Bill’s goods
book
whip
ball
bat
box
Jack’s goods
pen
toy
knife
hat

Utility to Bill
2
2
2
2
4

10

4

6

2

Utility to Jack
4
2
1
2
1

1

1

2

2

Table 2: Nash’s bargaining example [ 7].

 

Jack are supposed to trade them without the use of money, or any other common exchange medium.

 

Figure 1: Visualization of Nash’s original bargaining example [ 7].

Figure 2: Visualization of another case.

 

Nash visualized the situation in a plot as shown in Figure 1. The idea is that every possible solution to the problem will plot to a point (x, y) in the plane. Here x is the gain in utility the first individual could expect from that solution and y is the gain in utility the second individual could expect.

One possible solution to Nash’s example could be that Bill gives Jack the book and nothing else. It plots to (– 2, 4), denoting that Bill loses two utility units and Jack gains four. This solution can directly be looked up in Table 2, but there is no need to restrict Bill and Jack to trading only one of the goods. Bill and Jack could also agree to a trade like Bill giving Jack the book and Jack giving Bill the knife. Here Bill loses two utility units, because he has to give away the book, and he gains six, because he gets the knife, leaving him with an overall gain of four. Jack loses two, because of the knife, and gains four, because of the book, leaving him with an overall gain of two. This solution would then plot to ( 4, 2). Another solution could be that Jack gives everything to Bill, which plots to I( 22,– 6).

 

I_Bill(b, j)

I_Jack(b, j)

F (b, j)

M(b, j)

N(b, j)

b is a minimum
j is a minimum
b+ j is the minimum also satisfying b= j
b + j is a maximum
b j is a maximum

Table 3: Possible solutions to a bargaining problem.

 

If Bill and Jack’s primary motivation were to find a “fair” trade, they could go for a solution in F( 6, 6). For example, Bill could give Jack the book, whip, bat, and box, and Jack could give Bill the pen and the knife. In this case, they would both gain the same amount of utility, which is six. The two of them would then be 12 utility units better off than before. (Here, a global measure of utility is used, obtained by arithmetically adding the two individual’s utilities.) But can they do better?

If they wanted to gain as much “global utility” as possible, they could go for a solution in M( 18,0), which would even have a global value of 18 utility units. One of these would be that Bill gives Jack the book, and Jack gives Bill the pen, toy, and knife. Probably Jack would not like the fact that he has to give away all of these items, and not gain anything for himself. Still, from a collectivist standpoint, it is clearly a better solution, because if Bill has more use for the items, then why should Jack possess them?

The solution suggested by Nash [ 7] is very interesting. It can be found in N( 12, 5). In this case, Bill gives Jack the book, whip, ball, and the bat, and Jack gives Bill the pen, toy, and the knife. This is the outcome Nash expects, when two idealized rational individuals bargain. Here it is assumed that the individual who has more potential to benefit will also be the stronger one in the bargaining process.

Figure 2 shows another possible bargaining situation and Table 3 summarizes the possible solutions, and their criteria. These criteria are the results of different philosophies applied to the same problem, and may help to show some of the concrete impact, that concepts as abstract as individualistic and collectivistic ethics have on everyday problems like trading.

 

Conclusion

Question three was approached by considering bargaining problems. That the maximum “net wealth” of a group of people is not necessarily reached, when each individual tries to maximize its own wealth was shown by considering the divide-the-dollar problem. The discussion was then extended to show that similar principles apply for the more general formulation of bargaining situations as studied by Nash.