Balanced per capita contributions
and levels structure of
cooperation
María Gómez-Rúa
Juan Vidal-Puga
Universidade de Vigo
The situation
A group of agents with independent interests may benefit if they
collaborate.
The agents form a priori coalitions (exogenously or for the
purpose of bargaining in a better position).
Question: How will these agents share the benefits of
cooperation when that occurs?
What is the rational expectations of the agents?
How does the coalition structure affect these expectations?
Assumption: The Shapley value is the “right” answer when there
is no coalition structure.
Possible answer: The Owen value (Owen, 1977)
The model
N = {1,2,...,n} set of players
v: 2N  IR characteristic function, where v(S) is the worth of
coalition S  N
f (N,v)  IRN value
C = {C1,...,Cp} coalition structure (partition of N), Cq  N
f (N,v,C)  IRN coalitional value
Game among coalitions: (C,v/C) with (v/C)(Q) = v(CqQ Cq)
Owen value
Given Cq C, the reduced game (Cq,vq) is
given by the Shapley value in the game
among coalitions, assuming that Cq\T is not
present.
vq(T) = Sh(C|N\(Cq\T),v/C)
Owen value:
Ow(N,v,C) = Sh(Cq,vq)
Levy-McLean value
Levy y McLean (1989): intercoalitional
symmetry may not be a reasonable
requirement for a value
Given   IRC,
vq (T) = Sh(C|N\(Cq\T),v/C)
Levy-McLean value:
LM(N,v,C) = Sh(Cq,vq )
Levy-McLean value
It is reasonable the weights be the size of the
coalitions Cr = |Cr| (Kalai and Samet
(1987))
However:
vq(T) = Sh(C|N\(Cq\T),v/C)
If Cq\T are not present...
the size of coalition Cq changes!
A new coalitional value
Given Cq C, define the game (Cq,vq*):
vq*(T) = Sh(C|N\(Cq\T),v/C)
with Cq = |T| and Cr = |Cr|.
Coalitional value:
(N,v,C) = Sh(Cq,vq*)
Some properties
Property 1:  coincides with the weighted
Shapley value in (C,v/C), with weights
given by the size of the coalition.
Property 2: When the game is convex, no
players are worse off by forming a coalition.
Axiomatic characterization
Theorem (Myerson, 1981): The Shapley value is
the only efficient value that satisfies balanced
contributions.
• Balanced contributions: For any two players,
their values are equally modified when the other
player leaves the game.
fi(N,v)  fi(N\{j},vN\{j}) = fj(N,v)  fj(N\{i},vN\{i})
for all i,jN.
Axiomatic characterization
Theorem (Amer and carreras, 1995; Calvo et al, 1996): The Owen value is
the only efficient coalitional value that satisfies balanced contributions
inside a coalition and among coalitions.
• Balanced contributions inside a coalition: For any two players in the
same coalition, their values are equally modified when the other player
leaves the game.
fi(N,v,C)  fi(N\{j},vN\{j},CN\{j})
= fj(N,v,C)  fj(N\{i},vN\{i},CN\{i})
for all i,jCqC.
• Balanced contributions among coalitions: For any two coalitions, their
aggregate values are equally modified when the other coalition leaves the
game.
∑iCq fi(N,v,C)  ∑iCq fi(N\Cr,vN\Cr,CN\Cr)
= ∑iCr fj(N,v,C)  ∑iCr fj(N\Cq,vN\Cq,CN\Cq)
for all Cq,CrC.
Axiomatic characterization
• Balanced per capita contributions among coalitions: For
any two coalitions, their average values are equally
modified when the other coalition leaves the game.
[∑iCq fi(N,v,C)  ∑iCq fi(N\Cr,vN\Cr,CN\Cr)]/|Cq|
= [∑iCr fj(N,v,C)  ∑iCr fj(N\Cq,vN\Cq,CN\Cq) ]/|Cr|
for all Cq,CrC.
Main result:  is the only efficient coalitional value that
satisfies balanced contributions inside a coalition and
balanced per capita contributions among coalitions.
The end
• Thank you for your attention.