Lecture 13: Bargaining
Bargaining is about persons who have opportunity to cooperate
for mutual benefit.
Bargaining problem is defined by:
1. set of players {1, 2,…, N} that participate in the
negotiation,
2. set of utility functions assigning player’s utility to all
solutions,
3. set of all possible solutions,
4. a disagreement point that represents solution if no
agreement is reached.
Example 1: Bill and Jack may barter goods but have no money
with which to facilitate exchange.
Bill’s goods
Utility to Bill
Utility to Jack
Book
2
4
Whip
2
2
Ball
2
1
Bat
2
2
Box
4
1
Pen
10
1
Toy
4
1
Knife
6
2
Hat
2
2
Jack’s goods
Current utilities: Bill 12, Jack 6 (disagreement point).
What is optimal bargaining solution for Jack and Bill?
Problem comes from paper: John F. Nash, The Bargaining
Problem, Econometrica, 1950, Vol. 18, No. 2, pp. 155-162.
Nash axiomatic bargaining solution – a set of characteristics
(axioms) that an optimal bargaining solution should satisfy and
a proof that there is only one such solution.
Definition: rationality (utility maximization and equality of
bargaining skills), full knowledge of the preferences of other
player, lottery is allowed.
Nash assumptions:
 Pareto efficiency. one could not be better off without making
other person worse off (non-dominated solutions are
considered only). E.g. if utilities are (6, 8) in one solution, and
(7, 9) in other one, then the former solution cannot be
optimal bargaining solution.
 Symmetry. Both [u(x), v(y)] and [v(y), u(x)] are bargaining
solutions, and in disagreement point u(x0)=v(y0), then
problem is symmetric.
 Invariance to equivalent payoff representations. If u is utility
function, then also is au + b, i.e. solution can be transformed
to point (1, 1).
 Independence of irrelevant alternatives. You have problems
P and Q, (Q is subset of P). If [u(x*), v(y*)] is solution of P and
lies in Q, then it is also a bargaining solution of Q.
Finding solution:
Maximizing Nash Product [u(x*) - u(x0)][v(y*) - v(y0)].
Note: after transformation maximum is [1-0][1-0]=1.
Example 2:
Peter and Paul think about investing their money.
Peter’s utility function is:
u(x) = x
if x > -20
4x + 60 otherwise
Paul’s utility function is:
v(x) = x
if x > -30
3x + 60 otherwise
Investment is offered to Peter. Peter can make
investment 60 and earn 160 (net profit is 100) or nothing
with equal probabilities.
u(x) = 0.50*u(100) + 0.50*u(-60) = -40
Peter rejects and investment is offered to Paul.
v(x) = 0.50*v(100) + 0.50*v(-60) = -10
Paul rejects and suggests to invest 60% (that is 36) and
offered Peter to invest 40% (ie. 24). Revenue will be also
divided by the 40/60 ratio.
Does Peter accept this offer?