J.F. AAS.  Oct., 1998

Notes on Schelling

What is the "situation" of Schelling's Ch. 7? He does not use the language but calls the subject of

the chapter "binary choices with externalities." He then notes the characteristics of MPD. Goods

with these characteristic produce payoff curves for the two classes of choosers which have the

following characteristics: The extremities of the payoff to the individual choosing the "left"

alternative are always higher than the payoff to the individual choosing the "right" alternative. (It

is perhaps better not to use the term "preferred choice" since it makes us think that individuals

choosing one or the other have different preferences while they could have identical preferences.

As Schelling uses the term, "preferred choice = dominant choice, i.e. what is better to do

regardless of what the other person does.) Also both curves rise and do not cross. This shifts

attention from the physical good to how the features of the good result in payoff curves of a

particular sort. High exclusion goods produce exactly these sort of payoff functions.

The Total Payoff Curve

Schelling discusses a total value curve (dotted line) on p. 219 and p. 222. He first notes that it

may not always be possible to produce such a curve if there is no common measure. But he notes

that a simple total might be constructed where everybody puts the same values on the same thing.

The total curve is not a summation of the two curves since they are each for one person. We

need to multiply the payoff times the number of people receiving it and then sum it and divide by

the total number of people for the average. The total curve must coincide with the left axis where

everybody chooses "left" and the right axis when everyone chooses "right." When the two payoff

curves are parallel, the total curve will be a straight line as shown in Figure 14A. In this case

everyone must cooperate if the total is to be maximized. If they diverge, the total will reach a

maximum and then decline to the right axis as shown in 14B. Here, not everyone must cooperate

to maximize the payoff. This creates tough decision problems of deciding who should cooperate

and who should not and how to split the total. (See how the theory tells us the character of the

interdependence that the institutional structure must deal with.) If the curves converge, the total

will decline into negative territory and then rise to the right axis as in 14C. Here in the early

stages, the cooperators though few in number lose so much that the total curve is <0 for awhile

before coming back to a positive value.

Consider the diverging case where n = 100 people:

Payoff to one choosing left:                        Payoff to one choosing right"

If   0 choose right =  0                                If   1 chooses right =  -2

If 25 choose right = 1                                 If 25 choose right =   -1

If 50 choose right = 4                                 If 50 choose right =    0  =  k

If 75 choose right = 6                                 If 75 choose right =   1

.                                                                 If 100 choose right =   2  = average

So the calculation of the total is as follows:

If 25 choose right the total is:

(25 x -1 = -25) plus (75 x 1 = 75) = 50. Average is 50/100 = .5

At "k" where the payoff to right is 50 people x 0 + payoff to left, 50 x 4 = 200. Average is
200/100 = 2

If 75 choose right the total is:

(75 people x 1) = 75 + (25 x 6) = 225. Average = 225/100 = 2.25

If all 100 people choose right, the total is:

100 x 2 = 200. Average is 200/100 = 2

The total curve reaches a maximum before it reaches the right axis and then declines to the right

axis. So if we could get agreement, it would be better if 75 choose right and 25 were high rolling

free riders. The 25 could be allowed to take their return of 6 while the others get 1 each if the

others are not resentful. Or everyone could get the average. Good luck!

A more elegant mathematical representation has been worked out by Jean-Marie Codron and is available on request.

If you have any questions or comments, please email schmid@msu.edu

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