Peters, H.J.M.

Tijs, S.H.

English

Research Papers in Economics

International Journal of Game Theory, Vol. 14 Issue 4, page 219-228 
Characterization  of all Individually Monotonic  Bargaining Solutions 
By H.J.M. Peters, Maastricht I , and S.H.  Tijs, Nijmegen  2 
Abstract: A description is given of the class of all individually monotonic bargaining solutions by 
associating with each of these solutions a monotonic curve in the triangle of R 2 with vertices 
(1, 0), (0, 1) and (1, 1). Also the family of globally individually monotonic bargaining solutions 
is characterized with the aid of monotonic curves in the unit square of R 2. 
1  Introduction 
In 1950Nash introduced the two-person bargaining problem. In such a problem 
two bargainers are involved who can agree upon one of the points in a set S of fea- 
sible utility pairs or who can disagree,  in which case the payoff is a utility pair d, 
called the disagreement point.  The pair (S, d) determines the problem. 
In the following we only look at bargaining pairs (S, d) where S is a compact con- 
vex subset of R 2 , d E S and such that s l  >  all, s2 >  d2  for some (s l, s2) E S. The 
family of these bargaining pairs is denoted by B_. By a solution of the bargaining 
problem (bargaining solution) we mean a map ~b : _B -+ R 2 , having the following pro- 
perties: 
(P.1)  4~ (S, d) ~>d for all (S, d) EB (IndividualRationality), 
(P.2)  q~ (S, d) EP(S) where P (S) =  (x ES; Vy ES [y ~>x ~y  =x]} is the Pareto 
boundary ofS (Pareto Optimality), 
(P.3)  for each (S, d) C _B and each transformationA : R 2 -+ R 2 of the form 
A  (xl,x2)=(al  xl  +bl,a2 x2 +b2)forall(xl,x2)CR2,whereal  >0, 
a2 >  0, b l, b2 E R 2 , we have 4~ (A (S), A (d)) = A  (~ (S, d)) (Independence 
of equivalent utility representations). 
t Faculty of Economics, University of Limburg, P.O. Box 616, 6200 MD Maastricht, The Nether- 
lands 
2 Department  of Mathematics, University of Nijmegen, Toernooiveld, 6225 ED Nijmegen, The 
Netherlands 220  H.J.M. Peters and S.H. Tijs 
The bargaining solution F 1/2 : _B --> R 2 , proposed by Nash, is the unique solution r 
with the following two additional properties: 
(P.4)  for each (S, d) E_B with d~ = d2 and S =  ((s2, sl) E R 2 ;(sl, s2) ES) we have 
r  (S, d) = ~2  (S, d) (Symmetry), 
(P.5)  for all (S, d), (T, e) ~ _B we have r  (S, d) = r  (T, e) ifd = e, S c  T and ~ (T, e) 
E S  (Independence of irrelevant alternatives). 
Maps Ft: _B --> R 2 (t E (0, 1)), satisfying (P. 1), (P.2), (P.3) and (P.5), were considered 
by Harsanyi and Selten, and Kalai. 
Here,F  t (S, d) is the unique point of (x EP (S);x/> d} in which the function 
(x l , x 2 ) -+ (x l  -  d l ) t (x 2 -- d 2 ) l "t attains its maximum. 
Also the solutions F ~ and F 1 , where for (S, d) E _B, F ~ (S, d) (F 1 (S, d)) is the 
lexicographical  minimum (lexicographical maximum) of (x EP (S); x 1> d), satisfy 
these axioms. In de Koster, Peters,  Tijs and lCakker it is proved that ~b satisfies (P.1), 
(P.2), (P.3) and (P.5) if and only if~b E  (Ft; t E  [0, 1]~. 
Because of criticism on property (P.5) by many authors, Kalai and Smorodinsky 
proposed to look at solutions with the individual monotonicity property, which 
property is described in the next section. Kalai and Smorodinsky  proved that there is 
a unique bargaining solution satisfying the symmetry property and the individual 
monotonicity property. The question arose whether there are more individually 
monotonic solutions. The purpose of section 2 of this paper is to characterize all 
these solutions. 
In Kalai and Rosenthal a symmetric bargaining solution was introduced having the 
property of global individual monotonicity, which property we introduce in section 
3. Also in section 3, all solutions having this property, are described. 
In the last section we look at the continuity and risk sensitivity of (globally) in- 
dividually monotonic solutions. 
2  Individually Monotonic Solutions 
We start with some notations. Let S be a compact convex set in R 2 and d ES. 
Then S d  =  {x E S; x  >~ d }. The utopia point of S (ideal point of S) is the point u (S) 
= (ul (S), u2 (S)) with u i (S) = max {x i E R;x ES} for i E  { 1, 2}. The d-ideal 
point of  S, denoted by u  (Sd), is the utopia point of the set S d. So, ifS is the set of 
attainable utility pairs of a bargaining problem (S, d), then u i (Sd) is the maximal 
attainable utility for player i if only utility pairs x are considered with x i> d. The 
smallest comprehensive set containing a set C, is denoted by C*. Hence, C* = 
=  {x E R 2 ;x ~<y for some y E C}. Characterization of all Individually Monotonic  221 
Now we introduce the following three partial orders on B: 
(S, d) C 1 (T, e) if (Sd) * C (Te) *, d = e and u2  (S  d) = u2 (Te), 
(S, d) C2  (T, e) if (Sd) * C (Te) *, d = e and ul  (S  d) = ul  (Te), 
(S, d) C 12 (T, e) if (S, d) C i (7, e) for i E  ( 1,2 ). 
Defmition 
A bargaining solution ~b : _B -+ R  2  is called an individually monotonic solution if 
for all (S, d), (T, e) E _B we have: 
(IM1) if(S, d) C 1 (T, e), then ~bl (S, d) ~< ~bl (T, e), 
(1M:) if(S, d) C2 (T, e), then ~2 (S, d) ~< ~2 (T, e). 
Note that, for an individually monotonic solution ~b, the following properties hold: 
(Q. 1)  (S, d) C 12 (T, e) =~ r  (S, d) ~< r  (T, e), 
(Q.2)  (S, d) c 12 (T, e), ~ (T, e) E S =~ r  (S, d) =  ~ (T, e), 
(Q.3)  (S,d)C12 (T,e),C(S,d)~Y(1)~C(S,d)=r 
In (ii) of the next proposition a nice property called restricted monotonicity 
[cf. Roth, p. 101], is given. This property proves to be equivalent to the individual 
monotonicity property. Hence, the proposition says that in case two bargaining 
pai~s have the same disagreement point d  and the same d-ideal point, and if the set of 
feasible utility pairs in the first problem contains that of the second problem, then in 
the first bargaining problem an individually monotonic solution assigns better utilities 
to the players than in the second one. 
Proposition 1 
Let r  : _B -~ R 2 be a bargaining solution. Then the following two statements are 
equivalent. 
(i)  r  is an individually monotonic solution. 
(ii)  For all (S, d) and (T, e) in _B with d = e, u (Sd) = u (Te) and S C T, we have 
(S, d)~<~b (T, e). 
Proof 
From (Q. 1) it follows immediately that (i) implies (ii). Suppose that (ii) holds. We 
have to show that (IM1) and (IM2) hold. We only prove (IM2). Take (S, d) and (T, d) 
in _B such that (Sd) * C (Td) * and U 1 (Sd)  =  U  1  (Td). We have to show that ~2 (S, d) 
<~ ~  (T, a). 222  H.J.M. Peters and S.H. Tijs 
First, let m i =  min {x  i E R; x E T) for i E  ( 1,2 ), and let V =  {x E (Td) *; ml ~< 
<~ x l , m2 <~x2 <. u~  (Sd) ). Then S d C V and u (Vd) = u (S  d). Hence, by property 
(ii), q~ (Sd, d) <. ~ (V, d). Furthermore, by applying property (ii) again to (S  d, d) and 
(S, d), we obtain r  (S  d, d) <~ (p (S, d) which by (P.2) yields ~b (Sd, d) = ~ (S, d). Hence 
we have 
(S, d) ~< q~ (V, d).  (2.1) 
Now, letA :R 2 ~R 2 be the transformation withA (xa,x2) = (xl,d2  + (u2  (Sd)-- 
--d2) "1 (u2  (T  d) --d2) (x2 --d2)) for allx ER a . YhenA (d) = d, TCA  (If) and 
u ((A (v)) a) --- u (Td).  Hence, by (ii) and (P.3) we haveA (q~ (V, d)) =  q~ (A (V), 
A  (d)) =  r  (A (1/3, d)/> q~ (T, d). Hence, 
q~, (V, d) =  (A ~ (V, d))1  ~> ~1  (T, d).  (2.2) 
Since r  (V, d) CP (T) and ~ (T, d) EP (T), (2.2) implies 
~2 (V, a) ~< q~2 (T, a).  (2.3) 
Combining (2.1) and (2.3), we obtain r  (S, d) ~< q~ (T, d). 
In section 1 we have already noted that Kalai and Smorodinsky proved there 
exists exactly one symmetric, individually monotonic bargaining solution G. The so- 
lution G assigns to a bargaining pair (S, d) the unique point ofP (S), lying on the line 
segment with endpoints d and u (Sd). Note that also F ~ and F 1 are individually mono- 
tonic. 
Our purpose is to find all individually monotonic bargaining solutions. Therefore 
we look at maps X: [1, 2] -+/~ where A =conv {(1, 0), (0, 1), (1, 1)), with the follo- 
wing property. 
(C) For all s, tE[1,2]withs<~t:  X (s)-..< ~ (t) and X~ (s)+ X2 (s) = s. 
Note that from (C) follows that ~t is a continuous map, in the following way. Let 
s, tE[1,2]  and llx lll  -- lxx  I+lx2lforxER2.ThenllX(s)--X(t)lla= 
=  I X1 (s) -- X1  (t) I +  [ ~2 (s) -- X2 (t)[ =  I (X~ (s) +  X2 (s)) -  (X~  (t) +  X2 (t))[ = 
=Is-  tl. 
The family of maps satisfying (C) is denoted by A and the elements are called mono- 
tonic curves. 
With each X E A we now associate a bargaining solution 7r x which is individually 
monotonic. Let (S, d) E _B. Ifd = (0, O) and u (Sd) =  (1, 1), then rr x (S, d) is defi- 
ned as the unique point ofP (S) which lies on the curve {X (t); t E [1, 2]). Ifd~: 
(0, 0) or u (Sd) 4: (1, 1), then construct a map A  : R 2 --, R 2 as in (P.3) such that 
A (d) = (0, 0) and u ((A (S))o) =  (1, 1) and put 7r x (S, d):=A -1 Qr x (A (S),A (d))), Characterization  of all Individually  Monotonic  223 
where rr x  (,4  (S),A  (d)) is the unique point ofP (A (S)) lying on  {X (t); t E  [1, 2]). 
We call rr x the solution  corresponding  to the curve X. It is then obvious that rr x 
satisfies (P.1), (P.2) and (P.3).  Furthermore, 
Theorem 2 
zr x is an individually monotonic solution. 
Proof 
In view of Prop.  1 and the definition ofrr x we only have to show that for (S, d) 
and (T, e) C _B with d  =  e  =  (0, 0), u  (Sd) = u  (Te) =  (1, 1) and S  C T, we have 
rr x (S, dr) ~< rr x (T, e). 
Let s C [1, 2] and t c  [1, 2] be such that rr x (S, d) =  )t (s) and rr x (T, d) =  )t (t). 
If s >  t, then in view of (6") : )t (s)/> )t (t) and )t (s) :/: )t (t), a contradiction with 
S  C T and (P.2). So, s <~ t  and rr x (S, d) =  )t (s) ~< )t (t) =  rr ;t (T, d).  [] 
Note that the Kalai-Smor0dinsky solution G corresponds to the curve )ta C Awith 
)ta (t) =  (1/2 t,  1/2 t) for t C [1, 2]. Further, the solutions F ~ and F 1 correspond to 
)to and Xl  in A where X0 (t) =  (t -- 1,  1) and Xl  (t) =  (1, t -- 1) for each tc  [1, 2]. 
The main result of this section is the following theorem, which states that each 
individually monotonic solution corresponds to a curve )t CA. 
Theorem 3 
Let q~ : _B ~  R 2 be an individually monotonic solution. Then there exists X CA 
such that r  =  ~r x. 
Proof 
For each t E  [1, 2],let  V (t) =  conv  {(0, 0), (1, 0), (1, t-  1), (t-  1, 1), (0,  1)}. 
Define X : [1,2] -+ R 2 by X  (t) =  ~a (V (t), 0) for all t E  [ 1, 2]. For 1 ~< s <  t ~< 2 we 
have X (s) =  ~ (V(s),  O) <~ ~) (V(t),  0) =  X (t) by (Q.1). Furthermore, for each 
t E  [1, 2], X (t) EP (v(t))  =  conv {(1, t -  1), (t -  1, 1)}, so Xl  (t) +  X2 (t) =  t. 
Hence X E A. Note that 
r  (V(t),  0) =  rr x  (V(t),  0) for each t E  [1, 2].  (2.4) 
We want to prove that r  =  ~r x. In view of (P.3) it is sufficient to show that r  (S,  0) = 
=  rr x  (S, 0) if(S, 0) E_B and u (So) =  (1,  1). Let s =  rr) (S,  0) +  rr x  (S,  0) and let 
lq =  cony {(0, 0), (1,0), rr x (S, 0), (0, 1)). Then, in view of (2.4), 
~  (w, o) = ~  (s, o) = ~x (V(s), o) = ~ (v (s), o) cP(w) nP(s) n 
e (v (s)).  (2.s) 224  H.J.M. Peters and S.H. Tijs 
In view of (Q.2), (2.5) and (W, 0) C12 (V(s), 0), we obtain 
~ (w, o) = ~ (V (s), o). 
In view of (Q.3), (2.5), (2.6) and (I4/, O) C12 (S, 0), we obtain 
~(w, 0) = ~ (s, 0). 
Combining (2.5), (2.6) and (2.7) we conclude that r  (S, 0) =  zr x (S, 0). 
(2.6) 
(2.7) 
3  Globally Individually Monotonic Bargaining Solutions 
In the solutions of section 2 the utopia point ofS  d played an important role. Now 
we consider solutions, where the utopia point of S is important. More precisely, we 
look at bargaining solutions which behave well with respect to the following partial 
orders on _B 
(S, d) <<-i (T, e) if S* C T*, d =  e and u2 (S) = u2 (T), 
(S, d)<<.2 (T, e) if S* C T*, d =  e and ul (S) = ul (T), 
(S, d) ~< a2 (T, e) if (S, d) ~<i (T, e) for i E  { 1, 2 }. 
Definition 
A bargaining solution q~ : _B -+ R 2 is called a globally individually monotonic 
(g.i.m.) solution if for all (S, d) and (T, e) in _B, we have for i E  { 1, 2 ): 
(GIMi) if (S, d) ~<i (T, e), then r  (S, d) ~< q~i (T, e). 
Many results in section 2, concerning individually monotonic solutions, can be modi- 
fied to g.i.m, solutions. E.g., we have modifications of (Q. 1) -  (Q.3) and 
Proposition 4 
is a g.i.m, solution iff for all (S, d), (T, d) E _B with u (S) =  u (T) and S C T, 
we have q~ (S, d) ~< ~ (T, e). 
In Kalai, Rosenthal the solution K : _B ~  R 2 was considered, where K (S, d) is the 
unique Pareto point of S, lying on the line segment with endpoints d  and u (S). Ob- 
viously, K is a g.i.m, solution. Moreover, by small modifications of the proof in 
Kalai, Smorodinsky one obtains 
Theorem 5 
K is the unique symmetric g.i.m, bargaining solution. 
In this section we want to describe all g.i.m, bargaining solutions. Characterization of all Individually Monotonic  225 
Let Q be the unit square with vertices (0, 0), (1, 0), (0, 1) and (1, 1). Let |  be the 
family of maps 0 : [0, 2] ~  Q with the property: (D) For all s, t E [0, 2], 0 (s) 
~< 0 (t) ifs <~ t, and 01 (s) +  02 (s) =  s. 
Note that,just as in section 2 for X E A, from (D) it follows that 0 ~ |  is a con- 
tinuous map. 
For 0 E O, let ~k~ _B ~  R 2 be the solution which assigns to an (S, d) E _B with 
d =  (0, 0) and u (S) =  (1, 1), the unique point ofP (S), lying on the curve  {0 (t); 
t E [0, 2]}. Then one easily verifies that C ~ is a g.i.m, solution. The following theorem 
says that all g.i.m, solutions are of this form. 
Theorem 6 
Let $ : _B ~  R 2 be a globally individually monotonic bargaining solution. Then 
there exists a O E O such that r  =  ff o. 
Proof 
For each t E [0, 1] letL (t) =conv  ((-- 1, 1), (0, t), (t, 0), (1,-  1)), and for each 
tE [1,2] letL (t) =  cony ((-- 1, 1), (t-- 1, 1),(1, t-- 1), (1,-- 1)). Define 
0 : [0, 2] ~  R 2 by 0 (t) =  0 (L (t), 0) for each t E [0, 2]. Similarly as in the proof of 
theorem 3, we obtain 0 E O and r  (L (t), 0) =  C ~ (L (t), 0) for all t E [0, 2]. 
Now take an arbitrary (S, d) with u (S) =  (1, 1) and d =  (0, 0). We prove that 
0 (s, o) =  ~ 0 (s, o). 
For i E  (1,2), let m i =  min  {x  i E R; x  E S} and n i =  min (-- 1, m i). Let T be 
the triangle with vertices (nl, 1), (1, n2) and ~k ~ (S, 0), and let s =  C ~ (S, 0) + 
+  ~ 0 (S, 0). Then 
r  (L (s), 0) =  ~ o (L (s), 0) =  ff o (S, 0) =  ~ o (T, 0) E P (L (s)) n e  (S) 
P(T).  (3.1) 
From (3.1) and (T, 0) <12  (L (s), 0), we obtain 
(T, 0) =  r  (L (s), 0).  (3.2) 
From (3.1), (3.2) and (T, 0) <12 (S, 0), we obtain 
(r, o) =  ~ (s, 0).  (3.3) 
Combining (3.1), (3.2) and (3.3) we can conclude that r  (S, 0) =  ~b ~ (S, 0). 
Note that the Kalai-Rosenthal solution K  corresponds to the curve O  K  with 226  H.J.M. Peters and S.H. Tijs 
O  K (t) =  (1/2 t, I/2 t) for all t E [0, 2], while F 1 and F ~ correspond to 01  and 
0o with 
01  (s)= (s, 0)fors E[0,  1] and 01  (s)= (1,s--1)  forsE[1,  2], 
00 (s) =  (0, s) for s E [0, 1] and 00 (s) =  (s -- 1, 1) for s E [1, 2]. 
4  Some Remarks 
In section 2 and 3 we have characterized all individually monotonic and globally 
individually monotonic solutions, respectively. It is now easy to derive some other 
theorems. Most of the proofs of these theorems are straightforward and left to the 
reader. 
(a) First we look at (globally) individually monotonic solutions, which also satisfy 
the property of independence of irrelevant alternatives. 
Theorem 7 
(i) The only solutions satisfying (P.5), (IM1) and (IM2), are F ~ and F 1 . 
(ii) The only solutions satisfying (P.5), (GIMt) and (GIM2), are F ~ and F I . 
(b) In Jansen, Tijs a systematic study of continuity properties of bargaining solutions 
is made. In the following theorem we characterize all continous (globally) indivi- 
dually monotonic solutions. 
Theorem 8 
(i) An individually monotonic solution r  is continous iff the corresponding curve 
E A satisfies the following condition: 
~1  (t)<  1 and ~,2 (t)<  1 for all t E [1,2). 
(ii) A globally individually monotonic solution •  is continuous iff the corresponding 
curve 0 E O satisfies the condition: 
01  (t) <  1 and 02 (t) <  1 for all t E [ 1, 2). 
(iii) For a (globally) individually monotonic solution ~b, at least one of the functions 
~1  and ~2  is continuous. 
(c) In Kihlstrom, Roth and Schmeidler and also in Peters, Tijs risk sensitivity of bar- 
gaining solutions is studied. For our purpose it is sufficient to say that a solution is 
risk sensitive if for each increasing concave transformation k : R -> R  and each (S, d) 
E _B, we have 
(RS~) O, (K ~ (S), K2 (d))~  (S,d), 
(RS2) ~2 (  Ka (S), K~  (d))~O2  (S, d), Characterization of all Individually Monotonic  227 
where K 1 (s) =  (k (s 1), s2) and K 2 (s) =  (s l,  k (s2)) for each s E S, and K i (S) = 
=  {Ki(s);s~S}  foriE  {1,2}. 
Theorem 9 
(i) Let ~b be an individually monotonic bargaining solution. Then r  is risk sensitive. 
(ii) Let r  be a globally individually monotonic bargaining solution. Then r  is the 
risk sensitive. 
Proof 
We only prove (i). In view of property (P.3), for the proof of (RS1), it is suffi- 
cient to show: for (S, d) E _B with u  (Sd) =  (1, 1) and d  =  (0, 0), and an increasing 
concave transformation k with k (0) =  0, k (1) =  1, we have ~b 1 (K 2 (S), 0) ~> ~bl  (S,  0). 
Now, for such k we have k (x) >~x ifx E  [0, 1]. This implies that 
(S, 0)C12  (K 2 (S), 0). Hence, by (Q.1), ~bl  (S,  0)~< ~bl  (K 2 (S), 0). 
Similarly, one proves (RS2).  n 
In Kihlstrom, Roth and Schmeidler it was already proved that G is risk sensitive and 
in Peters, Tijs that K  is risk sensitive. 
(d) In Thomson, a method of replication of bargaining pairs to n-person bargaining 
pairs is proposed. This replication method gives rise to an interpretation  of the non- 
symmetry of solutions belonging to a subclass of individually monotonic solutions. 
Acknowledgement 
The authors are indebted to P. Wakker for some helpful comments, which im- 
proved the presentation of this paper. 
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Received March 1982 
(Revised version July 1984) 

A Generalized Nash Solution for Two-Person Bargaining Games with Incomplete Information.

Arbitration of Two-Pary Disputes Under Ignorance.

Axiomatic Models of Bargaining.

Continuity of Bargaining Solutions.

Nonsymmetric Nash Solutions and Replications of 2-Person Bargaining.

Replication Invariance of Bargaining Solutions.

Revised version

Risk Aversion and Solutions to Nash's Bargaining Problem. In: Game Theory and Mathematical

Risk Sensitivity of Bargaining Solutions.

The Bargaining Problem.

Characterization of all individually monotonic bargaining solutions.

Characterization of all individually monotonic bargaining solutions.

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