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An axiomatization of the core of a cooperative game
111
Economics Letters 20 (1986) 111-115 North-Holland
AN AXIOMATIZATION Hans KEIDING
OF THE CORE OF A COOPERATIVE
GAME
*
Uniuersity of Copenhagen, DK-1455 Copenhagen K, Denmark Received
9 August
1985
In this paper we present a set of axioms on outcomes minimality condition, characterize the core.
of cooperative
games without
side payments
which,
together
with a
1. Introduction The core of a cooperative game is a solution concept which has been intensely studied and applied to a broad variety of problems in economics, social choice and other fields. The core of a game consists of the outcomes which are feasible for the grand coalition of all players and undominated in the sense that no coalition by itself can obtain something which is better for all its members. The ideas behind this definition are rather intuitive, and probably this is the main reason for the widespread use of the core in applications. It might be argued, thus, that there is little need for an axiomatization of the core, the main function of such an approach being that it gives a better understanding of the concept. However, the axioms considered in the present paper can be said to offer a new perspective on the core, showing that antimonotonicity (powerful coalitions mean small solution sets) and independence of irrelevant alternatives are crucial properties which, together with some weak axioms of technical character, characterize the core. It should be mentioned that other axiomatizations than the present one are possible, and indeed Peleg (1985) has given a characterization of the core which relies on reduced game properties. The following notational conventions will be followed: for N a finite set, 1N 1 denotes the cardinality of N. We denote by RN the set of all functions from N to the real numbers. If N= {l,..., n}, we write elements of RN as x = (xl,..., x”), and for S c N, S f o’, we write xS for the restriction of x to S. Finally, if A and B are non-empty subsets of RN, then A + B is the set (x~R~Ix=a+b,
ac.4,
DEB}.
2. Definitions In this section we introduce the class of cooperative dences assigning final results to each game. * The author
is grateful
0165-1765/86/$3.50
to Robert
Aumann
games and the abstract
and Volker Bohm for valuable
0 1986, Elsevier Science Publishers
B.V. (North-Holland)
advice on various
outcome
stages of this work
correspon-
112
H. Keiding
/
Thecoreof a cooperdoe
game
a game) is a pair (N, V), where N Definition I. A cooperative game without side payment (shorthand: is a non-empty, finite set of players and V a function assigning to each non-empty subset (coalition) S of N a subset V(S) of RS, such that for each S, V(S) is non-empty and comprehensive, i.e., if x E V(S), y E RS, and y’ 5 x’ for all i E S, then y E V(S). In applications, a game will normally satisfy some further conditions [closedness and upper boundedness of the sets V(S)]. The argument to follow could be restricted to smaller classes of games; however, this would make the formalism more tedious. Some particular games will be used in the sequel: a trivial game is a game (N, V), where V(S)={x~RS~Vi~S,x’~O}forallS~N,Sf~, N.AcornergamewithrelationtoaER”isa game(N,V”),whereV/“(S)={x~RS~3i~S,x’
correspondence
V(N)IaScN,
yE
- and one of particular
V(S),
y’>x’,
importance
for the following
iES},
but it is obvious that the notion of an outcome correspondence is a very general one, and that examples abound. For YP an outcome correspondence and (N, V) a game, we define a family (D,V)( S), S c N, S # fl, N, inductively as follows: for each S with 1S 1 = 1 we put (D,V)(S) = clV( S). Suppose that (D*V)( S’) is defined for all subsets S’ of S; if (0$‘)(Y) = 0, some S’ c S, put (D,V)( S) = 0; otherwise, let (D,V)(S) be the closed comprehensive hull of \k(S, V’), where (S, V’) is the game defined by V’( S’) = (D,V)(S’) for S’ # S, V’(S) = V(S). If (D,V)( S) # fl for all S c N, we define the q-derived game (N, D,V) of (N, V) by D,V( S) = (D,V)(S) for all S c N, S f 0, N, D,V(N) = V(N). A discussion of the ‘P-derived game will be given in connection with Axiom 5’ in the next section.
3. Axioms for outcome correspondences Below we list several conditions Axiom
1 (‘triviality’).
on outcome
correspondences.
If (N, V) is trivial and V(N)n
Ry=
{0}, then !P(N, V)=
(0).
A game of the type considered in Axiom 1 is one where no conflict is present. Every coalition can do exactly what its members can do for themselves. In view of this the outcome prescription by YP seems a very reasonable one. Axiom 2 (‘strategic equivalence’). For all a E RN, if (N, V-t {a}) V(S)+{as}, SC N, then !P(N, V+ {a})= \E(N, V)+ {a}.
is defined
by (V+
{a})(S)=
If outcome vectors are interpreted as utility payoffs to each player, Axiom 2 states that adding a constant to the utility function of each player should not change the outcome in any essential way.
H. Keiding
/ The core
113
ofa cooperafivegame
If (N, V) and (N, U) are games such that V(S) Axiom 3 (‘antimonotonicity’). S+%,and V(N)= U(N), then\k(N, U)cq(N, V).
c
U(S)
for all S
This axiom again seems quite reasonable: large sets U(S) mean that coalitions are powerful fewer of the feasible payoff vectors in U(N) = V(N) may qualify as final outcomes. Axiom S#@,
If (N, V) and (N, U) are games such that clV(S) 4 (‘continuity’). and V(N)= U(N), then \k(N, V)= \k(N, U).
= clU(S)
c
N,
so that
for all S c N,
The axiom says that taking closures of the sets V(S) for S f 0, N does not alter the outcome. This is a very weak continuity property of 9; if our argument was given in a smaller class of games, the continuity properties would have to be strengthened. For the final axiom, there are several possibilities. Axiom 5’ (‘independence of q-irrelevant alternatives’). If (N, V) is a game such that the q-derived game (N, &I’) is defined, then q( N, D,V) = ‘k( N, V). To get an understanding of Axiom 5’, it is helpful to think of outcomes as results of a bargaining procedure, where coalitions S may object to payoffs x by reference to some y in V(S) which they can enforce by themselves. In order for such an objection to be credible, y must be ‘really’ enforceable by S, that is y must not in its turn be objected against by some subcoalition of S. By the logic of our approach !I? should be used repeatedly to decide which of the elements y of V(S) are ‘really’ enforceable in the above sense. The construction of the ‘P-derived game keeps exactly such elements and excludes the non-enforceable. Thus, the axiom says that the non-enforceable options of V(S) do not count when ‘k( N, V) is determined. Another, less abstract, axiom to be used instead of Axiom 5’ is the following: Axiom 5”. Let (N, I”) Then 9( N, VO) = (0).
be a corner
game with relation
This axiom points to the important role played approach. However, the axiom itself may be difficult Theorem.
Let 9 be an outcome correspondence
to 0, and suppose
that VO( N) n It:=
(0).
by the class of corner games in the present to justify on intuitive considerations.
satisfying
either
(a) Axioms l-5’, or (6) Axioms 2-4, 5”, and such that in either case, \k is minimal for inclusion among outcome correspondences with this property. Then ?P = Core on all games (N, V) satisfying the non-levelness assumption [x E V( N)\int V(N)]
*[({x}+R,y)n V(N)= (x}]. The proof of the theorem
will be given in the next section.
4. Proof of the theorem We start by showing that the outcome some games) satisfies the axioms.
correspondence
Core (which
may have empty
values
for
114
Lemma
H. Keiding
1.
The outcome correspondence
/ The core of a cooperative gmne
Core satisfies Axioms
I-4,
5’, and 5”.
Proof
We leave the verification of Axioms 1-4 and 5”, which is quite straightforward, to the reader and show that Core satisfies Axiom 5’. Let (N, V) be a game such that (N, D,,,, V) is defined. It is clear (since Core satisfies Axiom 3) that Core(N, V) c Core(N, Dc,,, V). Suppose that x E Core( N, Dc,,,V), x P Core(N, V). Then there must be S c N, S # 0, N, and y E V(S) such that y’ > x’ for all i E S, but y @ Dc,,,V(S). Since y E V(S) and y 6GDc,,, V(S), there must be some S’ c S, S’ # 0, S, and z E Dc,,,V( S’) such that z’ > y’, all i E S’. But then z’ > x’, all i E S’, and we have that x G Core( N, D,,,,V), a contradiction. It follows that Core( N, D,,,,V) = Core( N, I’). Q.E.D. Lemma
2.
Let \k be an outcome correspondence
satisfying Axioms
1 and 5’. Then \k satisfies Axiom
5”. Proof
Let (N, V”) be a corner game with relation to 0 such that V’(N) n Ry= (0). We show that the q-derived game of (N, V”) is defined and that (D.J’)(S) = {x E RS 1x’ 6 0, i E S} for all S c N, S # 0, N. For S a one-player coalition, say S = { i }, we have (D,&‘“)({i})=clVo({i})= {X’E R]x’gO}. Suppose that ( DqVo)(S’) is defined and equal to {x E RS’ ) x’ 5 0, i E S’} for all proper subcoalitions S’ of a coalition S # N. Then the game (S, V’) with v’(Y) = ( DuVo)(S’), S’ # S, V’(S) = V(S), is trivial, and V’(S)~IRT= (0). By Axiom 1 it follows that q(S, V’)= {0}, so (D@“)(S)= {XE Rs]x’sO,
iES}.
Since (N, D,V”) is a trivial game with D,VO( N) n Ry= {0}, Axiom 1 gives that q( N, D,V”) (0). Applying Axiom 5 we finally get that \k( N, I”) = ‘k( N, D,V”) = (0). Q.E.D. By Lemma 2, case (a) of the theorem This is done in the following lemma:
=
implies case (b), so we need to prove it only in the latter case.
Lemma 3. Let \k be an outcome correspondence satisfying Axioms 2-4, satisfying the non-levelness assumption. Then Core( N, V) c ‘k( N, V).
5”, and let (N,
V) be a game
Proof If Core(N, V) = 0 there is nothing to prove. Suppose that Core(N, V) # fl and let z E Core(N, V) be arbitrary. Then z E V( N)\int V(N). Consider the game (N, V’) defined by V’(S)={~ER~]~~ES, x’
It can be seen from the proof that minimal satisfying Axioms 1-4, 5’, or of the core of (N, V). Alternatively, continuity axiom [closedness of the
the theorem can be given an alternative formulation: if \k is 2-4, 5”, then ‘k( N, V) consists of the Pareto optimal elements the non-levelness assumption might be replaced by a stronger correspondence \k for variations in V(N) at constant V(S),
S#N].
Finally, we mention that the idea behind the present approach - approximation by corner games can be used to characterize the games (N, V) for which the core is non-empty. The reader is referred to Keiding and Thorlund-Petersen (1985) for a similar argument.
H. K&ding / The core of a cooperative game
115
References Keiding, H. and L. Thorlund-Petersen. 1985, The core of a cooperative game without side payment, Discussion paper, Feb. (University of Copenhagen. Copenhagen). Peleg, B., 1985, An axiomatization of the core of cooperative games without side payment, Research memo. no. 66, Feb. (The Hebrew University of Jerusalem, Jerusalem).
Economics Letters 20 (1986) 111-115 North-Holland
AN AXIOMATIZATION Hans KEIDING
OF THE CORE OF A COOPERATIVE
GAME
*
Uniuersity of Copenhagen, DK-1455 Copenhagen K, Denmark Received
9 August
1985
In this paper we present a set of axioms on outcomes minimality condition, characterize the core.
of cooperative
games without
side payments
which,
together
with a
1. Introduction The core of a cooperative game is a solution concept which has been intensely studied and applied to a broad variety of problems in economics, social choice and other fields. The core of a game consists of the outcomes which are feasible for the grand coalition of all players and undominated in the sense that no coalition by itself can obtain something which is better for all its members. The ideas behind this definition are rather intuitive, and probably this is the main reason for the widespread use of the core in applications. It might be argued, thus, that there is little need for an axiomatization of the core, the main function of such an approach being that it gives a better understanding of the concept. However, the axioms considered in the present paper can be said to offer a new perspective on the core, showing that antimonotonicity (powerful coalitions mean small solution sets) and independence of irrelevant alternatives are crucial properties which, together with some weak axioms of technical character, characterize the core. It should be mentioned that other axiomatizations than the present one are possible, and indeed Peleg (1985) has given a characterization of the core which relies on reduced game properties. The following notational conventions will be followed: for N a finite set, 1N 1 denotes the cardinality of N. We denote by RN the set of all functions from N to the real numbers. If N= {l,..., n}, we write elements of RN as x = (xl,..., x”), and for S c N, S f o’, we write xS for the restriction of x to S. Finally, if A and B are non-empty subsets of RN, then A + B is the set (x~R~Ix=a+b,
ac.4,
DEB}.
2. Definitions In this section we introduce the class of cooperative dences assigning final results to each game. * The author
is grateful
0165-1765/86/$3.50
to Robert
Aumann
games and the abstract
and Volker Bohm for valuable
0 1986, Elsevier Science Publishers
B.V. (North-Holland)
advice on various
outcome
stages of this work
correspon-
112
H. Keiding
/
Thecoreof a cooperdoe
game
a game) is a pair (N, V), where N Definition I. A cooperative game without side payment (shorthand: is a non-empty, finite set of players and V a function assigning to each non-empty subset (coalition) S of N a subset V(S) of RS, such that for each S, V(S) is non-empty and comprehensive, i.e., if x E V(S), y E RS, and y’ 5 x’ for all i E S, then y E V(S). In applications, a game will normally satisfy some further conditions [closedness and upper boundedness of the sets V(S)]. The argument to follow could be restricted to smaller classes of games; however, this would make the formalism more tedious. Some particular games will be used in the sequel: a trivial game is a game (N, V), where V(S)={x~RS~Vi~S,x’~O}forallS~N,Sf~, N.AcornergamewithrelationtoaER”isa game(N,V”),whereV/“(S)={x~RS~3i~S,x’
correspondence
V(N)IaScN,
yE
- and one of particular
V(S),
y’>x’,
importance
for the following
iES},
but it is obvious that the notion of an outcome correspondence is a very general one, and that examples abound. For YP an outcome correspondence and (N, V) a game, we define a family (D,V)( S), S c N, S # fl, N, inductively as follows: for each S with 1S 1 = 1 we put (D,V)(S) = clV( S). Suppose that (D*V)( S’) is defined for all subsets S’ of S; if (0$‘)(Y) = 0, some S’ c S, put (D,V)( S) = 0; otherwise, let (D,V)(S) be the closed comprehensive hull of \k(S, V’), where (S, V’) is the game defined by V’( S’) = (D,V)(S’) for S’ # S, V’(S) = V(S). If (D,V)( S) # fl for all S c N, we define the q-derived game (N, D,V) of (N, V) by D,V( S) = (D,V)(S) for all S c N, S f 0, N, D,V(N) = V(N). A discussion of the ‘P-derived game will be given in connection with Axiom 5’ in the next section.
3. Axioms for outcome correspondences Below we list several conditions Axiom
1 (‘triviality’).
on outcome
correspondences.
If (N, V) is trivial and V(N)n
Ry=
{0}, then !P(N, V)=
(0).
A game of the type considered in Axiom 1 is one where no conflict is present. Every coalition can do exactly what its members can do for themselves. In view of this the outcome prescription by YP seems a very reasonable one. Axiom 2 (‘strategic equivalence’). For all a E RN, if (N, V-t {a}) V(S)+{as}, SC N, then !P(N, V+ {a})= \E(N, V)+ {a}.
is defined
by (V+
{a})(S)=
If outcome vectors are interpreted as utility payoffs to each player, Axiom 2 states that adding a constant to the utility function of each player should not change the outcome in any essential way.
H. Keiding
/ The core
113
ofa cooperafivegame
If (N, V) and (N, U) are games such that V(S) Axiom 3 (‘antimonotonicity’). S+%,and V(N)= U(N), then\k(N, U)cq(N, V).
c
U(S)
for all S
This axiom again seems quite reasonable: large sets U(S) mean that coalitions are powerful fewer of the feasible payoff vectors in U(N) = V(N) may qualify as final outcomes. Axiom S#@,
If (N, V) and (N, U) are games such that clV(S) 4 (‘continuity’). and V(N)= U(N), then \k(N, V)= \k(N, U).
= clU(S)
c
N,
so that
for all S c N,
The axiom says that taking closures of the sets V(S) for S f 0, N does not alter the outcome. This is a very weak continuity property of 9; if our argument was given in a smaller class of games, the continuity properties would have to be strengthened. For the final axiom, there are several possibilities. Axiom 5’ (‘independence of q-irrelevant alternatives’). If (N, V) is a game such that the q-derived game (N, &I’) is defined, then q( N, D,V) = ‘k( N, V). To get an understanding of Axiom 5’, it is helpful to think of outcomes as results of a bargaining procedure, where coalitions S may object to payoffs x by reference to some y in V(S) which they can enforce by themselves. In order for such an objection to be credible, y must be ‘really’ enforceable by S, that is y must not in its turn be objected against by some subcoalition of S. By the logic of our approach !I? should be used repeatedly to decide which of the elements y of V(S) are ‘really’ enforceable in the above sense. The construction of the ‘P-derived game keeps exactly such elements and excludes the non-enforceable. Thus, the axiom says that the non-enforceable options of V(S) do not count when ‘k( N, V) is determined. Another, less abstract, axiom to be used instead of Axiom 5’ is the following: Axiom 5”. Let (N, I”) Then 9( N, VO) = (0).
be a corner
game with relation
This axiom points to the important role played approach. However, the axiom itself may be difficult Theorem.
Let 9 be an outcome correspondence
to 0, and suppose
that VO( N) n It:=
(0).
by the class of corner games in the present to justify on intuitive considerations.
satisfying
either
(a) Axioms l-5’, or (6) Axioms 2-4, 5”, and such that in either case, \k is minimal for inclusion among outcome correspondences with this property. Then ?P = Core on all games (N, V) satisfying the non-levelness assumption [x E V( N)\int V(N)]
*[({x}+R,y)n V(N)= (x}]. The proof of the theorem
will be given in the next section.
4. Proof of the theorem We start by showing that the outcome some games) satisfies the axioms.
correspondence
Core (which
may have empty
values
for
114
Lemma
H. Keiding
1.
The outcome correspondence
/ The core of a cooperative gmne
Core satisfies Axioms
I-4,
5’, and 5”.
Proof
We leave the verification of Axioms 1-4 and 5”, which is quite straightforward, to the reader and show that Core satisfies Axiom 5’. Let (N, V) be a game such that (N, D,,,, V) is defined. It is clear (since Core satisfies Axiom 3) that Core(N, V) c Core(N, Dc,,, V). Suppose that x E Core( N, Dc,,,V), x P Core(N, V). Then there must be S c N, S # 0, N, and y E V(S) such that y’ > x’ for all i E S, but y @ Dc,,,V(S). Since y E V(S) and y 6GDc,,, V(S), there must be some S’ c S, S’ # 0, S, and z E Dc,,,V( S’) such that z’ > y’, all i E S’. But then z’ > x’, all i E S’, and we have that x G Core( N, D,,,,V), a contradiction. It follows that Core( N, D,,,,V) = Core( N, I’). Q.E.D. Lemma
2.
Let \k be an outcome correspondence
satisfying Axioms
1 and 5’. Then \k satisfies Axiom
5”. Proof
Let (N, V”) be a corner game with relation to 0 such that V’(N) n Ry= (0). We show that the q-derived game of (N, V”) is defined and that (D.J’)(S) = {x E RS 1x’ 6 0, i E S} for all S c N, S # 0, N. For S a one-player coalition, say S = { i }, we have (D,&‘“)({i})=clVo({i})= {X’E R]x’gO}. Suppose that ( DqVo)(S’) is defined and equal to {x E RS’ ) x’ 5 0, i E S’} for all proper subcoalitions S’ of a coalition S # N. Then the game (S, V’) with v’(Y) = ( DuVo)(S’), S’ # S, V’(S) = V(S), is trivial, and V’(S)~IRT= (0). By Axiom 1 it follows that q(S, V’)= {0}, so (D@“)(S)= {XE Rs]x’sO,
iES}.
Since (N, D,V”) is a trivial game with D,VO( N) n Ry= {0}, Axiom 1 gives that q( N, D,V”) (0). Applying Axiom 5 we finally get that \k( N, I”) = ‘k( N, D,V”) = (0). Q.E.D. By Lemma 2, case (a) of the theorem This is done in the following lemma:
=
implies case (b), so we need to prove it only in the latter case.
Lemma 3. Let \k be an outcome correspondence satisfying Axioms 2-4, satisfying the non-levelness assumption. Then Core( N, V) c ‘k( N, V).
5”, and let (N,
V) be a game
Proof If Core(N, V) = 0 there is nothing to prove. Suppose that Core(N, V) # fl and let z E Core(N, V) be arbitrary. Then z E V( N)\int V(N). Consider the game (N, V’) defined by V’(S)={~ER~]~~ES, x’
It can be seen from the proof that minimal satisfying Axioms 1-4, 5’, or of the core of (N, V). Alternatively, continuity axiom [closedness of the
the theorem can be given an alternative formulation: if \k is 2-4, 5”, then ‘k( N, V) consists of the Pareto optimal elements the non-levelness assumption might be replaced by a stronger correspondence \k for variations in V(N) at constant V(S),
S#N].
Finally, we mention that the idea behind the present approach - approximation by corner games can be used to characterize the games (N, V) for which the core is non-empty. The reader is referred to Keiding and Thorlund-Petersen (1985) for a similar argument.
H. K&ding / The core of a cooperative game
115
References Keiding, H. and L. Thorlund-Petersen. 1985, The core of a cooperative game without side payment, Discussion paper, Feb. (University of Copenhagen. Copenhagen). Peleg, B., 1985, An axiomatization of the core of cooperative games without side payment, Research memo. no. 66, Feb. (The Hebrew University of Jerusalem, Jerusalem).