Jozsef Sakovics

English

Research Papers in Economics

A Meaningful Two-Person Bargaining Solution
Based on Ordinal Preferences∗
József Sákovics†
February 11, 2004
1 Introduction
Bargaining solutions based on ordinal preferences are of interest for two reasons. On
o n eh a n d ,t h ec a r d i n a lp r e f e r e n c e so ft h ep l a y e r sm a yb ed i ﬃcult to elicit, leaving us
having to do with ordinal ones. On the other hand, we may want the solution to be
robust to certain variations (say, in attitudes towards risk) in the cardinal preferences,
leading us to the use of ordinal ones instead. Notwithstanding this interest, there are
no known reasonable ordinal bargaining solutions. The ultimate cause for this hiatus
in bargaining theory is Shapley’s (1969) famous non-existence proof (for the two-
person1 bargaining problem). In my opinion, we have given up all too quickly. In this
note, I scrutinize Shapley’s argument and propose an alternative interpretation of it,
which is more robust and also opens the way for a two-person, ordinal bargaining
∗I thank Ahmed Anwar, Olivier Compte, Marco Mariotti, Herakles Polemarchakis and Santiago
Sanchez-Pages for helpful discussions.
†Edinburgh School of Economics, 50 George Square, EDINBURGH, EH8 9JY United Kingdom.
E-mail: jozsef.sakovics@ed.ac.uk
1For bargaining games with more players Shapley’s result does not follow. See e.g. Shubik (1982)
and Safra and Samet (2002).
1solution. For completeness’ sake, I also propose such a solution, together with a
non-cooperative implementation.
2 Shapley’s impossibility result
Take a compact and strictly convex bargaining set, S,a n dad i s a g r e e m e n tp o i n t ,d,
(in utility space, of course). Assume that there exists an ordinal bargaining solution,
B(S,d) ∈ S.
First, note that, since the solution is based on ordinal preferences, any monotonic
(that is, order-preserving) transformation applied to the utilities should leave the
solution unchanged, in terms of physical outcome. In other words, the “new” solution
in utility terms should correspond to the transformed utility of the “old” solution.
Let the extreme points of the (individually rational part of the) Pareto frontier be
denoted by, a and b. Now, we can apply the following sequence of two monotonic
transformations (one for each player’s utility) to all the points in S that strictly
dominate the coordinate-wise minimum2 of a and b: ﬁr s t ,m o v et h ep o i n t st ot h e
right, maintaining the order with respect to the extreme point, which is not moving.
Next move the points down, in such a way that the ﬁnal bargaining set looks identical
to the old one. Figure 1 depicts these transformations. Shapley then argues that the
solution (in terms of utilities) of the transformed problem should be the same as in the
original problem (since the new problem is identical to the old one). However, for any
(feasible and strictly individually rational) solution, following the transformations the
original utility vector will correspond to a diﬀerent physical outcome from the original
one. Contradiction.3
2Sometimes called the “meet.”
3See Osborne and Rubinstein (1990), p. 25, for a slight modiﬁcation of this argument.
2Figure 1: Shapley’s transformations
3 Are we ranking the outcomes or their utilities?
I believe that Shapley’s result is wholly misleading. Note, that it hinges on the
assertion that the (twice) transformed utilities lead to the very same problem that
we started out with. I would like to convince you otherwise.
Consider the following (discrete4) bargaining problem: there are ﬁve possible loca-
tions (say, for an airport between Glasgow and Edinburgh) denoted by A,B,C,D,E.
Based on preliminary reports, Glasgow and Edinburgh city councils’ preferences are
antagonistic. Glasgow’s preferences are: ug(A) >u g(B) >u g(C)=ug(D)=ug(E),
while Edinburgh’s preferences are: ue(A) <u e(B) <u e(C)=ue(D)=ue(E). There
are two possible ways of approaching this problem — in the absence of information on
the intensities of preferences — in order to ﬁn da“ f a i r ”c o m p r o m i s e . O n ei st os a y
that eﬀectively there are only three possible outcomes, since C,D and E are “equiv-
alent”. Consequently, the fair solution is B. Alternatively, we can say that there are
ﬁve candidates, so the solution should be (one) in the middle: C,D or E.
Let me introduce a realistic procedure5 to determine the solution: the players take
turns in vetoing an alternative. The outcome of this mechanism clearly supports the
4The discreteness of the example is only used to simplify the exposition and has nothing to do
with the underlying argument.
5This is a simpler version of the veto game in Anbarci (1993).
3latter interpretation. This is not surprising, since the rules of the game are in terms
of outcomes rather than utilities. But that IS the point: bargaining parties think in
terms of alternatives, NOT the associated utilities.6
Alternatively, assume that further studies have broken the indiﬀerence and the
updated preferences are: ug(A) >u g(B) >u g(C) >u g(D) >u g(E), and ue(A) <
ue(B) <u e(C) <u e(D) <u e(E). Now, either interpretation would lead to C as the
solution. That is, just by breaking the tie between C,D and E,the solution would
move from B to C, according to the “utility-centric” view. But, is it reasonable that
a reordering of outcomes that do not include the solution — so that the rank of the
solution is unchanged for both players — should have an eﬀect on it? I claim that it
isn’t.7 Note that the “axiom” I am advocating is in the spirit of — but much weaker
than — Nash’s Independence of Irrelevant Alternatives, since I do not ask for all those
alternatives not to matter, just their internal ranking not to make a diﬀerence.
Once we accept that bargaining is over the physical outcomes and not their utili-
ties, we can keep working in utility space, but keeping track of the number of physical
outcomes that give the same utility to both8 players. In other words, it is not suﬃ-
cient to deﬁne a bargaining problem in terms of the support of the utility distribution
— as it is usually done — rather we need to take into account the “density” of phys-
6My arguments are similar to Sen’s (1979) and Roemer’s (1986) critique of the “welfarist” (i.e.
utility based) nature of axiomatic bargaining theory. On one hand, I feel the need to repeat these
in a game theory journal, since the message seems not to have ﬁltered across (witness the Osborne-
Rubinstein (1990) treatment of Shapley’s result). On the other hand, there are two diﬀerences
between my argument and theirs. First, I am not appealing to any subjective motive (like “social
justice”), just to common sense. Second, I am not rejecting the idea of carrying out the analysis in
utility space, I am only requiring that we make sure to “carry over” all the necessary information.
7Yet another way of making the same argument is to require that the solution should satisfy con-
tinuity to small perturbations. That is, the solution to ug(A) >u g(B) >u g(C)=ug(D)=ug(E),
ue(A) <u e(B) <u e(C)=ue(D)=ue(E) should be consistent with the solution to
limε→0 {ug(A) >u g(B) >u g(C)+ε >u g(D) >u g(E) − ε,u e(A) <u e(B) <u e(C) − ε <u e(D) <u e(E)+ε}.
However, continuity in ordinal utilities is of limited meaning, so I am not stressing this argument.
8If the indiﬀerence is only for one player, one of the two outcomes is Pareto dominated.
4ical outcomes in utility space. As a consequence, we have already refuted Shapley’s
argument, since the bargaining problem after the two utility transformations will not
be identical to the original one, since we have moved “utility mass” down and to the
right.
The above observation should really not come as a surprise, since for a ﬁnite
number of agreements it is easy to see (and well known) that Shapley’s trick does not
work even according to the old interpretation: there is no way of using a monotone
transformation to reach the same Pareto frontier. Now, why should we expect that
the existence of an ordinal solution depend on the number of agreements? Note that
the very justiﬁcation of using the continuum approach is that it is the limit of the
discrete one. Consequently, we need continuity in the solution (and especially in its
existence) at the limit as the number of alternatives grows towards inﬁnity.
4 Constructing a solution
The above argument may lead you to believe that I am advocating a radically new
deﬁnition of the bargaining problem. For the purposes of the current paper I am
not.9 Notice that all we need to do is not to throw away information that we have
already got. Any real-life bargaining problem — obviously — is deﬁned in terms of
the available physical alternatives. The modeler should simply not lose sight of the
density of the feasible agreements corresponding to each utility vector.
In order to construct an ordinal solution, recall, that in an ordinal world of nego-
tiation, what identiﬁes an agreement is its rank according to bargainers’ preferences.
Therefore, I propose the following candidate for a “reasonable” ordinal bargaining
solution:
Deﬁnition 1 (Ordinal Bargaining Solution) Take the set of Pareto eﬃcient (physi-
cal) agreements most preferred by Player 1 among those that she prefers to at most
9See Esteban and Sákovics (2003) for a proposal to redeﬁne the nature of “the” bargaining
problem.
550%o fa l lP a r e t oe ﬃcient (physical) agreements. Similarly, take the set of Pareto
eﬃcient (physical) agreements most preferred by Player 2 among those that he prefers
to at most 50%o fa l lP a r e t oe ﬃcient (physical) agreements. If these coincide, that is
the solution. Otherwise, pick Player 1’s preferred outcome among them.
This is a well-deﬁned, and obviously ordinal, solution — even10 when there are a
continuum of feasible agreements. Note as well, that while the agreement selected by
this solution may not be unique, both players are indiﬀerent between all solutions.
That is, the utility vector is uniquely determined. Finally, it is implemented by the
veto game introduced in the previous section.
5C o n c l u s i o n
The purpose of this note was to draw attention to the dangers of forgetting that
even though we carry out our analysis in utility space, bargaining is conducted over
physical agreements. This observation happens to overthrow the widely held belief
that (meaningful) two-person ordinal bargaining solutions are not feasible. I have
made no eﬀort here11 to come up with a full-ﬂedged analysis of ordinal two-person
bargaining problems. The example of such a solution that I provided is just that: an
example.
References
[1] Anbarci, N. (1993), “Noncooperative Foundations of the Area Monotonic Solu-
tion,” Quarterly Journal of Economics,1 0 8 ,2 4 5 - 2 5 8 .
[2] Esteban, J. and J. Sákovics (2003), “Endogenous bargaining power,” working
paper #644, Universitat Pompeu Fabra.
10In fact, especially, since we have no tie-breakingworries when there are a continuum of feasible
agreements.
11I do take a crack at that in a companion paper, though.
6[3] Osborne, M., and A. Rubinstein (1990), Bargaining and markets, Academic Press.
[4] Roemer, J. (1986), “The mismarriage of bargaining theory and distributive jus-
tice,” Ethics, 97, 88-110. Reprinted in Egalitarian perspectives (by J. Roemer),
Cambridge University Press, 1996, pp. 202-218.
[5] Safra, Z. and D. Samet (2002), “An Ordinal Solution to Bargaining Problems with
Many Players,” IIBR Working Paper No. 11/2002, Tel-Aviv.
[6] Sákovics, J. (2004), “A fresh look at bargaining with complete information”,
mimeo, Edinburgh School of Economics.
[7] Sen, A. (1979), “Utilitarianism and welfarism,” Journal of Philosophy, 76, 463-
489.
[8] Shapley, L.S. (1969), “Utility Comparison and the Theory of Games,” pp. 251-263
in La décision: agrégation et dinamique des ordres de préférence (G.T. Guilbaud
ed.), Editions du CNRS, Paris. Reprinted in The Shapley value,( A . E .R o t he d . ) ,
Cambridge University Press, 1988, pp. 307-319.
[9] Shubik, M. (1982), Game theory in the social sciences: concepts and solutions,
MIT Press, Cambridge.
7

A fresh look at bargaining with complete information”, mimeo,

An Ordinal Solution to Bargaining Problems with Many Players,”

Endogenous bargaining power,” working paper #644, Universitat Pompeu Fabra. 10In fact, especially, since we have no tie-breakingworries when there are a continuum of feasible agreements. 11I do take a crack at that in a companion paper,

A Meaningful Two-Person Bargaining Solution Based on Ordinal Preferences

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