di¤erential marginality (Casajus, 2011), which demands equal productivity di¤erentials to translate into equal
payo¤ di¤erentials.4 Any of the afore-mentioned properties together with e¢ ciency and the
null player property characterizes the Shapley value (van den Brink, 2001, Theorem 2.5;
Casajus, 2011, Corrollary 4).
In this paper, we consider a substantial relaxation of di¤erential marginality, which we
call weak di¤erential marginality. Di¤erential marginality can be rephrased as that equal
changes in two players’ productivities should lead to equal changes in their payo¤, which
obviously implies that both payo¤s change in the same direction. Weak di¤erential marginality relaxes di¤erential marginality in this vein. Equal changes in two players’productivities
should entail that their payo¤s change in the same direction.
As our main result, we considerably improve the characterization of the Shapley value
by van den Brink (2001) and Casajus (2011). For games with more than two players, we
show that the Shapley value can be characterized by e¢ ciency, the null player property,
and weak di¤erential marginality (Theorem 2). Moreover, we provide a counterexample for
games with two players (Appendix B).
The remainder of this paper is organized as follows. In Section 2, we give basic de…nitions
and notation. In Section 3, we present our main result. Some remarks conclude this paper.
An appendix contains the proof of our main result and some complementary …ndings.
2. Basic de…nitions and notation
A (…nite TU) game on a non-empty and …nite set of players N is given by a coalition
function v 2 V (N ) := f : 2N ! R j f (;) = 0 ; where 2N denotes the power set of N .
Subsets of N are called coalitions; v (S) is called the worth of coalition S. Since we deal
2
Ju and Wettstein (2009) suggest a class of bidding mechanisms that implement several solution concepts
for TU games including the Shapley value.
3
As already mentioned by Young (1985), strong monotonicity can be relaxed into marginality, i.e., a
player’s payo¤ only depends on her own productivity.
4
Casajus (2011, Proposition 4) shows that di¤erential marginality coincides with fairness (van den Brink,
2001) on the full domain of games, for example.
2
with a …xed player set N , the latter mostly is dropped as an argument. For v; w 2 V; 2 R;
the coalition functions v + w 2 V and
v 2 V are given by (v + w) (S) = v (S) + w (S)
and ( v) (S) =
v (S) for all S N: The game 0 2 V given by 0 (S) = 0 for all S N
is called the null game. For T
N; T 6= ;; the game uT 2 V, uT (S) = 1 if T
S and
uT (S) = 0 otherwise, is called a unanimity game. Any v 2 V can be uniquely represented
by unanimity games, i.e.,
X
v=
(1)
T (v) uT ;
T N :T 6=;
where the coe¢ cients
T
(v) can be determined recursively via
X
v (S) =
for all S N:
T (v)
(2)
T S:T 6=;
Player i 2 N is called a dummy player in v 2 V if v (S [ fig) v (S) = v (fig) for all
S
N n fig ; player i 2 N is called a null player in v 2 V if v (S [ fig) = v (S) for all
S N n fig; players i; j 2 N are called symmetric in v 2 V if v (S [ fig) = v (S [ fjg) for
all S N n fi; jg.
A value on N is a mapping ' : V ! RN : The Shapley value (Shapley, 1953), Sh, is
given by
X
jT j 1 T (v)
for all v 2 V and i 2 N:
(3)
Shi (v) :=
T N :i2T
3. Weak di¤erential marginality and the Shapley value
The Shapley value satis…es a very natural fairness condition due to van den Brink (2001).
Fairness, F. For all v; w 2 V and i; j 2 N such that i and j are symmetric in w; we have
'i (v + w)
'i (v) = 'j (v + w)
'j (v) :
Fairness guarantees that if a game changes by adding another game in which two players are
symmetric, then both players’payo¤s change by the same amount. Since adding such a game
changes these players’productivities by the same amount, this property is a rather natural
requirement. Equal productivity di¤erentials should translate into equal payo¤ di¤erentials.
Casajus (2011) states this idea directly with his di¤erential marginality axiom, which is
equivalent to the fairness property on the full domain of games (his Proposition 4).5
Di¤erential marginality, DM. For all v; w 2 V and i; j 2 N such that
v (S [ fig)
v (S [ fjg) = w (S [ fig)
w (S [ fjg)
for all S
N n fi; jg ;
we have
'i (v)
'j (v) = 'i (w)
5
'j (w) :
Casajus and Huettner (2013) consider a strengthened version of di¤erential marginality called strong
di¤erential monotonicity: For all v; w 2 V and i; j 2 N such that v (S [ fig) v (S [ fjg) w (S [ fig)
w (S [ fjg) for all S N n fi; jg ; we have 'i (v) 'j (v) 'i (w) 'j (w) :
3
Together with two standard properties, e¢ ciency and the null player property, these axioms
characterize the Shapley value.
P
E¢ ciency, E. For all v 2 V; we have `2N '` (v) = v (N ) :
Null player, N. For all v 2 V and i 2 N such that i is a null player in v; we have 'i (v) = 0:
Theorem 1 (van den Brink, 2001; Casajus, 2011). The Shapley value is the unique
solution that satis…es e¢ ciency (E), the null player property (N), and fairness (F)/di¤erential
marginality (DM).
In the following, we suggest a substantial relaxation of di¤erential marginality. The
implication of di¤erential marginality can be rewritten as
'i (v)
'i (w) = 'j (v)
'j (w) :
Further, the hypothesis of di¤erential marginality is satis…ed if and only if players i and j
are symmetric in v w: Hence, di¤erential marginality can be phrased as that equal changes
in two players’productivities should translate into equal changes of their payo¤s. Of course,
this implies that their payo¤s change in the same direction, i.e.,
'i (v) ? 'i (w)
if and only if
'j (v) ? 'j (w) :
Therefore, di¤erential marginality implies the following considerably weaker property.
Weak di¤erential marginality, DM . For all v; w 2 V and i; j 2 N such that
v (S [ fig)
we have
v (S [ fjg) = w (S [ fig)
'i (v) ? 'i (w)
w (S [ fjg)
if and only if
for all S
N n fi; jg ;
'j (v) ? 'j (w) :
As our main result, we show that one can replace di¤erential marginality with weak
di¤erential marginality in Theorem 1 for jN j =
6 2:
Theorem 2. Let jN j =
6 2: The Shapley value is the unique value that satis…es e¢ ciency (E),
the null player property (N), and weak di¤erential marginality (DM ).
The proof of Theorem 2 can be found in Appendix A. Appendix B contains the counterexample to our characterization for jN j = 2: The non-redundancy of our characterization for
jN j > 2 is indicated in Appendix C.
If the null player property is strenghtened into the dummy player property in Theorem 2,
then it also holds for jN j = 2: It is straightforward to show this using the techniques employed
for establishing the induction basis within the proof of Theorem 2.
Dummy player, D. For all v 2 V and i 2 N such that i is a dummy player in v; we have
'i (v) = v (fig).
4
4. Concluding remarks
Although di¤erential marginality or di¤erential monotonicity can be used to characterize
the Shapley value, these properties show their full potential when it comes to solutions that
fail marginality or strong monotonicity. Consider, for example, the egalitarian Shapley values
(Joosten, 1996), which are the convex mixtures of the Shapley value and the equal division
value. For games with more than two players, Casajus and Huettner (2013) characterize
this class via e¢ ciency, strong di¤erential monotonicity, the null player in a productive
environment property, where the latter requires a null player to obtain a non-negative payo¤
whenever the worth generated by the grand coalition is non-negative. In view of our main
result, it seems to be interesting to explore whether in this characterization strong di¤erential
monotonicity can be relaxed in the same vein as weak di¤erential monotonicity relaxes
di¤erential monotonicity. For example, van den Brink et al. (2013) and Casajus and Huettner
(2014) use weak monotonicity, which is a relaxation of strong monotonicity, in order to
characterize the class of egalitarian Shapley values.
5. Acknowledgements
We are grateful to Yukihiko Funaki and Frank Huettner for valuable comments on this
paper. Financial support by the Deutsche Forschungsgemeinschaft for André Casajus (grant
CA 266/4-1) is gratefully acknowledged. Financial support by the Japan Society for the
Promotion of Science (JSPS) for Koji Yokote is gratefully acknowledged.
Appendix A. Proof of Theorem 2
Proof. Existence: It is well-known that Sh satis…es E and N. Since DM implies DM ,
Casajus (2011, Corollary 5) entails that Sh obeys DM .
Uniqueness: Let the solution ' meet E, N, and DM . If jN j = 1; then E already implies
' = Sh. Let now jN j > 2: For v 2 V; set
T>1 (v) := fT
N j jT j > 1 and
T
(v) 6= 0g :
We show ' = Sh by induction on jT>1 (v)j : For T 2 T>1 (v) ; let vT 2 V be given by
vT := v
T
(v) uT +
By construction, (*) jT>1 (vT )j = jT>1 (v)j
jT j > 1, let uT 2 V be given by
(v) X
uf`g :
jT j
`2T
T
1 and (**) v (N ) = vT (N ) : Further, for T
uT := jT j uT
Note that Shi (uT ) = 0 for all i 2 N:
5
X
`2T
uf`g :
(A.1)
N;
(A.2)
v
Induction basis: Let v 2 V be such that jT>1 (v)j 1: There are
and T v N; jT v j > 1 such that
X
v
v = v uT v +
` uf`g :
2 RN and
v
2 R,
`2N
Set Rv := fi 2 N j
a number of claims.
v
Claim 1, C1: If
v
i
6= 0g : We show that ' (v) = Sh (v) for all v 2 V with jT>1 (v)j
1 by
= 0 and Rv 6= N , then ' (v) = Sh (v) :
There exists some k 2 N n Rv : By E and N, we have
v
i
'k (v) = 0 = 'k
ufig = Shk (v)
for all i 2 N n fkg
(A.3)
and
'i
v
i
for all i 2 N:
(A.4)
for all i 2 N n fkg :
(A.5)
ufig = Shi (v)
By (A.3) and DM applied to i and k, we have
v
i
'i (v) = 'i
ufig
Now, the claim drops from (A.3), (A.4), and (A.5).
v
Claim 2, C2: If
= 0, then ' (v) = Sh (v) :
Suppose ' (v) 6= Sh (v). By E, there are i; j 2 N such that
'i (v) > Shi (v)
and
Let k 2 N n fi; jg : By DM and since i and j are symmetric in
'i (v) ? 'i v
C1
k
(A.6)
'j (v) < Shj (v) :
ufkg = Shi v
k
ufkg , we have
k
ufkg = Shi (v)
k
ufkg = Shj (v) ;
if and only if
'j (v) ? 'j v
C1
k
ufkg = Shj v
which contradicts (A.6).
Claim 3, C3: If jRv j = 0, then ' (v) = Sh (v) :
Note that v =
v
uT v : By N, we have
for all i 2 N n T v :
'i (v) = 0 = Shi (v)
(A.7)
Since all players in T v are pairwise symmetric in v and by N and DM ; we thus have
'i (v) ? 'i (0) = 0 if and only if 'j (v) ? 'j (0) = 0
By
v
for all i; j 2 T v :
uT v (N ) = 0; E, (A.7), and (A.8), we have 'i (v) = 0 = Shi (v) for all i 2 T v :
6
(A.8)
Claim 4, C4: If Rv [ T v 6= N; then ' (v) = Sh (v) :
We prove the claim by induction on jRv j :
Induction basis C4 : For jRv j = 0; the claim holds by C3.
Induction hypothesis C4 : Let the claim hold for jRv j t:
Induction step C4 : Let jRv j = t + 1: Suppose ' (v) 6= Sh (v). By E, there are i; j 2 N
such that
'i (v) > Shi (v)
and
'j (v) < Shj (v) :
(A.9)
By N, we have i; j 2 Rv [ T v :
Case (i): Suppose i; j 2 Rv n T v or i; j 2 T v : By C2, we have
' (v
v
uT v ) = Sh (v
v
uT v ) :
(A.10)
By (A.9) and (A.10), we further have
'i (v)
'i (v
v
uT v ) > Shi (v)
Shi (v
v
uT v ) = Shi (
v
uT v ) = 0
'j (v)
'j (v
v
uT v ) < Shj (v)
Shj (v
v
uT v ) = Shj (
v
uT v ) = 0:
and
v
Since i and j are symmetric in
uT v ; this contradicts DM .
Case (ii): Suppose, w.l.o.g., i 2 Rv n T v and j 2 T v : By Induction hypothesis C4 , we
have
v
v
'i v
(A.11)
i ufig = Shi v
i ufig :
By assumption, there exists k 2 N n (Rv [ T v ) : By (A.9) and (A.11), we have
'j (v)
'j v
v
i
ufig < Shj (v)
Shj v
v
i
ufig = Shj
v
i
ufig = 0:
(A.12)
By N, we have
v
i
'k (v) = 0 = 'k v
Since j and k are symmetric in
Claim 5, C5: If jT v n Rv j
v
i
(A.13)
ufig :
ufig ; (A.12) and (A.13) contradict DM :
1; then ' (v) = Sh (v) :
Suppose ' (v) 6= Sh (v). By E, there are i; j 2 N such that
and
'i (v) > Shi (v)
(A.14)
'j (v) < Shj (v) :
By N, we have i; j 2 Rv [ T v :
Case (i): Suppose i; j 2 Rv n T v or i; j 2 T v : By C2, we have
' (v
v
uT v ) = Sh (v
v
uT v ) :
v
uT v ) = Shi (
(A.15)
By (A.14) and (A.15), we further have
'i (v)
'i (v
v
uT v ) > Shi (v)
Shi (v
7
v
uT v ) = 0
and
'j (v)
'j (v
v
uT v ) < Shj (v)
v
Shj (v
uT v ) = Shj (
v
uT v ) = 0:
v
Since i and j are symmetric in
uT v ; this contradicts DM .
Case (ii): Suppose, w.l.o.g., i 2 Rv n T v and j 2 T v :
v
v
Case (ii-a): Suppose j 2 T v n Rv : Let w := v
uT v
u(T v nfjg)[fig : By C4, we
have
' (w) = Sh (w) :
(A.16)
By (A.14) and (A.16), we further have
'i (v)
'i (w) > Shi (v)
Shi (w) = Shi
v
uT v +
v
u(T v nfjg)[fig = 0
'j (v)
'j (w) < Shj (v)
Shj (w) = Shj
v
uT v +
v
u(T v nfjg)[fig = 0:
and
v
v
Since i and j are symmetric in
uT v
u(T v nfjg)[fig ; this contradicts DM .
v
v
Case (ii-b): Suppose j 2 T \ R : By assumption, there exists k 2 T v n Rv such that
k 6= i and k 6= j: By C2, we have
v
'j (v
v
uT v ) = Shj (v
Since j and k are symmetric in
'j (v)
'j (v
v
and
uT v )
v
v
'k (v
v
uT v ) = Shk (v
uT v ) :
(A.17)
uT v ; by (A.14) and C2, we have
uT v ) < Shj (v)
v
Shj (v
uT v ) = Shj (
v
uT v ) = 0:
By DM ; we further have
'k (v)
Since
v
'k (v
v
'k (v
uT v ) < 0:
(A.17)
v
uT v ) = Shk (v
uT v ) = Shk (v) ;
we obtain
(A.18)
'k (v) < Shk (v) :
Let z = v
'i (v)
v
uT v
v
u(T v nfkg)[fig : By (A.14), (A.18), and C4, we have
'i (z) > Shi (v)
Shi (z) = Shi
'k (z) < Shk (v)
Shk (z) = Shk
v
uT v +
v
u(T v nfkg)[fig = 0
(A.19)
u(T v nfkg)[fig = 0:
(A.20)
and
'k (v)
v
uT v +
v
Since i 2 Rv n T v and k 2 T v n Rv ; i and k are symmetric in
Hence, (A.19) and (A.20) contradict DM :
Claim 6, C6: If Rv [ T v = N; then ' (v) = Sh (v) :
8
v
uT v
v
u(T v nfkg)[fig :
If jT v n Rv j 1; then the claim drops from C5. Let now jT v n Rv j = 0: By assumption,
we have Rv = N: Suppose ' (v) 6= Sh (v). By E, there are i; j 2 N such that
'i (v) > Shi (v)
and
(A.21)
'j (v) < Shj (v) :
Let k 2 N n fi; jg : If k 2 N n T v ; then C4 implies
' v
k
ufkg = Sh v
k
(A.22)
ufkg :
If k 2 T v ; then C5 implies (A.22). By (A.21) and (A.22), we have
'i (v)
'i v
k
ufkg > Shi (v)
Shi v
k
ufkg = Shi
k
ufkg = 0
'j (v)
'j v
k
ufkg < Shj (v)
Shj v
k
ufkg = Shj
k
ufkg = 0:
and
Since i and j are symmetric in
k ufkg ; this contradicts DM :
Note that the induction basis is proved by C4 and C6.
Induction hypothesis: Let the claim hold for all v 2 V such that jT>1 (v)j
t; t 2 N;
1:
Induction step: Let now v 2 V be such that jT>1 (v)j = t + 1: There exist S; T 2 T>1 (v)
such that S 6= T: By (3), (A.1), (*), and the induction hypothesis, we have
t
' (vS ) = Sh (vS ) = Sh (v) = Sh (vT ) = ' (vT ) :
(A.23)
Case (i): S \ T 6= ;: W.l.o.g., S n T 6= ;: Let i 2 S \ T and j 2 S n T: By (A.23) and
DM , we have
'` (v) ? Sh` (v)
'` (v) ? Sh` (v)
'` (v) ? Sh` (v)
if and only if 'i (v) ? Shi (v)
if and only if 'i (v) ? Shi (v)
if and only if 'j (v) ? Shj (v)
for all ` 2 S;
for all ` 2 T;
for all ` 2 N n T;
and therefore
'` (v) ? Sh` (v)
if and only if 'i (v) ? Shi (v)
for all ` 2 N:
(A.24)
Case (ii): S [ T 6= N: W.l.o.g., S n T 6= ;: Let i 2 N n (S [ T ) and j 2 S n T: By (A.23)
and DM , we have
'` (v) ? Sh` (v)
'` (v) ? Sh` (v)
'` (v) ? Sh` (v)
if and only if 'i (v) ? Shi (v)
if and only if 'i (v) ? Shi (v)
if and only if 'j (v) ? Shj (v)
for all ` 2 N n S;
for all ` 2 N n T;
for all ` 2 S;
and therefore
'` (v) ? Sh` (v)
if and only if 'i (v) ? Shi (v)
9
for all ` 2 N:
(A.25)
Case (iii): S \ T = ; and S [ T = N: Hence, T>1 (v) = fS; T g : Let i 2 S, j 2 T; and
w 2 V be given by
w = vS
S
(v) u(Snfig)[fjg +
(v)
jSj
S
X
`2(Snfig)[fjg
(A.26)
uf`g :
By construction, we have T>1 (w) = f(S n fig) [ fjg ; T g and (****) v (N ) = w (N ) : In view
of Case (i), we have (*****) ' (w) = Sh (w) :
Since i and j are symmetric in v w and by DM and (A.23), we have
(*****)
(*****)
'j (v) ? 'j (w) = Shj (v) if and only if 'i (v) ? 'i (w) = Shi (v) ;
'` (v) ? Sh` (v) if and only if 'i (v) ? Shi (v)
for all ` 2 S;
'` (v) ? Sh` (v) if and only if 'j (v) ? Shj (v)
for all ` 2 T;
and therefore
'` (v) ? Sh` (v)
if and only if 'i (v) ? Shi (v)
for all ` 2 N:
(A.27)
Finally, (A.24), (A.25), (A.27), and E imply ' (v) = Sh (v) :
Appendix B. Counterexample to Theorem 2 for jN j = 2
by
Theorem 2 fails for jN j = 2: Let N = f1; 2g : Consider the solution '~ : V ! R2 given
8
(Sh1 (v) ; Sh2 (v)) ;
>
>
>
>
Sh2 (v) Sh2 (v)
>
>
;
Sh1 (v) +
>
>
>
2
2
>
>
>
Sh
(v)
Sh
(v)
>
1
1
>
; Sh2 (v) +
<
2
2
~
'~
1 (v) ; '2 (v) =
(Sh
(v)
;
Sh
(v))
;
>
1
2
>
>
>
Sh
>
2 (v) Sh2 (v)
>
Sh1 (v) +
;
>
>
>
2
2
>
>
>
Sh
(v)
Sh
(v)
>
1
1
>
; Sh2 (v) +
:
2
2
Sh1 (v)
0; Sh2 (v)
0;
; Sh1 (v) > 0; Sh2 (v) < 0; v (N )
0;
; Sh1 (v) > 0; Sh2 (v) < 0; v (N ) < 0;
Sh1 (v)
0 Sh2 (v)
0;
; Sh1 (v) < 0, Sh2 (v) > 0; v (N )
0;
; Sh1 (v) < 0, Sh2 (v) > 0; v (N ) < 0
for all v 2 V: One easily checks that '~ 6= Sh and that '~ inherits E, N, and DM from
Sh.
Appendix C. Non-redundancy of Theorem 2 for jN j > 2
Our characterization is non-redundant for jN j > 2. The value 'E given by 'E
i (v) = 0
for all v 2 V and i 2 N satis…es N and DM but not E. The value 'N given by 'N
i (v) =
10
v (N ) = jN j for all v 2 V and i 2 N satis…es E and Mo but not N. For v 2 V; let
N0 (v) := fi 2 N j i is a null player in vg : The value 'DM given by
8
v (N )
<
; i 2 N n N0 (v) ;
DM
(v) =
'i
for all v 2 V and i 2 N
jN n N0 (v)j
:
0;
i 2 N0 (v)
satis…es E and N but not DM .
References
van den Brink, R., 2001. An axiomatization of the Shapley value using a fairness property. International
Journal of Game Theory 30, 309–319.
van den Brink, R., Funaki, Y., Ju, Y., 2013. Reconciling marginalism with egalitarism: Consistency,
monotonicity, and implementation of egalitarian Shapley values. Social Choice and Welfare 40, 693–714.
Calvo, E., Santos, J. C., 1997. Potentials in cooperative TU games. Mathematical Social Sciences 34, 175–
190.
Casajus, A., 2011. Di¤erential marginality, van den Brink fairness, and the Shapley value. Theory and
Decision 71 (2), 163–174.
Casajus, A., Huettner, F., 2013. Null players, solidarity, and the egalitarian Shapley values. Journal of
Mathematical Economics 49, 58–61.
Casajus, A., Huettner, F., 2014. Weakly monotonic solutions of cooperative games. Journal of Economic
Theory 154, 162–172.
Hart, S., Mas-Colell, A., 1989. Potential, value, and consistency. Econometrica 57 (3), 589–614.
Joosten, R., 1996. Dynamics, equilibria and values. Ph.D. thesis, Maastricht University, The Netherlands.
Ju, Y., Wettstein, D., 2009. Implementing cooperative solution concepts: a generalized bidding approach.
Economic Theory 39, 307–330.
Myerson, R. B., 1980. Conference structures and fair allocation rules. International Journal of Game Theory
9, 169–182.
Ortmann, K. M., 1998. Conservation of energy in value theory. Mathematical Methods of Operations Research 47, 423–449.
Pérez-Castrillo, D., Wettstein, D., 2001. Bidding for the surplus : A non-cooperative approach to the Shapley
value. Journal of Economic Theory 100 (2), 274–294.
Roth, A. E., 1977. The Shapley value as a von Neumann-Morgenstern utility. Econometrica 45, 657–664.
Shapley, L. S., 1953. A value for n-person games. In: Kuhn, H., Tucker, A. (Eds.), Contributions to the
Theory of Games. Vol. II. Princeton University Press, Princeton, pp. 307–317.
Young, H. P., 1985. Monotonic solutions of cooperative games. International Journal of Game Theory 14,
65–72.
11
ISSN 1864-4562 (Online version)
© HHL Leipzig Graduate School of Management, 2016
The sole responsibility for the content of the HHL Working Paper lies with the author/s.
We encourage the use of the material for teaching or research purposes with reference to the
source. The reproduction, copying and distribution of the paper for non-commercial purposes
is permitted on condition that the source is clearly indicated. Any commercial use of the document or parts of its content requires the written consent of the author/s.
For further HHL publications see www.hhl.de/publications

Download Report