Institute for Economics and
Mathematics at St. Petersburg
Russian Academy of Sciences
Vladimir Matveenko
Bargaining Powers, Weights of Individual
Utilities, and Implementation of the
Nash Bargaining Solution
10th International Meeting of the Society for Social Choice and Welfare
Moscow, July 21-24, 2010
n-person bargaining problem with bargaining powers
bi , i  1,..., n
•
S  Rn - the feasible set of utilities
•
d=0
•
Asymmetric Nash bargaining solution (N.b.s.):
- the disagreement point
x  arg max x1b1 ...xnbn
xS
Axiomatized by Roth, 1979, Kalai, 1977
Nash Program / Implementation (Support for the solution)
Plan:
1. Introduction: Relations between weights
of individual utilities and the bargaining
powers.
2. A 2-stage game:
I. Formation of a surface of weights Λ
II. An arbitrator finds
max min {i xi }
xS , 
i
The solution is the asymmetric N.b.s. x
Utilitarian, Egalitarian, and Nash Solutions
THEOREM.
• Let the set S be restricted by coordinate planes and by a surface
 ( x1 ,..., xn )  0 where  is a smooth strictly convex function, and
let b1 ,..., bn
be positive bargaining powers. Then the following
two statements are equivalent:
1. For a point x such weights 1 ,..., n exist that x is
simultaneously:
(a) a “utilitarian” solution arg max ( 1 x1  ...   n x n ) with
xS


b

weights
i
i i
i xi
(b) an “egalitarian” solution arg max min
i
xS
(what is the same, 1 x1  ...  n xn )
2. x is an asymmetric N.b.s. with bargaining powers b1 ,..., bn
• Shapley, 1969:
“(III) An outcome is acceptable as a “value of the game”
only if there exist scaling factors for the individual
(cardinal) utilities under which the outcome is both
equitable and efficient”.
Binmore, 2009: “A small school of psychologists who work
on “modern equity theory” [Deutsch, Kayser et al., Lerner,
Reis, Sampson, Schwartz, Wagstaff, Walster et al.] ...
They find experimental support for Aristotle’s ancient
contention that “what is fair is what is proportional”. More
precisely, they argue that an outcome is regarded as fair
when each person’s gain over the status quo is
proportional to that person’s “social index”.
The “social index” of the player i is the number
inverse to the weight i : i xi   j x j
Geometric characterization of the asymmetric
N.b.s
x  1 X 1   2 X 2  ...   n X n
bi
i 
b1  ...  bn
bi
1 x1  ...   n xn 
i xi 
b1  ...  bn
i  1,..., n
 i - transfer coefficients
bi - bargaining powers
A case of the (asymmetric) Shapley value for
TU and NTU games
(Kalai and Samet, 1987, Levy and McLean, 1991)
Does the iteration sequence converge?
 1 
 1 
     
 2 t 1
  2 t
• THEOREM. A stability condition is the inequality
E 1
where E is the elasticity of
substitution of function  ( x1 , x2 ) in the solution
point
A new approach to the N.b.s.
A 2-stage game:
On the 1st stage the players form a
surface of weights Λ.
On the 2nd stage an arbitrator in a
concrete situation chooses weights (1 ,..., n )  
and an outcome x following a Rawlsian
principle.
Formation of the surface of weights
(which can be used in many bargains)
• Under this mechanism, the utility is negatively
connected with the “own” weight and positively - with
the another’s.
• That is why each participant is interested in
diminishing the weight of his “own” utility and in
increasing the weight of another’s. However
participant i agrees in a part of the surface on a
decrease in another’s weight at the expense of an
increase in his own weight, as soon as a partner
similarly temporizes in another part of surface Λ .
• So far as the system of weights is essential only to
within a multiplier, the participants may start
bargaining from an arbitrary vector of weights and
then construct the surface on different sides of the
initial point.
Formation of the surface of weights
(which can be used in many bargains)
• On what increase of his own weight (under a decrease
in the another’s weight) will the participant agree?
• Bargaining powers become apparent here. We
suppose that a constancy of bargaining powers of
participants mean a constancy of elasticities of i in
respect to  j . The more is the relative bargaining
power of the participant the less increase in his utility
can he achieve.
d2 1
b1
   const
Differential equation:
d1 2
b2
Its solution is the curve of weights:
The arbitrator’s problem:
2   const
b2
b1
1
max min{ 1 x1 , 2 x2 }
 , xS
THEOREM: it is the asymmetric N.b.s.
Curves of weights with different relative
bargaining powers
INTERPRETATION:
A role of a community (a society) in decision
making
Community (society) serves as an arbitrator
takes into account moral-ethical valuations
confesses a maximin (Rawlsian) principle of fairness
is manipulated
Participants form moral-ethical valuations
using all accessible means (propaganda through media, Internet,
meetings, rumours)
Bargaining powers of the participants depend on their military power,
access to media and to government, on their propagandist and
imagemaking talent, and earlier reputation
Moral-ethical valuations are not one-valued. Although in any bargain
concrete valuations act, in general the society is conformist and
there is a whole spectrum of valuations which can be used in case
of need. There is a correspondence between acceptable outcomes
and vectors of valuations (weights).
• THEOREM
max max min{ 1 x1 ,..., n xn } 
xS

 max min max{ 1 x1 ,..., n xn }
xS

 max min
xS

N
 x ,
i 1
i i
i  bi i
• A mathematical comment is provided in:
V.Matveenko. 2009. Ekonomika i Matematicheskie
Metody
Mechanism (simultaneous change in weights
and in a supposed outcome)
Another version of the model: in a concrete situation of bargaining a process
of changing valuations and, correspondingly, assumed outcomes.
The society encourages those valuations (and those outcomes) which
correspond to the maximin principle of fairness.
Mechanism (simultaneous change in weights
and in a supposed outcome)
  , if 1 x1  1 x1  0 
1  0

x


x
1 
1
1
1

   , if 1 x1  1 x1  0
1 
 x   x  0
0
,
if

1 1
1 1

x2  g ( x1 )

1
2
1 
1
(1   ) g( x1 ) x1  g ( x1 )  0
The egalitarian solution (Kalai) and
the Kalai-Smorodinsky solution
No 1st stage of the game:
For these solutions the question of the
choice of weights is solved already in a
definite way
Thank you
Спасибо
[email protected]