Institute for Economics and

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• 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

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n-person bargaining problem with bargaining
powers

• - the feasible set of utilities
• d 0 - the disagreement point
• Asymmetric Nash bargaining solution (N.b.s.)
• Axiomatized by Roth, 1979, Kalai, 1977

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Plan

• Introduction Relations between weights of
  individual utilities and the bargaining powers.
• A 2-stage game
• I. Formation of a surface of weights ?
• II. An arbitrator finds
• The solution is the asymmetric N.b.s.

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Utilitarian, Egalitarian, and Nash Solutions

• THEOREM.
• Let the set S be restricted by coordinate planes
  and by a surface
• where is a smooth
  strictly convex function, and let
  be positive bargaining powers. Then the
  following two statements are equivalent
• 1. For a point such weights
  exist that is simultaneously
• (a) a utilitarian solution
  with weights
• (b) an egalitarian solution
• (what is the same,
  )
• 2. is an asymmetric N.b.s. with bargaining
  powers

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• 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 Aristotles ancient
  contention that what is fair is what is
  proportional. More precisely, they argue that an
  outcome is regarded as fair when each persons
  gain over the status quo is proportional to that
  persons social index.
• The social index of the player i is the number
  inverse to the weight

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Geometric characterization of the asymmetric
N.b.s
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• - transfer coefficients
• - bargaining powers
• A case of the (asymmetric) Shapley value for TU
  and NTU games
• (Kalai and Samet, 1987, Levy and McLean, 1991)

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Does the iteration sequence converge?

• THEOREM. A stability condition is the inequality
  
• where E is the elasticity of
  substitution of function in
  the solution point

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• A new approach to the N.b.s.
• A 2-stage game
• On the 1st stage the players form a surface of
  weights ?.
• On the 2nd stage an arbitrator in a concrete
  situation chooses weights and
  an outcome x following a Rawlsian principle.

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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 anothers.
• That is why each participant is interested in
  diminishing the weight of his own utility and
  in increasing the weight of anothers. However
  participant i agrees in a part of the surface on
  a decrease in anothers weight at the expense of
  an increase in his own weight, as soon as a
  partner similarly temporizes in another part of
  surface ? .
• 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.

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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 anothers 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
  in respect to . The more is the relative
  bargaining power of the participant the less
  increase in his utility can he achieve.
• Differential equation
• Its solution is the curve of weights
• The arbitrators problem
• THEOREM it is the asymmetric N.b.s.

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Curves of weights with different relative
bargaining powers
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• 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).

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• THEOREM
• A mathematical comment is provided in
• V.Matveenko. 2009. Ekonomika i Matematicheskie
  
• Metody

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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.
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Mechanism (simultaneous change in weights and in
a supposed outcome)
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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

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