Jun 26, 2009 - yields #e'#s',l($2*,e(#s',l($ 2 *$. Here, agent i could disagree with this payoff by arguing that if she changed her action to l', then she could inflict ...
The Value for Action-set Games Emilio Calvo Amparo Urbano University of Valencia. Departament of Economic Analysis and ERI-CES. Spain June 26, 2009
Abstract Action-set games are games where the set of players is …nite, every player has a …nite set of actions, and the worth of the game is a function of the actions taken by the players. In this setting a rule determines individual payo¤s for each combination of actions. Following an axiomatic approach, we de…ne the set of Consistent Bargaining Equilibria.
Keywords: Action-set games, Shapley value, Prekernel, Consistent Bargaining Equilibria.
1
Action-set games
One of the features common to most economic situations is that the interaction among agents in activities like production, exchange, etc. generates bene…ts that are shared among the participating agents. Moreover, the productivity gains from specialization in di¤erent tasks and roles (labor division) are important within any type of production process, ranging from pin manufacture to legal practice and medical care. When the production process is performed by the combination of di¤erent tasks, cooperative agents must agree on how to assign tasks to agents and how to share the outcome between them. These two decisions are not independent because the rewards assigned to each task will condition that which would like to be chosen by each agent. We illustrate this setting with the help of the following example. Example 1. Suppose that two friends, say i and j, go to deer hunting. Each of them can either line driving (action l), i.e., ‡ushing deers toward the hunter, or shooting deers (action s). The best hunting strategy is that one of them line drives and the other shoots; if both line drive, then no deer will be shooted, and, …nally, if both shoot deers, then less deers will be shooted, because nobody will ‡ush them to the hunters. Suppose in addition that they only own one ri‡e; that, for simplicity, the number of deers shooted is proportional to the shooting time, and that if both want to shoot they must agree on the redistribution of time. Payo¤s are represented by the following matrix. v
sj
lj
si
4
8
li
6
0
1
The matrix is asymmetric, stressing that the hunting team combination (si ; lj ) is better than (li ; sj )1 . An example of a rule that, a priori, could be appealing is the equal payo¤ division , which in principle is a good candidate for a fair rule. Such a rule yields the payo¤ matrix: ( i; j )
sj
lj
si
(2; 2)
(4; 4)
li
(3; 3)
(0; 0)
Suppose now that the agents choose the pair of actions (si ; lj ), where the equal payo¤ division rule yields ( i (si ; lj ) = 4; j (si ; lj ) = 4). Here, agent i could disagree with this payo¤ by arguing that if she changed her action to li , then she could in‡ict a loss on agent j of : j (si ; lj )
j (li ; lj )
=4
0=4
This objection cannot be balanced by a counterobjection of j, consisting of a deviation from lj to sj , because the loss on agent i’s payo¤s that agent j could in‡ict is: i (si ; lj )
If we wish a rule
i (si ; sj )
=4
2=2
satisfying that the above kind of objections and counterobjections be balanced
at (si ; lj ), the payo¤s must be
i (si ; lj )
= 5;
j (si ; lj )
= 3 . Moreover, at (li ; sj ) the agents are in reverse
position and therefore the payo¤s ( i (li ; sj ) = 3; j (li ; sj ) = 3) must change to
i (li ; sj )
= 2;
j (li ; sj )
=4
in order to be balanced. Finally, at (si ; sj ) and (li ; lj ) no player can threat to in‡ict a payo¤ loss on another player by a unilateral deviation of her action, so there is no reason to readjust the corresponding payo¤s. To sum up, the payo¤ matrix [ (x)] should be sj
lj
si
(2; 2)
(5; 3)
li
(2; 4)
(0; 0)
i;
j
We would like to stress that the determination of such a rule
is an independent and previous issue to
what pro…le of actions will be …nally taken by the players. In our example, perhaps the players’priority is to maximize the number of deers shooted and then they will choose (si ; lj ). Then, our claim is that the redistribution of deers will be (si ; lj ) = (5; 3). Alternatively, we can think that they are risk-neutral utility maximizers and that they agree to play a fair lottery [ (si ; lj ); (1 were the lottery played, the payo¤s would be
)(li ; sj )]. In that case if (li ; sj )
(li ; sj ) = (2; 4).
Once [ (x)] has been obtained, we may try then to determine what action pro…le will …nally be taken, based either on evolutionary dynamic arguments, or on utilitarian/egalitarian considerations, or on strategic bargaining models. Our goal is to show the existence of a rule
such that given a payo¤ matrix [v], the payo¤s at each
action pro…le x are balanced against individual deviations as described above. For that purpose we follow an axiomatic approach, that is, we impose some properties which rationalize the behavior of the agents involved in the bargaining process. 1 This
example can be seen as an illustration of a two-agent cooperation problem under a simple joint production
technology with two heterogeneous working types.
2
2
Axiomatics
Formally, an action-set game is any
= N; Ai
i2N
i
; v , where N is a …nite set of players, with jN j = n,
and for any player i in N , A is a …nite non empty set of actions available for player i. An action pro…le a = (ai )i2N is a combination of actions that the players in N might choose, and denote the set of all Ai by A.
possible action pro…les i2N
For any action pro…le x = (xi )i2N in A, the real number v (x) represents the total worth that players would get if x where the combination of the actions taken by the players. Therefore, v is a mapping v : A ! R+ . Given an action-set game
= (N; A; v), a value solution
is a mapping that speci…es a payo¤ vector
for every action pro…le, that is : A ! RN The number
i (x; v)
represents the payo¤ that player i 2 N receives when action pro…le x 2 A is taken
in . When no confusion arises, we denote such a number by P payo¤ vector (x) is feasible if i2N i (x) v(x).
i (x).
As utility is totally transferable, a
The three main features characterizing our setting are:
1.- Unanimity. The …nal agreements must be unanimous; hence partial cooperation is not allowed and strict subcoalitions of agents play no role in this setting. 2.- No disagreement point. The agents cannot take any particular action to guarantee themselves a minimum payo¤. Whatever the actions taken by the players, they must agree on the redistribution of the outcome. 3.- Transferable utility. The outcome is a totally divisible good which can be redistributed among agents. The …rst requirement that a rule De…nition 1 E¢ ciency:
P
i2N
i (x)
must satisfy is the budget constraint: = v(x), for all x 2 A
To illustrate the next property suppose that players are involved in bargaining at some x, and some agent i disagrees with the proposal at hand. She can reject that proposal by threatening to change her action to a di¤erent yi 2 Ai , where v(x i ; yi ) < v(x). In this way, she can impose a loss on the other players as far as they do not take her claim into account. Note however that, if players were in action pro…le x where it is impossible for a player to lower any other player’s payo¤ by an action change, then no player would make this kind of threat. Assuming that players have equal bargaining skills, payo¤s must be the same for all of them at the worst situation. De…nition 2 Equal minimum rights: Let x 2 A such that v(x) = M inx2A fv(x)g, then
i (x)
=
j (x),
for all i; j 2 N . Therefore, individual payo¤s at the worst possible outcome of the game act as a reference point for the remaining possible agreements. 3
Moreover, since unanimity of any agreement is a desirable property, then it is required that every player obtains at least as much as she obtains at the reference point. De…nition 3 Individual rationality: Let x 2 A such that v(x) = M inx2A fv(x)g, then for all x 2 A, i (x)
i (x),
for all i 2 N .
Finally, we are looking for a rule with the property that at any action pro…le, agent i’s threat to agent j -by unilaterally deviating from her action- is balanced by agents j’s counterthreat to agent i. This property is called Equal Punishments. We de…ne the concept of "punishment" as follows: De…nition 4 Given a value
, the punishment that player i can in‡ict on player j at x, for each x 2 A,
is given by Pij (x) :=
j (x)
min
yij 2Ai
j (x i ; yij ):
The amount Pij (x) measures the maximum payo¤ losses that player i can in‡ict on player j at x. Note that since minyij 2Ai Pij (x)
j (x i ; yij )
j (x),
then Pij (x)
0, for all x 2 A, i; j 2 N . The di¤erence
Pji (x) can also be interpreted as the adjustment that player i could claim at (x) against player
j based on their punishment comparison De…nition 5 Total equal punishments:
X
Pij (x) =
j2N ni
X
j2N ni
Pji (x), for all i; j 2 N and x 2 A.
We o¤er a partial justi…cation of why the total equal punishment property should hold in a bargaining equilibrium2 . This justi…cation is based on strategic considerations. Suppose an action-set game (N; A; v) where the distribution payo¤ matrix [ ] is already given. Now at certain x 2 A the players wish to reconsider how to share v(x). For that purpose consider an o¤er bargaining process as follows: There is an in…nite sequence of rounds. At the beginning of each round a player out of N is chosen as the proposer being equally likely to be selected. If player i is the proposer, then she has to simultaneously P announce both a feasible o¤ er g i (x) 2 RN , that is j2N gji (x) v(x), and an action threat yij 2 Ai
(possibly yij = xi ) for every responder j 2 N ni. Now players in N ni are asked if they accept or reject the o¤er following a prespeci…ed order. If the o¤er is accepted for all players, the game ends with payo¤s g i (x). If at least a player j rejects the o¤er, then with a given probability , 0 < starts; and with probability (1
) the game ends with payo¤s
1, a new round
(x i ; yij ), where player j has been the
…rst player in the order which rejects the o¤er. As usual we will focus on the stationary subgame perfect equilibria of the game. Denote by gi (x) = X j 1 i 1 gi (x) for all i 2 N . The equilibrium o¤er that player i would make at each round if she n gi (x) + n j2N ni
were the proposer is computed by letting to every j indi¤erent between accepting or rejecting the o¤er3 , 2 This
is only an informal argument and it is not intended to be a complete description of a noncooperative bargaining
model, yielding the matrix [ ] as the equilibrium payo¤s. This complementary strategic approach excedes the purpose of this note. 3 We wish to remark that this is a partial analysis, and thus we assume that o¤ers are only made to obtain payo¤s at x as big as possible. Hence, those i’s o¤ers designed to be rejected for sure by the other players, and with the purpose to have a chance to obtain a better payo¤ at (x
i ; yij ),
are excluded.
4
i.e. gji (x) = gj (x) + (1
)
j (x i ; yij ),
and taking the rest for herself, i.e. gii (x) = v(x)
We have then the following chain of equalities: 0 1 X X 1 i 1 X j 1 1 gi (x) = gji (x)A + gi (x) + gij (x) = @v(x) gi (x) n n 2 n j2N ni j2N ni j2N ni X X ngi (x) = v(x) gj (x) + (1 ) j (x i ; yij ) + [ gi (x) + (1 ) j2N ni
ngi (x) n(1
)gi (x)
2
4gi (x) +
= v(x) =
(1
)v(x)
X
j2N ni
(1
)
i j2N ni gj (x).
i (x j ; yji )]
j2N ni
3
gj (x)5 + n gi (x)
X
P
j (x i ; yij )
(1
)
X
j (x i ; yij )
j2N ni
+ (1
X
)
j2N ni
+ (1
)
X
j2N ni
i (x j ; yji )
j2N ni
which …nally leads to gi (x) =
1 X n
1 v(x) n
j (x i ; yi )
+
j2N ni
1 X n
i (x j ; yj )
(1)
j2N ni
By equation (1) the only way that i has to rise her expected payo¤ gi (x) is by choosing an action threat vector (yij )j2N ni which minimizes
j (x i ;
) for every j 2 N ni. This translates to saying that
player i’s payo¤s are higher the higher her ability to punish the other players and the lower players j’s ability to punish player i. Note also that from (1) we can further obtain ngi (x) =
X
X
gj (x)
j2N
(n
The di¤erence
1) gi (x) X
j (x i ; yij )
j2N ni
X
i (x j ; yji )
=
j2N ni
(gi (x)
j2N ni
(gi (x)
can in‡ict on i at x, and similarly,
X
i (x j ; yji ))
=
X
i (x j ; yji )
j2N ni
gj (x)
j2N ni
j2N ni
P
X
+
X
j (x i ; yij )
j2N ni
gj (x)
j (x i ; yij )
j2N ni i (x P
j ; yji ))
j2N ni
is the sum of the punishments that the remaining players
gj (x)
j (x i ; yij )
is the sum of the punishments that i
can in‡ict of the remaining players. Hence in equilibrium the total punishments have to be balanced. Obviously, we are looking for a rule recontracting process, i.e.
3
yielding payo¤s
(x) which cannot be beaten by the above
(x) = g(x), for every x 2 A.
Equilibrium
The equilibrium notion is de…ned axiomatically: De…nition 6 A value
is a Consistent Bargaining Equilibrium if it satis…es e¢ ciency, equal minimum
rights, individual rationality and total equal punishments. Given
= (N; A; v), denote by CE(N; A; v) the set of all consistent bargaining equilibrium values.
The next property is useful for proving the existence of a consistent bargaining equilibrium. Let = (N; A; v) and
0
= (N; A; w), where for all x 2 A; w(x) = v(x) + a; a 2 R. 5
i (x j ; yji )
De…nition 7 A rule
veri…es strategic equivalence i¤
+
a a n ; :::; n
2 CE(N; A; w) when
2 CE(N; A; v).
This property says that when the worth of the game is raised in the same amount at every action x 2 A, i.e., w(x) = v(x) + a , the increase in the “productivity” cannot be attributed to any player in particular, so that it should be equally redistributed among all the players. It is not di¢ cult to show that any
2 CE(N; A; v) satis…es strategic equivalence.
Theorem 1 CE(N; A; v) 6= ? Proof. Consider a game
= (N; A; v). By strategic equivalence, we can assume without loss of
generality that minx2A fv(x)g = 0. Let
[v] be the set: [v] :=
By construction
(
g
X
[g(x)]x2A :
gi (x) = v(x) and gi (x)
i2N
0, i 2 N
)
[v] is a compact and convex set.
De…ne the following (adjusting) mapping f : fi (g(x)) = gi (x) +
1 n
1
for all x 2 A and for all i 2 N .
[v] ! [v], by 2 3 X g X g 4 Pij (x) Pji (x)5 j2N ni
j2N ni
By the de…nition of punishments Pijg it follows that f is a continuous mapping. Now we show that f (g) 2
[v]. Note that by construction: X
Pijg (x)
0, for all g 2
j2N ni
[v] , and x 2 A
Therefore, fi (g(x)) can be at least: fi (g(x))
gi (x) +
= gi (x) + gi (x) + Moreover, since
P
i2N
P
j2N ni
1
2
1
2
1 n 1 n 1 n
Pijg (x)
1
40 4
j2N ni
X
3
g Pji (x)5
gi (x) +
j2N ni
[ (n
P
for all x 2 A. Hence, the mapping f goes from
X
j2N ni
=
X
gi (x
j2N ni
1) gi (x) + 0] = 0 g Pji (x)
[v] into
3 5
j ; yji )
= 0 it follows that [v].
P
i2N
fi (g(x)) = v(x)
Then, by Brouwer’s …xed point Theorem, there exists g , such that f (g ) = g . We show now that g 2 CE(N; A; v). Firstly, e¢ ciency follows by construction of f (g). Secondly, for all x 2 A such that v(x) = 0, it holds that gi (x) = 0, for all i and then Equal Minimum Rights is veri…ed. 6
Moreover, by construction gi (x) f (g ) = g if and only if:
0, for all x 2 A and then Individual Rationality holds. Finally, X
X
Pijg (x)
j2N ni
g Pji (x) = 0
j2N ni
and then the total equal punishments property is satis…ed. For instance, in Example 1, the following payo¤ matrix: aj
bj
ai
(2; 2)
(5; 3)
bi
(2; 4)
(0; 0)
corresponds to an equilibrium. In particular: Pij (ai ; bj ) = Pji (ai ; bj ) = 3 Pij (bi ; aj ) = Pji (bi ; aj ) = 2 Pij (ai ; aj ) = Pji (ai ; aj ) = Pij (bi ; bj ) = Pji (bi ; bj ) = 0 In general, this equilibrium concept gives rise to a set of equilibria, for example the next payo¤ division is also an equilibrium: aj
bj
ai
(1; 3)
(4:5; 3:5)
bi
(1:5; 4:5)
(0; 0)
Example 2. Let us consider a three player game, where N = fi; j; kg. The set of actions are:
Player i chooses between two matrices ti and bi , Ai = fti ; bi g. Player j chooses between rows uj and dj , Aj = fuj ; dj g, and Player k chooses between columns lk and rk , Ak = flk ; rk g. The matrix [v] is given by
v
lk
rk
uj
0
0
dj
0
8
lk
rk
uj
0
0
dj
3
8
ti
bi
7
The consistent bargaining equilibrium payo¤s are
(ψ 1 , ψ , ψ )
lk
rk
uj
(0,0,0)
(0,0,0)
dj
(0,0,0)
(2/6,24/6,22/6)
lk
rk
uj
(0,0,0)
(0,0,0)
dj
(7/6,9/6,2/6)
2
3
ti
bi (3/6,27/6,18/6)
It can be checked that at x = (bi ; dj ; lk ) it holds that h
i h i h i h i Pij (x) = 9=6 + Pik (x) = 2=6 = Pji (x) = 7=6 + Pki (x) = 4=6 h i h i h i h i Pji (x) = 7=6 + Pjk (x) = 2=6 = Pij (x) = 9=6 + Pkj (x) = 0 h i h i h i h i Pki (x) = 4=6 + Pkj (x) = 0 = Pik (x) = 2=6 + Pjk (x) = 2=6
Every di¤erence Pij (x)
Pji (x) can be interpreted as the adjustment that player i could claim at
(x) against player j based on their punishment comparison. Notice also that although i could make an objection against j because Pij (x) = 9=6 > Pji (x) = 7=6 and claim to adjust her payo¤s by raising with respect to
j (x),
i (x)
there is a chain of objections among the players at x, such that, at the end, the
initial objection goes back against player i. In particular, as Pjk (x) = 2=6 > Pkj (x) = 0, player j wishes to raise raise
j (x)
k (x)
with respect to
with respect to
Denote by i from i1 to im at Theorem 2 Let
(x)
k (x),
i (x).
and because Pki (x) = 4=6 > Pik (x) = 2=6, then player k wishes to
This a general property that this equilibrium concept satis…es4 .
j i¤ Pij (x) > Pji (x), and say that m players fi1 ; i2 ; :::; im g form an objection chain
(x) if i1
(x) i2
(x)
(x) im .
2 CE(N; A; v) and suppose that k
(x)
l for some k; l 2 N , jN j
3 and x 2 A.
Then there exists an objection chain from l to k. Proof. Let
(x) be an equilibrium payo¤ and suppose that k
(x)
l. Assume on the contrary that
there does not exist an objection chain from l to k and let S = flg [ fi 2 N nk : 9 a chain from l to ig. By the total equal punishment condition X X
i2S j2N ni
4 This
Pij (x)
Pji (x) =
XX
Pij (x)
Pji (x) = 0
i2S j 2S =
chain of binary punishments de…nes a binary relation that is similar to the relation for NTU games of "justi…ed
objections" between any two players, and which leads to the ordinal bargaining set of Asscher (1976)
8
For every i 2 S and j 2 = S, Pij (x) Plk (x) < Pkl (x). Therefore:
Pji (x), otherwise j will be in S. Moreover by assumption,
XX
Pij (x)
Pji (x) < 0
i2S j 2S =
which is a contradiction.
4
Concluding remarks
4.1
Independence of the axiomatic system
The axiomatic system given by e¢ ciency, equal minimal rights, individual rationality and total equal punishments, is independent. Indeed: (i) For all action-set game (N; A; v), de…ne v := minx2A v(x); and let i (x; v)
=
v n,
be the rule de…ned by
1
for all i 2 N . Then F (N; A; v) := f g satis…es all axioms, but not e¢ ciency.
(ii) For all action-set game (N; A; v), let
be the equal payo¤ division rule de…ned by
i (x; v)
=
v(x) n ,
for all i 2 N . Then F 2 (N; A; v) := f g satis…es all axioms, but not total equal punishments. (iii) Let
1
be the action-set game of Example 1, and let the payo¤ matrix [ ] be
Let F 3 be de…ned by
sj
lj
si
(2; 2)
(4:5; 3:5)
li
(1:5; 4:5)
( 1; 1)
8 < CE( 1 ) [ f g; when F 3 ( ) := : CE( ); Otherwise.
=
1
;
Then F 3 satis…es all axioms, but not equal minimal rights. (iv) Let
1
be the action-set game of Example 1, and let the payo¤ matrix [ ] be
Let F 4 be de…ned by
sj
lj
si
( 1; 5)
(3:5; 4:5)
li
(0:5; 5:5)
(0; 0)
8 < CE( 1 ) [ f g; when F 4 ( ) := : CE( ); Otherwise.
=
1
;
Then F 4 satis…es all axioms, but not individual rationality.
4.2
Symmetry and e¢ ciency
It is important to emphasize that ex-ante symmetric environments, where players have the same skills to undertake di¤erent jobs, may give rise to asymmetric ex-post individual payo¤s with players being asymmetrically rewarded as a function of the di¤erent performance roles in labor division. 9
To illustrate this aspect, let us consider a slight variation of the hunters’game in Example 1 with a symmetric matrix [v] (at action-pairs (si ; lj ) and (li ; sj )) and equilibrium payo¤ matrix [ ]: v
sj
lj
si
4
8
li
8
0
and
sj
lj
si
(2; 2)
(5; 3)
li
(3; 5)
(0; 0)
In this ex-ante symmetric setting, the ex-post asymmetric payo¤s at (si ; lj ) and (li ; sj ) are generated by the players’location inside the matrix. In the action-pair (li ; sj ) the bargaining position of i is worse than that of j, and viceversa. This is so because the hunter with the shooter’s role can always threaten the line driver with doing the same activity and then getting no deer, whereas the line driver hunter’s thread of becoming a shooter is less dramatic, since it only means hunting less deers. Thus, the di¤erent players’ bargaining power at (li ; sj ) and (si ; lj ) play a key role in the asymmetric ex-post equilibrium payo¤s. Additionally, if the target were to maximize the total number of deers shooted, then this would imply choosing between (li ; sj ) and (si ; lj ), with their corresponding unequal rewards. However, if the players were not able to agree on unequal rewards, then they would be forced to play (si ; sj ). Therefore, the conclusion is that equality and e¢ ciency are incompatible in this example. Of course, it is always possible to make e¢ ciency and ex-ante equality compatible by agreeing to play the fair lottery L [1=2(li ; sj ); 1=2(si ; lj )], which yields expected payo¤s
4.3
i (L)
=
j (L)
= 4.
Related Literature
The Prekernel. The prekernel was introduced for the class of transferable utility (TU) games in Davis and Maschler (1965), and extended to the class of non-transferable utility (NTU) games in Moldovanu (1990) and Serrano (1997). The prekernel consists of those e¢ cient payo¤s x in which each player is in a "bilateral equilibrium" with any other player. This balanced condition is expressed in terms of the individual excess 5 of player i against player j, eij (x), that is eij (x) = eji (x) for all i; j 2 N . When considering NTU-games, these excesses must be weighted by the normal vector components at x, (x), so that the balanced condition translates to
i (x)eij (x)
=
j (x)eji (x),
for all i; j 2 N . For two-person problems,
this solution coincides with the Nash bargaining solution. For three or more players, as pointed out in Moldovanu (1990) and Serrano (1997), the prekernel is often an empty set. To overcome this impossibility result, Orshan and Zarzuelo (2000) de…ne the bilateral consistent prekernel, by imposing the equilibrium condition that the average (aggregate) excesses of a player against all the others must be equal to that of all of the others against her: X
j2N ni
i (x)eij (x)
=
X
j (x)eji (x);
j2N ni
for all i 2 N:
This property is similar to the total equal punishment property used in our work. 5 The
reader is referred to the cited bibliography for the formal de…nitions.
10
The Shapley value. The Shapley value was introduced for the class of TU-games in Shapley (1953). Given a TU-game v with player set N , the value
i (N; v)
is the expectation of what player i will obtain in v if, for
any possible order in which players arrive to the game, all equally likely, she is paid according to her marginal contribution to her predecessors. Myerson (1980) gives a characterization of the Shapley value by imposing the property of balanced contributions, which is very close in spirit to our equal punishment property. Suppose that player i leaves the game and then compute the value in the subgame (N ni; v) for the remaining players. The di¤erence
j (N; v)
j (N ni; v)
is just the variation in j’s payo¤s due to
player i’s decision of leaving the game. The balanced contribution axiom says that a value
satis…es
this property if these di¤erences are balanced for every pair of players, i.e. i (N; v)
i (N nj; v)
=
j (N; v)
j (N ni; v),
for all i; j 2 N:
Myerson proves that the Shapley value is the unique value satisfying e¢ ciency and balanced contributions. Trying to impose the balanced contributions property to NTU-games, means running into the same existence problem than that of the prekernel. For two-person NTU-games (fi; jg; V ), it su¢ ces to consider both a payo¤ vector
(fi; jg; V ) = a and a vector of weights ( i ;
E¢ ciency:
i ai
+
j aj
i bi
+
j bj ,
j)
satisfying
for all feasible (bi ; bj );
and Balanced contributions: where di =
i (fig; V
)
i (fi; jgnj; V
i
(ai
di ) =
j
(aj
dj ) ,
). As is well known, the Nash bargaining solution (Nash, 1950) is
the only one satisfying these two properties (Harsanyi, 1963). Unfortunately again, in coalitional games with three or more players the condition i
(
i (N; V
)
i (N nj; V
)) =
j
j (N; V
)
j (N ni; V
) , for all i; j 2 N;
cannot be imposed, because the existence of such payo¤ vectors is not guaranteed. Remarkably, replacing the above property by the average (aggregate) balanced contributions: X
j2N ni
j
j (N; V
)
j (N ni; V
) =
X
j2N ni
i
(
i (N; V
)
i (N nj; V
)) , for all i 2 N;
as in Hart and Mas-Colell (1996) makes it possible to show the existence of payo¤ allocations satisfying this property jointly with e¢ ciency; this is the set of Consistent allocations. This concept was previously (and independently) introduced by Maschler and Owen (1989) for hyperplane games, and later de…ned in Maschler and Owen (1992) for the more general setting of NTU-games. Acknowledgments: The authors would like to thank the Spanish Ministry of Science and Technology and the European Feder Funds for …nancial support under project SEJ2007-66581. E. Calvo also thanks partial support from "Subvención a grupos consolidados del Gobierno Vasco". We would also like to thank two anonymous referees for their helpful comments and suggestions that help us to considerably improve the presentation of this article. 11
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References
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