1. Desired Properties
- invariant to coordinate-wise affine transformation
- symmetry-preserving
- efficient
- monotone
2. Problem Statement
We consider a two-person bargaining problem formulated as follows
: to every two-person game we associated a pair
, where
is a point in the plane and
is a subset of the plane.
- The pair
has the following intuitive interpretation:
where
is the level of utility that player
receives if the two players do not cooperate with each other.
- Every point
represents the level of utility for players 1 and 2 that can be reached by an outcome of the game which is feasible for the two players when they do cooperate.
3. Assumption
- Assumption 1: There is at least one point
such that
. In other words, bargaining may prove worthwhile for both players.
- Assumption 2:
is convex. This is justified under the assumption if two outcomes of the game give raise to points
and
in
, then randomization of these two outcomes give raise to all convex combinations of
and
.
- Assumption 3:
is compact.
- Assumption 4:
for every
. If this is not the case, we can disregard all the points of
that fail to satisfy this condition because it is impossible that both players will agree to such a solution.
4. Axioms
We let denote the set of pairs
that satisfying these four conditions, and we call an element in
a bargaining pair.
- Axiom 1: Pareto Optimality
- For every
there is no
such that
and
.
- Axiom 2: Symmetry
- We let
be defined by
and we require that for every
,
.
- Axiom 3: Invariance with Respect to Affine Transformation of Utility
- A is an afine transformation of utility if
,
, and the maps
are of the form
for some positive constant
and some constant
. We require that for such a transformation
,
.
- Axiom of Independence of Irrelavant Alternatives
- If
and
are bargaining pairs such that
and
, then
.
- Interpretation: given a bargaining pair
, for every point
, consider the product
. Then
is the unique point in
that maximizes this product.
- Many objectives are raised to Nash’s axiom of independence of irrelevant alternatives.
- Let
be a function defined for
in the following way
if
is the Pareto of
.
if there is no such
.
- thus
is the maximum player 2 can get if player 1 get at least x.
- By assumption 1 in the definition of a bargaining pair
.
- By the compactness of
,
and
are finite and are attained by points in
.
- A pair
will be called normalized if
and
. Clearly every game can be normalized by a unique affine transformation of the utilities.
Example objective to Nash’s Solution: consider the following two normalized pair and
where
= convex hull,
and
= convex hull,
- Nash’s solution for
is
, and
for
.
- Limitations pf Nash’s solution: Player 2 has good reasons to demand that he get more in the bargaining pair
than he does in
.






- This axiom states that if, for every utility level that player 1 may demand, the maximum feasible utility level that player 2 can simultaneously reach is increased, then the utility level assigned to player 2 according to the solution should also be increased.








- to normalize the utility function of each agent in such a way that it is worth zero at the status quo and one at this agent’s best outcome — given that all others get at least their status quo utility level
- to sharing equally the benefits of cooperation. In other words, this solution equalizes the relative benefit from status quo or equivalently the relative frustration until the shadow optimum.
6. Solution
- Independence of irrelevant alternatives can be substituted with a monotonicity condition. It is the point which maintains the ratios of maximal gains. In other words, if player 1 could receive a maximum of
with player 2’s help (and vice versa for
), then the bargaining solution would yield the point
on the Pareto frontier such that
.
References
[1] Bargaining problem, wiki
[2] Other solutions to Nash’s bargaining problem, by Ehud Kalai, Meir Smorodinsky, in STOR 1975