J . Conlisk et al. Mathematical Social Sciences 40 2000 197 –213
201
people to HIGH. Similarly, at LOW, most people reject the norm; hence imitation attracts people to LOW. The MID equilibrium involves a balance of opposite attractions;
hence a push off the dashed line in either direction tips the scale and thus sends pt to either HIGH or LOW. Although the population distribution pt thus converges,
individuals in the population continue to move among the adherence levels q , q , q ,
1 2
3
with two important exceptions. If the example had q 5 1 instead of q , 1, then the
3 3
HIGH equilibrium would be at the upper left corner of the triangle, where everyone would have the single adherence rate q 5 1. Similarly, if the example had q 5 0 instead
3 1
of q . 0, the LOW equilibrium would be at the lower right corner of the triangle.
1
Two adjustment paths for pt are shown on Fig. 1: a path from p0 5 0.1, 0.7, 0.29 heading toward the HIGH equilibrium, and a path from p0 5 0.5, 0.45, 0.059
heading toward the LOW equilibrium. Intuitively, Fig. 1 invites us to visualize a ‘turnpike’ running through all three equilibria. For each of the two sample paths shown,
the model’s initial movements are as if a search for the turnpike, with repeated, partial overshooting in the process the initially jagged pattern. Once near the turnpike, the
model moves more smoothly along it toward the relevant equilibrium. If the initial point p0 were on the dashed line itself, but away from the MID equilibrium, then pt
would stay on the dashed line, jumping back and forth from one side of MID to the other. Despite the overshooting, the jumps would get smaller, and pt would converge
to MID.
Fig. 1 suggests fragility in social behavior. Two societies with identical parameters and very similar starting points can display similarly erratic initial behavior, but
nonetheless tip to opposite equilibria if their starting points are on opposite sides of the dashed line. A one-time social shock which perturbs a society from the HIGH or LOW
equilibrium to the opposite side of the dashed line can tip the society to the opposite equilibrium, with some erratic movement on route. The HIGH or LOW equilibrium is
more or less fragile according to whether it is close or far from the dashed line. Further, as discussed below, a perturbation of parameters can change the number of equilibria.
Since a central concern of the model is that people may have different degrees of commitment to a norm, the n 5 3 case is not adequate. We think of n as sizable. Thus,
we need results for any n 3. The general results to follow can be previewed relative to Fig. 1. So long as q , q , . . . , q have sufficient spread, there will be exactly three
1 2
n
equilibria, as on Fig. 1. Further, the regions of attraction for the high and low equilibria will extend all the way to the middle equilibrium, which is thus a tipping point. There
will never be more than three equilibria, but there may be two or one. The number of equilibria will be found by counting real roots of an nth degree polynomial.
3. Equilibria and stability
An equilibrium of 2 is a solution p of p 5 Pp 9p . Associated with an equilibrium distribution p is its average adherence probability f 5 q9p , the
probability that a person selected at random will adhere to the norm. Since the right side of p 5 Pp 9p maps continuously from the unit simplex to itself, we know by fixed
point logic that there must be at least one equilibrium.
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. Conlisk et al. Mathematical Social Sciences 40 2000 197 –213
However, for the particular P-matrix 1, we can say much more. In the following theorem and below, a distribution p will be said to ‘strictly dominate’ a distribution
p, written p ss p, if and only if the cumulative dominance inequalities are all strict:
p . p , p 1 p . p
1 p
, . . . ,
1 1
1 2
1 2
p 1 ? ? ? 1 p .
p 1 ? ? ? 1
p .
1 n 21
1 n 21
Equilibrium theorem. Assume the model 1–2, and define an equilibrium to be a solution p of p 5 Pp 9p .
1. Number of equilibria. The number of equilibria equals the number of distinct real roots f, in the interval 0 f 1, of the polynomial
n i 21
n 2i
p f 5
O
f 2 q f 1 2 f
.
i i 51
That number is one, two, or three. When there is one such root, p9 f is positive at the root. When there are two such roots, p9 f is positive at one and zero at the
other. When there are three such roots, p9 f is positive at the smallest and largest roots and negative at the middle root.
2. Benchmark cases. If q 5 0, then p 5 1, 0, . . . , 09 is an equilibrium. If q 5 1, then
1 n
p 5 0, . . . , 0, 19 is an equilibrium. If q 5 0 and q 5 1, there is a third equilibrium
1 n
with all elements of p positive. 3. Form of an equilibrium. The equilibrium distribution p corresponding to a root f
is 1
f
2 n 21
]]]]]]] ]]
p 5 1, x, x , . . . , x
9, where x 5
.
S D
2 n 21
1 2 f 1 1 x 1 x 1 ? ? ? 1 x
If f 5 1, this equation is taken to mean the limit as f →
1, namely p 5 0, . . . , 0, 19. A root f and the corresponding p satisfy f 5 q9p ; hence the root is an
equilibrium average adherence probability. 4. Dominance ranking. Multiple equilibria have a strict dominance ranking. That is, if
p and p are two equilibria with average adherence probabilities f 5 q9p and f 5 q9p , then either i p ss p and f . f or ii p aa p and
f , f .
In verbal summary: Equilibria correspond to roots of the polynomial p f in the interval 0 f 1; there are one, two, or three; the number is three in the benchmark
case when total adherence and total rejection are both possible when q 5 0 and q 5 1;
1 n
each equilibrium p is determined by its root according to the formula in Part 3; and multiple equilibria are dominance ranked. In addition, the theorem gives information on
p9 f at equilibria, which will be useful in the stability analysis below. Examples of
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203
Fig. 2. Illustration of three, two, and one root cases.
triple, double, and single equilibrium cases are q 5 0.1, 0.4, 0.5, 0.99, q 5 0.125, 0.5, 0.5, 1.09, and q 5 0.2, 0.4, 0.5, 0.89. Fig. 2, used momentarily in the proof, illustrates
the qualitative look of p f in these cases. As Fig. 2 suggests, a double equilibrium involves a tangency root and is thus a knife-edge case; a change in any q will bump the
i
tangency up or down, thus eliminating the root or changing it to two roots.The intuition of the three- equilibrium case was discussed in the context of Fig. 1 above. The intuition
of other cases can be thought of as perturbations from this case. As we make q larger
1
and larger, the low equilibrium will sooner or later be lost because the lowest adherence rates won’t be low enough to support a low equilibrium. Similarly, as we make q
n
smaller and smaller, the high equilibrium will be lost.
Proof of the equilibrium theorem. The transition matrix Pp from 1 depends on p only through f 5 q9p. Rewrite Pp as P f for this paragraph only. Solving the
equilibrium condition p 5 Pp9p for p can be thought of as solving p 5 P f 9p and f 5 q9p simultaneously for p and f
. Asterisks, as equilibrium indicators, are suppressed during this proof. The equation p 5 P f 9p can be solved by linear means for p as a
function of f, call the solution p 5 a f . The ith element is
i 2
n
a f 5 x [x 1 x 1 ? ? ? 1 x ], where x 5 f1 2 f . 3
i
Combining p 5 a f and f 5 q9p yields f 5 q9a f as a scalar equation to solve for equilibrium values of f subject to 0 f 1. Premultiplying f 2 q9a f 5 0 by
i 21 n 2i
o f 1 2 f
and rearranging yields the polynomial equation p f 5 0 from Part 1
i
of the Equilibrium Theorem. Thus, there is an equilibrium p 5 a f associated with each root of p f in the unit interval. Next show that the number of such roots is one,
two, or three. Count roots on the boundary of the unit interval f 5 0 or f 5 1 separately from roots in the interior 0 , f , 1.
Boundary roots. By careful inspection of the polynomial p f , f 5 0 is a root if and only if q 5 0, and f 5 1 is a root if and only if q 5 1. Thus, there may be zero, one, or
1 n
two boundary roots depending on whether neither, one, or both of the conditions q 5 0
1
and q 5 1 hold.
n n
Interior roots. Dividing p f 5 0 by 1 2 f and using the transform x 5 fl 2 f allows p f 5 0 to be rewritten
2
1 2 q 1 1 2 q 2 q x 1 1 2 q 2 q x 1 ? ? ?
1 1
2 2
3 n 21
n
1 1 2 q 2 q x
1 1 2 q x 5 0. 4
n 21 n
n
Since x 5 f1 2 f is increasing in f, and since any positive x 5 fl 2 f can be realized
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. Conlisk et al. Mathematical Social Sciences 40 2000 197 –213
by an f in the interval 0 , f , 1, the number of interior roots of p f 5 0 equals the number of positive roots x of 4. That number can be bounded using Descartes’ rule of
signs. Moving across the coefficients of 4 from left to right reveals three possible sign changes: i a possible sign change from negative to positive between the first two
coefficients 1 2 q and 1 2 q 2 q ; ii one possible sign change from positive to
1 1
2
negative among the middle n 2 1 coefficients 1 2 q 2 q to 1 2 q 2 q ; and iii a
1 2
n 21 n
possible sign change from negative to positive between the last two coefficients 1 2 q
2 q and 1 2 q . Regarding ii, there cannot be more than one sign change
n 21 n
n
among the middle n 2 1 coefficients because they are declining in size since q q
1 2
? ? ? q . Thus, three is the maximum number of sign changes. By Descartes’ rule of
n
signs, the number of positive roots x with multiple roots counted multiply can be no greater than the number of sign changes. It follows that there are at most three distinct
interior roots for f. However, if q 5 0 and thus f 5 0 is a root, a sign change of type i
1
is ruled out. Similarly, if q 5 1 and thus f 5 1 is a root, a sign change of type iii is
n
ruled out. Therefore, the maximum number of interior roots is three minus the number of boundary roots. That is, the total number of roots cannot exceed three. Numerical
examples easily establish that all three possibilities do occur. To complete proof of Part 1, find the signs of p9 f at the roots by considering the
graph of p f over 0 f 1, delineating cases. Consider first the case when q . 0 and
1
q , 1. Then the graph must be qualitatively as on the illustrations of Fig. 2. To see why,
n
verify from the definition of p f that p0 5 2 q , 0 and p1 5 1 2 q . 0. That is,
1 n
the left end of the graph has negative intercept at f 5 0, and the right end has positive intercept at f 5 1, as illustrated on Fig. 1. But then the only ways to finish the graph with
a maximum of three real roots in the unit interval are qualitatively like the cases shown. Since a tangency intersection always represents a pair of identical roots, two tangency
intersections would require more than three roots and thus cannot occur. The sign claims about p9 f then follow. When q 5 0, similar considerations apply, except that
1
the left intersection is at f 5 0. When q 5 1, the right intersection is at f 5 1.
n
Part 3 follows immediately from 3 above, and Part 4 follows from Part 3. To verify Part 2, recall from the discussion of boundary roots that the graph of p f has an
intersection at f 5 0 when q 5 0 and an intersection at f 5 1 when q 5 1. The first two
1 n
sentences of Part 2 then follow from Part 3. To verify the third sentence, first verify that p90 . 0 and p91 . 0 when q 5 0 and q 5 1. It follows that the graph of p f must
1 n
have a third intersection with 0 , f , 1, and it then follows from Part 3 that, at the corresponding equilibrium, p has all positive elements.
h Now turn to stability and regions of attraction. We will refer to an equilibrium in p as
‘locally stable’ if and only if pt converges to p from any direction, so long as pt is close enough to p to begin with. Formally, local stability means that, for some positive
number ´, pt 2 p 9pt 2 p , ´ implies pt →
p as t →
`. For the case of three equilibria, the intuition of Fig. 1 suggests that the attractive force of imitation will
make the low and high equilibria locally stable, with regions of attraction extending all the way to the middle equilibrium. Part 1 of the Stability Theorem confirms this
intuition. Part 2 of the Theorem states that, when we move from three to two equilibria,
J . Conlisk et al. Mathematical Social Sciences 40 2000 197 –213
205
one of the stable equilibria drops off, leaving one stable and one unstable equilibrium. Part 3 states that, when we move from two to one equilibrium, the unstable equilibrium
drops off, leaving one stable equilibrium.
Stability theorem. Distinguish the three main cases:
1. Triple equilibrium. Assume a triple equilibrium p , p , p with the dominance order p aa p aa p . Then the low equilibrium p is locally stable, with
region of attraction including every vector strictly dominated by the middle equilibrium. That is, pt
→ p if p0 aa p. Similarly, the high equilibrium in
p is locally stable, with region of attraction including every vector strictly dominating the middle equilibrium. That is, pt
→ p if p0 ss p. The middle
equilibrium is therefore not locally stable. 2. Double equilibrium. Assume a double equilibrium p , p , with adherence rates
f , f , ordered such that p aa p and f , f . i If p9 f 5 0 and p9 f . 0, then p is not locally stable, and p is locally stable with region of
attraction including every vector strictly dominating p [that is, pt →
p if p0 ss p]. ii In the reverse case, when p9 f . 0 and p9 f 5 0, the reverse
conclusions hold. 3. Single equilibrium. A single equilibrium p is globally stable; it has region of
attraction including every possible starting point. That is, pt →
p for any p0. For a triple equilibrium, Part 1 asserts not just local stability for the low and high
equilibria, but also regions of attraction extending all the way to the middle, or ‘tipping’, equilibrium. Nonetheless, this is not a complete result. Compare the n 5 3 example of
Fig. 1 to Part 1 of the theorem. On Fig. 1, the entire space of pt was partitioned into regions of attraction for the three equilibria – above the dashed line, below the dashed
line, and the dashed line itself. A corresponding result for the general n 3 case would say that there is a surface in the space of pt such that the regions of attraction for the
three equilibria are all initial points to one side of the surface, all to the other side of the surface, and all on the surface itself. We suspect that some such partition exists, but we
have not found it. Part 1 is less complete since it does not cover all initial points; rather it covers only points which dominate or are dominated by the middle equilibrium.
Nonetheless, proof of the Stability Theorem is long and difficult. See Appendix A. The approach of the proof is to transform the model to terms of the cumulative distribution
Pt and to show that, under the model’s dynamic, Pt11 is weakly increasing in every element of Pt. Roughly, more or less adherence one period means more or less the
next period. Convergence results from the literature, based on monotonicity, then become available. The intuition is buried in the details, but these results allow us to
conclude that, within each region specified in the theorem, Pt converges monotonically element by element to the relevant equilibrium.
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. Conlisk et al. Mathematical Social Sciences 40 2000 197 –213
4. Conclusion
