AN ORDINAL SOLUTION TO BARGAINING
PROBLEMS WITH MANY PLAYERS
by
Zvi Safra
Dov Samet
Working Paper No 11/2002
May 2002
IIBR working papers are intended for preliminary circulation of tentative research
results. Comments are welcome and should be addressed directly to the authors.
This paper was partially financed by the Israel Institute of Business Research.
Research Proposal - 2002
Ordinal Solutions to Bargaining Problems
Dov Samet and Zvi Safra
Solutions to bargaining problems can be classified by their symmetries with
respect to the utility functions that represent the players’ preferences, symmetries being the transformations of the utility functions with which the solution
is covariant.
For example, the Nash solution to bargaining problems is covariant with
respect to linear, positive transformations of the players’ utility functions. In
fact, this covariance is one of the axioms used by Nash (1950) to characterize
this solution. This group of symmetries reflects the fact that the Nash solution
presumes Von Neumann-Morgenstern utilities which are determined up to such
transformations.
The symmetries of the egalitarian solution studied by Kalai (1977) are of
a different type. Any order-preserving transformation of utility is allowed, but
the same transformation should be applied to the utilities of all the players.
This group of symmetries reflects the interpersonal comparison of utilities which
underlies the egalitarian solution.
The larger the group of symmetries of a solution, the less the assumptions
made on the nature of the utility functions. It is natural then to look for
a solution with a large group of symmetries: the group of order-preserving
transformations, where different transformations are applied to different players.
Such a solution is said to be ordinal, (see Shubik (1982) for further discussion
of the symmetries of solutions).
Obviously, there are ordinal solutions. Consider for example the solution in
which all players except player 1 are bound to their disagreement payoff, while
player 1 receives her bliss payoff—the highest payoff possible, given the payoffs
of the others. When order-preserving transformations are applied to the utility
functions of the players, this point is transformed to a point of the same nature,
that is, to the solution of the transformed problem. But this discriminatory
solution is not appealing.
Another ordinal solution is one in which each player receives her bliss payoff.
This solution treats all player on an equal footing, but it is infeasible.
Finally, assigning to each problem its disagreement point is also an ordinal
solution which fails only the efficiency test.
Shapley (1969) has shown that to two-player bargaining problems there is
no ordinal solution which is also Pareto efficient and non-discriminating. Indeed
his proof shows that the only ordinal solutions are the four solutions mentioned
1
above: the two discriminatory ones, the infeasible one, and the inefficient one.
Figure
fig:twoplayers sketches the proof.
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....................................... B
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The disagreement point is the origin. The curved arrows along the axes depict
two order-preserving transformations of the utilities. On each axis the transformation maps the coordinate of the disagreement point, and the coordinate of the
discriminatory point to themselves. The point A on the Pareto surface moves
to B, B moves to C and C to D. The bargaining problem remains the same
under the transformations. Therefore the solution for this problem should be a
point that is mapped to itself. The only points that are mapped to themselves
are depicted by large dots.
Figure 1: The four ordinal solutions to two-player problems
For three players there are eight simple, but not attractive, ordinal solutions,
similar to the four solutions to the two-player case. But in this case it is possible
to construct an ordinal solution which is also efficient and symmetric. The
construction is based on the following observation. Suppose that (a1 , a2 , a3 ) is
the disagreement point of a bargaining problem with a Pareto surface S. Then
there exists a unique point x, the projections of which, (a1 , x2 , x3 ), (x1 , a2 , x3 )
and (x1 , x2 , a3 ), are all in S.
The solution that assigns to each bargaining problem the point x, thus defined, is infeasible, but it is non-discriminating, and most importantly, it is
ordinal. To prove the ordinality, note that the three projections are characterized by three properties: each player i receives her disagreement payoff ai in
one of the points; for each player there are two points in which her payoffs are
the same; all the points are on the Pareto surface of the problem. Since these
three properties are preserved by order-preserving transformation of utility, so
too is the unique point defined by them.
To construct a solution which is also on the Pareto surface of the bargaining
problem we note that x is closer in each coordinate to the Pareto surface than
the disagreement point a. We now solve the problem starting with x as a
disagreement point.1 Continuing this way, we generate a sequence of points that
converge to a point on the Pareto surface. This point is the desired solution.
1 The
point x is above the Pareto surface, which makes it a strange disagreement point.
2
3
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(x1 , a2 , x3 ) ...
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(x1 , x2 , a3 )
1
The unique point x for which the three projections on the planes x1 = a1 , x2 =
a2 , and x3 = a3 , are all on the Pareto surface. Assigning the point x to each
problem is an ordinal and symmetric solution, but it is infeasible.
Figure 2: The first step in Shapley’s construction
Unfortunately, this construction does not lend itself easily to an extension
to problems with more than three players. In such problems there can be more
than one point for which the projections on the planes xi = ai are all on the
Pareto surface of the problem. An example is given in Sprumont (2000). It is not
clear how one of these points can be selected in covariance with order-preserving
transformations. A recent survey of bargaining theory, Thomson (1994), reports
on an ordinal solution to still only three-player problems. Sprumont (2000), who
used Shapley’s construction for the classification of Pareto surfaces, also refers
to the difficulties with more than three players.
Our purpose it to extend Shapley’s three-player solution to more than three
players and further construct an infinite family of ordinal, efficient, and symmetric solutions, for three and more players, which includes the extension of
Shapley’s solution. The strategy of the construction will be based on pathvalued solutions, which assign a path in the utility space, rather than a single
point, to each bargaining problem.
Path-valued solutions were suggested by O’Neill, Samet, Winter, and Weiner
(2001) as a solution to gradual bargaining, in which players face not a single
bargaining problem, but a family of increasing bargaining problems ordered by
time. The natural solution for such a problem specifies a path, parameterized
by time, which provides a solution to each of the problems in the given family.
O’Neill et al. (2001) give an axiomatic characterization of such a solution which
turns out to be covariant with respect to order-preserving transformations of
One may try to give it a game theoretic interpretation, or simply treat it as a technical step
in constructing a game theoretically meaningful solution.
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both utility and time for any number of players.
This idea can in principle be employed in a single bargaining problem as
follows. We single out one player i and consider the family of bargaining problems faced by the rest of the players when i’s utility is fixed at various levels.
A gradual solution specifies a path of agreements parameterized by i’s utility. Employing a gradual solution which is ordinal with respect to utilities and
time results here in a path valued solution which is ordinal with respect to the
utilities of the player other than i, and that of i too, whose utility plays the
role of time in O’Neill et al. (2001). Thus it is covariant with order-preserving
transformations of all players.
The problem with this approach is that the ordinal solution obtained in this
way are highly not symmetric. But if we consider a set of such solutions—one
for each player—this set can be symmetric. The problem now is to create from
this set a single ordinal, symmetric solution. We expect that the minimum and
maximum operators can play an important role here, as they commute with
ordinal transformations. The main task that remains is to show that a process
thus defined converges to an efficient solution.
References
Kalai, E., (1977). Proportional Solutions to Bargaining Situations: Interpersonal Utility Comparison, Econometrica, 45, 1623-1630.
Nash, J., (1950). The Bargaining Problem, Econometrica, 18, 155-162.
O’Neill, B., D. Samet, Z. Wiener, and E. Winter, (2001) Bargaining Under
Agenda, mimeo.
Shapley, L., (1969), Utility Comparison and the Theory of Games. In La
Décision: Agrégation et dynamique des orders de préférence, G. Th. Guilbaud ed., Editions du CNRS, Paris.
Shubik, M., (1982). Game Theory in the Social Sciences: Concepts and Solutions, MIT Press, Cambridge.
Sprumont, Y., (2000). A Note on Ordinally Equivalent Pareto Surfaces, Journal
of Mathematical Economics, 34, 27-38.
Thomson, W., (1994). Cooperative models of bargaining. In Handbook of Game
Theroy 2, R. J. Aumann and S. Hart eds., North Holland, p.1237-1248.
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Dov Samet
List of publications and working papers since 1999
Publications
“Coherent Beliefs are not Always Types,” (with A. Heifetz), Journal of Mathematical Economics, vol. 32, 1999, pp.475-488.
“Bayesianism without Learning,” Research in Economics, vol. 53, 1999, pp.227242.
“Hierarchies of Knowledge: an Unbounded Stairway” (with A. Heifetz), Mathematical Social Sciences, vol. 38, 1999, pp.157-170.
“Quantified Beliefs and Believed Quantities,” Journal of Economic Theory, vol
95, 2000, pp. 169-185.
“Between Liberalism and Democracy,” (with D. Schmeidler), Journal of Economic Theory, forthcoming 2001.
Working Papers
“Bargaining with an Agenda,” (with B. O’Neill, E. Winter and Z. Wienner).
“Learning to Play Games in Extensive Form by Valuation,” (with P. Jehiel).
“Utilitarian Aggregation of Beliefs and Tastes,” (with I. Gilboa, and D. Schmeidler).
“Derivation of Knowledge from Belief,” (with E. Segev).
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