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.
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