1. Theory of Games and Bargaining
Theoretical Part of Negotiation(ii)

2. Cooperative Games
It seems that threats actually work in many instances, but the theory says they may not work. True in theory, but not in practice?
If the game is not played just once, but repeated over time as in repeated Prisoner’s Dilemma, threats may work.
In labor disputes, breakdown of negotiation does not mean closing down the factory. Just a loss of one week’s operation.
Can we assume that commitments are actually carried out? Or what if we allow side-payment?
If they work together to find a better solution, and commit themselves to implement the solution, we may end up better solutions. This leads us to cooperative solution.
TPN by KJ Sung

3. Example of Tom and Sue: how many possible ways of division are there?
Each item may go to Tom or Sue: there are 25=32 ways
Item
(Tom)
(Sue)
Tom
Sue
12345 -
100 12
345 59 45
1234 5 87 18 13
245 50 58
1235 4 15 14
235 48 55
1245 3 85
234 52
1345 2 76 25 23
145 39 63
2345 1 65 30 24
135 37 60
123 74 33
134 57
124 35 72 34
125 28 73 27 70 43 26 67 40 61 75 88 42 82 41
Looks quite reasonable..
We may ignore these inferior ones…
TPN by KJ Sung

4. Example of Tom and Sue: how many possible ways of division are there?
Each item may go to Tom or Sue: there are 25=32 ways
TPN by KJ Sung

5. Divorce Game?
Suppose a married couple, Michael and Becky, agreed to divorce, and they have to divide their belongings. Further suppose that there are 10 such items as follows.
How would they divide these items?
Can we ask them to assign utilities for each item? After careful consideration, they did it as follows. Existence of utility function…
What would you do? 1 2 3 4 5 6 7 8 9 10
house
Car I
Paintings
yacht
membership
cash
audio
stocks
horse
Car II 1 2 3 4 5 6 7 8 9 10
Michael 21 16 11 23
Becky 28 24 40 32 20
TPN by KJ Sung

6. Axiomatic Approach I Nash Bargaining Solution
Whatever the mechanism may be, it should at least satisfy the following properties.
1. Efficiency
if everybody prefers A over B, then A should be chosen…
2. Symmetry
name of the player should not matter, or if two are exactly the same, then they should get the same…
3. Linear Invariance
if utilities are linearly transformed, the solution should not change…
4. Independence of Irrelevant Alternatives (IIA)
if irrelevant alternatives are removed, solution should not change…
TPN by KJ Sung

7. Axiomatic Approach I (cont’d)
Nash Bargaining Solution
If you buy all the properties, then there is only one way to distribute.
Find x and y that maximize UA(x)UB(y) ( x is given to A, y to B)
x,y are distributions; UA(x), UB(y) are utilities
Example: if x=(1,2,3) and y=(4,5), then
UTom((1,2,3))=74 and USue((4,5))=33
Example: if x=(1,3) and y=(2,4,5), then
UTom((1,3))=50 and USue((2,4,5))=58
TPN by KJ Sung

8. Example of Tom and Sue: how many possible ways of division are there?
Each item may go to Tom or Sue: there are 25=32 ways
Item (Tom)
Sue
Tom
12345
100 12
345 59 45
2,655
1234 5 87 18
1,566 13
245 50 58
2,900
1235 4 15
1,215 14
235 48 55
2,640
1245 3 85
1,020
234 52
2,496
1345 2 76 25
1,900 23
145 39 63
2,457
2345 1 65 30
1,950 24
135 37 60
2,220
123 74 33
2,442
134 57
2,109
124 35 72
2,160 34
125 28 73
2,044 27
1,944 70
1,960 43
2,709 26 67
1,742 40
2,520
2,450 61
2,257 75
1,800 88
1,320
2,340
1,105 42
2,100 82
1,066 41
2,255
TPN by KJ Sung

9. Axiomatic Approach I (cont’d)
Nash Bargaining Solution
If you buy all the properties, then there is only one way to distribute.
Find x and y that maximize UA(x)UB(y) ( x is given to A, y to B)
x,y are distributions; UA(x), UB(y) are utilities
Example
Suppose you and your friend picked up $100 on the street. How would you divide?
Further suppose that your utility function is given by U(x)=x, where x is the amount of money.
TPN by KJ Sung

10. Axiomatic Approach I (cont’d)
Nash Bargaining Solution : simple example
How to divide $100 between the rich and the poor? Here U(x)=x
Money terms
Utility U(. )
UR(x)UP(y)
Rich
Poor
100 75 25 73
5,475 50 90
4,500 98
2,450
Is this solution socially ‘unfair’?
TPN by KJ Sung

11. Axiomatic Approach I (cont’d)
Nash Bargaining Solution: suppose U(x)=x
UB(y), not y
100 50
UA(x)UB(y)
Utility
Possibility
Frontier
UA(x), not x 50
100
skip
TPN by KJ Sung

12. Axiomatic Approach I (cont’d)
What if the utility functions look different? For example, if U(x)=X1/2?
Which one do you prefer? Sure $2 or lottery?
U(x) 2
U(x)=x1/2
1.4 1
Risk Averse !!
X, not U(x) 2 4
TPN by KJ Sung

13. Axiomatic Approach I (cont’d)
Then, Nash solution becomes different. First, utility possibility frontier is changed. This is important. The rest is the same.
UB(y)
Thus, B gets $67.7,
but A gets only $33.3!
100
If you are risk averse,
then you get less!
67.7
UA(x)UB(y)
UA(x) 10
5.8
UPF
TPN by KJ Sung

14. Axiomatic Approach I (cont’d)
Nash Bargaining Solution : simple example
How to divide $100 between the rich and the poor?
Money terms
Utility U(. )
UR(x)UP(y)
Rich
Poor
100 75 25 92 73
5,475
6,716 50 85 90
4,500
7,650 98
2,450
4,900
Is this solution socially ‘unfair’?
What is unfair is not the solution, but the utility function…
TPN by KJ Sung

15. Axiomatic Approach I (cont’d)
Divorce Example
Are you ready?
What is the utility possibility frontier?
There are 210 = 1,024 possibilities!
TPN by KJ Sung

16. Axiomatic Approach I (cont’d)
Divorce Example 1 2 3 4 5 6 7 8 9 10
Michael 21 16 11 23
Becky 28 24 40 32 20
Michael has all…now..which one to give to Becky? In order for Becky to have 1 unit of utility, how much Michael should give up? 1 2 3 4 5 6 7 8 9 10
Michael 21 16 11 23
Becky 14 12 20
Ratio
0.14
0.58
0.25
7.00
0.06
1.60
1.10
1.13
2.50
4.60
Order
Order =
TPN by KJ Sung

17. Axiomatic Approach I (cont’d)
This is the short cut….
Order =
Michael
Becky
M's Util
B's Util
M'U * B'U
100 - 5 99 16
1,584
5-1 97 30
2,910
5-1-3 92 50
4,600 85 62
5,270 74 72
5,328 65 80
5,200
9-10-4 49 90
4,410
10-4 44
4,048 4 21
2,037
TPN by KJ Sung

18. Axiomatic Approach I (cont’d)
And the graph looks like this…
skip
TPN by KJ Sung

19. Kalai-Smorodinski Solution
Axiomatic Approach II
Kalai-Smorodinski Solution
Suppose your friend cannot take more than $60. What would be your suggestion?
According to Nash, still both get $50.
What if we introduce the concept “monotonicity”? That is, I should be better of if UPP expands to my favor….
Then, again, there is only one solution…
It is the intersection between UPP, and the line from the origin to (max UA(x), max UB(y))
TPN by KJ Sung

20. Axiomatic Approach II (cont’d)
Kalai-Smorodinski Bargaining Solution
UB(y)
100 60 50
(max UA(x), max UB(y))
37.5
Utility
Possibility
Frontier
UA(x) 50
100
62.5
TPN by KJ Sung

21. Axiomatic Approach III
Split the difference
M=money on the table, x & y = amount of money asked by player A & B, f(x) & f(y) are money to be given
1. Efficiency, symmetry as before
2. Non-imposition
If (x+y)<M, then x & y will be given
3. Linearity
If (f(x),f(y)) for M, and (f(x’),f(y’)) for M’, then (f(x)+f(x)’,f(y)+f(y’)) for M+M’
Then, again, you have only one solution;
f(x) = x – (x+y-M)/2
But note that as x gets larger, f(x) gets lager…
TPN by KJ Sung

22. Axiomatic Approach III (cont’d)
Too extreme a solution?
“last-best-offer arbitration”
Difference between arbitration and mediation
Arbitration is binding while mediation is not
Each submits own offer
Arbitrator cannot make adjustments, but chooses one of the two
Reasonable outcome is possible with risk averse parties : used in MLB
TPN by KJ Sung

23. Candidate A will be elected, but 60%(15+15) prefers C over A!
A little digression
What if we vote?
Majority voting is normal
Three candidates(A,B,C) & groups of voters(1,2,3)…
Group 1(20 members): A>B>C
Group 2(15 members): B>C>A
Group 3(15 members): C>A>B
Candidate A will be elected, but 60%(15+15) prefers C over A!
Are there any rules where this does not happen?
TPN by KJ Sung

24. A little digression(cont’d)
Arrow’s Impossibility Theorem
Desirable Properties….
Unanimity
Transitivity
IIA (independence of irrelevant alternatives)
No dictatorship
Then, there is NO such voting system!
TPN by KJ Sung

25. Axiomatic Approach IV Shapley Value
Desirable properties of Efficiency, Symmetry, Linearity and Null Player
Once again, there is only one formula…
Shap(i) = SS ((s-1)!(n-s)!/n!)[n(S)- n(S-{i})] (S: group, s: number of members in S)
(s-1)!(n-s)!/n! is the probability that player i will join the group S as the sth member
n(S)- n(S-{i}) is the value ith player contributes to the group s
Simple majority voting: n({1,2})= n({1,3})=n({2,3})=n({1,2,3})=1, O/W=0
Example: only 2,3 do matter: n({1,2})= n({1,3})=0, ({2,3})=1
Shap(1)=0, Shap(2)=Shap(3)=1/2
Majority voting by 4 players, but the fourth player has two votes...
Shap(1)= Shap(2)=Shap(3)=1/6, Shap(4)=1/2
Korean National Assembly?
TPN by KJ Sung

26. When everyone has equal votes: 3 persons, majority voting
Ordering
Coalition for 1
Contribution by 1
Contribution by 2
Contribution by 3
1-2-3
{1} 1
1-3-2
2-1-3
{2,1}
2-3-1
{2,3,1}
3-1-2
{3,1}
3-2-1
{3,2,1}
Shapley Value
2/6
TPN by KJ Sung

27. What if v(1,2)=v(1,3)=0, while v(2,3)=1 Or v(
What if v(1,2)=v(1,3)=0, while v(2,3)=1 Or v(.)=1 only when 2,3 are included
Ordering
Coalition for 2
Contribution by 1
Contribution by 2
Contribution by 3
1-2-3
{1,2} 1
1-3-2
{1,3,2}
2-1-3
{2}
2-3-1
3-1-2
{3,1,2}
3-2-1
{3,2}
Shapley Value
0/6
3/6
TPN by KJ Sung

28. UN Security Council with veto power…
How many sequences are there to list 15 members? 15! Ways
Among these, which sequences would give value 1 to Korea (for an example) when it is placed at the 9th position? The probability to be in the 9th position is 1/15.
1) all 5 veto-powers should be included
2) choose 3 non-veto powers out of 9; how many? 9C3= 84 ways
3) let 3 non-veto and 5 veto stand in a row, and let the remaining 6 in a row…; how many? 8! times 6!
Since the 9th position is fixed, thus there are 14! ways.
Thus, (1/15)*(84*8!*6!)/(14!) or (1/15)*(9C3 / 14C8)= 0.2%
98% of powers are shared by the veto-powers; about 20%!
TPN by KJ Sung

29. Thank you for your cooperation… Was the confusion worthwhile?
TPN by KJ Sung