Nash Bargaining
How to divide a dollar between two individuals?
Suppose philanthropist gives a dollar to two individuals, if they can
agree upon a division of this dollar. If they cannot agree on any
division of this dollar, the philanthropist dies give the dollar.
Let (m1 , m2 ) be the division of one dollar received by individual 1
and 2 respectively.
The set of possible divisions, M = {m = (m1 , m2 ) : m1 + m2 ≤ 1}.
M is also the feasible set. The feasible set is shown in figure 1.
The two individuals have to agree on any point in the set M.
The bargaining problem is to choose a particular point from this
feasible set.
Another example can be wage bargaining between an owner of a
firm and the worker.
The bargaining problem is defined in terms of utility of individuals.
Each individual receives utility or attain satisfaction from the
possession of a share of the dollar they are dividing. There is
function which map the amount of currency ( share of
dollar)received an individual into utility level of that individual.
ui (mi ) is a function which gives the amount of utility attained by
individual i when it gets mi amount of currency.
We assume that ui (mi ) is strictly increasing in mi and
differentiable in mi .
We assume that individuals are risk averse. The assumption of risk
averse gives a particular type of strictly increasing differentiable
utility function. The utility function is concave in nature. It is
shown in figure 2.
What do we mean by concave utility function in this context?
Suppose a risk averse individual has m0 amount of currency. With
currency this guy can buy a lottery. The lottery is of following
nature:
It gives m1 with probability p
It gives m2 with probability (1 − p).
Suppose m0 = m1 p + m2 (1 − p), this is called the expected return
of the lottery.
0 < m1 < m0 < m2 .
These are shown in figure 2.
A risk averse person is such that
u(m0 = m1 p + m2 (1 − p)) > u(m1 ) × p + u(m2 ) × (1 − p).
This risk averse person is going to buy this lottery when its price is
m00 as shown in figure 3.
