On the Actual Inefficiency of Efficient Negotiation Methods

Abstract

In this contribution we analyze the effect that mutual information has on the
actual performance of efficient negotiation methods. Specifically, we start by
proposing the theoretical notion of Abstract Negotiation Method (ANM) as a map
from the negotiation domain in itself, for any utility profile of the parties.
ANM can face both direct and iterative negotiations, since we show that ANM
class is closed under the limit operation. The generality of ANM is proven by
showing that it captures a large class of well known in literature negotiation
methods. Hence we show that if mutual information is assumed then any Pareto
efficient ANM is manipulable by one single party or by a collusion of few of
them. We concern about the efficiency of the resulting manipulation. Thus we
find necessarily and sufficient conditions those make manipulability equivalent
to actual inefficiency, meaning that the manipulation implies a change of the
efficient frontier so the Pareto efficient ANM converges to a different, hence
actually inefficient, frontier. In particular we distinguish between strong and
weak actual inefficiency. Where, the strong actual inefficiency is a drawback
which is not possible to overcome of the ANMs, like the Pareto invariant one,
so its negotiation result is invariant for any two profiles of utility sharing
the same Pareto frontier, we present. While the weak actual inefficiency is a
drawback of any mathematical theorization on rational agents which constrain in
a particular way their space of utility functions. For the weak actual
inefficiency we state a principle of Result's Inconsistency by showing that to
falsify theoretical hypotheses is rational for any agent which is informed
about the preference of the other, even if the theoretical assumptions, which
constrain the space of agents' utilities, are exact in the reality, i.e. the
preferences of each single agent are well modeled.

Full text

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arXiv:1201.5478v1 [math.OC] 26 Jan 2012
On the Actual Inefficiency of Efficient Negotiation
Methods
Luca Barzanti
Department of Mathematics
for Economic and
Social Sciences
University of Bologna
luca.barzanti@unibo.it
Marcello Mastroleo
Department of Mathematics
for Economic and
Social Sciences
University of Bologna
marcello.mastroleo@unibo.it
January 27, 2012
Abstract
In this contribution we analyze the effect that mutual information
has on the actual performance of efficient negotiation methods. Specif-
ically, we start by proposing the theoretical notion of Abstract Negoti-
ation Method (ANM) as a map from the negotiation domain in itself,
for any utility profile of the parties. ANM can face both direct and
iterative negotiations, since we show that ANM class is closed under
the limit operation.
The generality of ANM is proven by showing that it captures a large
class of well known in literature negotiation methods like: the Nash’s
bargaining solution, all the methods derived by multi criteria decision
theory and, in particular, the ones based on Lagrange multipliers, the
Single Text Negotiation which was used in the Camp David Accords,
the Improving Direction Methods, and so on.
Hence we show that if mutual information is assumed then any
Pareto efficient ANM is manipulable by one single party or by a col-
lusion of few of them. At this point we concern about the efficiency
of the resulting manipulation. Thus we find necessarily and sufficient
conditions those make manipulability equivalent to actual inefficiency,
meaning that the manipulation implies a change of the efficient fron-
tier so the Pareto efficient ANM converges to a different, hence actually
inefficient, frontier.
In particular we distinguish between strong and weak actual inef-
ficiency. Where, the strong actual inefficiency is a drawback which is
not possible to overcome of the ANMs, like the Pareto invariant one, so
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its negotiation result is invariant for any two profiles of utility which
share the same Pareto frontier, we present. While the weak actual
inefficiency is a drawback of any mathematical theorization on ratio-
nal agents which constrain in a particular way their space of utility
functions.
For the weak actual inefficiency we then state a principle of Re-
sult’s Inconsistency by showing that to falsify theoretical hypotheses
is rational for any agent which is informed about the preference of the
other, even if the theoretical assumptions, which constrain the space
of agents’ utilities, are exact in the reality, i.e. the preferences of each
single agent are well modeled. In essence we show that, under weak ac-
tual inefficiency assumption, any mathematical model which correctly
capture the reality, it produces inconsistent results.
Keywords. Negotiation, Pareto Efficiency, Mutual Information, Manipu-
lation, Collusion, Result’s Inconsistency by Rationality.
Contents
1 Introduction3
2Abstract Negotiation Methods
2.1 Pareto Efficiency . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Iterative ANMs and Limit Efficiency . . . . . . . . . . . . . .
6
8
9
3 Remarkable Examples
3.1 Bargaining Solutions . . . . . . . . . . . . . . . . . . . . . . .
3.2 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . .
3.3 Improving Direction Method
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12
13
13. . . . . . . . . . . . . . . . . .
4Pareto Efficiency Implies Manipulability
4.1Efficient Manipulability . . . . . . . . . . . . . . . . . . . . .
15
17
5 Actual Inefficiency
5.1 Weak Actual Inefficiency . . . . . . . . . . . . . . . . . . . . .
19
20
6Conclusions22
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Engage people with what they expect; it is what they are able to discern and confirms their
projections. It settles them into predictable patterns of response, occupying their minds while
you wait for the extraordinary moment - that which they cannot anticipate. (Sun Tzu)
1 Introduction
The rise of efficiency by rationality is not trivial when the interaction be-
tween two or more Decision Makers (DM’s) is considered. Clearly in finding
a trade-off between parties preferences, the crucial role is always played by
mutual information for example, in [Ker01], the author shows how knowing
mutual utilities allows to reach efficiency by means of the straightforward
illustration of the “orange”.
Nevertheless in a conflict situation the idea of sharing proper preferences
with the opponents is impracticable, thus the rise of the necessity in hav-
ing an extern trusted party which can find the best settlement by knowing
everything of everyone.
For this reason in the last century, the problem of how to implement such
extern trusted party has been widely approached under different perspectives
giving rise to the Negotiation Analysis, an interdisciplinary effort which aims
to draw a prescriptive theory for negotiators, see [Rai82] of Raiffa which is
a pioneer of this field.
Between all the theories of interest, the two most prolifically used are
Game Theory and Multiple Criteria Decision Making, see [Seb08] for a punc-
tual survey of the last half century evolution of the Negotiation Analysis
concept.
In game theoretical approach to negotiation, see [Bra90] for a wide range
of negotiation techniques, there is not a real extern party but its function
is instead achieved by the same negotiating parties which coordinates them
selfs by means of a predetermined scheme of interaction called game.
In the other approach, instead, the real extern party, which is called
negotiator, privately collects informations related to individual preferences
and then he implements a multi criteria decision making algorithm to find
a “win-win” settlement which is Pareto efficient.
The choice of these two theory is natural since the first one models
interaction dynamics while the second one trade-offs.
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Moreover, due to the flourish of electronic transactions in the last decade,
negotiation techniques have the new perspective to became automated ser-
vices those can be worldwide offered to find Pareto efficient gains in several
contexts see, for example: [TWW94] for a classification of modeling aids,
[RT03] for a report on the development of interactive electronic negotiations
in the supply chain, [GBP00] for an artificial intelligence perspective of ne-
gotiating techniques, and [EKH01] for a potentially universal negotiation
platform.
Even if both game theoretical and multi criteria decision making ap-
proaches to negotiation achieve the same results, the techniques, which are
based on the latter theory, are widely imposing as the best candidates to
automated negotiations, since they open to the whole literature about op-
timization and control, due to their numerical and analytical root. Never-
theless there are tacit differences between the two problems and the most
relevant is that in multi criteria decision making there is one single DM,
who possesses multiple conflicting interests, while in negotiation each utility
is associated to a different DM. This slight discrepancy is determinant in
practices because a single rational DM has no incentive to be inefficient,
while the same DM is pushed by rationality to pursue his own good even if
this would imply a global inefficiency of the negotiation dynamic, whenever
that global inefficiency leads him a payoff which is better than the one he
would get in the efficient case.
In fact rationality is a local property that is owned by each DM, which
does all his best to pursuit his own good, whereas efficiency is a global
property in the sense that it is referred to the outcome of their interaction.
It is not difficult to see that fully rational DM’s often fail to reach an efficient
interaction if they negotiate without an external aid.
However the growth of computational power, that caused the born of
electronic negotiations, is itself a potential danger for this field of applica-
tions, in fact it is not difficult to imagine a software specifically designed to
exploit eventual lacks of a negotiation protocol. For example in [Vet09], the
authors propose a volume based method to learn mutual information during
the negotiation so to reach efficient settlements, which can be reversed engi-
neered to be used, within any other negotiation method, by a party to learn
mutual informations and to exploit them during the negotiation process.
The rule of mutual information is determinant and well studied, for ex-
ample, in the Social Choice theory, which is close to negotiation, where
the well-known result due to Arrow, in [Arr50], states that it is impossi-
ble to aggregate personal welfare functions efficiently and independently to
irrelevant alternatives if the absence of a dictatorial party is assumed. More-
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over Gibbard, in [Gib73], and Satterthwaite independently showed that by
renouncing to the independence of irrelevant alternatives in seeking of non-
dictatorship leads to manipulable welfares aggregations. Hence a party can
obtain a better ending welfare by making-up his own declarations so to
exploit the knowledge of the other social parties.
Both results are originally formulated into a finite space of alternatives
and other differences subsist between social choice and negotiation problems,
in fact a negotiation can default and the status quo can be an inner point
of the negotiation domain, for example in iterative negotiation methods
or in the improving ones. However there are several evidences that Social
Choice like results could hold also within negotiation. In fact, it is well
known that some of the most diffused Pareto efficient negotiation methods
are sensible to manipulatory or collusive behaviors, like, for example, the
Adjusted Winner procedure ([BT95]), which is not truthful as detailed in
[Rai82], and the Improving Direction Method ([EVH99]) which is proven to
be both not truthful and information leaking in [BMed].
Moreover in [BM11], the authors shows that if the status quo is exoge-
nous to the negotiation domain, like in Social Choice theory, both Arrow’s
and Gibbard-Satterthwaite’s results apply to negotiation over continuous
issues, thus in this contribution we decided to explore what mutual infor-
mation implies if the status quo is in the negotiation domain, like in iterative
negotiation methods.
In the first part of Section 2 we fix the notions which we are going to
use later in the section to define the class of Abstract Negotiation Methods.
This general framework allows us to move the analysis of a negotiation
methods from a prescriptive point of view to an analytical one, giving us
the possibility to state general results about what can be really achieved in
a negotiation process. In Subsection 2.1 we explore some notions of Pareto
efficiency for ANMs and in Subsection 2.2 we show the equivalence between
direct Pareto efficient, or “one-shot” Pareto efficient, and improving acyclic
iterative negotiations by showing that each iterative ANM has a limit which
is an ANM that achieve the same results in just one iteration.
We show, in Section 3, that the hypotheses, in particular the continuity
ones, used to define ANMs are minimal since they are able to generalize a
wide class of different approach to negotiations, namely: in Subsection 3.1
the Egalitarian and the Nash’s Bargaining solution, as presented respectively
in [Kal77] and [Nas50], Multi-Criteria Decision Making strategies based on
Lagrange Multipliers, like the ones used in [Hei99, JSJJ08], in Subsection 3.2
and, in Subsection 3.3, the Single Text Negotiation framework, introduced
by Fisher in [Fis78], and implemented in the Improved Direction Method
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