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\begin{document}

\title{The Nash- and Kalai-Smorodinsky Bargaining Solution for Decision Weight
Utility Functions\thanks{%
\ The authors like to thank Gerard van der Laan and Gijs van Lomwel for
comments on an earlier draft. Discussions with William Thomson and Peter
Wakker were also helpfull. }}
\author{Harold Houba \and Xander Tieman \and Rene Brinksma}
\maketitle

\begin{center}
Department of Econometrics\\ Free University\\ De Boelelaan 1105\\ 1081 HV
Amsterdam\\ The Netherlands\\ \  \\ Correspondence to: Harold Houba,
HHouba@econ.vu.nl\\
\end{center}

\clearpage
\section{Introduction}

Almost all economic theories are based upon the axioms underlying expected
utility preferences and many experiments have shown that the preference
relation of individuals often violate at least one of these axioms.
Therefore, these economic theories have to be extended to the domain of
non-expected utility (NEU) preferences. For axiomatic bargaining theory this
has been done in Rubinstein, Safra and Thomson (1992). In this article both
the Nash bargaining solution (NBS), as proposed in Nash (1950), and the
Kalai-Smorodinsky bargaining solution (KSBS), as proposed in Kalai and
Smorodinsky (1975), are redefined in terms of NEU preference relations and
for each bargaining solution a new axiomatization is given on the domain of
NEU preferences. The interpretations of these new definitions for both
bargaining solutions are very attractive.

Despite these nice positive and general results, there is a real possibility
that the formulation of this NEU axiomatic bargaining theory in terms of
preference relations will not be taken up by applied economists. The reason
is that applied economists prefer to work with utility functions rather than
preference relations. In order to encourage applied economists to use NEU
preference relations economic theory has to provide NEU utility functions
and formulas that are simple to apply. The popularity of the expected
utility NBS and KSBS in many economic applications stems from the fact that
both are easy to apply.

A significant part of the literature on NEU preferences (e.g. Gul, 1991,
Tversky and Wakker, 1995, and the references therein) deals with decision
weight utility function (DW) preferences, i.e. preferences that can be
represented jointly by the utility function that measures the subjective
value on the domain of outcomes (or prizes) and by the decision weights that
capture the attitude toward risk or uncertainty. The list of references in
Tversky and Wakker (1995) of experimental studies in favour of the S-shaped
decision weight function is impressive. It is the class of DW preferences
that is considered in this note, although we will not impose the S-shape. By
doing so, we also allow for the disappointment averse (DA) utility functions
axiomatized in Gul (1991).

The two aims of this note are modest. First, we want to demonstrate how
easily the new definitions of the NBS and KSBS in terms of NEU preference
relations can be applied to the subclass of DW utility functions. Second, we
want to fully characterize these two bargaining solutions for this subclass
and to provide simple formulas. For each bargaining solution an elegant
formula is derived which extends the standard expected utility formula in a
straightforward manner. Especially, the new formula of the NBS is strikingly
similar to the well known asymmetric NBS for expected utility functions
(e.g. Roth, 1979), provided the bargaining weights are properly chosen.
These bargaining weights are uniquely determined by the derivative of the
players' decision weight functions evaluated at the ``no risk'' point. The
subclass of DA utility functions serves as a ``numerical'' example.

\section{The Bargaining Problem}

Consider the bargaining problem over one dollar with NEU preferences, i.e.
the set $X$ of alternatives is equal to $\left\{ \underline{x}\in \R_{+}^2
\left| x_1+x_2=1\right. \right\} $, the NEU preference relation is $%
\succeq _i$, $i=1,2$, of player $i$ over lotteries and the disagreement
outcome is $\underline{0}=\left( 0,0\right) $. The NEU definition of the NBS
and KSBS are sated in terms of lotteries with at most two prizes. Therefore,
only these lotteries are considered. Denote $p\cdot \underline{x}$ as the
lottery which assigns probability $p$ to the partition $\underline{x}\in X$,
and the probability $1-p$ to the disagreement outcome.

Each player's NEU preference relation $\succeq _i$, $i=1,2$, is represented
by a DW utility function as analyzed in Tversky and Wakker (1995). This
means that the NEU utility function for player $i$, $i=1,2$, over the
lottery $p\cdot \underline{x}$ has the form: $w_i\left( p\right) u_i\left(
x_i\right) ,$ where $u_i\left( 0\right) $ is normalized to $0$. The strictly
increasing function $u_i\left( x\right) $, $i=1,2$, measures the utility
over prizes and $w_i\left( p\right) $ is the decision weight function, where 
$w_i\left( 0\right) =0$, $w_i\left( 1\right) =1$ and $w_i\left( p\right) $
is strictly increasing in $p$. Therefore, the inverse function of $w_i\left(
p\right) $, $i=1,2$, denoted as $w_i^{inv}\left( q\right) $, exists, is
strictly increasing and twice differentiable. For simplicity, we assume that
the functions $u_i\left( x_i\right) $, $i=1,2$, and $w_i\left( p\right) $
are twice continuously differentiable in $x_i$ respectively $p$ and $\ln
\left( u_i\left( x_i\right) \right) $ is concave in $x_i$. We denote the
derivatives $\frac{\partial u_i\left( x_i\right) }{\partial x_i}$, $i=1,2$,
and $\frac{\partial w_i\left( p\right) }{\partial p}$ as $u_i^{^{\prime
}}\left( x_i\right) $ respectively $w_i^{^{\prime }}\left( p\right) $.

The class of DW preferences contains three interesting subclasses of NEU
preferences. First of all the subclass of expected utility preferences which
correspond to $w_i\left( p\right) =p$, $i=1,2$. Second, the subclass of DA
utility functions as axiomatized in Gul (1991) for which $w_i\left( p\right)
=\frac p{1+\left( 1-p\right) \beta _i}$, $i=1,2$, where the parameter $\beta
_i>-1$ measures the disappointment aversion of player $i$. If $\beta _i>0$ $%
(\beta _i<0)$, then $w_i\left( p\right) $ is strictly convex (concave). Note
that $w_i\left( p\right) =p$, $i=1,2$, is equivalent with $\beta _i=0$.
Third, the subclass of S-shaped functions $w_i\left( p\right) $, $i=1,2$,
that are characterized in Tversky and Wakker (1995), i.e. $w_i^{^{\prime
}}\left( 0\right) ,w_i^{^{\prime }}\left( 1\right) \geq 1$, $w_i^{^{\prime
\prime }}\left( 0\right) \leq 0$, $w_i^{^{\prime \prime }}\left( 1\right)
\geq 0$ and there exists an $\delta \geq 0$ such that $w_i^{^{\prime
}}\left( p\right) \leq \min \left\{ w_i^{^{\prime }}\left( 0\right)
,w_i^{^{\prime }}\left( 1\right) \right\} $ for $p\in \left( \delta
,1-\delta \right) $. These S-shaped functions are consistent with
experimental data.

\section{The Nash Bargaining Solution}

In this section the NBS is fully characterized in theorem \ref{necNBS} as
the maximizer of the function $\left[ u_1\left( x_1\right) \right]
^{w_2^{^{\prime }}\left( 1\right) }\left[ u_2\left( x_2\right) \right]
^{w_1^{^{\prime }}\left( 1\right) }$. Furthermore, this formula is briefly
discussed, which includes a comparison to the asymmetric NBS for expected
utility functions and the relation to the interpretation of the symmetry
axiom as given in Nash (1953). We also argue that the alternating offer
model with DW utilities does not satisfy the stationary preference property.
But first we give the new definition of the NBS for NEU bargaining problems
in terms of preference relations, as proposed in Rubinstein, Safra and
Thomson (1992).

\begin{definition}
\label{NBS}The partition $\underline{x}^{*}\in X$ is a NBS if for every $%
p\in \left[ 0,1\right] $ and $\underline{x}\in X$: 
\[
\left[ p\cdot \underline{x}\succ _11\cdot \underline{x}^{*}\Rightarrow
p\cdot \underline{x}^{*}\succeq _21\cdot \underline{x}\right]
\mbox{ and }%
\left[ p\cdot \underline{x}\succ _21\cdot \underline{x}^{*}\Rightarrow
p\cdot \underline{x}^{*}\succeq _11\cdot \underline{x}\right] .
\]
\end{definition}

\noindent In Rubinstein, Safra and Thomson (1992) it is shown that the NBS
exists and is unique.

The following theorem states our main result for the NBS.

\begin{theorem}
\label{necNBS}If $\underline{x}^{*}$ is the NBS, then $\underline{x}^{*}$
maximizes $\left[ u_1\left( x_1\right) \right] ^{w_2^{^{\prime }}\left(
1\right) }\left[ u_2\left( x_2\right) \right] ^{w_1^{^{\prime }}\left(
1\right) }$.
\end{theorem}

\noindent {\em Proof}\\ Consider $\underline{x}=\left( x,1-x\right) $, $%
x\in \left( x^{*},1\right] $. Then for all $p\in \left[ 0,1\right] $ it
holds that $p\cdot \underline{x}\prec _21\cdot \underline{x}^{*}$ and,
therefore, for all $p\in \left[ 0,1\right] $ the condition $\left[ p\cdot 
\underline{x}\succ _21\cdot \underline{x}^{*}\Rightarrow p\cdot \underline{x}%
^{*}\succeq _11\cdot \underline{x}\right] $ holds trivially. Next, it holds
that 
\begin{equation}
p\cdot \underline{x}\succ _11\cdot \underline{x}^{*}\Leftrightarrow
w_1\left( p\right) u_1\left( x\right) >u_1\left( x^{*}\right)
\Leftrightarrow p>w_1^{inv}\left( \frac{u_1\left( x^{*}\right) }{u_1\left(
x\right) }\right)   \label{eq1}
\end{equation}
and 
\begin{equation}
p\cdot \underline{x}^{*}\succeq _21\cdot \underline{x}\Leftrightarrow
w_2\left( p\right) u_2\left( 1-x^{*}\right) \geq u_2\left( 1-x\right)
\Leftrightarrow p\geq w_2^{inv}\left( \frac{u_2\left( 1-x\right) }{u_2\left(
1-x^{*}\right) }\right) .  \label{eq2}
\end{equation}
The condition $\left[ p\cdot \underline{x}\succ _11\cdot \underline{x}%
^{*}\Rightarrow p\cdot \underline{x}^{*}\succeq _21\cdot \underline{x}%
\right] $ requires that the lower bound on $p$ in (\ref{eq2}) should not
exceed the lower bound on $p$ in (\ref{eq1}). Thus, 
\[
w_1^{inv}\left( \frac{u_1\left( x^{*}\right) }{u_1\left( x\right) }\right)
\geq w_2^{inv}\left( \frac{u_2\left( 1-x\right) }{u_2\left( 1-x^{*}\right) }%
\right) .
\]
Rewriting yields 
\[
\frac{u_1\left( x^{*}\right) }{u_1\left( x\right) }\geq w_1\left(
w_2^{inv}\left( \frac{u_2\left( 1-x\right) }{u_2\left( 1-x^{*}\right) }%
\right) \right) .
\]
This weak inequality holds with an equal sign at $x=x^{*}$ and both the LHS
and the RHS are decreasing in $x$. In order for this inequality to hold for $%
x>x^{*}$ it should hold that the LHS decreases less rapidly than the RHS in $%
x^{*}$, i.e. 
\begin{equation}
-\frac{u_1^{^{\prime }}\left( x^{*}\right) }{u_1\left( x^{*}\right) }\geq -%
\frac{w_1^{^{\prime }}\left( 1\right) }{w_2^{\prime }\left( 1\right) }\cdot 
\frac{u_2^{^{\prime }}\left( 1-x^{*}\right) }{u_2\left( 1-x^{*}\right) }.
\label{eq3}
\end{equation}
Next, consider $x\in \left[ 0,x^{*}\right) $. Then similar arguments yield 
\[
w_1\left( w_2^{inv}\left( \frac{u_2\left( 1-x^{*}\right) }{u_2\left(
1-x\right) }\right) \right) \geq \frac{u_1\left( x\right) }{u_1\left(
x^{*}\right) }.
\]
This weak inequality holds with an equal sign at $x=x^{*}$ and both the LHS
and the RHS are increasing in $x$. In order for this inequality to hold for $%
x<x^{*}$ it should hold that the LHS increases less rapidly than the RHS in $%
x^{*}$, i.e. 
\begin{equation}
\frac{w_1^{^{\prime }}\left( 1\right) }{w_2^{\prime }\left( 1\right) }\cdot 
\frac{u_2^{^{\prime }}\left( 1-x^{*}\right) }{u_2\left( 1-x^{*}\right) }\leq 
\frac{u_1^{^{\prime }}\left( x^{*}\right) }{u_1\left( x^{*}\right) }
\label{eq4}
\end{equation}
Combining equations (\ref{eq3}) and (\ref{eq4}) and rewriting yields 
\[
w_2^{\prime }\left( 1\right) \frac{u_1^{^{\prime }}\left( x^{*}\right) }{%
u_1\left( x^{*}\right) }-w_1^{^{\prime }}\left( 1\right) \frac{u_2^{^{\prime
}}\left( 1-x^{*}\right) }{u_2\left( 1-x^{*}\right) }=0,
\]
which is the FOC for maximizing the concave function $w_2^{^{\prime }}\left(
1\right) \ln \left( u_1\left( x\right) \right) +w_1^{^{\prime }}\left(
1\right) \ln \left( u_2\left( 1-x\right) \right) $. Finally, the stated
result follows after taking the monotonic transformation $f\left( y\right)
=e^y$.\hfill $\Box $\\ 

The derived formula is strikingly similar to the standard asymmetric NBS for
expected utility preferences (e.g. Roth, 1979). Recall that for some given $%
\alpha \in \left( 0,1\right) $ this asymmetric NBS maximizes the function $%
\left[ v_1\left( x\right) \right] ^\alpha \left[ v_2\left( 1-x\right)
\right] ^{1-\alpha }$, where the function $v_i$, $i=1,2$, corresponds to
expected utility preferences. For $\alpha =\frac{w_2^{^{\prime }}\left(
1\right) }{w_1^{^{\prime }}\left( 1\right) +w_2^{^{\prime }}\left( 1\right) }
$ and $v_i\left( x_i\right) =u_i\left( x_i\right) $, $i=1,2$, it follows
that after a monotonic transformation the new formula for the NBS with DW
utilities coincides with the formula for the asymmetric NBS. Moreover, the
decision weight functions $w_1$ and $w_2$ that are part of the DW utility
functions uniquely determine the weight $\alpha $. For $w_1^{^{\prime
}}\left( 1\right) =w_2^{^{\prime }}\left( 1\right) $ we obtain the
``symmetric'' NBS. Furthermore, an increase in $w_i^{^{\prime }}\left(
1\right) $, $i=1,2$, weakens player $i$'s bargaining position. For S-shaped
functions $w_i\left( p\right) $, $i=1,2$, a sufficient condition for this to
happen is an increase in player $i^{\prime }$s ``subadditivity'' (e.g.
Tversky and Wakker, 1995).

The decision weight functions $w_1\left( p\right) $ and $w_2\left( p\right) $
uniquely determine the players' weights and only the marginal attitudes
around $p=1$ matter, i.e. only $w_1^{^{\prime }}\left( 1\right) $ and $%
w_2^{^{\prime }}\left( 1\right) $ matter. Note that even if $w_1\left(
p\right) \neq w_2\left( p\right) $ for all $p\in \left( 0,1\right) $ it is
still possible that $w_1^{^{\prime }}\left( 1\right) =w_2^{^{\prime }}\left(
1\right) $, i.e. $\alpha =\frac 12$. The role of marginal attitudes against
risk can be explained as follows. In Rubinstein, Safra and Thomson (1992) it
is argued that the NBS should be robust in the sense that every objection
made by one of the players should be credibly counter objected by the other
player. In particular this should hold for an objection in which one player
wants an incremental higher share. For such an incremental extra share this
player can only afford lotteries with a low risk of breakdown, i.e. $%
p\approx 1$. Otherwise, the costs of losing exceed the incremental gain.
Similarly, for the other player, who can only afford credible counter
objections with a low risk of breakdown. Thus for both players there risk
attitudes around $p=1$ matter and this is reflected in the formula.

In Nash (1953) it is argued that the symmetry axiom means that the only
differences between the players can be differences in preferences. The
formula of theorem \ref{necNBS} nicely reflects this point of view. If both
players have equal attitudes towards risk around $p=1$, i.e. $w_1^{^{\prime
}}\left( 1\right) =w_2^{^{\prime }}\left( 1\right) $, then only the
asymmetries in the functions $u_i$, $i=1,2$, can explain asymmetries in the
players' shares. This case includes the standard expected utilities case,
i.e. $w_1^{^{\prime }}\left( 1\right) =w_2^{^{\prime }}\left( 1\right) =1$.
Similarly, if the functions $u_i$, $i=1,2$, are identical, then only the
differences in risk attitudes $w_1^{^{\prime }}\left( 1\right) $ and $%
w_2^{^{\prime }}\left( 1\right) $ matter.

A final remark concerns the alternating offer model with DW preferences.
This model does not have the property of stationary preferences. Let $\left( 
\underline{z},\tau \right) $ denote the outcome in which the players agree
upon $\underline{z}\in X$ at time $\tau $ and let $\delta \in \left(
0,1\right) $ be the probability of a next bargaining round, i.e. $1-\delta $
is the probability of breakdown. Then $\left( \underline{z},\tau \right) $
induces the lottery $\delta ^\tau \cdot \underline{z}$ and stationary
preferences require 
\[
w_i\left( \delta ^t\right) u_i\left( x_i\right) >w_i\left( \delta
^{t+s}\right) u_i\left( y_i\right) \Leftrightarrow u_i\left( x_i\right)
>w_i\left( \delta ^s\right) u_i\left( y_i\right) .
\]
Hence, stationary DW preferences require $w_i\left( \delta ^{t+s}\right)
=w_i\left( \delta ^t\right) w_i\left( \delta ^s\right) $ for all $t$ and $s$%
. Clearly, this latter relation does not hold in general and DW preferences
fail the property of stationarity. The consequences are that the alternating
offer model with DW preferences is a non-stationary bargaining game and for
general functions $w_i\left( p\right) $ and $u_i\left( x_i\right) $ one can
only state recursive relations between proposals of subsequent bargaining
rounds that cannot be solved. This simple observation casts serious doubts
whether it would be possible to extend the relation between the unique SPE
in the alternating offer model and the NBS known for the standard expected
utility case.

The following example discusses the NBS for the special case of DA utility
functions.

\begin{example}\ \\
For DA utility functions we have $w_i^{^{\prime }}(1)=1+\beta _i>0$, $i=1,2$.
Thus, the NBS maximizes $u_1(x_1)^{1+\beta _2}u_2(x_2)^{1+\beta _1}$ and
equal weights is equivalent to $\beta _1=\beta _2$. In Gul (1991) it is
shown that {\em risk aversion} is equivalent to $\beta _1,\beta _2>0$ and
$u_i$, $i=1,2$, concave. Risk aversion weakens a player's bargaining
position, a result that is already shown in Rubinstein, Safra and Thomson
(1992). For DA utility functions an alternative derivation of this result
can be given by replicating the arguments in Osborne and Rubinstein (1990).
There are two sources that can lead to an increased risk aversion for player 
$i$, $i=1,2$, namely either an increase in $\beta _i$ or a concave
transformation of the utility function $u_i$ (or both). The negative effect
for player $i$, $i=1,2$, of an increase in $\beta _i$ is nicely illustrated
in the new formula where the weight given to player $j$, $j=1,2$ and
$j\neq i$, will increase.
\end{example}

\section{The Kalai-Smorodinsky Bargaining Solution}

In this section an elegant formula is derived that fully characterizes the
KSBS. But first the definition of the KSBS for NEU bargaining problems, as
proposed in Rubinstein, Safra and Thomson (1992), is given.

\begin{definition}
\label{KSBS}The partition $\underline{y}^{*}\in X$ is the KSBS if there
exists a $p\in \left[ 0,1\right] $ such that 
\[
p\cdot \underline{e}_1\sim _11\cdot \underline{y}^{*}\mbox{ and }p\cdot
\underline{e}_2\sim _21\cdot \underline{y}^{*},
\]
where $\underline{e}_i$, $i=1,2$, denotes the $i$-th unit vector in $\R^2$.
\end{definition}

The following theorem gives the full characterization of the KSBS.

\begin{theorem}
\label{necKSBS}If $\underline{y}^{*}=\left( y^{*},1-y^{*}\right) $ is the
KSBS, then $y^{*}$ solves 
\[
\frac{u_1\left( y^{*}\right) }{u_1\left( 1\right) }=w_1\left(
w_2^{inv}\left( \frac{u_2\left( 1-y^{*}\right) }{u_2\left( 1\right) }\right)
\right) .
\]
\end{theorem}

\noindent {\em Proof}\\Making use of DW utility functions yields the
two-equation system 
\[
w_1\left( p^{*}\right) u_1\left( 1\right) =u_1\left( y^{*}\right) \mbox{ and
}w_2\left( p^{*}\right) u_2\left( 1\right) =u_2\left( 1-y^{*}\right) , 
\]
where $p$ and $y^{*}$ are the two unknowns. Rewriting yields 
\[
p^{*}=w_1^{inv}\left( \frac{u_1\left( y^{*}\right) }{u_1\left( 1\right) }%
\right) \mbox{ and }p^{*}=w_2^{inv}\left( \frac{u_2\left( 1-y^{*}\right) }{%
u_2\left( 1\right) }\right) . 
\]
Equating both expressions and rewriting yields the stated expression for the
KSBS.\hfill $\Box $\\

In Rubinstein, Safra and Thomson (1992) existence and uniqueness of the KSBS
has been proved. The derived expression in the latter theorem can be used to
construct a simple alternative proof of this result based upon the mean
value theorem and the increasing utility in a player's own share.

For $w_1\left( p\right) =w_2\left( p\right) $ it simply follows that $\frac{%
u_1\left( 1\right) }{u_1\left( y^{*}\right) }=\frac{u_2\left( 1\right) }{%
u_2\left( 1-y^{*}\right) }$, which includes the special case of expected
utility, i.e. $w_1\left( p\right) =w_2\left( p\right) =p$. This once more
illustrates the interpretation of the symmetry axiom as given in Nash (1953)
already discussed extensively for the NBS. For $w_1\left( p\right) \neq
w_2\left( p\right) $ the new formula is a straightforward and simple
extension of the well known standard formula for expected utility. The
concluding example states the KSBS for the special case of DA utility
functions.

\begin{example}  \ \\
The KSBS $\left( y^{*},1-y^{*}\right) $ for bargaining problems in which the
players have DA utility functions solves 
\[
\left( 1+\beta _2\right) \frac{u_1\left( 1\right) }{u_1\left( y^{*}\right) }%
=\left( 1+\beta _1\right) \frac{u_2\left( 1\right) }{u_2\left(
1-y^{*}\right) }+\left( \beta _2-\beta _1\right) .
\]
\end{example}

\section{References}

\begin{verse}
Gul, F. (1991): ``A Theory of Disappointment Aversion'', {\em Econometrica,
59}, 667-686.

Kalai, E. and M. Smorodinsky (1975): ``Other Solutions to Nash's Bargaining
Problem'', {\em Econometrica, 43}, 513-518.\\

Nash, J. (1950): ``The Bargaining Problem'', {\em Econometrica, 18},
155-162.\\

Nash, J. (1953): ``Two-Person Cooperative Games'', {\em Econometrica, 21},
128-140.\\

Osborne, M., and A. Rubinstein (1991): {\em Bargaining and Markets}. San
Diego: Academic Press.\\

Roth, A. (1979): {\em Game-Theoretic Models of Bargaining}. Berlin:
Springer-Verlag.\\

Rubinstein, A., Z. Safra and W. Thomson (1992): ``On the Interpretation of
the Nash Bargaining Solution and its Extension to Non-Expected Utility
Preferences'', {\em Econometrica, 60}, 1171-1186.\\

Tversky, A. and P. Wakker (1995): ``Risk Attitudes and Decision weights'', 
{\em Econometrica, 63}, 1255-1280.\\
\end{verse}

\end{document}