A DISCRETE NONLINEAR AND NON–AUTONOMOUS MODEL OF CONSENSUS FORMATION U. Krause Fachbereich Mathematik und Informatik, Universit¨at Bremen, Bremen, Germany e–mail:
[email protected]–bremen.de Abstract Consensus formation among n experts is modeled as a positive discrete dynamical system in n dimensions. The well–known linear but non–autonomous model is extended to a nonlinear one admitting also various kinds of averaging beside the weighted arithmetic mean. For this model a sufficient condition for reaching a consensus is presented. As a special case consensus formation under bounded confidence is analyzed. Key words: discrete dynamical systems, Markov chains, averaging, consensus formation, bounded confidence. AMS Subject Classification: 39A11, 60J20, 92H30
1
The Model
Consider a group of experts who have to make a joint assessment of a certain magnitude. Each of the experts has his own opinion but is open to some extent to revise it when being informed about the opinions of all the other experts. Knowing the revisions may lead to further revisions and the question then is if this iterative process of changing opinions will tend to a consensus among the experts concerning the value of the magnitude. Denote by xi (t) ≥ 0 the assessment made by expert i ∈ {1, . . . , n} at time t ∈ N = {0, 1, 2, . . . } of the nonnegative magnitude under consideration. Suppose that expert i arrives at a revision xi (t + 1) by taking the assessments xj (t) of the other experts into account with certain weights aij . If A denotes the row–stochastic matrix of the weights aij , i.e., aij ≥ 0 and n P aij = 1, and x(t) the column vector of the xi (t) this amounts to j=1
x(t + 1) = Ax(t) for all t ∈ N.
(1)
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Ulrich Krause
This model has been put foward in [6] where standard limit theorems for Markov chains are used to formulate conditions on the weights under which a consensus c will be reached, that is lim xi (t) = c for all i ∈ {1, . . . , n}. t→∞
(See [9] for a discussion of this model). A similar model, though in continuous time, has been developed in [1] for the formation of attitudes within a group of persons influencing each other. (In [7] this model is considered in a setting where time as well as space are discrete.) The model from [6] has been extended in [3] and [4] by admitting the weights aij to depend on time t. (See also [5] for weights decreasing in time.) In the present paper we extend this model further by admitting the weights to depend also on the assessments made by the experts which makes system (1) nonlinear. Moreover, we shall allow also more general kinds of averaging than the weighted arithmetical mean underlying system (1) as, e.g., a geometric mean or a harmonic mean. This will be achieved in the following framework. Denote by K the cone of all (column) vectors x ∈ Rn with strictly positive components and let A(x, t) be a row–stochastic matrix of weights for every x ∈ K, t ∈ N. Let F : K −→ Rn be a mapping with the property that for each x ∈ K and each t ∈ N there exists a uniquely determined x∗ = x∗ (x, t) ∈ K such that F (x∗ ) = A(x, t)F (x).
(2)
By Eqn. (2) the mapping F induces various kinds of averaging with the weighted arithmetic mean as special case for F being the identity map. (See Corollary 1.) Taking some x(0) ∈ K as starting point by ¡ ¢ ¡ ¢ ¡ ¢ F x(t + 1) = A x(t), t F x(t) for t ∈ N (3) a discrete dynamical system on the cone K is defined which is nonlinear and non–autonomous. This system has the peculiar feature that it is positive. (Cf. [8] for those systems.) In Section 2 a sufficient condition is given for the system (3) to reach a consensus. In Section 3 a simple but appealing nonlinear system is analyzed. This system portrays bounded confidence for placing weights to others and exhibits, albeit a rather special case of (2), already a complicated dynamics.
2
Convergence to a Consensus
For x in the cone Rn+ of all vectors in Rn with nonnnegative components let v(x) = max xi − min xi = max (xi − xj ). 1≤i≤n
1≤i≤n
1≤i,j≤n
The following statement for row–stochastic matrices is well–known (cf. [10, Theorem 3.1]), but for the readers’ convenience we supply a short proof.
CONSENSUS FORMATION
3
Lemma row stochastic ´ matrix then ³ 1 If A is aP n v(Ax) ≤ 1 − min min{aik , ajk } v(x) for all x ∈ Rn+ . 1≤i,j≤n k=1
Proof. By definition of v(·) v(Ax) = max i,j
X
(aik − ajk )xk .
k
¢ ¢ P The assertion then P ¡ follows from P¡ (aik − ajk )xk = aik − min{aik , ajk } xk − ajk − min{aik , ajk } xk k ¡ k k ¢¡ ¢ P ≤ 1 − min{aik , ajk } max xk − min xk . 3 k
k
k
Theorem 1 Suppose for the model given by (3) that there exist numbers ∞ n P P 0 ≤ δt ≤ 1 such that δt = ∞ and min{aik (x(t), t), ajk (x(t), t)} ≥ δt t=0
for all i, j ∈ {1, . . . , n}, all t ∈ N.
k=1
¡ ¢ ¡ ¢ (i) For each x(0) ∈ K there exists a number a x(0) such that lim Fi x(t) = t→∞ ¡ ¢ a x(0) for all i ∈ {1, . . . , n}, Fi (x) being the i–th component of F (x). (ii) Assume that F is given by Fi (x) = f (xi ) for all x ∈ K, all i ∈ {1, . . . , n} where f : P −→ R for P = {r ∈ R | r > 0} is continuous, injective and such that f (P ) is a convex cone in R; assume further that lim f (rn ) ≥ f (s) for some s ∈ P implies lim rn ∈ P . n→∞ n→∞ ¡ ¢ Then for each¡ x(0)¢ ∈ K there exists a number c x(0) such that lim xi (t) = c x(0) for all i ∈ {1, . . . , n}. t→∞
¡ ¢ Proof. (i) Let y(t) = F x(t) . From Eqn. (3) we have that ¡ ¢ y(t + 1) = A x(t), t y(t) for all t ∈ N. (4) ¡ ¢ By the and the assumptions of the Theorem v y(t + 1) ≤ (1 − ¡ Lemma ¢ −r δt )v y(t) for¡t ∈ N. The ¢ mean ¡value¢theorem gives (1 − r) ≤ e for r ≥ 0 −δt and, hence, v y(t + 1) ≤ e v y(t) . By iteration t P
− δi ¡ ¡ ¢ ¡ ¢ ¢ v y(t + 1) ≤ e−δt e−δt−1 . . . e−δ0 v y(0) = e i=0 v y(0) .
¡ ¢ δt = ∞ implies lim v y(t) = 0 for any x(0) ∈ K. Let t→∞ t=0 ¡ ¢ x(0) ∈ K be arbitrary but fixed. Since A x(t), t is row–stochastic from Eqn. (4) we have
The assumption
∞ P
min yj (t) ≤ y(t + 1)i ≤ max yj (t) for all i. j
j
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Ulrich Krause
This implies for a(t) = min yj (t), b(t) = max yj (t) that a(t) ≤ a(t + 1) ≤ j
j
b(t + 1) ≤ b(t) for all t and, therefore, lim a(t) = a and lim b(t) = b exist. t→∞ t→∞ ¡ ¢ ¡ ¢ Because of 0 = lim v y(t) = lim b(t) − a(t) we must have that t→∞ t→∞ ¡ ¢ b = a. From a(t) ≤ y(t + 1)i ≤ b(t) for all i it follows that lim Fi x(t) = t→∞
lim yi (t) = a for all i.
t→∞
(ii) PSince f (P ) is a convex cone it follows for x ∈ K, t ∈ N given that aij (x, t)f (xj ) ∈ f (P ) for all i. This together with the injectivity j
of f implies (2) for ¡ property ¢ ¡ F .¢ From (i) we have for x(0) ∈ K fixed that lim f xi (t) = lim Fi x(t) = a for all i. Also by (i), a ≥ a(0) = t→∞ t→∞ ¡ ¢ min f xj (0) = f (s) for some s ∈ P . The assumptions of (ii) imply for each j
i that ci = lim xi (t) exists and ci ∈ P . Continuity of f implies f (ci ) = a t→∞ for all i and by injectivity of f we must have that ci = c for all 3 Corollary 1 Let A(x, t) for x ∈ K, t ∈ N be a row–stochastic matrix n P satisfying the condition min{aik (x, t), ajk (x, t)} ≥ δt for all x ∈ K, all k=1
i, j ∈ {1, . . . , n}, all t ∈ N and numbers δt ≥ 0 such that
∞ P t=0
δt = 0.
Then a consensus will be reached for consensus formation by a geometric mean, i.e., xi (t + 1) =
n Q j=1
and a power mean, i.e., xi (t + 1) =
xj (t)aij (x(t),t)
³P n j=1
´ α1 ¡ ¢ aij x(t), t xj (t)α , α 6= 0
with arithmetic mean and harmonic mean as special cases for α = 1 and α = −1, respectively. (Thereby, i ∈ {1, . . . , n}, t ∈ N, A(x, t) = (aij (x, t))1≤i,j≤n ). Proof. The statements follow from Theorem 1 with f (r) = lnr for the geometric mean and f (r) = rα (α 6= 0) for the power mean. 3 Remarks In a different way the above results have been obtained in [2]. For weights depending not on x but on t only the result for the arithmetic means goes back to [3] and [4]. If the weights are constant, i.e., depend neither on x nor on t, the condition on the weights in the Corollary is satisfied if any two experts place jointly a positive weight to some expert. A corresponding result for the arithmetic mean gives [6].
CONSENSUS FORMATION
3
5
Consensus formation under bounded confidence
A meaningful variation of the most simple model using the arithmetic mean is the one where each expert forms an arithmetic mean only of those opinions which are not too far from his own opinion. Suppose expert i takes at a profile x of opinions only those experts j into account for which |xi −xj | ≤ ²i where ²i > 0 is a certain level of confidence employed by experts i. For I(i, x) = {1 ≤ j ≤ n | |xi − xj | ≤ ²i } this means that aij (x) is 0 for j ∈ / I(i, x) and equals |I(i, x)|−1 for j ∈ I(i, x), where |M | denotes the number of elements in a finite set M . The matrix A(x) of the weights aij (x) is still row–stochastic. Disregarding any explicit time dependence of the weights and taking F the identity map Eqn. (3) then becomes X ¡ ¢ xi (t + 1) = |I i, x(t) |−1 xj (t). (5) ¡ ¢ j∈I i,x(t)
This is a simple model the dynamics of which, however, seems to be rather complex due to the nonlinearity embodied in Eqn. (5). For reaching consensus in this model from Theorem 1 we obtain a sufficient condition which in the special case of n = 2 turns out to be also necessary. Corollary 2 For the model given by Eqn. (5) consider the condition that for some t0 ∈ N I(i, x(t)) ∩ I(j, x(t)) 6= ∅ for all i, j ∈ {1, . . . , n}, all t ≥ t0
(6)
(i) Condition (6) implies consensus, i.e., for each x(0) ∈ K there exists a number c(x(0)) such that lim xi (t) = c(x(0)) for all i ∈ {1, . . . , n}. t→∞
(ii) For n = 2 condition (6) is also necessary to reach consensus and it is equivalent to |x1 (0) − x2 (0)| ≤ max{²1 , ²2 }. Furthermore, if this condition holds then consensus is reached after finitely many steps m + 1 where m = plog2
|x1 (0) − x2 (0)| q. min{²1 , ²2 }
(paq denotes the smallest natural number above a.)
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Ulrich Krause
Proof. (i) Condition (6) implies that
n P
min{aik (x), ajk (x)} ≥
k=1
1 n
for all x ∈ K,
all i, j ∈ {1, . . . , n}. Hence, Theorem 1 (ii) with f (r) = r implies consensus. (ii) Obviously, for t ∈ N fixed I(1, x(t)) ∩ I(2, x(t)) 6= ∅ is equivalent to |x1 (t) − x2 (t)| ≤ max{²1 , ²2 }. If |x1 (0) − x2 (0)| > max{²1 , ²2 } then we must have x1 (t) = x1 (0) and x2 (t) = x2 (0) for all t ∈ N. On the other hand, from |x1 (0) − x2 (0)| ≤ max{²1 , ²2 } it follows by induction that |x1 (t) − x2 (t)| ≤ max{²1 , ²2 }. Therefore, I(1, x(t)) ∩ I(2, x(t)) 6= ∅ for some t0 and all t ≥ t0 iff |x1 (0) − x2 (0)| ≤ max{²1 , ²2 }. Furthermore, suppose that |x1 (0) − x2 (0)| ≤ max{²1 , ²2 } and let ²1 ≤ ²2 . By induction it follows that |x1 (t) − x2 (t)| ≤ ²2 for all t and, hence, x2 (t+1) = 21 (x1 (t)+x2 (t)) for all t. Since a consensus will be reached there exists a smallest m ∈ N such that |x1 (m) − x2 (m)| ≤ ²1 and the consensus is reached after m + 1 steps. In particular, x1 (t) = x1 (0) for all t ≤ m and, hence, |x1 (t) − x2 (t)| = 21 |x1 (t − 1) − x2 (t − 1)| for all t ≤ m. Induction yields |x1 (m) − x2 (m)| = 21m |x1 (0) − x2 (0)| and, 2 (0)| therefore, 21m |x1 (0) − x2 (0)| ≤ ²1 and log2 |x1 (0)−x ≤ m. ²1 3 Arithmetic means are preformed over multi–sets, i.e., sets admitting equal elements or, equivalently, sequences for which the ordering is irrelevant. For a (nonempty) finite multi–set A of real numbers let |A| denote the number P of elements of A and let MA denote the arithmetic mean a. The following Lemma collects some useful properties of MA = |A|−1 a∈A
the arithmetic mean. Lemma 2. (i) If U and V are (nonempty) finite multi–sets of real numbers with max U ≤ min V , then MU ≤ MU ∪V ≤ MV . (ii) Let A be a (nonempty) finite multi–set of real numbers and for i = 1, 2 let ai ∈ A, ²i ≥ 0 and Ai = {a ∈ A | |a − ai | ≤ ²i }. If a2 − a1 ≥ |²1 − ²2 | then MA1 ≤ MA2 . (iii) If a2 − a1 > max{²1 , ²2 } then MA1 ≤ max{a ∈ A | a < a2 } and MA2 ≥ min{a ∈ A | a1 < a}.
CONSENSUS FORMATION
7
Proof. U +|V |MV (i) From MU ∪V = |U |M and max U ≤ min V it follows that |U |+|V | MU ≤ max U ≤ min V ≤ MV and, hence, MU ≤ MU ∪V ≤ MV .
(ii) In a first step we show the following two implications (*): If a ∈ A1 If a ∈ A1 \A2
and and
b ∈ A2 \A1 b ∈ A2
then then
a < b. a < b.
(A\B denotes the multi–set of elements in A which are not in B.) Suppose the first implication does not hold, that is there exist a ∈ A1 and b ∈ A2 \A1 such that a ≥ b. It follows that b ≤ a ≤ a1 + ²1 and, because of b ∈ A2 \A1 , a2 − ²2 ≤ b < a1 − ²1 . Hence, a2 − a1 < ²2 − ²1 which contradicts a2 − a1 ≥ |²1 − ²2 |. Similarly, if the second implication does not hold then there exist a ∈ A1 \A2 and b ∈ A2 such that a ≥ b. It follows that a2 − ²2 ≤ b ≤ a and a2 + ²2 < a ≤ a1 + ²1 . Hence, a2 − a1 < ²1 − ²2 which contradicts a2 − a1 ≥ |²1 − ²2 |. Now, if A1 ∩ A2 = ∅ or A1 \A2 = ∅ or A2 \A1 = ∅ then from (*) together with (i) we obtain MA1 ≤ MA2 . Hence, suppose the multi– sets A1 ∩ A2 , A1 \A2 and A2 \A1 are all nonempty. Then (*) implies MA1 \A2 ≤ MA1 ∩A2 ≤ MA2 \A1 . Since MA1 =
|A1 ∩ A2 |MA1 ∩A2 + |A1 \A2 |MA1 \A2 , |A1 ∩ A2 | + |A1 \A2 |
and similarly for MA2 , it follows that MA1 ≤ MA1 ∩A2 ≤ MA2 . (iii) If a ∈ A1 then by assumption a ≤ a1 + ²1 ≤ a1 + max{²1 , ²2 } < a2 . Similarly, a ∈ A2 implies that a ≥ a2 − ²2 ≥ a2 − max{²1 , ²2 } > a1 and, hence, mA1 ≥ min{a ∈ A | a1 < a}. 3 Definition For ² ≥ 0 a vector x ∈ Rn+ is called an ²–profile if there exists an ordering xi1 ≤ xi2 ≤ . . . ≤ xin of the components of x such that two adjacent components have a distance less or equal to ², i.e. xik+1 − xik ≤ ² for all 1 ≤ k ≤ n − 1. Whereas Corollary 2 presents a sufficient condition for reaching a consensus the following Theorem gives a necessary condition in the case that all confidence levels are equal. Theorem 2 For a consensus to be reached in the model given by Eqn. (5) with ²i = ² for all i ∈ {1, . . . , n} it is necessary that x(t) is an ²–profile for all t ∈ N.
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Ulrich Krause
Proof. Suppose x(t) is not an ²–profile for some t0 ∈ N. By relabelling the components of x(t) we may assume that x1 (t0 ) ≤ . . . ≤ xn (t0 ) and xk+1 (t0 ) − xk (t0 ) > ² for some k ∈ {1, . . . , n}. Let A be the multi–set of the components x(t) and let for i < j fixed a1 = xi (t0 ), a2 = xj (t0 ). Lemma 2 (ii) implies for ²1 = ²2 = ² that xi (t0 + 1) = MA1 ≤ MA2 = xj (t0 + 1). This shows that x1 (t0 + 1) ≤ . . . ≤ xn (t0 + 1). Now, choose from the multi–set A the elements a1 = xk (t0 ), a2 = xk+1 (t0 ). By assumption a2 − a1 > ² and Lemma 2 (iii) implies for ²1 = ²2 = ² that xk+1 (t0 + 1) − xk (t0 + 1) = MA2 − MA1 ≥ ≥ min{xi (t0 ) | xk (t0 ) < xi (t0 )} − max{xi (t0 ) | x >i (t0 ) < xk+1 (t0 )}. By the ordering of the components of x(t0 ) it follows that xk+1 (t0 + 1) − xk (t0 + 1) ≥ xk+1 (t0 ) − xk (t0 ) > ². Thus, the step which lead from t0 to t0 +1 can be iterated which implies that xk+1 (t) − xk (t) > ² for all t ≥ t0 . This shows that a consensus cannot be reached. 3 The following Theorem shows that the ²–profile condition is also sufficient for n ≤ 4 but not for n ≥ 5. Theorem 3 For the model given by Eqn. (5) with ²i = ² for all i ∈ {1, . . . , n} the following statements hold. (i) For 2 ≤ n ≤ 4 consensus is reached if and only if x(0) is an ²–profile. (ii) For n ≥ 5 the condition on x(0) to be an ²–profile is only necessary but not sufficient for reaching a consensus. Proof. (i) By Theorem 2 it is sufficient to show that for an ²–profile x(0) consensus is reached. Suppose x(0) to be an ²–profile with x1 (0) ≤ . . . ≤ xn (0) without restriction. Lemma 2 (i) implies as in the proof of Theorem 2 that x1 (t) ≤ . . . ≤ xn (t) for all t ∈ N. Define ∆(t) = xn (t) − x1 (t) for t ∈ N. For n = 2 we have x1 (1) = 12 (x1 (0) + x2 (0)) = x2 (1) and, hence, ∆(1) = 0. Thus, a consensus is rached for t = 1. For n = 3 Lemma 2 (i) implies that x1 (1) ≥ 21 (x1 (0) + x2 (0) = and x3 (1) ≤ 21 (x2 (0) + x3 (0)) and, hence. ∆(1) ≤ 12 (x3 (0) − x1 (0)) ≤ ². Thus, ∆(2) = 0 and consensus is reached for t = 2. Consider now the case n = 4. By Lemma 2 (i) we have x4 (t + 1) ≤ 12 (x3 (t) + x4 (t)) and x1 (t + 1) ≥ 12 (x1 (t) + x2 (t)) which implies ∆(t + 1) ≤
1 1 (x3 (t) − x2 (t)) + ∆(t). 2 2
By Lemma 2 (i) again x3 (t + 1) ≤ 13 (x2 (t) + x3 (t) + x4 (t)) and x2 (t + 1) ≥ 1 1 3 (x1 (t)+x2 (t)+x3 (t)) which implies x3 (t+1)−x2 (t+1) ≤ 3 (x4 (t)−x1 (t)).
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Putting together, we obtain the recursion inequality ∆(t + 1) ≤
1 (3∆(t) + ∆(t − 1)) for all t ≥ 1. 6
Because of ∆(0) = x4 (0) − x1 (0) ≤ 3² and ∆1 ≤ 12 (x3 (0) − x2 (0) + 21 ∆(0) ≤ 1 3 2 ² + 2 ² = 2² we compute by using the recursion inequality ∆(2)
≤
1 6 (3∆(1)
+
∆(0))
≤
3 2 ²,
∆(3)
≤
1 6 (3∆(2)
+
∆(1))
≤
1 9 6(2
+ 2)²
=
13 12 ²,
3 57 ∆(4) ≤ 16 (3∆(3) + ∆(2)) ≤ 16 ( 39 12 + 2 )² = 72 ² < ². This implies ∆(5) = 0 and consensus is reached in the case n = 4 for t = 5. This proves (i).
(ii) By Theorem 2 the condition on x(0) to be an ²–profile is necessary. Consider for n ≥ 5 and ² = 1 the profile given by xi (0) = 1 for 1 ≤ i ≤ 4 and xi (0) = 4 for 5 ≤ i ≤ n. Obviously, x(0) is an ²–profile. For t = 1 one obtains x1 (1) = 22 , x2 (1) = 2, x3 (1) = and x1 (1) =
3+4(n−3) n−2
=
4n−9 n−2
2+3+4(n−3) n−1
=
4n−7 n−1
for 4 ≤ i ≤ n.
Therefore, x2 (1) − x1 (1) < ², x3 (1) − x2 (1) = 2n−5 n−1 > ² and x1 (1) − 4n−7 x3 (1) = 4n−1 − < ² for 4 ≤ i ≤ n. n−2 n−1 This shows that x(1) is no longer an ²–profile. Therefore, by Theorem 2 a consensus cannot be reached. Actually, for t ≥ 2 one has x1 (t) = x2 (t) and x3 (t) = x4 (t) = . . . = xn (t) but x3 (t) − x2 (t) > ². That is, there are two different subgroups of experts such that consensus is reached within each group but not across the groups. 3 By the above Theorem a uniform confidence level ² together with an initial ²–profile does not guarantee that a consensus will be reached. Instead, as the proof of part (ii) of the Theorem illustrates, an opinion pattern emerges showing consensus within subgroups and persistent dissent between these groups. This is an interesting phenomenon which seems quite likely for real processes of opinion formation. This point has been discussed in [1] where (albeit within a different setting) the question is raised how to describe this phenomenon by resorting from linear models to nonlinear ones. For a nonlinear model very similar to that given by Eqn. (5) detailed computer simulations performed in [7] demonstrate that segregated opinion patterns of the above kind are to be expected stable outcomes of opinion formation processes.
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Ulrich Krause
References [1] R.P. Abelson, Mathematical models of the distribution of attitudes under controversy, Contributions to mathematical psychology, (eds. N. Frederiksen, H. Gulliksen), New York 1964; Holt, Rinehart and Winston, pp.142–160. [2] Th. Beckmann, Starke und schwache Ergodizit¨ at in nichtlinearen Konsensmodellen, Diploma thesis, Universit¨at Bremen, 1997. [3] S. Chatterjee, Reaching a consensus: some limit theorems, Proc. Int. Statist. Inst., 1975, pp. 159–164. [4] S. Chatterjee and E. Seneta, Toward consensus: some convergence theorems on repeated averaging, J. Appl. Prob. 14, 1977, pp. 89–97. [5] J. Cohen, J. Hajnal, and C.M. Newman, Approaching consensus can be delicate when positions harden, Stochastic Proc. and Appl. 22, 1986, 315–322. [6] M. H. De Groot, Reaching a consensus, J. Amer. Statist. Assoc. 69, 1974, pp. 118–121. [7] R. Hegselmann, A. Flache and V. M¨oller, Cellular automata models of solidarity and opinion formation: sensitivity analysis, unpublished paper, Universit¨ at Bayreuth, 1998. [8] U. Krause, Positive nonlinear systems: some results and applications, Proceedings of the First World Congress of Nonlinear Analysts ’92, (ed. V. Lakshmikantham), Berlin 1996, Walter de Gruyter, pp. 1529–1539. [9] K. Lehrer and C.G. Wagner, Rational consensus in science and society, D. Reidel Publ. Co., Dordrecht, 1981. [10] E. Seneta, Non–negative matrices and Markov chains, 2nd ed., Springer Verlag, Berlin, 1980.
