Arithmetic–geometric discrete systems - CiteSeerX

Arithmetic–geometric discrete systems Ulrich Krause Fachbereich Mathematik und Informatik Universit¨at Bremen 28334 Bremen, Germany e–mail: [email protected] Abstract Analyzing a consensus in opinion dynamics one meets interesting open problems concerning the iteration of various means. Whereas in two dimensions the consensus can be computed by the arithmetic–geometric mean of Gauss not very much is known in higher dimensions. AMS No. 39A10 Keywords Arithmetic–geometric mean, opinion dynamics, consensus computation

1. Background Consider the following two difference equations where the first equation is given by an arithmetic mean and the second equation by a geometric mean

(1)

p 1 x1 (t + 1) = (x1 (t) + x2 (t)), x2 (t + 1) = x1 (t)x2 (t), t = 0, 1, 2, . . . 2

with initial conditions x1 (0) > 0, x2 (0) > 0. It is not difficult to show that lim x1 (t) and x2 (t) always exist and coincide. What is t→∞

t→∞

more difficult is to determine the common value c(x1 (0), x2 (0)). On May 30th, 1799, Gauss found that the common value is given, surprisingly, by a complete elliptic integral, that is π

(2)

c(x1 (0), x2 (0)) =

1 π with I(a, b) = 2 I(x1 (0), x2 (0))

Z2

dϕ

p 0

a2 cos2 ϕ

+ b2 sin2 ϕ

(see [2, p. 5], [3]). 2. Problems Consider now the following generalization of the above classical arithmeticgeometric mean c(·, ·) to a finite system of difference equations, the right hands of which are given either by a – weighted – arithmetic or geometric mean. More precisely, let

(3)

xi (t + 1) = fi (x1 (t), . . . , xn (t)), t = 0, 1, 2, . . .

where for every i and given weights aij > 0,

n P

i ∈ I, given by the weighted arithmetic mean geometric mean

n Q

aij = 1, the function fi is either, say for

j=1

n P

aij xj or, for i 6∈ I, by the weighted

j=1 a

xj ij . Again, it is not too difficult (see [2, 4, 6, 7]) to show that

j=1

lim xi (t) = c(x1 (0), . . . , xn (0)) for all 1 ≤ i ≤ n. What can be said, however, about the

t→∞

following obvious questions? Problem 1 Find an explicit formula for c(x1 (0), . . . , xn (0)). Such a formula seems to be unkown even for n = 2 in the case of weighted means. Problem 2 Since an explicit formula seems to be hard to find, obtain properties (monotonicity properties; inequalities; functional equation etc.) of the function c(r1 , . . . , rn ), 0 < ri for 1 ≤ i ≤ n. Problem 3 Determine the dependence of the common limit c(x1 (0), . . . , xn (0)) on the stochastic matrix of weights A = (aij ) (for fixed initial conditions). Even for n = 2 an answer seems not to be known. Remark Following the work of Gauss various attempts have been made to find explicit formulas for n = 3, 4 and weighted means which are composed of arithmetic and geometric mean. For example, Gauss’ formula was generalized in [1] to n = 4 and functions fi as follows √ 1 1 √ f1 (x) = (x1 + x2 + x3 + x4 ), f2 (x) = ( x1 x2 + x3 x4 ) 4 2 √ √ 1 √ 1 √ f3 (x) = ( x1 x3 + x2 x4 ) , f4 (x) = ( x1 x4 + x2 x3 ) . 2 2 3. Application: Computing a consensus 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 (cf. [5, 6, 7]). The experts may, in particular, change opinions by taking averages or means over opinions in the previous period which models a compromizing behavior. More specifically, let A be a real n × n–matrix with n P entries 0 < aij such that aij = 1 for all i. Suppose a subset I ⊂ {1, . . . , n} of experts j=1

2

average by taking a weighted arithmetic mean with weights given by the i–rows of A for i ∈ I and all other experts take a weighted geometric mean with weights given by the other rows. This yields the following model of opinion dynamics (4)

xi (t + 1) =

n X

aij xj (t) for i ∈ I, xi (t + 1) =

j=1

n Y

xj (t)aij for i 6∈ I,

j=1

for periods t = 0, 1, 2, . . . and initial opinions x1 (0) > 0, . . . , xn (0) > 0. Obviously, (4) is equivalent to (3). Approaching a consensus means that lim xi (t) = c(x1 (0), . . . , xn (0)) for t→∞

all 1 ≤ i ≤ n. Two extreme cases are I = {1, . . . , n} and I = ∅, respectively. In the first case all experts apply weighted arithmetic means and the consensus c(x1 (0), . . . , xn (0)) n P can be computed by the Basic Limit Theorem for Markov Chains as vj xj (0) where v = (v1 , . . . , vn

)T

is uniquely determined by

AT v

= v,

n P

j=0

vj = 1. The second extreme

j=1

case where all experts apply a weighted geometric mean can be reduced to the first case by taking the logarithm. (See [6].) What is difficult, however, is the computation of a consensus in the mixed case, where some experts apply arithmetic and others apply geometric means. For n = 2, this is answered by formula (2) given by Gauss in 1799. For general n, if not an explicit formula for the consensus, it would be helpful to obtain properties about the dependence of the consensus on the initial conditions. Remark. Compromizing behavior in opinion dynamics may be modeled also by other means like the# harmonic mean #1 " " −1 p n n P 1 P p −1 n or weighted verxj (t) or the power mean with parameter p 6= 0, n xj j=1

j=1

sions thereof. For these and many other means the convergence to a consensus can be shown (see [5, 6]).

References [1] C.W. Borchardt: Theorie des arithmetisch–geometrischen Mittels aus vier Elementen. Math. Abh. Berliner Akad. Wiss. 1878, 33–96. [2] J.M. Borwein and P.B. Borwein: Pi and the AGM. A Study in Analytic Number Theory and Computational Complexity. John Wiley & Sons, New York, 1987. [3] B.C. Carlson: Algorithms involving arithmetic and geometric means. American Mathematical Monthly, 78(1971), 496–505. [4] C.J. Everett and N. Metropolis: A generalization of the Gauss limit for iterated means. Advances in Mathematics, 7 (1971), 297–300. [5] R. Hegselmann and U. Krause: Opinion dynamics driven by various ways of averaging. Computational Economics, 25 (2005), 381–405. 3

[6] U. Krause: A discrete nonlinear and non–autonomous model of consensus formation. In S. Elaydi, G. Ladas, J. Popenda and J. Rakowski (Eds.), Communication in Difference Equations. Gordon and Breach Publ., Amsterdam, 2000, 227–236. [7] U. Krause: How to compromise if you must? Preprint 2005, submitted.

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