Nash Bargaining Solution
Non-c oop e rativ e Bargaining T he ory
Strate gic M ov e s in Bargaining
A rb itration and M od e ration m od e ls
Nash (1950) has shown that
I if a solu tion to a b arg aining p rob le m satisfi e s IR , P E , IR , and
II, the n it only d e p e nd s on som e (u nob se rv ab le )
α ∈
[0
,
1].
I
T he NB S se ts e q u al the slop e of the Iso-we lfare -c u rv e and the
“ ratio of the d iff e re nc e s” .
I
If, in a sy m m e tric b arg aining p rob le m , the solu tion satisfi e s
IR , P E , IR , and II, the n the solu tion is e v e n u niq u e and an e q u al sp lit of the d iff e re nc e b e twe e n c ak e siz e and su m of thre at p oints.
I
S u c h an e q u al sp lit is, in this the ory, not im p lie d b y fairne ss c onsid e rations (fairne ss d oe s not ap p e ar in the ax iom s), b u t is d e riv e d from d e sirab le p rop e rtie s (c olle c tiv e rationality ) of the solu tion.
P D D r. R oland K irste in m d @ roland k irste in.d e Bargaining, A rb itration, M e d iation

Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
The NBS reflects an invisible hand idea: no party intends to maximize the Nash product, but obeying the few axioms leads to collective rationality (maximizing the Nash welfare function).
F or positive economics , Nash’s insights are of limited value:
I most (interesting) bargaining problems lack symmetry,
I the A NBS does not provide falsifiable point predictions, as any observed bargaining outcome can be explained by some value of
α
.
PD Dr. Roland Kirstein md@rolandkirstein.de
Bargaining, Arbitration, Mediation

Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
The ANBS is often used as a positive theory for
I predicting the outcome of asymmetric bargaining situations, under the simplifying assumption α = 1 / 2
⇒ maximization of the symmetric Nash product ( it is rather tradition than p rec ision to c all this S N B S ;
I comparative static analysis of different bargaining situations, assuming that α is unaffected by the modifications of, e.g., the threat point.
Recall that the idea of comparative statics is to evaluate how the result of a ( static ) model is affected by the modification of one parameter. Either you can use calculus, or you solve the model under both parameter settings and compare the outcome.
PD Dr. Roland Kirstein md@rolandkirstein.de
Bargaining, Arbitration, Mediation

Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
If we have no hint on different (unexplained) bargaining power, we can use the ANBS with α = 1 / 2 as a simple positive theory to analyze the impact of
I different utility functions
I modified threat points on the bargaining outcome, even if the bargaining situation is not symmetric (it necessarily lacks symmetry before/ after the modification of just one threat point)
I comparative statics analysis of strategic moves
I combines elements of axiomatic and non-cooperative game theory.
PD Dr. Roland Kirstein md@rolandkirstein.de
Bargaining, Arbitration, Mediation

Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
(Carmichael) Consider a bilateral monopoly between S and F.
Factor supplier S seeks to maximize S ( w ) = wL , where
I w denotes the agreed upon factor price,
I
L is the amount of the factor bought by F, hence L = L ( w )
(for the moment, we assume L > 0 to be constant in case of agreement), and a firm F which maximizes F = R
− wL , where R = p ( Y ) Y ( L ) are the product market revenues (also assumed to be constant for the moment). Assume that p ( Y ) is high enough that F is able to bid w > 0. If no agreement takes place, then L = 0.
The two parties bargain over w . D erive the bargaining set, the threat points, the Nash Product and the NBS (with α = 0 .
5).
PD Dr. Roland Kirstein md@rolandkirstein.de
Bargaining, Arbitration, Mediation

Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
Bargaining set : if w = 0, then F collects R = pY ( L ). If w is maximal (such that F just remains operating in the product market), then the union may “skim” the whole revenues pY .
M aximum w satisfies R
− wL = 0, hence w = R / L .
Recall that (in theory) makes no difference whether the parties bargain over w
∈
[0 , w ] or over the distribution of the producer’s profit via w .
Both parties are risk-neutral. H ence, the bargaining set is a linear
(affi ne) function F = pY
−
S with slope
−
1.
Threat points : if parties fail to agree, then F = 0 and L = 0, thus
S = 0.
Thus, the Nash product (with α = 0 .
5) is [ R
− wL ] wL .
PD Dr. Roland Kirstein md@rolandkirstein.de
Bargaining, Arbitration, Mediation

Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
Solution FOC:
− wL = [ R
− wL ] L
⇒ w ∗ = R / 2 L w ∗ = R / 2 L
⇒ the parties agree upon a wage rate (payment per unit of labor) that shares evenly the average revenue (per unit of labor).
E q uation of an iso-w elfare line : F
−
S
− combinations which yield a constant level of “social welfare” V = FS , hence F = V / S
(each hyperbola represents a different welfare level).
PD Dr. Roland Kirstein md@rolandkirstein.de
Bargaining, Arbitration, Mediation

Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
F
6 iso-welfare lines w ∗
R
L d = (0 , 0)
@
@
@
@ u
R
− u
@
@
@@ w ∗ L
R
-
S
PD Dr. Roland Kirstein md@rolandkirstein.de
Bargaining, Arbitration, Mediation

Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
Now consider a factor supplier S who maximizes
S ( w ) = wL + ( M
−
L ) w
0
, where
I w and L denote the agreed upon factor price and the amount of the factor bought by F,
I
M is the factor stock,
I and w
0 the best alternative sales price (e.g., social security benefits).
F maximizes F = R
− wL , and is able to bid w > w
0
. If no agreement takes place, then L = 0.
Again, the two parties bargain over w . Derive the bargaining set, the threat points, the Nash Product and the NBS (with α = 0 .
5).
PD Dr. Roland Kirstein md@rolandkirstein.de
Bargaining, Arbitration, Mediation

Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
Bargaining set : intercepts are again pY for both parties. Both parties are risk-neutral. Hence, the bargaining set is given by
{ ( S , F ) | F ≤ pY − S } .
Threat points : if parties fail to agree, then F = 0, L
=
0, but now
U = Mw
0
.
Thus, the Nash product (with α = 0 .
5) here is
[ R − wL ][ wL + ( M − L ) w
0
− Mw
0
] = [ R − wL ][ wL − Lw
0
]
PD Dr. Roland Kirstein md@rolandkirstein.de
Bargaining, Arbitration, Mediation

Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
We expect the parties to agree upon the NBS (with α = 0 .
5), i.e., w ∗ = arg max[ R − wL ][ w − w
0
] L
FOC: − L 2 [ w − w
0
] + [ R − wL ] L
!
= 0
⇔ 2 Lw − Lw
0
!
= R
⇔ w ∗ =
R + Lw
0
2 L
=
R
2 L
+ w
0
2
PD Dr. Roland Kirstein md@rolandkirstein.de
Bargaining, Arbitration, Mediation

Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
Note that
∂ w ∗
∂ w
0
= 1 / 2 .
An increase in the outside option of S increases the bargaining power of S and, therefore, the symmetric bargaining result.
Hence, the government may influence wage negotiations in the labor markets by setting social benefits for the unemployed
⇒ political externality, scope for lobbying !
PD Dr. Roland Kirstein md@rolandkirstein.de
Bargaining, Arbitration, Mediation

Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
The predicted outcome of negotiations depends on three factors:
I the shape of the bargaining set,
I the parties’ threat points,
I the relative bargaining (unexplained) power.
BATNA = best alternative to a negotiated approach
(instead of“getting to yes”) .
PD Dr. Roland Kirstein md@rolandkirstein.de
Bargaining, Arbitration, Mediation

Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
Bargaining z one : the range for the bargaining parameter(s) in which individual rationality is obeyed.
E.g.: bargaining over π , threat point of player i ∈ { A ; B } is d i
The bargaining range (if x denotes A’s share) is [ d
A
, π − d
B
].
.
If the bargaining zone is non-empty , then there is scope for an agreement which is Pareto-superior to the non-agreement.
However, if the bargaining zone is empty , then no Pareto-superior agreement exists.
The bargaining range [ d
A
, π − d
B d
A
> π − d
B or, equivalently,
] is non-empty if, and only if d
A
+ d
B
> π
(if the collective value of the threat points exceeds the pie size).
PD Dr. Roland Kirstein md@rolandkirstein.de
Bargaining, Arbitration, Mediation

Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
Bazermann/Neale, ch. 9, prescriptions for rational bargaining:
1.
What will you (and your current opponent) do if you don’t reach an agreement? ( Outside options )
2.
What is your bargaining zone?
3 .
What are the true issues in negotiation?
4 .
How important is each issue to you (the value of an agreement) and your opponent?
5.
Which trade-offs exist? ( Shape of b argaining set )
PD Dr. Roland Kirstein md@rolandkirstein.de
Bargaining, Arbitration, Mediation