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Boltzmann-like and Boltzmann-Fokker-Planck equations as a foundation of behavioral models
Physica A 196 (1993) 546-573 North-Holland SDI: 0378-4371(93)E0027-C
Boltzmann-like and Boltzmann-Fokker-Planck equations as a foundation of behavioral models Dirk Helbing II. Institute for Theoretical
Physics, University of Stuttgart, W-7000 Stuttgart 80, Germany
Received 18 January 1993
It is shown that the Boltzmann-like equations allow the formulation of a very general model for behavioral changes. This model takes into account spontaneous (or externally induced) behavioral changes and behavioral changes by pair interactions. As most important social pair interactions, imitative and avoidance processes are distinguished. The resulting model turns out to include as special cases many theoretical concepts of the social sciences. A Kramers-Moyal expansion of the Boltzmann-like equations leads to the BoltzmannFokker-Planck equations, which allows the introduction of “social forces” and “social fields”. A social field reflects the influence of the public opinion, social norms and trends on behavioural changes. It is not only given by external factors (the environment) but also by the interactions of the individuals. Variations of the individual behavior are taken into account by diffusion coefficients.
1. Introduction
The methods of statistical physics have shown to be very fruitful in physics, but in the last decades they also have become increasingly important in interdisciplinary research. For example, the master equation has found many applications in thermodynamics [l], chemical kinetics [2], laser theory [3] and biology [4]. Moreover, Weidlich and Haag have successfully introduced it for the description of social processes [5,6] like opinion formation [7], migration [8], agglomeration [9] and settlement processes [lo]. Another kind of wide-spread equations are the Boltzmann equations, which have been developed for the description of the kinetics of gases [ll] and of chemical reactions [12]. However, Boltzmann-like equations [13] play also an important role for quantitative models in the social sciences: it turns out (cf. section 2.2) that the logistic equation for the description of limited growth processes 114,151, the so-called gravity model for spatial exchange processes [ 161, and the game dynamical equations modelling competition and cooperation processes [17,18] are special cases of Boltzmann-like equations. Moreover, 0378-4371/93/$06.00 0
1993 - Elsevier Science Publishers B.V. All rights reserved
D. Helbing
I Boltzmann-like
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Boltzmann-like models have recently been suggested for avoidance processes of pedestrians [19,20] and for attitude formation by direct pair interactions of individuals occurring in discussions [ 19,211. In this paper we shall show that Boltzmann-like equations and BoltzmannFokker-Planck equations [13] are suited as a foundation of quantitative behavioral models. For this purpose, we shall proceed in the following way: in section 2 the Boltzmann-like equations will be introduced and applied to the description of behavioral changes. The model includes spontaneous (or externally induced) behavioral changes and behavioral changes by pair interactions of individuals. These changes are described by transition rates. They reflect the results of mental and psychical processes, which could be simulated with help of Osgood and Tannenbaum’s congruity principle [22], Heider’s balance theory [23] or Festinger’s dissonance theory [24]. However, it is sufficient for our model to determine the transition rates empirically (section 5). The ansatz used for the transition rates distinguishes imitative and avoidance processes, and assumes utility maximization of the individuals (section 2.1). It is shown that the resulting Boltzmann-like model for imitative processes implies as special cases many generally accepted theoretical approaches in the social sciences (section 2.2). In section 3 a consequent mathematical formulation related to an idea of Lewin [25] is developed, according to which the behavior of individuals is guided by a social field. This formulation is achieved by a Kramers-Moyal expansion of the Boltzmann-like equations leading to a kind of diffusion equations: the so-called Boltzmann-Fokker-Planck equations [13]. In these equations the most probable behavioral change is given by a vectorial quantity that can be interpreted as social force (section 3.1). The social force results from external influences (the environment) as well as from individual interactions. In special cases the social force is the gradient of a potential. This potential reflects the public opinion, social norms and trends, and will be called the social field. By diffusion coejficients an individual variation of the behavior (the “freedom of will”) is taken into account. In section 4 representative cases are illustrated by computer simulations. The Boltzmann-Fokker-Planck model for the behavior of individuals under the influence of a social field shows some analogies with the physical model for the behavior of electrons in an electric field (e.g. of an atomic nucleus) [13,19] (cf. Hartree’s self-consistent field ansatz [26]). Especially, individuals and electrons influence the concrete form of the effective social, resp. electric field. However, the behavior of electrons is governed by a different equation: the Schrodinger equation. In physics, the Boltzmann-Fokker-Planck equations can be used for the description of diffusion processes [27].
D. Helbing
548
I Boltzmann-like equations as foundation of behavioral models
2. The Boltzmann-like
equations
Let us consider a system consisting of a great number N 4 1 of subsystems. These subsystems are in one state x of several possible states combined in the set 0. Due to Jluctuafions one cannot expect a deterministic theory for the temporal change dxldt of the state x(t) to be realistic. However, one can construct a stochastic model for the change of the probability distribution P(x, t) of states r(t) within the given system (P(x, t) 2 0, CxER P(x, t) = 1). By introducing an index a we may distinguish A different types a of subsystems. If N, denotes the number of subsystems of type a, we have Xi=, N, = N, and the following relation holds:
P(x, t) =
Ii $TPJX, t) .
(1)
a=1
Our goal is now to find a suitable equation for the probability distribution Pa(x, t) of states for subsystems of type a (P,(x, t) 2 0, C,,, P,(x, t) = 1). If we neglect memory effects (cf. section 6.1), the desired equation has the form of a muster equation [13,19],
-$PAX,t) = x, [Wa(+‘; 4 PO@‘,t) -
w”(x’lx; t) P&, t)] .
(2)
(x’tx)
wa(x’ Ix; t) is the effective transition rare from state x to x’ and takes into account the fluctuations. Restricting the model to spontaneous (or externally induced) transitions and transitions due to pair interactions, we have [13,19]
w’(x’Ix; t):= w,(x’Ix;
t) + 2 b=l
c
2
PER
y’ER
&G&Y’,
~‘1x7 Y; t) P,(Y, t).
(3)
w,(x) Ix; t) describes the rate of spontaneous (respectively externally induced) transitions from x to x’ for subsystems of type a. Gab(x’, y’lx, y; t) is the transition rate for two subsystems of types a and b to change their states from x and y to x’ and y’ due to pair interactions. Inserting (3) into (2), we now obtain the so-called Boltzmunn-like equations [13,191, ;
P,(x, t) =
c
[w,(xl x’; t) P,(x’, t) - w,(x’ Ix; t) P,(x, t)l
X’Ef2
+
b$, x&y;o ,zu
w,b(x,
Y’ 1x’,
Y;
l)
pb(Y,
‘1 ‘atx’, ‘!
(44
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
549
A
b=l
X’EO
yEL’y’Ef2
with
w&(x’, y’ 1x, y; t, := NbG)ab(x’, Y’ 1x, Y;
(5)
t, .
Obviously, (4b) depends nonlinearly on the probability distributions P,(x, t), which is due to the interaction processes. Neglecting spontaneous transitions (i.e., w,(x’ 1x; t) = 0), the Boltzmannlike equations agree with the Boltzmann equations, which originally have been developed for the description of the kinetics of gases [ll]. A more detailed discussion can be found in [19]. In order to apply the Boltzmann-like equations to behavioral changes we now have to take the specifications which are given in table I (cf. [19]). It is possible to generalize the resulting behavioral model to simultaneous interactions of an arbitrary number of individuals (i.e., higher order interactions) [13,19]. However, in most cases behavioral changes are dominated by pair interactions. Many of the phenomena occurring in social interaction processes can already be understood by the discussion of pair interactions. 2.1. The form of the transition rates In the following we have to find a concrete form of the effective transition rates:
Wa(X’IX; ‘) = w,(x’Ix; t) + i
b=l
W&‘IX,
y; t):=
c
2 &,(X’IX, y; t)
yE0
w,&‘, y’lx,
(64
p,(y, t),
(6’3)
Y; t>
Y’EO
Table I Specification Statistical
of the notions physics
system subsystems states types of subsystems transitions interactions fluctuations
used in statistical
physics
for an application
Behavioral
to behavioral
models.
models
population individuals behaviors (concerning a special topic of interest) types of behavior (subpopulations) behavorial changes imitative processes, avoidance processes “freedom of will”
D. Helbing
550
(cf.
which
(3))
I Boltzmann-like
is suitable
equations
for
the
as foundation
description
of behavioral
models
of behavioral
changes.
Walh(x’ 1x, y; t) is the rate of pair interactions,
XlZX)
(7)
where an individual of subpopulation a changes the behavior from x to x’ under the influence of an individual of subpopulation b showing the behavior y. There are only two important kinds of social pair interactions:
x’
tix
(x’ # x) .
(8b)
Obviously, the interpretation of the above kinds of pair interactions is the following: - The interactions (8a) describe imitative processes (processes of persuasion), that means, the tendency to take over the behavior x’ of another individual. -The interactions (8b) describe avoidance processes, where an individual changes the behavior when meeting another individual showing the same behavior x. Processes of this kind are known as aversive behavior, defiant behavior or snob effect. The corresponding
transition rates are of the general form
where the term (9a) describes imitative processes and the term (9b) describes avoidance processes. aYyxhas the meaning of the Kronecker function. By inserting (9) into (6) we arrive at the following general form of the effective transition rates: d(x’
1x; t) := v,(t) R,(x’ 1x; t) + i
vab(t) [Rib@’
b=l
For behavioral R;,(x’)x;
In (1%
Ix; t) P,(x’, t) + R:,(x’lx; t>f’,(x, t)] . VW
models one often assumes t) := fib(t)
R”(x’lx;
t) .
(lob)
D. Helbing
I Boltzmann-like
equations as foundation
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551
- v,(t) is a measure for the rate of spontaneous (or externally induced) behavioral changes within subpopulation a. - R,(x’ 1x; t) [R”(x’ 1x; t)] is the readiness for an individual of subpopulation a to change the behavior from x to x’ spontaneously [in pair interactions]. - Gus = Nb Fab(f) is the interaction rate of an individual of subpopulation a with individuals of subpopulation b. - ftb(t) is a measure for the frequency of imitative processes. - fib(t) is a measure for the frequency of avoidance processes. A more detailed discussion of the different kinds of interaction processes and of ansatz (10) is given in [19,28]. For R’(x’ 1x; t) we take the quite general form x.
Ra(x,,
,
t)
=
expVW 9 - UYx,91
with Da(x’, x; t) = Da(x, x’; t) > 0 for an individual of subpopulation the greater, - the greater the difference of the - the smaller the incompatibility haviors x and x’ is. Similar to (lla) R
(x,
a
IX’
q
3
=
(114
’
Da(X’, x; t)
(cf. [8,19]). Then, the readiness Ra(x’ 1x; t) u to change the behavior from x to x’ will be utilitiesU”(. . . , t) of behaviors x’ and x is, (“distance”)
Do(x’, x; t) between
the be-
we use exp[U,(x’, t) - Un(x,
D,(x),x; t)
t)l
’
(lib)
and, therefore, allow the utility function U,(x, t) for spontaneous (or externally induced) behavioral changes to differ from the utility function V(x, t) for behavioral changes in pair interactions. Ansatz (11) is related to the multinomiul logit model [29,30] and assumes utifity maximization with incomplete information about the exact utility of a behavioral change from x to x’, which is, therefore, estimated and stochastically varying (cf. [19]). Computer simulations of the Boltzmann-like equations (2), (lo), (11) are discussed and illustrated in [19,21,28] (cf. also section 4). 2.2.
Special fields of application in the social sciences
The Boltzmann-like equations (2)) (10) include a variety of special cases, which have become very important in the social sciences: (i) The logistic equation [14,15] describes limited growth processes. Let us consider the situation of two behaviors xE {1,2} (i.e., P,(l, t) = 1- P,(2, t))
D. Helbing
552
I Boltzmann-like
equations as foundation
of behavioral models
and one subpopulation (A = 1). x = 2 may, for example, have the meaning to apply a certain strategy, and x = 1 not to do so. If only imitative processes, 2&l)
(12)
and processes of spontaneous
replacement,
lt2.
(13)
are considered, ;
one arrives at the logistic equation
P,(2, t>= -q(t) R,(W; t>P,(2,4 +
q,(t)
f:,(t) R’(211;t) [l- P,G f>l P,(2, t>
= A(t) P,(2, t> [B(t) - P,(2, t)] .
(14)
(ii) The gravity model [16] describes processes of exchange between different places x. It results for R,(x’ 1x; t) = 0, fib(t) = 1, fzb(t) = 0, and A = 1 in
x P(x, t) P(x’, t) . Here, we have dropped the index a because of a = 1. P(x, of being at place x. The absolute rate of exchange from x to the probabilities P(x, t) and P(x’, t) at the places x and chosen as a function of the metric distance ]Jx - x’]] D(x, x’) = D(](x - x’]]).
(15) t) is the probability to x’ is proportional x’. D(x, x’) is often between x and x’:
(iii) The behavioral model of Weidlich and Haag [5,6,8] assumes spontaneous transitions due to indirect interactions, which are, for example, induced by the media (TV, radio, or newspapers). We obtain this model for fib(t) = 0 =fib(t) and UJX, t) := 6a(x, t) +
2 ‘%bPb(X, t) .
b=l
(16)
6,(x, t) is the preference of subpopulation a for behavior x. K,~ are coupling parameters describing the influence of the behavioral distribution within subpopulation b on the behavior of subpopulation a. For K,~ > 0, K,~ reflects the social pressure of behavioral majorities.
D. Helbing
I Boltzmann-like
(iv) The game fib(t) = 0, and
dynamical
R’(n’ 1n; t) := max[E,(x’,
equations as foundation
equations
[17-19,311
553
of behavioral models
result
for fib(t) = Sob,
t) - E,(x, t), 0]
(17)
(cf. [19,28]). Their explicit form is $ P,(x, 0 = X~OIw,(xIx’;
+ I,,
t) PAX’, t)-
W&k
t) P&,
91
PAX,4 [K?k 4 - (&I)I .
(184 (18b)
Whereas (18a) again describes spontaneous behavioral changes (“mutations”, innovations), (18b) reflects competition processes leading to a “selection” of behaviors with a success E,(x, t) that exceeds the average success (E,) := c E,(x’, t) P,(x’, t) . X’ER The success E,(x, t) is connected
(19) with the so-called payoff matrices A,, =
(A,b(x, Y)) by
-%(x,4 := A,(x) +
c c A.b(x, Y> P,(Y,
b=l
yER
t)
(20)
]I9,281. A,( x ) means the success of behavior x with respect to the environment. Since the game dynamical equations (18) agree with the selection mutation equations [17] they are not only a powerful tool in social sciences and economy [18,31-331, but also in evolutionary biology [34-371. 3. The Boltzmann-Fokker-Planck
equations
We shall now assume the set L? of possible behaviors to build a continuous space. The n dimensions of this space correspond to different characteristic aspects of the considered behaviors. In the continuous formulation, the sums in (2), (3) have to be replaced by integrals:
$ P,(x,
d”x’ [ wyx1x’; t) P,(x’, t) - wU(x’ 1x; t) Pa(x, t)]
t) =
0 { WO[X’ 1x - x’; t] P,(x - x’, t) - w@[x’1x; t] PJX, t)}
)
(214
D. Helbing
554
I Boltzmann-like
equations as foundation
of behavioral models
where
wU[x’ - x 1x; t] : = Wyr’ (x; t) : = w,(x’
lx; t) + i
j- d”y j- d”y’ &W,&‘,
b=l 0
y’1-v
Y; t) P,(Y,
t> .
(21b)
R
A reformulation of the Boltzmann-like equations (21) via a Kramer+Moyal expansion [38,39] (second order Taylor approximation) leads to a kind of diffusion equations, the so-called Boltzmann-Fokker-Planck equations [13]:
(224 with the effective drift coefficients Q,(x, t) :=
I R
d”x’ (x;
x;)
wyx )x;
t)
Pb)
and the effective diffusion coefjicients#’ Q,,(x,
t) : = j- d”x’ (x: - xJ(x;
- Xj)Wa(X’ 1x; t) .
VW
Whereas the drift coefficients &(x, t) govern the systematic change of the distribution P,(x, t), the diffusion coefficients Qeij(x, t) describe the spread of the distribution P,(x, t) due to fluctuations resulting from the individual variation of behavioral changes. For ansatz (lo), the effective drift and diffusion coefficients can be split into contributions due to spontaneous (or externally induced) transitions (k = 0), imitative processes (k = l), and avoidance processes (k = 2):
k=O
” In [13] the expression (22). However, they make 1191).
for Q,,(x, t) contains additional terms due to another no contributions since they result in vanishing surface
derivation integrals
of (cf.
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
555
where
K,Oi(X, t) ‘= I, K&, t) :=
I
d”x’ (xl -
2 vab(t) fib(t) b=l
Xi)R,(X’
(x;
t)
,
d”x’ (xi - xJZ?‘(x’
WI
I
A
Kzi(x, t) :=
c
vab(t) ffb(f)
b=l
d”x’ (x; - x;)R’(x’
I
and Qiij(x,
t) : = v,(t) j dnx’ (xl - x,)(x; - xj)R,(x’
Qi,
vab(t)
fLb(‘)
/
1x; t) ,
dnx’ (xi - xi)(x] - x,)R”(x
1x; t) Pb(x’ t) ,
(23~)
Q,‘ij‘= The behavioral and Qiij(x,t).
ISI
'abet) fzb(l)
b=l
1d”x’ (x; - xi)@; - xj)Ra(x’
changes induced by the environment
1x; t) Pb(x, t) .
are included in Kii(x, t)
3.1. Social force and social field The Boltzmann-Fokker-Planck equations chastic equations (Langevin equations) dx. $ = &i(x> t) + SI G,ij(x, t) tj(t)
(22a) are equivalent
to the sto-
(244
7
j=l
with
Kai(X,‘>= ‘ai(X>‘I+ i
j$,(& G,ij(x, t))Gajk(X,t)
(24b)
and
Qaij
(24~) a the vector
5,(x, t) with the
D. Helbing I Boltzmann-like equations as foundation of behavioral models
556
describes the contribution to the change of behavior x that is caused by behavioral fluctuations t(t) (which are assumed to be delta-correlated and Gaussian [19]). Since the diffusion coefficients Q,,(x, t) and the coefficients Gaij(x, t) are usually small quantities, we have Fai(x, t) = KOi(x, t) (cf. (24b)), and (24a) can be put into the form dx - = Kn(x, t) + fluctuations . dt
(26)
Whereas the fluctuation term describes individual behavioral variations, the vectorial quantity KO(x, t) drives the systematic change of the behavior x(t) of individuals of subpopulation a. Therefore, it is justified to denote K,(x, t) as social force acting on individuals of subpopulation a. The social force influences the behavior of the individuals, but, conversely, due to interactions, the behavior of the individuals also influences the social force via the behavioral distributions PO(x, t) (cf. (21b), (22b)). That means &(x, t) is a function of the social processes within the given population. Under the integrability conditions
$
Kai(xl ‘1 = I
&I
Kaj(xY
there exists a time-dependent V,(x, t) := -
dx’*K,(x’,
t,
for all i, j ,
(27)
potential
t) ,
so that the social force is given by its gradient V, K,(X, t) = -VV,(x,
t) .
(29)
The potential V,(x, t) can be understood as social field. It reflects the social influences and interactions relevant for behavioral changes: the public opinion, trends, social norms, etc. 3.2. Discussion
of the concept
of force
Clearly, the social force is no force obeying the Newtonian laws of mechanics. Instead, the social force K=(x, t) is a vectorial quantity with the following properties:
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
K,(x, t) drives the temporal change drldt of another vectorial namely the behavior x(t) of an individual of subpopulation a. The component Kab(x, t) := vohb(t) dnx’ (x’ - x)[f;,(t) I n
+ &(t)
Pb(X,
P,(x’,
41mx Ix; t)
551
quantity
t)
(30)
of the social force P&(X, t) describes the reaction of subpopulation a on the behavioral distribution within subpopulation b and usually differs from Kbu(x, t), which describes the influence of subpopulation a on subpopulation b. Neglecting fluctuations, the behavior x(t) does not change if Ka(x, t) vanishes. Ka(x, t) = 0 corresponds to an extremum of the social field VO(x, t), because it means VV,(x, t) = 0 .
(31)
We can now formulate our results in the following form related to Lewin’s “field theory” [25]: Let us assume that an individuals objective is to behave in an optimal way with respect to the social field, that means, he or she tends to a behavior corresponding to a minimum of the social field. If the behavior x does not agree with a minimum of the social field VO(x, t), this evokes a psychical tension (force)
K,(x, 4 = -Vv,(x, t) ,
(32)
that is given by the gradient of the social field V,(x, t). The psychical tension K,(x, t) is a vectorial quantity that induces a behavioral change according to dx dt = K(x,
t) .
(33)
The behavioral change dxldt drives the behavior x(t) towards a minimum xz of the social field Va(x, t). When the behavior has reached a minimum x: of the social field V,(x, t), it holds VV,(x, t) = 0
(34)
558
D. Helbing
! Boltzmann-like
equations as foundation
of behavioral models
and, therefore, &(x, t) = 0, that means the psychical tension vanishes. - If the psychical tension K,(x, t) vanishes, except for fluctuations no behavioral changes take place, in accordance with (33). In the special case, where an individual’s objective is the behavior xz, one would expect behavioral changes according to dx - = y(xZ - x) ) dt which corresponds
(35)
to a social field
vn(x,0 = lY(x: - 4’
7
with a minimum at ~2. Examples for this case are discussed in [40]. Note, that the social fields V,(x, t) of different subpopulations a usually have different minima xz. That means, individuals of different subpopulations a will normally feel different psychical tensions &(x, t). This shows the psychical tension Ka(x, t) to be a “subjective” quantity.
4. Computer simulations In the following, the Boltzmann-Fokker-Planck equations for behavioral changes will be illustrated by representative computer simulations. We shall examine the case of A = 2 subpopulations, and situations for which the interesting aspect of the individual behavior can be described by a certain position x E [l/20,1] on a one-dimensional continuous scale (i.e., II = 1, x = x). Then, the integrability conditions (27) are automatically fulfilled, and the social field x
V,(x, t) = -
I
dx’ K,(x’, t) - c,(t)
(37)
x0 is well defined. The parameter c,(t) can be chosen arbitrarily. We will take for the value that shifts the absolute minimum of Va(x, t) to zero, that means
c,(t)
x
c,(t) := m;ln
(38) XC1
Since we will restrict the simulations to the case of imitative or avoidance processes, the shape of the social field V,(x, t) changes with time only due to
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
559
changes of the probability distributions Z’,(x, t) (cf. (23)), that means due to behavioral changes of the individuals (see figs. l-6). In the one-dimensional case one can find the formal stationary solution
P,(x) = Pa(xo)
$$ff$exp(2
1
dx’ s)
,
(39)
*0
which we expect to be approached in the limit of large times r+ ~0. Due to the dependence of K,(x) and Q,(X) on P,(X), eqs. (39) are only implicit equations. However, from (39) we can derive the following conclusions: (i) If the diffusion coefficients are constant (Q,(x) = Q,), (39) simplifies to
that means the stationary solution P,(x) is completely determined by the social field V,(x). Especially, P,(x) has its maxima at the positions x:, where the social field V,(x) has its minima (see fig. 1). The diffusion constant Q, regulates the width of the behavioral distribution P,(x). (ii) If the diffusion coefficients Q,(x) are varying functions of the position x, the behavioral distribution P,(x) is also influenced by the concrete form of Q,(x). From (39) one expects high behavioral probabilities P,(x) where the diffusion coefficients Q,(x) are small (see fig. 2, where the probability distribution PI(x) cannot be explained solely by the social field V,(x)). (iii) Since the stationary solution P,(x) depends on both, K,(x) and Q,(x), different combinations of K,(x) and Q,(x) can lead to the same probability distribution P,(x) (see fig. 4 in the limit of large times). For the following simulations, we shall use the ansatz Ra(x,
lx.
t)
=
expWYx’,0 - UX, 01
7
Da@‘,x; t>
for the readiness ZY(x’ 1x; t) to change from x to x’ (cf. (lla)). function UU(x, t):= -i(F)*)
I, := LO/20 ) a
(414 With the utility
(41b)
subpopulation a prefers behavior x,. L, means the indifference of subpopulation a with respect to variations of the position x. Moreover, we take v*bb(0
D,(x’,x;t)
-I+‘-xl/r
:=e
’
r = R/20,
(414
560
D. Helbing
1 Boltzmann-like
equations as foundation
of behavioral models
L,= L2= R=
\
Fig. 1. Imitative processes to behavioral changes.
in the case of one-sided
sympathy
and low indifference
I
L, with respect
D. Helbing
/ Boltzmann-like
equations as foundation
Fig. 2. As fig. 1, but for high indifference
of behavioral models
L, with respect to behavioral changes.
561
562
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
Fig. 3. Imitative processes for mutual sympathy and low indifference
L, in both subpopulations
D.
“,(X*0
n
Helbing
/ Lkkvnann-like
equations as foundation of behavioral mod&
n
x
Fig. 4. As fig. 3, but for h’~gh m d‘ff I erence
22, in subpopulation
2.
564
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
where R can be interpreted as measure for the range of interaction. According to (41c), the rate of behavioral changes is the smaller the greater the changes are. Only small changes of the position (i.e., between neighboring positions) contribute with an appreciable rate. 4.1. Sympathy and interaction frequency Let sob(t) be the degree of sympathy which individuals of subpopulation a feel towards individuals of subpopulation b. Then, one expects the following: whereas the frequency f ah(t) of imitative processes will be increasing with sob(t), the frequency fib(t) of avoidance processes will be decreasing with sab(t). This functional relationship can, for example, be described by f :b(t) := f l(t) s&(t) >
f:bw
:= f f(t) [1 - s&(t)1 3
(42)
with (43)
Oss,,(t)~l.
f i(t) is a measure for the frequency of imitative processes within subpopulation a, f t(t) is a measure for the frequency of avoidance processes. If we assume
the sympathy between individuals of the same subpopulation we have sll(t) = 1 = sz2(t). 4.2.
Imitative processes
to be maximal,
(f f(t) = 1, f z(t) -0)
In the following simulations of imitative processes we assume the preferred With positions to be X, =&andx,=$$.
(S&At))= (f
:b(t)) := (A
;)
7
(44)
the individuals of subpopulation a = 1 like the individuals of subpopulation a = 2, but not the other way round. That means, subpopulation 2 influences subpopulation 1, but not vice versa. One could say, the individuals of subpopulation 2 act as trendsetters. As expected, in both behavioral distributions P=(x, t) there appears a maximum around the preferred behavior x,. In addition, due to imitative processes of subpopulation 1, a second maximum of P,(x, t) develops around the preferred behavior SC*of the trendsetters. This second maximum is small, if the indifference L, of subpopulation 1 with respect to variations of the position x is low (see fig. 1). For high values of the indifference L, even the majority of individuals of subpopulation 1 imitates the behavior of the trendsetters (see fig. 2)!
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
565
We shall now consider the case
for which the subpopulations influence each other mutually with equal strengths. If the indifference L, with respect to changes of the position x is small in both subpopulations a, each probability distribution PO(x, t) has two maxima. The higher maximum is located around the prefered position x,. A second maximum can be found around the position prefered in the other subpopulation. The greater the indifference L, is, the higher this maximum is (see fig. 3). However, if L, exceeds a certain value in at least one subpopulation, only ooze maximum develops in each behavioral distribution P,(x, t)! Despite the fact that the social fields Va(x, t) and diffusion coefficients Q,(x, t) of the subpopulations a are different because of their different prefered positions X, (and different utility functions U(I(x, t)), the behavioral distributions P,(x, t) agree after some time! Especially, the maxima xz of the distributions Pll(x, t) are located at the same position x* in both subpopulations. x* is nearer to the position X, of the subpopulation a with the lower indifference L, (see fig. 4). 4.3. Avoidance
processes
(f:(t)
= 0, f:(t)
= 1)
For the simulation of avoidance processes we assume with x1 = & and x2 = $$ that both subpopulations nearly prefer the same behavior. Fig. 5 shows the case where the individuals of different subpopulations dislike each other,
Gab(t)):=
(A;> 7
i.e., (ffb(t)) = (y i) .
This corresponds to a mutual influence of one subpopulation on the respective other. The computational results prove: (i) The individuals avoid behaviors which can be found in the other subpopulation. (ii) The subpopulation a = 1 with the lower indifference L, < L, is distributed around the prefered behavior xi and pushes away the other subpopulation! Despite the fact that the initial behavioral distribution Pa(x, 0) agrees in both subpopulations, there is nearly no overlapping of P,(x, t) and Pz(x, t) after some time. This is a typical example for polarization phenomena in the society. In fig. 6, we assume that the individuals of subpopulation 2 like the individuals of subpopulation 1 and, therefore, do not react on the behaviors in subpopulation 1 with avoidance processes,
D. Helbing
566
! Boltzmann-like
equations as foundation
of behavioral models
L,= L,= R=
Fig. 5. Avoidance
processes
for mutual
dislike
of both subpopulations.
D. Helbing
/ Boltzmann-like
equations as foundation
of behavioral models
567
Fig. 6. Avoidance processes for one-sided dislike.
(S&(f)):= (;
y) 3
i.e., (fib(t)) = (8 i)
(47)
As a consequence, P2(x, t) remains unchanged with time, whereas P,(x, t) drifts away from the prefered behavior x1, due to avoidance processes. Surprisingly, the polarization effect is much smaller than in fig. 5! The distributions P,(x, t) and P&K, t) overlap considerably. This is because the slope of PZ(x, t) is smaller than in fig. 5 (and remains constant). As a consequence, the probability for an individual of subpopulation 1 to meet a disliked individual of subpopulation 2 with the same behavior x can hardly be decreased by a small behavioral change. One may conclude that polarization effects (which often lead to an escalation) can be reduced, if individuals do not return dislike by dislike.
5. Empirical
determination
of the model parameters
For practical purposes one has, of course, to determine the model parameters from empirical data. Therefore, let us assume to know empirically the distribution functions Pz (x,tr) (the interaction rates uzb (t,)) and the effective transition rates IV: (x’ 1x; t,) (x’ # x) for a couple of times t, E {I,, . . . , CL}. The corresponding effective social fields Vz (x,tr)and diffusion coefficients Q,‘, (x, tr) are, then, easily obtained as x
v,“(X,4) := - I
dx’ .~;(x’,
t,)
(484
568
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
with K,‘&, tr) :=
I n
d”x’ (xl - q)w;(x’
1x; t,) ,
(48b)
and
Q,e,(X, tl) '=I dnX’ (x:
- ~i)(~l -
Xi) W~(X’
1X; tr)
(49)
.
R Much more difficult is the determination of the utility functions u~(x, tr), U~(X, t,), the distance functions D,“(x’, X; t,), and the rates vz(t,), &t,) := G(G) f:;(G), v$(t,) := ~z~(t[) fff(t,). This can be done by numerical minimization of the error function
for example with the method introduced the abbreviation
of steepest descent [41]. In (50), we have
I$
g&‘, xi 4) := v,(f,>exp[u,(x’, 4) - U,(x, t,)] + h=l [~:~(t,)P;(x', tr> +
db(4)6(x, t,)] exp[U”(x’,
4) - V(X, t,)] .
(51)
It turns out (cf. [19]) that the rates a, have to be taken constant during the U,(x, t,), minimization process (e.g., va(tl) = l), whereas the parameters Ua(x, t,), I:,, := vzb(tl) fib(fl) and ua,(t,) := vzb(fl) f#?$(t,) are to be varied. For l/D,(x’, x; tr) one inserts 1
n,(x’, x; tr) D,(x’, x; tr) = C&(x’,x; tr) ’
(524
with ~&‘, x; tr) := wZ(x’ Ix; tJ g,(x’, x; t[) [PZ(x, t,)12
(52b)
+ WXXIx’; 4) g,(x, x’; ?[) [PZ(x’, t,)12 and &(x’, x; tr) := [g&r’, x; tr) PZ(x, tJ2 + (52)
[g,(x,
x’;
t,) PZ(x’,
t,)12 .
follows from the minimum condition for Da(x), x; tr) (cf. 1191).
(52~)
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
569
Since F may have a couple of minima due to its nonlinearity, suitable start parameters have to be taken. Especially, the numerically determined rates vi,(t,) and vzb(tl) have to be non-negative. If F is minimal for the parameters Ua(x, t,), U’(X, tl), D,(x’, x; tl), I,, vib(tl) and vi,(t,), this is (as can easily be checked) also true for the scaled parameters
In order to obtain unique results we put (54)
and
which leads to C,(l[) :=
c
4) ) 1 “‘g UAX,
C”(t,) :=
c
4) ) 1 XE; UYX,
XER
(56)
XER
and
1 -
D,(G)
:=
c 1 rzfld[D.(X~,X;tl)]-lx,x’ER ( (LF‘X)
iii
.
(57)
(X’ZX)i
C,(t,) and C”(t,) are mealz utilities, whereas D,(t[) is a kind of unit of dzhnce. The distances Dz(x’, x; t) are suitable quantities for multidimensional scaling [42,43]. They reflect the “physical structure” (physical topology) of individuals of subpopulation a since they determine which behaviors are more or less related (compatible). By the dependence on a, Dz(x’, x; t) distinguishes different physical structures resulting in different types a of behavior and, therefore, different “characters”.
570
D. Helbing
I Boltzmann-like
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of behavioral
models
6. Summary and outlook In this article, a behavioral model has been proposed that incorporates in a consistent way many models of social theory: the diffusion models, the multinomial logit model, Lewin’s field theory, the logistic equation, the gravity model, the Weidlich-Haag model, and the game dynamical equations. This very general model opens new perspectives concerning a theoretical description and understanding of behavioral changes, since it is formulated fully mathematically. It takes into account spontaneous (or externally induced) behavioral changes and behavioral changes due to pair interactions. Two important kinds of pair interactions have been distinguished: imitative processes and avoidance processes. The model turns out to be suitable for computational simulations, but it can also be applied to concrete empirical data. 6.1.
Memory
effects
of the model in the previous sections has neglected memory changes. However, memory effects can be easily included by generalizing the Boltzmann-like equations to The formulation
effects that may also influence behavioral
-
w;_,.(x‘Ix; t’) P,(x, t’)] ,
(584
with the effective transition rates W;_&
1x; t’) :=
wy(xf 1x;
t’)
+ic c w:;Ye b=l
and generalizing
yEtl
y’~0
the Boltzmann-Fokker-Planck f
6 I’&,
t) =
I ( dt’
‘0
with the effective drift coefficients
Y'
Ix, y; t’)
Pb(Y,
equations to
t’> >
(58b)
D. Helbing
K;;“(x,
t’) : =
I Boltzmann-like
I R
equations as foundation
of behavioral models
d”x’ (x; - x~)w;_~,(x’ 1x; t’) ,
571
(59b)
the effective diffusion coefficients
Q;;“(n, t’) :=
I
d “x’ (xi - Xi)(XI - xj)w;_,.(x’ Ix; t’) )
(59c)
0
and w;_,,(x’ 1x; t’) : =
W;y(X’ 1x; t’) +
g1 I d”yI R
d”y’ w;;“(x’, y’ 1x, y; t’) P,(y, t’)
R
(59d) Obviously, in these formulas there only appears an additional integration over past times t’ [19]. The influence of the past results in a dependence of w;_,,(x’ Ix; t’), K&T”(X, t’), and QLi”(x, t’) on (t - t’). The Boltzmann-like equations (4), resp. the Boltzmann-Fokker-Planck equations (22) used in the previous sections, result from (58), resp. (59), in the Markovian limit w;_r,(x’ 1x; t’) : = wU(x’ 1x; t) qt - t’)
(60)
of short memory (where 6( * - *) is the Dirac delta function). 6.2. Analogies
with chemical
reactions
The Boltzmann-like equations (4) can also be used for the description of chemical reactions, where the states x denote the different sorts of molecules (or atoms) and a distinguishes different isotopes or conformers. Imitative and avoidance processes correspond in chemistry to self-activatory and self-inhibitory reactions. Although the concrete transition rates will be different from (lo), (lla) in detail, there may be found analogous results for chemical reactions. Note that the Arrhenius formula for the rate of chemical reactions [44] can be put into a form similar to (lla) [19].
Acknowledgements
This work has been financially supported by the Volkswagen Stiftung and the Deutsche Forschungsgemeinschaft (SFB 230). The author is grateful to Prof.
D. Helbing
572
I Boltzmann-like
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of behavioral models
Dr. W. Weidlich and Dr. R. Reiner for valuable discussions and commenting on the manuscript.
References [l] [2] [3] [4] [5] [6] [7] [S] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [lS] [19]
[20]
R. Zwanzig, On the identity of three generalized master equations, Physica 30 (1964) 1109-1123. I. Oppenheim, K.E. Schuler and G.H. Weiss, eds., Stochastic Processes in Chemical Physics: The Master Equation (MIT Press, Cambridge, MA, 1977). H. Haken, Laser Theory (Springer, Berlin, 1984). L. Arnold and R. Lefever, eds., Stochastic Nonlinear Systems in Physics, Chemistry and Biology (Springer, Berlin, 1981). W. Weidlich and G. Haag, Concepts and Models of a Quantitative Sociology. The Dynamics of Interacting Populations (Springer, Berlin, 1983). W. Weidlich, Physics and social science - The approach of synergetics, Phys. Rep. 204 (1991) 1-163. W. Weidlich, The use of statistical models in sociology, Collect. Phenom. 1 (1972) 51-59. W. Weidlich and G. Haag, eds., Interregional Migration (Springer, Berlin, 1988). W. Weidlich and G. Haag, A dynamic phase transition model for spatial agglomeration processes, J. Regional Sci. 27 (1987) 529-569. W. Weidlich and M. Munz, Settlement formation, Part I: A dynamic theory, Ann. Regional Sci. 24 (1990) 83-106. L. Boltzmann, Lectures on Gas Theory (Univ. California Press, Berkeley, CA, 1964). F. Wilkinson, Chemical Kinetics and Reaction Mechanisms (Van Nostrand Reinhold, New York, 1980). D. Helbing, Interrelations between stochastic equations for systems with pair interactions, Physica A 181 (1992) 29-52. R. Pearl, Studies in Human Biology (Williams & Wilkins, Baltimore, MD, 1924). P.F. Verhulst, Nuov. Mem. Acad. R. Bruxelles 18 (1845) 1. G.K. Zipf, The PlPZ/D hypothesis on the intercity movement of persons, Am. Sociolog. Rev. 11 (1946) 677-686. J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems (Cambridge Univ. Press, Cambridge, 1988). P. Schuster, K. Sigmund, J. Hofbauer, R. Wolff, R. Gottlieb and Ph. Merz, Selfregulation of behavior in animal societies I-III, Biol. Cybern. 40 (1981) l-25. D. Helbing, Stochastische Methoden, nichtlineare Dynamik und quantitative Modelle soziakr Prozesse, PhD thesis (University of Stuttgart, Stuttgart, 1992) (Oldenbourg, Miinchen), submitted. D. Hclbing, A fluid dynamic model for the movement of pedestrians, Complex Syst., in print.
[21] D. Helbing, A mathematical model for attitude formation by pair interactions, Behav. Sci. 37 (1992) 190-214. [22] Ch.E. Osgood and P.H. Tannenbaum, The principle of congruity in the prediction of attitude change, Psychological Rev. 62 (1955) 42-55. [23] F. Heider, Attitudes and cognitive organization, J. Psychology 21 (1946) 107-112. [24] L. Festinger, A Theory of Cognitive Dissonance (Row & Peterson, Evanston, IL, 1957). [25] K. Lewin, Field Theory in Social Science (Harper & Brothers, New York, 1951). [26] D.R. Hartree, Proc. Cambridge Philos. Sot. 24 (1928) 111. [27] D. Montgomery, Brownian motion from Boltzmann’s equation, Phys. Fluids 14 (1971) 2088-2090.
D. Helbing
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of behavioral models
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[28] D. Helbing, A mathematical model for behavioral changes by pair interactions and its relation to game theory, Angew. Sozialforsch. 17 (1991/92) 179-194. [29] Th.A. Domencich and D. McFadden, Urban Travel Demand. A Behavioral Analysis (North-Holland, Amsterdam, 1975) pp. 61-69. [30] J. de D. Orttizar and L.G. Willumsen, Modelling Transport (Wiley, Chichester, UK, 1990). [31] D. Helbing, Stochastic and Boltzmann-like models for behavioral changes, and their relation to game theory, Physica A 193 (1993) 241-258. [32] R. Axelrod, The Evolution of Cooperation (Basic Books, New York, 1984). [33] J. von Neumann and 0. Morgenstern, Theory of Games and Economic Behavior (Princeton Univ. Press, Princeton, 1944). [34] M. Eigen, The selforganization of matter and the evolution of biological macromolecules, Naturwissenschaften 58 (1971) 465. [35] R.A. Fisher, The Genetical Theory of Natural Selection (Oxford Univ. Press, Oxford, 1930). [36] M. Eigen and P. Schuster, The Hypercycle (Springer, Berlin, 1979). [37] R. Feistel and W. Ebeling, Evolution of Complex Systems (Kluwer, Dordrecht, 1989). [38] H.A. Kramers, Physica 7 (1940) 284. [39] J.E. Moyal, J.R. Stat. Sot. 11 (1949) 151-210. [40] D. Helbing, A mathematical model for the behavior of pedestrians, Behav. Sci. 36 (1991) 298-310. [41] G.E. Forsythe, M.A. Malcolm and C.B. Moler, Computer Methods for Mathematical Computations (Prentice Hall, Englewood Cliffs, NJ, 1977). [42] J.B. Kruskal and M. Wish, Multidimensional Scaling (Sage, Beverly Hills, CA, 1978). [43] F.W. Young and R.M. Hamer, Multidimensional Scaling: History, Theory, and Applications (Lawrence Erlbaum, Hillsdale, NJ, 1987). [44] C.W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer, Berlin, 1985).
Boltzmann-like and Boltzmann-Fokker-Planck equations as a foundation of behavioral models Dirk Helbing II. Institute for Theoretical
Physics, University of Stuttgart, W-7000 Stuttgart 80, Germany
Received 18 January 1993
It is shown that the Boltzmann-like equations allow the formulation of a very general model for behavioral changes. This model takes into account spontaneous (or externally induced) behavioral changes and behavioral changes by pair interactions. As most important social pair interactions, imitative and avoidance processes are distinguished. The resulting model turns out to include as special cases many theoretical concepts of the social sciences. A Kramers-Moyal expansion of the Boltzmann-like equations leads to the BoltzmannFokker-Planck equations, which allows the introduction of “social forces” and “social fields”. A social field reflects the influence of the public opinion, social norms and trends on behavioural changes. It is not only given by external factors (the environment) but also by the interactions of the individuals. Variations of the individual behavior are taken into account by diffusion coefficients.
1. Introduction
The methods of statistical physics have shown to be very fruitful in physics, but in the last decades they also have become increasingly important in interdisciplinary research. For example, the master equation has found many applications in thermodynamics [l], chemical kinetics [2], laser theory [3] and biology [4]. Moreover, Weidlich and Haag have successfully introduced it for the description of social processes [5,6] like opinion formation [7], migration [8], agglomeration [9] and settlement processes [lo]. Another kind of wide-spread equations are the Boltzmann equations, which have been developed for the description of the kinetics of gases [ll] and of chemical reactions [12]. However, Boltzmann-like equations [13] play also an important role for quantitative models in the social sciences: it turns out (cf. section 2.2) that the logistic equation for the description of limited growth processes 114,151, the so-called gravity model for spatial exchange processes [ 161, and the game dynamical equations modelling competition and cooperation processes [17,18] are special cases of Boltzmann-like equations. Moreover, 0378-4371/93/$06.00 0
1993 - Elsevier Science Publishers B.V. All rights reserved
D. Helbing
I Boltzmann-like
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547
Boltzmann-like models have recently been suggested for avoidance processes of pedestrians [19,20] and for attitude formation by direct pair interactions of individuals occurring in discussions [ 19,211. In this paper we shall show that Boltzmann-like equations and BoltzmannFokker-Planck equations [13] are suited as a foundation of quantitative behavioral models. For this purpose, we shall proceed in the following way: in section 2 the Boltzmann-like equations will be introduced and applied to the description of behavioral changes. The model includes spontaneous (or externally induced) behavioral changes and behavioral changes by pair interactions of individuals. These changes are described by transition rates. They reflect the results of mental and psychical processes, which could be simulated with help of Osgood and Tannenbaum’s congruity principle [22], Heider’s balance theory [23] or Festinger’s dissonance theory [24]. However, it is sufficient for our model to determine the transition rates empirically (section 5). The ansatz used for the transition rates distinguishes imitative and avoidance processes, and assumes utility maximization of the individuals (section 2.1). It is shown that the resulting Boltzmann-like model for imitative processes implies as special cases many generally accepted theoretical approaches in the social sciences (section 2.2). In section 3 a consequent mathematical formulation related to an idea of Lewin [25] is developed, according to which the behavior of individuals is guided by a social field. This formulation is achieved by a Kramers-Moyal expansion of the Boltzmann-like equations leading to a kind of diffusion equations: the so-called Boltzmann-Fokker-Planck equations [13]. In these equations the most probable behavioral change is given by a vectorial quantity that can be interpreted as social force (section 3.1). The social force results from external influences (the environment) as well as from individual interactions. In special cases the social force is the gradient of a potential. This potential reflects the public opinion, social norms and trends, and will be called the social field. By diffusion coejficients an individual variation of the behavior (the “freedom of will”) is taken into account. In section 4 representative cases are illustrated by computer simulations. The Boltzmann-Fokker-Planck model for the behavior of individuals under the influence of a social field shows some analogies with the physical model for the behavior of electrons in an electric field (e.g. of an atomic nucleus) [13,19] (cf. Hartree’s self-consistent field ansatz [26]). Especially, individuals and electrons influence the concrete form of the effective social, resp. electric field. However, the behavior of electrons is governed by a different equation: the Schrodinger equation. In physics, the Boltzmann-Fokker-Planck equations can be used for the description of diffusion processes [27].
D. Helbing
548
I Boltzmann-like equations as foundation of behavioral models
2. The Boltzmann-like
equations
Let us consider a system consisting of a great number N 4 1 of subsystems. These subsystems are in one state x of several possible states combined in the set 0. Due to Jluctuafions one cannot expect a deterministic theory for the temporal change dxldt of the state x(t) to be realistic. However, one can construct a stochastic model for the change of the probability distribution P(x, t) of states r(t) within the given system (P(x, t) 2 0, CxER P(x, t) = 1). By introducing an index a we may distinguish A different types a of subsystems. If N, denotes the number of subsystems of type a, we have Xi=, N, = N, and the following relation holds:
P(x, t) =
Ii $TPJX, t) .
(1)
a=1
Our goal is now to find a suitable equation for the probability distribution Pa(x, t) of states for subsystems of type a (P,(x, t) 2 0, C,,, P,(x, t) = 1). If we neglect memory effects (cf. section 6.1), the desired equation has the form of a muster equation [13,19],
-$PAX,t) = x, [Wa(+‘; 4 PO@‘,t) -
w”(x’lx; t) P&, t)] .
(2)
(x’tx)
wa(x’ Ix; t) is the effective transition rare from state x to x’ and takes into account the fluctuations. Restricting the model to spontaneous (or externally induced) transitions and transitions due to pair interactions, we have [13,19]
w’(x’Ix; t):= w,(x’Ix;
t) + 2 b=l
c
2
PER
y’ER
&G&Y’,
~‘1x7 Y; t) P,(Y, t).
(3)
w,(x) Ix; t) describes the rate of spontaneous (respectively externally induced) transitions from x to x’ for subsystems of type a. Gab(x’, y’lx, y; t) is the transition rate for two subsystems of types a and b to change their states from x and y to x’ and y’ due to pair interactions. Inserting (3) into (2), we now obtain the so-called Boltzmunn-like equations [13,191, ;
P,(x, t) =
c
[w,(xl x’; t) P,(x’, t) - w,(x’ Ix; t) P,(x, t)l
X’Ef2
+
b$, x&y;o ,zu
w,b(x,
Y’ 1x’,
Y;
l)
pb(Y,
‘1 ‘atx’, ‘!
(44
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
549
A
b=l
X’EO
yEL’y’Ef2
with
w&(x’, y’ 1x, y; t, := NbG)ab(x’, Y’ 1x, Y;
(5)
t, .
Obviously, (4b) depends nonlinearly on the probability distributions P,(x, t), which is due to the interaction processes. Neglecting spontaneous transitions (i.e., w,(x’ 1x; t) = 0), the Boltzmannlike equations agree with the Boltzmann equations, which originally have been developed for the description of the kinetics of gases [ll]. A more detailed discussion can be found in [19]. In order to apply the Boltzmann-like equations to behavioral changes we now have to take the specifications which are given in table I (cf. [19]). It is possible to generalize the resulting behavioral model to simultaneous interactions of an arbitrary number of individuals (i.e., higher order interactions) [13,19]. However, in most cases behavioral changes are dominated by pair interactions. Many of the phenomena occurring in social interaction processes can already be understood by the discussion of pair interactions. 2.1. The form of the transition rates In the following we have to find a concrete form of the effective transition rates:
Wa(X’IX; ‘) = w,(x’Ix; t) + i
b=l
W&‘IX,
y; t):=
c
2 &,(X’IX, y; t)
yE0
w,&‘, y’lx,
(64
p,(y, t),
(6’3)
Y; t>
Y’EO
Table I Specification Statistical
of the notions physics
system subsystems states types of subsystems transitions interactions fluctuations
used in statistical
physics
for an application
Behavioral
to behavioral
models.
models
population individuals behaviors (concerning a special topic of interest) types of behavior (subpopulations) behavorial changes imitative processes, avoidance processes “freedom of will”
D. Helbing
550
(cf.
which
(3))
I Boltzmann-like
is suitable
equations
for
the
as foundation
description
of behavioral
models
of behavioral
changes.
Walh(x’ 1x, y; t) is the rate of pair interactions,
XlZX)
(7)
where an individual of subpopulation a changes the behavior from x to x’ under the influence of an individual of subpopulation b showing the behavior y. There are only two important kinds of social pair interactions:
x’
tix
(x’ # x) .
(8b)
Obviously, the interpretation of the above kinds of pair interactions is the following: - The interactions (8a) describe imitative processes (processes of persuasion), that means, the tendency to take over the behavior x’ of another individual. -The interactions (8b) describe avoidance processes, where an individual changes the behavior when meeting another individual showing the same behavior x. Processes of this kind are known as aversive behavior, defiant behavior or snob effect. The corresponding
transition rates are of the general form
where the term (9a) describes imitative processes and the term (9b) describes avoidance processes. aYyxhas the meaning of the Kronecker function. By inserting (9) into (6) we arrive at the following general form of the effective transition rates: d(x’
1x; t) := v,(t) R,(x’ 1x; t) + i
vab(t) [Rib@’
b=l
For behavioral R;,(x’)x;
In (1%
Ix; t) P,(x’, t) + R:,(x’lx; t>f’,(x, t)] . VW
models one often assumes t) := fib(t)
R”(x’lx;
t) .
(lob)
D. Helbing
I Boltzmann-like
equations as foundation
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551
- v,(t) is a measure for the rate of spontaneous (or externally induced) behavioral changes within subpopulation a. - R,(x’ 1x; t) [R”(x’ 1x; t)] is the readiness for an individual of subpopulation a to change the behavior from x to x’ spontaneously [in pair interactions]. - Gus = Nb Fab(f) is the interaction rate of an individual of subpopulation a with individuals of subpopulation b. - ftb(t) is a measure for the frequency of imitative processes. - fib(t) is a measure for the frequency of avoidance processes. A more detailed discussion of the different kinds of interaction processes and of ansatz (10) is given in [19,28]. For R’(x’ 1x; t) we take the quite general form x.
Ra(x,,
,
t)
=
expVW 9 - UYx,91
with Da(x’, x; t) = Da(x, x’; t) > 0 for an individual of subpopulation the greater, - the greater the difference of the - the smaller the incompatibility haviors x and x’ is. Similar to (lla) R
(x,
a
IX’
q
3
=
(114
’
Da(X’, x; t)
(cf. [8,19]). Then, the readiness Ra(x’ 1x; t) u to change the behavior from x to x’ will be utilitiesU”(. . . , t) of behaviors x’ and x is, (“distance”)
Do(x’, x; t) between
the be-
we use exp[U,(x’, t) - Un(x,
D,(x),x; t)
t)l
’
(lib)
and, therefore, allow the utility function U,(x, t) for spontaneous (or externally induced) behavioral changes to differ from the utility function V(x, t) for behavioral changes in pair interactions. Ansatz (11) is related to the multinomiul logit model [29,30] and assumes utifity maximization with incomplete information about the exact utility of a behavioral change from x to x’, which is, therefore, estimated and stochastically varying (cf. [19]). Computer simulations of the Boltzmann-like equations (2), (lo), (11) are discussed and illustrated in [19,21,28] (cf. also section 4). 2.2.
Special fields of application in the social sciences
The Boltzmann-like equations (2)) (10) include a variety of special cases, which have become very important in the social sciences: (i) The logistic equation [14,15] describes limited growth processes. Let us consider the situation of two behaviors xE {1,2} (i.e., P,(l, t) = 1- P,(2, t))
D. Helbing
552
I Boltzmann-like
equations as foundation
of behavioral models
and one subpopulation (A = 1). x = 2 may, for example, have the meaning to apply a certain strategy, and x = 1 not to do so. If only imitative processes, 2&l)
(12)
and processes of spontaneous
replacement,
lt2.
(13)
are considered, ;
one arrives at the logistic equation
P,(2, t>= -q(t) R,(W; t>P,(2,4 +
q,(t)
f:,(t) R’(211;t) [l- P,G f>l P,(2, t>
= A(t) P,(2, t> [B(t) - P,(2, t)] .
(14)
(ii) The gravity model [16] describes processes of exchange between different places x. It results for R,(x’ 1x; t) = 0, fib(t) = 1, fzb(t) = 0, and A = 1 in
x P(x, t) P(x’, t) . Here, we have dropped the index a because of a = 1. P(x, of being at place x. The absolute rate of exchange from x to the probabilities P(x, t) and P(x’, t) at the places x and chosen as a function of the metric distance ]Jx - x’]] D(x, x’) = D(](x - x’]]).
(15) t) is the probability to x’ is proportional x’. D(x, x’) is often between x and x’:
(iii) The behavioral model of Weidlich and Haag [5,6,8] assumes spontaneous transitions due to indirect interactions, which are, for example, induced by the media (TV, radio, or newspapers). We obtain this model for fib(t) = 0 =fib(t) and UJX, t) := 6a(x, t) +
2 ‘%bPb(X, t) .
b=l
(16)
6,(x, t) is the preference of subpopulation a for behavior x. K,~ are coupling parameters describing the influence of the behavioral distribution within subpopulation b on the behavior of subpopulation a. For K,~ > 0, K,~ reflects the social pressure of behavioral majorities.
D. Helbing
I Boltzmann-like
(iv) The game fib(t) = 0, and
dynamical
R’(n’ 1n; t) := max[E,(x’,
equations as foundation
equations
[17-19,311
553
of behavioral models
result
for fib(t) = Sob,
t) - E,(x, t), 0]
(17)
(cf. [19,28]). Their explicit form is $ P,(x, 0 = X~OIw,(xIx’;
+ I,,
t) PAX’, t)-
W&k
t) P&,
91
PAX,4 [K?k 4 - (&I)I .
(184 (18b)
Whereas (18a) again describes spontaneous behavioral changes (“mutations”, innovations), (18b) reflects competition processes leading to a “selection” of behaviors with a success E,(x, t) that exceeds the average success (E,) := c E,(x’, t) P,(x’, t) . X’ER The success E,(x, t) is connected
(19) with the so-called payoff matrices A,, =
(A,b(x, Y)) by
-%(x,4 := A,(x) +
c c A.b(x, Y> P,(Y,
b=l
yER
t)
(20)
]I9,281. A,( x ) means the success of behavior x with respect to the environment. Since the game dynamical equations (18) agree with the selection mutation equations [17] they are not only a powerful tool in social sciences and economy [18,31-331, but also in evolutionary biology [34-371. 3. The Boltzmann-Fokker-Planck
equations
We shall now assume the set L? of possible behaviors to build a continuous space. The n dimensions of this space correspond to different characteristic aspects of the considered behaviors. In the continuous formulation, the sums in (2), (3) have to be replaced by integrals:
$ P,(x,
d”x’ [ wyx1x’; t) P,(x’, t) - wU(x’ 1x; t) Pa(x, t)]
t) =
0 { WO[X’ 1x - x’; t] P,(x - x’, t) - w@[x’1x; t] PJX, t)}
)
(214
D. Helbing
554
I Boltzmann-like
equations as foundation
of behavioral models
where
wU[x’ - x 1x; t] : = Wyr’ (x; t) : = w,(x’
lx; t) + i
j- d”y j- d”y’ &W,&‘,
b=l 0
y’1-v
Y; t) P,(Y,
t> .
(21b)
R
A reformulation of the Boltzmann-like equations (21) via a Kramer+Moyal expansion [38,39] (second order Taylor approximation) leads to a kind of diffusion equations, the so-called Boltzmann-Fokker-Planck equations [13]:
(224 with the effective drift coefficients Q,(x, t) :=
I R
d”x’ (x;
x;)
wyx )x;
t)
Pb)
and the effective diffusion coefjicients#’ Q,,(x,
t) : = j- d”x’ (x: - xJ(x;
- Xj)Wa(X’ 1x; t) .
VW
Whereas the drift coefficients &(x, t) govern the systematic change of the distribution P,(x, t), the diffusion coefficients Qeij(x, t) describe the spread of the distribution P,(x, t) due to fluctuations resulting from the individual variation of behavioral changes. For ansatz (lo), the effective drift and diffusion coefficients can be split into contributions due to spontaneous (or externally induced) transitions (k = 0), imitative processes (k = l), and avoidance processes (k = 2):
k=O
” In [13] the expression (22). However, they make 1191).
for Q,,(x, t) contains additional terms due to another no contributions since they result in vanishing surface
derivation integrals
of (cf.
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
555
where
K,Oi(X, t) ‘= I, K&, t) :=
I
d”x’ (xl -
2 vab(t) fib(t) b=l
Xi)R,(X’
(x;
t)
,
d”x’ (xi - xJZ?‘(x’
WI
I
A
Kzi(x, t) :=
c
vab(t) ffb(f)
b=l
d”x’ (x; - x;)R’(x’
I
and Qiij(x,
t) : = v,(t) j dnx’ (xl - x,)(x; - xj)R,(x’
Qi,
vab(t)
fLb(‘)
/
1x; t) ,
dnx’ (xi - xi)(x] - x,)R”(x
1x; t) Pb(x’ t) ,
(23~)
Q,‘ij
ISI
'abet) fzb(l)
b=l
1d”x’ (x; - xi)@; - xj)Ra(x’
changes induced by the environment
1x; t) Pb(x, t) .
are included in Kii(x, t)
3.1. Social force and social field The Boltzmann-Fokker-Planck equations chastic equations (Langevin equations) dx. $ = &i(x> t) + SI G,ij(x, t) tj(t)
(22a) are equivalent
to the sto-
(244
7
j=l
with
Kai(X,‘>= ‘ai(X>‘I+ i
j$,(& G,ij(x, t))Gajk(X,t)
(24b)
and
Qaij
(24~) a the vector
5,(x, t) with the
D. Helbing I Boltzmann-like equations as foundation of behavioral models
556
describes the contribution to the change of behavior x that is caused by behavioral fluctuations t(t) (which are assumed to be delta-correlated and Gaussian [19]). Since the diffusion coefficients Q,,(x, t) and the coefficients Gaij(x, t) are usually small quantities, we have Fai(x, t) = KOi(x, t) (cf. (24b)), and (24a) can be put into the form dx - = Kn(x, t) + fluctuations . dt
(26)
Whereas the fluctuation term describes individual behavioral variations, the vectorial quantity KO(x, t) drives the systematic change of the behavior x(t) of individuals of subpopulation a. Therefore, it is justified to denote K,(x, t) as social force acting on individuals of subpopulation a. The social force influences the behavior of the individuals, but, conversely, due to interactions, the behavior of the individuals also influences the social force via the behavioral distributions PO(x, t) (cf. (21b), (22b)). That means &(x, t) is a function of the social processes within the given population. Under the integrability conditions
$
Kai(xl ‘1 = I
&I
Kaj(xY
there exists a time-dependent V,(x, t) := -
dx’*K,(x’,
t,
for all i, j ,
(27)
potential
t) ,
so that the social force is given by its gradient V, K,(X, t) = -VV,(x,
t) .
(29)
The potential V,(x, t) can be understood as social field. It reflects the social influences and interactions relevant for behavioral changes: the public opinion, trends, social norms, etc. 3.2. Discussion
of the concept
of force
Clearly, the social force is no force obeying the Newtonian laws of mechanics. Instead, the social force K=(x, t) is a vectorial quantity with the following properties:
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
K,(x, t) drives the temporal change drldt of another vectorial namely the behavior x(t) of an individual of subpopulation a. The component Kab(x, t) := vohb(t) dnx’ (x’ - x)[f;,(t) I n
+ &(t)
Pb(X,
P,(x’,
41mx Ix; t)
551
quantity
t)
(30)
of the social force P&(X, t) describes the reaction of subpopulation a on the behavioral distribution within subpopulation b and usually differs from Kbu(x, t), which describes the influence of subpopulation a on subpopulation b. Neglecting fluctuations, the behavior x(t) does not change if Ka(x, t) vanishes. Ka(x, t) = 0 corresponds to an extremum of the social field VO(x, t), because it means VV,(x, t) = 0 .
(31)
We can now formulate our results in the following form related to Lewin’s “field theory” [25]: Let us assume that an individuals objective is to behave in an optimal way with respect to the social field, that means, he or she tends to a behavior corresponding to a minimum of the social field. If the behavior x does not agree with a minimum of the social field VO(x, t), this evokes a psychical tension (force)
K,(x, 4 = -Vv,(x, t) ,
(32)
that is given by the gradient of the social field V,(x, t). The psychical tension K,(x, t) is a vectorial quantity that induces a behavioral change according to dx dt = K(x,
t) .
(33)
The behavioral change dxldt drives the behavior x(t) towards a minimum xz of the social field Va(x, t). When the behavior has reached a minimum x: of the social field V,(x, t), it holds VV,(x, t) = 0
(34)
558
D. Helbing
! Boltzmann-like
equations as foundation
of behavioral models
and, therefore, &(x, t) = 0, that means the psychical tension vanishes. - If the psychical tension K,(x, t) vanishes, except for fluctuations no behavioral changes take place, in accordance with (33). In the special case, where an individual’s objective is the behavior xz, one would expect behavioral changes according to dx - = y(xZ - x) ) dt which corresponds
(35)
to a social field
vn(x,0 = lY(x: - 4’
7
with a minimum at ~2. Examples for this case are discussed in [40]. Note, that the social fields V,(x, t) of different subpopulations a usually have different minima xz. That means, individuals of different subpopulations a will normally feel different psychical tensions &(x, t). This shows the psychical tension Ka(x, t) to be a “subjective” quantity.
4. Computer simulations In the following, the Boltzmann-Fokker-Planck equations for behavioral changes will be illustrated by representative computer simulations. We shall examine the case of A = 2 subpopulations, and situations for which the interesting aspect of the individual behavior can be described by a certain position x E [l/20,1] on a one-dimensional continuous scale (i.e., II = 1, x = x). Then, the integrability conditions (27) are automatically fulfilled, and the social field x
V,(x, t) = -
I
dx’ K,(x’, t) - c,(t)
(37)
x0 is well defined. The parameter c,(t) can be chosen arbitrarily. We will take for the value that shifts the absolute minimum of Va(x, t) to zero, that means
c,(t)
x
c,(t) := m;ln
(38) XC1
Since we will restrict the simulations to the case of imitative or avoidance processes, the shape of the social field V,(x, t) changes with time only due to
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
559
changes of the probability distributions Z’,(x, t) (cf. (23)), that means due to behavioral changes of the individuals (see figs. l-6). In the one-dimensional case one can find the formal stationary solution
P,(x) = Pa(xo)
$$ff$exp(2
1
dx’ s)
,
(39)
*0
which we expect to be approached in the limit of large times r+ ~0. Due to the dependence of K,(x) and Q,(X) on P,(X), eqs. (39) are only implicit equations. However, from (39) we can derive the following conclusions: (i) If the diffusion coefficients are constant (Q,(x) = Q,), (39) simplifies to
that means the stationary solution P,(x) is completely determined by the social field V,(x). Especially, P,(x) has its maxima at the positions x:, where the social field V,(x) has its minima (see fig. 1). The diffusion constant Q, regulates the width of the behavioral distribution P,(x). (ii) If the diffusion coefficients Q,(x) are varying functions of the position x, the behavioral distribution P,(x) is also influenced by the concrete form of Q,(x). From (39) one expects high behavioral probabilities P,(x) where the diffusion coefficients Q,(x) are small (see fig. 2, where the probability distribution PI(x) cannot be explained solely by the social field V,(x)). (iii) Since the stationary solution P,(x) depends on both, K,(x) and Q,(x), different combinations of K,(x) and Q,(x) can lead to the same probability distribution P,(x) (see fig. 4 in the limit of large times). For the following simulations, we shall use the ansatz Ra(x,
lx.
t)
=
expWYx’,0 - UX, 01
7
Da@‘,x; t>
for the readiness ZY(x’ 1x; t) to change from x to x’ (cf. (lla)). function UU(x, t):= -i(F)*)
I, := LO/20 ) a
(414 With the utility
(41b)
subpopulation a prefers behavior x,. L, means the indifference of subpopulation a with respect to variations of the position x. Moreover, we take v*bb(0
D,(x’,x;t)
-I+‘-xl/r
:=e
’
r = R/20,
(414
560
D. Helbing
1 Boltzmann-like
equations as foundation
of behavioral models
L,= L2= R=
\
Fig. 1. Imitative processes to behavioral changes.
in the case of one-sided
sympathy
and low indifference
I
L, with respect
D. Helbing
/ Boltzmann-like
equations as foundation
Fig. 2. As fig. 1, but for high indifference
of behavioral models
L, with respect to behavioral changes.
561
562
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
Fig. 3. Imitative processes for mutual sympathy and low indifference
L, in both subpopulations
D.
“,(X*0
n
Helbing
/ Lkkvnann-like
equations as foundation of behavioral mod&
n
x
Fig. 4. As fig. 3, but for h’~gh m d‘ff I erence
22, in subpopulation
2.
564
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
where R can be interpreted as measure for the range of interaction. According to (41c), the rate of behavioral changes is the smaller the greater the changes are. Only small changes of the position (i.e., between neighboring positions) contribute with an appreciable rate. 4.1. Sympathy and interaction frequency Let sob(t) be the degree of sympathy which individuals of subpopulation a feel towards individuals of subpopulation b. Then, one expects the following: whereas the frequency f ah(t) of imitative processes will be increasing with sob(t), the frequency fib(t) of avoidance processes will be decreasing with sab(t). This functional relationship can, for example, be described by f :b(t) := f l(t) s&(t) >
f:bw
:= f f(t) [1 - s&(t)1 3
(42)
with (43)
Oss,,(t)~l.
f i(t) is a measure for the frequency of imitative processes within subpopulation a, f t(t) is a measure for the frequency of avoidance processes. If we assume
the sympathy between individuals of the same subpopulation we have sll(t) = 1 = sz2(t). 4.2.
Imitative processes
to be maximal,
(f f(t) = 1, f z(t) -0)
In the following simulations of imitative processes we assume the preferred With positions to be X, =&andx,=$$.
(S&At))= (f
:b(t)) := (A
;)
7
(44)
the individuals of subpopulation a = 1 like the individuals of subpopulation a = 2, but not the other way round. That means, subpopulation 2 influences subpopulation 1, but not vice versa. One could say, the individuals of subpopulation 2 act as trendsetters. As expected, in both behavioral distributions P=(x, t) there appears a maximum around the preferred behavior x,. In addition, due to imitative processes of subpopulation 1, a second maximum of P,(x, t) develops around the preferred behavior SC*of the trendsetters. This second maximum is small, if the indifference L, of subpopulation 1 with respect to variations of the position x is low (see fig. 1). For high values of the indifference L, even the majority of individuals of subpopulation 1 imitates the behavior of the trendsetters (see fig. 2)!
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
565
We shall now consider the case
for which the subpopulations influence each other mutually with equal strengths. If the indifference L, with respect to changes of the position x is small in both subpopulations a, each probability distribution PO(x, t) has two maxima. The higher maximum is located around the prefered position x,. A second maximum can be found around the position prefered in the other subpopulation. The greater the indifference L, is, the higher this maximum is (see fig. 3). However, if L, exceeds a certain value in at least one subpopulation, only ooze maximum develops in each behavioral distribution P,(x, t)! Despite the fact that the social fields Va(x, t) and diffusion coefficients Q,(x, t) of the subpopulations a are different because of their different prefered positions X, (and different utility functions U(I(x, t)), the behavioral distributions P,(x, t) agree after some time! Especially, the maxima xz of the distributions Pll(x, t) are located at the same position x* in both subpopulations. x* is nearer to the position X, of the subpopulation a with the lower indifference L, (see fig. 4). 4.3. Avoidance
processes
(f:(t)
= 0, f:(t)
= 1)
For the simulation of avoidance processes we assume with x1 = & and x2 = $$ that both subpopulations nearly prefer the same behavior. Fig. 5 shows the case where the individuals of different subpopulations dislike each other,
Gab(t)):=
(A;> 7
i.e., (ffb(t)) = (y i) .
This corresponds to a mutual influence of one subpopulation on the respective other. The computational results prove: (i) The individuals avoid behaviors which can be found in the other subpopulation. (ii) The subpopulation a = 1 with the lower indifference L, < L, is distributed around the prefered behavior xi and pushes away the other subpopulation! Despite the fact that the initial behavioral distribution Pa(x, 0) agrees in both subpopulations, there is nearly no overlapping of P,(x, t) and Pz(x, t) after some time. This is a typical example for polarization phenomena in the society. In fig. 6, we assume that the individuals of subpopulation 2 like the individuals of subpopulation 1 and, therefore, do not react on the behaviors in subpopulation 1 with avoidance processes,
D. Helbing
566
! Boltzmann-like
equations as foundation
of behavioral models
L,= L,= R=
Fig. 5. Avoidance
processes
for mutual
dislike
of both subpopulations.
D. Helbing
/ Boltzmann-like
equations as foundation
of behavioral models
567
Fig. 6. Avoidance processes for one-sided dislike.
(S&(f)):= (;
y) 3
i.e., (fib(t)) = (8 i)
(47)
As a consequence, P2(x, t) remains unchanged with time, whereas P,(x, t) drifts away from the prefered behavior x1, due to avoidance processes. Surprisingly, the polarization effect is much smaller than in fig. 5! The distributions P,(x, t) and P&K, t) overlap considerably. This is because the slope of PZ(x, t) is smaller than in fig. 5 (and remains constant). As a consequence, the probability for an individual of subpopulation 1 to meet a disliked individual of subpopulation 2 with the same behavior x can hardly be decreased by a small behavioral change. One may conclude that polarization effects (which often lead to an escalation) can be reduced, if individuals do not return dislike by dislike.
5. Empirical
determination
of the model parameters
For practical purposes one has, of course, to determine the model parameters from empirical data. Therefore, let us assume to know empirically the distribution functions Pz (x,tr) (the interaction rates uzb (t,)) and the effective transition rates IV: (x’ 1x; t,) (x’ # x) for a couple of times t, E {I,, . . . , CL}. The corresponding effective social fields Vz (x,tr)and diffusion coefficients Q,‘, (x, tr) are, then, easily obtained as x
v,“(X,4) := - I
dx’ .~;(x’,
t,)
(484
568
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
with K,‘&, tr) :=
I n
d”x’ (xl - q)w;(x’
1x; t,) ,
(48b)
and
Q,e,(X, tl) '=I dnX’ (x:
- ~i)(~l -
Xi) W~(X’
1X; tr)
(49)
.
R Much more difficult is the determination of the utility functions u~(x, tr), U~(X, t,), the distance functions D,“(x’, X; t,), and the rates vz(t,), &t,) := G(G) f:;(G), v$(t,) := ~z~(t[) fff(t,). This can be done by numerical minimization of the error function
for example with the method introduced the abbreviation
of steepest descent [41]. In (50), we have
I$
g&‘, xi 4) := v,(f,>exp[u,(x’, 4) - U,(x, t,)] + h=l [~:~(t,)P;(x', tr> +
db(4)6(x, t,)] exp[U”(x’,
4) - V(X, t,)] .
(51)
It turns out (cf. [19]) that the rates a, have to be taken constant during the U,(x, t,), minimization process (e.g., va(tl) = l), whereas the parameters Ua(x, t,), I:,, := vzb(tl) fib(fl) and ua,(t,) := vzb(fl) f#?$(t,) are to be varied. For l/D,(x’, x; tr) one inserts 1
n,(x’, x; tr) D,(x’, x; tr) = C&(x’,x; tr) ’
(524
with ~&‘, x; tr) := wZ(x’ Ix; tJ g,(x’, x; t[) [PZ(x, t,)12
(52b)
+ WXXIx’; 4) g,(x, x’; ?[) [PZ(x’, t,)12 and &(x’, x; tr) := [g&r’, x; tr) PZ(x, tJ2 + (52)
[g,(x,
x’;
t,) PZ(x’,
t,)12 .
follows from the minimum condition for Da(x), x; tr) (cf. 1191).
(52~)
D. Helbing
I Boltzmann-like
equations as foundation
of behavioral models
569
Since F may have a couple of minima due to its nonlinearity, suitable start parameters have to be taken. Especially, the numerically determined rates vi,(t,) and vzb(tl) have to be non-negative. If F is minimal for the parameters Ua(x, t,), U’(X, tl), D,(x’, x; tl), I,, vib(tl) and vi,(t,), this is (as can easily be checked) also true for the scaled parameters
In order to obtain unique results we put (54)
and
which leads to C,(l[) :=
c
4) ) 1 “‘g UAX,
C”(t,) :=
c
4) ) 1 XE; UYX,
XER
(56)
XER
and
1 -
D,(G)
:=
c 1 rzfld[D.(X~,X;tl)]-lx,x’ER ( (LF‘X)
iii
.
(57)
(X’ZX)i
C,(t,) and C”(t,) are mealz utilities, whereas D,(t[) is a kind of unit of dzhnce. The distances Dz(x’, x; t) are suitable quantities for multidimensional scaling [42,43]. They reflect the “physical structure” (physical topology) of individuals of subpopulation a since they determine which behaviors are more or less related (compatible). By the dependence on a, Dz(x’, x; t) distinguishes different physical structures resulting in different types a of behavior and, therefore, different “characters”.
570
D. Helbing
I Boltzmann-like
equations
as foundation
of behavioral
models
6. Summary and outlook In this article, a behavioral model has been proposed that incorporates in a consistent way many models of social theory: the diffusion models, the multinomial logit model, Lewin’s field theory, the logistic equation, the gravity model, the Weidlich-Haag model, and the game dynamical equations. This very general model opens new perspectives concerning a theoretical description and understanding of behavioral changes, since it is formulated fully mathematically. It takes into account spontaneous (or externally induced) behavioral changes and behavioral changes due to pair interactions. Two important kinds of pair interactions have been distinguished: imitative processes and avoidance processes. The model turns out to be suitable for computational simulations, but it can also be applied to concrete empirical data. 6.1.
Memory
effects
of the model in the previous sections has neglected memory changes. However, memory effects can be easily included by generalizing the Boltzmann-like equations to The formulation
effects that may also influence behavioral
-
w;_,.(x‘Ix; t’) P,(x, t’)] ,
(584
with the effective transition rates W;_&
1x; t’) :=
wy(xf 1x;
t’)
+ic c w:;Ye b=l
and generalizing
yEtl
y’~0
the Boltzmann-Fokker-Planck f
6 I’&,
t) =
I ( dt’
‘0
with the effective drift coefficients
Y'
Ix, y; t’)
Pb(Y,
equations to
t’> >
(58b)
D. Helbing
K;;“(x,
t’) : =
I Boltzmann-like
I R
equations as foundation
of behavioral models
d”x’ (x; - x~)w;_~,(x’ 1x; t’) ,
571
(59b)
the effective diffusion coefficients
Q;;“(n, t’) :=
I
d “x’ (xi - Xi)(XI - xj)w;_,.(x’ Ix; t’) )
(59c)
0
and w;_,,(x’ 1x; t’) : =
W;y(X’ 1x; t’) +
g1 I d”yI R
d”y’ w;;“(x’, y’ 1x, y; t’) P,(y, t’)
R
(59d) Obviously, in these formulas there only appears an additional integration over past times t’ [19]. The influence of the past results in a dependence of w;_,,(x’ Ix; t’), K&T”(X, t’), and QLi”(x, t’) on (t - t’). The Boltzmann-like equations (4), resp. the Boltzmann-Fokker-Planck equations (22) used in the previous sections, result from (58), resp. (59), in the Markovian limit w;_r,(x’ 1x; t’) : = wU(x’ 1x; t) qt - t’)
(60)
of short memory (where 6( * - *) is the Dirac delta function). 6.2. Analogies
with chemical
reactions
The Boltzmann-like equations (4) can also be used for the description of chemical reactions, where the states x denote the different sorts of molecules (or atoms) and a distinguishes different isotopes or conformers. Imitative and avoidance processes correspond in chemistry to self-activatory and self-inhibitory reactions. Although the concrete transition rates will be different from (lo), (lla) in detail, there may be found analogous results for chemical reactions. Note that the Arrhenius formula for the rate of chemical reactions [44] can be put into a form similar to (lla) [19].
Acknowledgements
This work has been financially supported by the Volkswagen Stiftung and the Deutsche Forschungsgemeinschaft (SFB 230). The author is grateful to Prof.
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Dr. W. Weidlich and Dr. R. Reiner for valuable discussions and commenting on the manuscript.
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