Download Games Theory: Bargaining and cooperative games

Bargaining Game
Further Economic Analysis
Dr. Keshab Bhattarai
Hull Univ. Business School
March 30, 2011
Dr. Bhattarai
(Hull Univ. Business School)
Bargaining
March 30, 2011
1 / 14
Bargaining Game
The very common example for bargaining game is splitting a pie
between two individuals.
The sum of the shares of the pie claimed by both cannot exceed more
than 1, otherwise each will get zero.
If we denote these shares by θ i and θ j then θ i + θ j
1 is required for
a meaningful solution of the game where each get θ i
0 and θ j
0
payo¤. When θ i + θ j > 1 then and θ i = 0 and θ j = 0 .
Standard technique to solve this problem is to use the concept of
Nash Product.
Dr. Bhattarai
(Hull Univ. Business School)
Bargaining
March 30, 2011
2 / 14
Nash Product in Bargaining Game
max U = (θ i
0) ( θ j
0)
(1)
subject to
θi + θj
1
(2)
or by non-satiation property θ i + θ j = 1
Using a Lagrangian function
L (θ i , θ j , λ) = (θ i
Dr. Bhattarai
(Hull Univ. Business School)
0) ( θ j
Bargaining
0) + λ [1
θi
θj ]
March 30, 2011
(3)
3 / 14
First Order Conditions
First order conditions of this maximization problem are
L (θ i , θ j , λ)
= θj
∂θ i
λ=0
(4)
L (θ i , θ j , λ)
= θi
∂θ j
λ=0
(5)
L (θ i , θ j , λ)
= 1 θi θj = 0
(6)
∂λ
From the …rst two …rst order conditions θ j λ = θ i λ implies θ j = θ i
and putting this into the third …rst order condition θ j = θ i = 12 . This is
called focal point.
Thus Nash solution of this problem is to divide the pie symmetrically into
two equal parts. Any other solution of this not stable. Roy Gardner (2003)
and Charles Holt (2007) have a number of interesting examples on
bargaining game.
Dr. Bhattarai
(Hull Univ. Business School)
Bargaining
March 30, 2011
4 / 14
Application of Bargaining Game
Money to be divided between two players
M = u1 + u2
(7)
N = u1 u2
(8)
Nash product
The origin of this bargaining game is the disagreement point d (0, 0),
the threat point.
Here the utility of player one (u1 ) is plotted against the utility of
player two u2 and the line u1 u2 is the utility possibility frontier (UPF).
Starting of bargaining can be (0, M ) or (M, 0) where one player
claims all but other nothing.
But this is not stable.
O¤ers and counter o¤ers will be made until the game is settled at
u = 12 M, 12 M where each player gets equal share.
Dr. Bhattarai
(Hull Univ. Business School)
Bargaining
March 30, 2011
5 / 14
Numerical Example of Bargaining Game
Suppose there is 1000 in the table to be split between two players. What
is the optimal solution from a symmetric bargaining game if the threat
point is given by d(0,0)? Using a Lagrangian function for constrained
optimisation
L (u1, u2 , λ) = u1 u2 + λ [1000
u1
u2 ]
(9)
First order conditions of this maximization problem are
L (u1, u2 , λ)
= u2
∂u1
λ=0
(10)
L (u1, u2 , λ)
= u1
∂u2
λ=0
(11)
L (u1, u2 , λ)
= 1000 u1 u2 = 0
(12)
∂λ
From the …rst two …rst order conditions u2 λ = u1 λ implies u2 = u1
and putting this into the third …rst order condition u2 = u1 = 1000
2 = 500.
Dr. Bhattarai
(Hull Univ. Business School)
Bargaining
March 30, 2011
6 / 14
Numerical Example of Bargaining Game
The Nash bargaining solution is the values of u1 and u2 that maximise the
value of the Nash product u1 u2 subject to the resource allocation
constraint,u1 + u2 = 1000.
This bargaining solution ful…ls four di¤erent properties: 1) symmetry 2)
e¢ ciency 3) linear invariance 4) independence of irrelevant alternatives
(IIA).
1
Symmetry implies that equal division between two players
2
e¢ ciency implies no wastage of resources u1 + u2 = M or
maximisation of the Nash product, u1 u2 .
3
Linear invariance refers to the location of threat point as (200, 200).
If u is a solution to the bargaining game then u + d is a solution to
the bargaining problem with disagreement point d.
4
IIA implies irrelevant alternatives are not discussed in the game.
Dr. Bhattarai
(Hull Univ. Business School)
Bargaining
March 30, 2011
7 / 14
Another example
Dr. Bhattarai
(Hull Univ. Business School)
Bargaining
March 30, 2011
8 / 14
Numerical Example of Bargaining Game
L (u1, u2 , λ) = (u1
d1 ) (u2
d2 ) + λ [50000
u1
u2 ]
(13)
u1
u2 ]
(14)
Suppose the player 1 has side payment d1 = 15000
L (u1, u2 , λ) = (u1
15000) (u2
d2 ) + λ [50000
First order conditions of this maximization problem are
L (u1, u2 , λ)
= u2
∂u1
L (u1, u2 , λ)
= u1
∂u2
15000
L (u1, u2 , λ)
= 50000
∂λ
Dr. Bhattarai
(Hull Univ. Business School)
λ=0
Bargaining
u1
(15)
λ=0
(16)
u2 = 0
(17)
March 30, 2011
9 / 14
Numerical Example of Bargaining Game
From the …rst two …rst order conditions u2
implies u2 = u1
λ = u1
15000
λ
15000 and
putting this into the third …rst order condition
u2 + 15000 = u1 ; u2 =
32500.
50000 15000
2
= 17500; u1 = 15000 + u2 =
Then u1 + u2 = 17500 + 32500 = 50000.
Dr. Bhattarai
(Hull Univ. Business School)
Bargaining
March 30, 2011
10 / 14
Risk and Bargaining
A risk averse person loses in bargaining but the risk neutral person gains.
Suppose the utility functions of risk averse person is given byu2 = (m2 )0.5
but the risk neutral person has a linear utility u1 = m1 .
m 1 + m2 = M
.u1 + u22 = 100
Using a Lagrangian function for constrained optimisation
L (u1, u2 , λ) = u1 u2 + λ 100
u1
u22
(18)
First order conditions of this maximization problem are
L (u1, u2 , λ)
= u2
∂u1
L (u1, u2 , λ)
= u1
∂u2
L (u1, u2 , λ)
= 100
∂λ
Dr. Bhattarai
(Hull Univ. Business School)
λ=0
(19)
2λu2 = 0
(20)
u1
Bargaining
u22 = 0
(21)
March 30, 2011
11 / 14
Numerical Example of Bargaining Game
λ
From the …rst two …rst order conditions uu1,2 = 2λu
implies u1 2u22 and
2
putting this into the third …rst order condition .3u22 = 100 ;
u22 = 100
3 = 33.3 ; u2 = 5.77
u1 = 2u22 = 2 (5.77)2 = 66.6
u1 + u22 = 66.6 + 33.3 = 100
Thus the risk neutral player’s utility is 66.7 and risk averse player’s utility
is only 5.7.
Morale: do not reveal anyone if you are risk averse, otherwise you will lose
in the bargaining.
Dr. Bhattarai
(Hull Univ. Business School)
Bargaining
March 30, 2011
12 / 14
Coalition Possibilities
2N -1 rule for possible coalition
Consider Four Players A,B,C,D
A, B, C, D
AB, AC, AD
BC, BD, CD
ABC, ABD,ACD, BCD,
ABCD
16 -1=15
What is core of a coalition? It is a coalition that cannot be blocked by any
coalition. Detected by the shapley value.
Shapley value is the amount a particular player can bring in the team di¤erence a particular player makes in the game. Weakest link game.
What is empty core? When non-cooperatively playing coalition partners
all end up in a zero payo¤.
Examples of empty core. Global warming game; Why no constitution in
Nepal?
Dr. Bhattarai
(Hull Univ. Business School)
Bargaining
March 30, 2011
13 / 14
References
Binmore K. (1990) Fun and Games: A text on Game Theory, Lexington,
Heath.
Cripps, M.W.(1997) Bargaining and the Timing of Investment, International
Economic Review, 38:3 :Aug.:527-546
Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton.
Gardener R (2003) Games of Business and Economics, Wiley, Second Edition.
Holt Charles (2007) Markets, Games and Strategic Behaviour, Pearson, .
Mailath G. J. and L. Samuelson (2006) Repeated Games and Reputations:
long run relationship, Oxford.
Kreps D. M. (1990) A Course in Microeconomic Theory, Princeton.
Osborne M.J. and A. Robinstein (1994) A course in game theory, MIT Press.
Perlo¤ J. M. (2008) Microeconomics: Theory and Applications with
Calculus, Pearson.
Rasmusen E(2007) Games and Information, Blackwell.
Varian HR (2010) Intermediate Microeconomics: A Modern Approach,
Norton,8th ed..
Dr. Bhattarai
(Hull Univ. Business School)
Bargaining
March 30, 2011
14 / 14