COLLECTIVE DYNAMICS OF MANY-FACETED AGENTS Ulrich Krause Department of Mathematics, University of Bremen [email protected]

1. Introduction This paper considers many-faceted agents, that is, agents who employ many points of view according to the different aspects or facets a choice between alternatives may have. Therefore, this type of agents may, instead of single real numbers, value the alternatives by whole bundles of numbers. Actually, the paper introduces agents with quite general valuations of which numerical utility functions are just a special case. For these agents a collective dynamics is formulated which is driven by certain procedures of forming a mean. Among those procedures are on the one hand abstract means, which include all known concrete means, and convex means on the other hand, which seem most appropriate when dealing with values in higher dimensions. Thus, the agents considered are heterogeneous in a twofold way: Agents can differ in their valuations of states and they can differ in the mean procedure applied. A simple but not trivial example for this kind of collective dynamics is provided by the joint iteration of the arithmetic and the geometric mean for two agents which yields as a limit the so-called arithmetic – geometric mean of Gauss. In the general framework of collective dynamics as treated in this paper, the above example is extended to the general concept of a Gauss soup. There exist various kinds of a Gauss soup according to the different kinds of heterogeneity among the agents. The paper closes with four results on the dynamics of Gauss soups which specify conditions under which the dynamics lead to a consensus among the agents.

2. Many-faceted agents “It is a common place that a given individual can feel the force of conflicting considerations, some telling for and some against a particular action, whether it be a trivial one or one of great significance.” ([9, p. 197]) This important feature of decision-making is not taken into account by the common model of the homo oeconomicus or rational agent which is widely used in the social sciences. Instead, neglecting any process of finding a decision in spite of conflicts, the agent is simply equipped with some final, unifying preference ordering or utility function. According to [9], a manyfaceted agent or a multi-faceted agent is an agent who considers a decision from many points of view according to the different aspects or facets a choice between alternatives may have. (In [9] the reader may find several examples as well as a discussion of the history of this subject. For a more formal analysis see [5].) Obviously, the crucial question then arises how an agent, in spite of the diversity of facets, can arrive at an conclusive decision at all. For this in [9, Section IV] the model of an agent’s character is developed which addresses the domain 1

of conflict admitted by the agent and the particular rule applied by the agent to aggregate the different points of view into one overall consistent picture. The process of aggregating reminds of the formally similar aggregation problem in social choice theory. Without going into the details of the model of an agent’s character, we take from [9] the concept of a many-faceted agent and allow agents to express opinions in several dimensions according to the several points of view. Actually, we will allow agents to express opinions which belong to some general space V of values. To extend the opinion dynamics developed in [3, 4, 6] for opinions in one dimension to such a general space V one needs in V a notion of “compromise” or “mean formation”. This can be done for so-called mixture spaces or convex spaces. Beside many-faceted agents, our model described in the next section takes care of the phenomenon that an opinion is an observable expression of an attitude which itself need not be observable. An attitude may be considered also as a state of an agent which is perceived by another agent as an opinion or some more general kind of signal by a process of valuation. Thus, if S denotes the set of all states which might be relevant then a valuation of an agent is a mapping v : S → V which yields for a state x in S the value v ( x) meaning state x as observed or valued by the agent. The domain of values V may be a “big” one as a multi-dimensional space or even a space of functions or a “small” one as the set of real or natural numbers or even the set of two elements. Valuations leading to real numbers are, for example, the utility functions in economics, valuations leading to natural numbers occur in sociology when valuations are given by rankings and the two element value domain occurs if the valuation consists simply in checking a state if it is “on” or “off”. Concerning the communication of states among agents one can introduce further distinctions (see [10]) which, however, will not be relevant for the dynamics in the next section.

3. Collective dynamics of agents in the joint state space Consider a group of n agents where each agent I can be in a state within a subset Si of the set S of all states. Each agent is able to evaluate states by an individual valuation vi : S → Vi where his set of values Vi is a subset of the set V of all values. The collective dynamics we consider among these agents will be described in the following. Let for each agent j a state x j given in S j and let x = ( x1 , , xn ) be the collection of states in the joint state space S1 × × Sn . Fix an agent i and denote by xi∗ the state agent i will choose on his valuations vi ( x1 ), , vi ( xn ) of the present states of all agents (including his own). The dynamics then is defined by the rule that agent i chooses xi∗ in such a way that his value of this state is equal to a mean of his values of the states of the other agents. That is, xi∗ is chosen by agent i such that vi ( xi∗ ) = M i ( vi ( x1 ),l , vi ( xn ) ) (1)

where M i : V1 × × Vn → Vi describes a certain mean procedure applied by agent i . Depending on how the value domains Vi are specified this mean procedure may be a concrete mean like the arithmetic or the geometric mean of numbers or it might be an abstract mean M in the sense that min{v j |1 ≤ j ≤ n} ≤ M (v1 , , v n ) ≤ max{v j |1 ≤ j ≤ n} (2) To formulate such sandwich inequalities (see [4]) we want the values to be a lattice that is a set which is partially ordered by a relation ≤ and such that minima and maxima of finitely 2

many elements do exist. In the particular case, where the value domains V j consist of real numbers, the inequalities (2) define just the ordinary abstract means as considered in [1] und [4]. Beside the abstract mean as defined by (2) there is another general type of mean which also extends the one dimensional ordinary abstract mean and which seems more appropriate for higher dimensions. For value domains Vi which are convex subsets of some finite dimensional space V, a convex mean M is defined by the property that M (v1 , , v n ) ∈ conv {v1 , , vn } for all (v1 , , v n ) ∈ V1 × × Vn

(3)

n n where conv {v1 , , v n } denotes the set of all convex combinations ∑ α j v j |0 ≤ α j , ∑ α j = 1 j =1 j =1 (see[7] for the concept of a convex mean). Dealing with abstract means for real numbers one needs that the mean is strict. Similarly, the more abstract mean defined by (2) is called strict if an inequality becomes an equality only in the case that all v j are equal. A convex mean

defined by (3) is called strict if M (v1 , , v n ) is in the relative interior of conv {v1 , , v n } . (Instead of a finite dimensional space V one could consider also a general vector space which for the formulation of “strict” has to be assumed a topological vector space.) All these notations coincide with the ordinary ones for the particular case that value domains Vi are sets of real numbers. As for the ordinary abstract means, also for the means defined by (2) and (3), respectively, a partial version can be defined which fits to model bounded confidence within the corresponding context. In this paper, however, we will abstain from that and take up the issue of bounded confidence for value domains Vi in a forthcoming paper. Now, equation (1) induces a dynamics on the joint state space

x(t + 1) = f ( x(t ) ) for t = 0,1, 2,

(4)

where x(⋅) is an element of the joint state space S1 × × Sn and f is a selfmapping of this space with the i -th component f i given by xi∗ = f i ( x1 , , xn ) for xi ∈ Si . To define f in this way, however, requires some assumption. We will assume that all vi are bijections on their value domains. (Otherwise, we could treat f not as a mapping in the common sense but as a set-valued mapping.) In other words, the dynamics following (1) is given by

xi (t + 1) = vi−1 ( M i ( vi ( x1 (t ),l , xn (t ) ) )

(5)

(See [6] for a similar model.) In the next section we will present examples for these dynamics together with some general results.

4. The Gauss soup and its dynamics A simple example of the collective dynamics considered in the previous section which, however, is not trivial at all, is the following. Consider two agents, n = 2 , with state spaces S1 = S2 = + , the set of (strict) positive real numbers. Let the value domains be V1 = + , V2 = and let agent 1’s valuation be given by v1 (r ) = r and agents 2’s valuation by v2 (r ) = log r . Thus, we are considering two heterogeneous agents which differ in their valua-

3

tions which may be looked at as utility functions. Furthermore, as mean procedure choose simply for both agents the arithmetic mean. The dynamics given by (5) means that

( x1 (t ) + x2 (t ) ) x2 (t + 1) = log −1 ( 12 ( log x1 (t ) + log x2 (t ) ) ) x1 (t + 1) =

1 2

(

(6)

)

= log −1 log x1 (t ) x2 (t ) = x1 (t ) x2 (t )

That is, in the joint state space 2+ agent 1 acts by the arithmetic mean, whereas agent 2 acts by the geometric mean. The joint iteration of the arithmetic and the geometric mean has been explored already by Carl Friedrich Gauss, who found out the iteration converges to the socalled arithmetic-geometric mean which is given by an elliptic integral. In the light of the previous section there is a huge variety of possible generalizations of Gauss’ arithmetic-geometric mean. Instead of two agents we may consider an arbitrary number of n of agents. As with the arithmetic-geometric mean we may consider agents who all apply an arithmetic mean but differ in choosing arbitrarily between identity and logarithm as their valuations. A collection of n agents (or particles) interacting in this manner we call a Gauss soup. For n = 2 the Gauss soup reduces essentially to the setting of Gauss’ arithmetic-geometric mean. (The two other possible cases where both agents choose identity or logarithm, respectively, are trivial.) There are, however, much more ways in speaking of a Gauss soup. For example, instead of identity and logarithm we may consider other valuations admitting that all are different from each other. Or, we consider agents applying all the same valuations but different mean procedures. (Of course, the setting of the arithmetic-geometric mean can be viewed in this way.) In general, we can consider a Gauss soup of n interacting heterogeneous agents who differ in valuations as well as in mean procedures. Moreover, we can consider also a Gauss soup where valuations and/or means procedures change with time (see below). As interesting as these Gauss soups of heterogeneous agents may be, as difficult are they to explore. An important tool, of course, are computer simulations. (Some results in [3, 4] can be viewed as dealing with particular Gauss soups.) Much more difficult seem analytical investigations to be. The first question to answer is whether the dynamics converges to some limit pattern, in particular, to a consensus. In the following we present four results in this direction. All results state condition under which a collective dynamics as modelled by equation (5) converges to a consensus c, that is lim xi (t ) = c for all 1 ≤ i ≤ n where c ∈ S can be a real x →∞

number or a collection (vector) of real numbers. To determine the value of the consensus c, however, is in general a task much more formidable. The following result is a classical result for Markov chains where it is easy to determine the value of c. (See also [2, Theorem 3], [3, Theorem 1].)

Result 1 • • •

The state spaces as well as the value domains of all agents are equal to + and the valuations of all agents are given by identity. All agents apply weighted arithmetic means which may be different means. The means are such that any two agents put a positive weight jointly on some third agent.

4

The next result includes the previous one and may also be considered to be a classical result. It applies also to the arithmetic-geometric means of Gauss which is not covered by Result 1. A proof of Result 2 can be found in [1, Theorem 8.8]. In [4, Corollary] , Result 2 is obtained as a special case of a more general result on partial abstract means (PAM).

Result 2 • •

The state spaces as well as the value domains of all agents are equal to + and the valuations of all agents are given by identity. All agents apply arbitrary strict abstract means (which, of course, can be different).

The following result deals with valuations that are not given by identity. This result appears as a consequence of a more general theorem in [6, Corollary 1].

Result 3 •

The state space of all agents is equal to

•

equal to . The valuations v for all agents are given by logarithm or by v(r ) = r α for some α ≠ 0 . All agents apply weighted arithmetic means with weights which may change with time. There is a condition on the weights to decrease not “too fast” with time.

•

+

and the valuation domain of all agents is

The final result deals with multi-dimensional values and the convex mean as mean procedure in the sense of (3). For the special case of one dimension only see [6, Theorem 1]. For higher dimensions see [7, Theorem 1] and also [3, Theorem 4]. For bounded confidence in higher dimensions see [8].

Result 4 • • •

The state spaces as well as the value domains of all agents are given by a closed convex subset in a finite-dimensional real space. The valuations of all agents are given by identity. All agents apply strict convex means which may be different from each other and with coefficients changing in time. There is a condition on the weights to decrease not “too fast” (in particular, the coefficients are constant in time).

References [1]

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.

[2]

J.C. Dittmer, Consensus formation under bounded confidence, Nonlinear Analysis 47 (2001), pp. 4615 – 4621.

5

[3]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: models, analysis, and simulation. Journal of Artificial Societies and Social Simulation 5 (2002), 33 pages. http://jasss.soc.surrey.ac.uk/5/3/2.html.

[4]

R. Hegselmann and U. Krause, Collective dynamics of interacting agents when driven by PAM. Paper presented at the Conference Complexity 2003, Aix-en-Provence, 2003.

[5]

U. Krause, Hierarchical structures in multicriteria decision making. In J. Jahn, W. Krabs (Eds.), Recent Advances and Historical Development of Vector Optimization, Springer, Berlin etc. 1987, 183-193.

[6]

U. Krause, A discrete nonlinear and non-autonomous model of consensus formation. In S. Elaydi, G. Ladas, J. Popenda, and J. Rakowski (Eds.), Communications in Difference Equations, Gordon and Breach Publ., Amsterdam 2000, 227-236.

[7]

U. Krause, Positive particle interaction. To appear in the Proceedings of the Conference on Positive Systems, Theory and Applications, Rome, 2003.

[8]

J. Lorenz, Multidimensional opinion dynamics when confidence changes. Paper presented at the Conference Complexity 2003, Aix-en-Provence, 2003.

[9]

I. Steedman and U. Krause, Goethe’s Faust, Arrow’s Possibility Theorem and the individual decision taker. In J. Elster (Ed.), The Multiple Self, Cambridge University Press, Cambridge, 1986, 197-231.

[10]

D. Urbig, Attitude dynamics with limited verbalisation capacities, Journal of Artificial Societies and Social Simulation 6 (2003), 22 pages. http://jasss.soc.surrey.ac.uk./6/1/2.html.

6

1. Introduction This paper considers many-faceted agents, that is, agents who employ many points of view according to the different aspects or facets a choice between alternatives may have. Therefore, this type of agents may, instead of single real numbers, value the alternatives by whole bundles of numbers. Actually, the paper introduces agents with quite general valuations of which numerical utility functions are just a special case. For these agents a collective dynamics is formulated which is driven by certain procedures of forming a mean. Among those procedures are on the one hand abstract means, which include all known concrete means, and convex means on the other hand, which seem most appropriate when dealing with values in higher dimensions. Thus, the agents considered are heterogeneous in a twofold way: Agents can differ in their valuations of states and they can differ in the mean procedure applied. A simple but not trivial example for this kind of collective dynamics is provided by the joint iteration of the arithmetic and the geometric mean for two agents which yields as a limit the so-called arithmetic – geometric mean of Gauss. In the general framework of collective dynamics as treated in this paper, the above example is extended to the general concept of a Gauss soup. There exist various kinds of a Gauss soup according to the different kinds of heterogeneity among the agents. The paper closes with four results on the dynamics of Gauss soups which specify conditions under which the dynamics lead to a consensus among the agents.

2. Many-faceted agents “It is a common place that a given individual can feel the force of conflicting considerations, some telling for and some against a particular action, whether it be a trivial one or one of great significance.” ([9, p. 197]) This important feature of decision-making is not taken into account by the common model of the homo oeconomicus or rational agent which is widely used in the social sciences. Instead, neglecting any process of finding a decision in spite of conflicts, the agent is simply equipped with some final, unifying preference ordering or utility function. According to [9], a manyfaceted agent or a multi-faceted agent is an agent who considers a decision from many points of view according to the different aspects or facets a choice between alternatives may have. (In [9] the reader may find several examples as well as a discussion of the history of this subject. For a more formal analysis see [5].) Obviously, the crucial question then arises how an agent, in spite of the diversity of facets, can arrive at an conclusive decision at all. For this in [9, Section IV] the model of an agent’s character is developed which addresses the domain 1

of conflict admitted by the agent and the particular rule applied by the agent to aggregate the different points of view into one overall consistent picture. The process of aggregating reminds of the formally similar aggregation problem in social choice theory. Without going into the details of the model of an agent’s character, we take from [9] the concept of a many-faceted agent and allow agents to express opinions in several dimensions according to the several points of view. Actually, we will allow agents to express opinions which belong to some general space V of values. To extend the opinion dynamics developed in [3, 4, 6] for opinions in one dimension to such a general space V one needs in V a notion of “compromise” or “mean formation”. This can be done for so-called mixture spaces or convex spaces. Beside many-faceted agents, our model described in the next section takes care of the phenomenon that an opinion is an observable expression of an attitude which itself need not be observable. An attitude may be considered also as a state of an agent which is perceived by another agent as an opinion or some more general kind of signal by a process of valuation. Thus, if S denotes the set of all states which might be relevant then a valuation of an agent is a mapping v : S → V which yields for a state x in S the value v ( x) meaning state x as observed or valued by the agent. The domain of values V may be a “big” one as a multi-dimensional space or even a space of functions or a “small” one as the set of real or natural numbers or even the set of two elements. Valuations leading to real numbers are, for example, the utility functions in economics, valuations leading to natural numbers occur in sociology when valuations are given by rankings and the two element value domain occurs if the valuation consists simply in checking a state if it is “on” or “off”. Concerning the communication of states among agents one can introduce further distinctions (see [10]) which, however, will not be relevant for the dynamics in the next section.

3. Collective dynamics of agents in the joint state space Consider a group of n agents where each agent I can be in a state within a subset Si of the set S of all states. Each agent is able to evaluate states by an individual valuation vi : S → Vi where his set of values Vi is a subset of the set V of all values. The collective dynamics we consider among these agents will be described in the following. Let for each agent j a state x j given in S j and let x = ( x1 , , xn ) be the collection of states in the joint state space S1 × × Sn . Fix an agent i and denote by xi∗ the state agent i will choose on his valuations vi ( x1 ), , vi ( xn ) of the present states of all agents (including his own). The dynamics then is defined by the rule that agent i chooses xi∗ in such a way that his value of this state is equal to a mean of his values of the states of the other agents. That is, xi∗ is chosen by agent i such that vi ( xi∗ ) = M i ( vi ( x1 ),l , vi ( xn ) ) (1)

where M i : V1 × × Vn → Vi describes a certain mean procedure applied by agent i . Depending on how the value domains Vi are specified this mean procedure may be a concrete mean like the arithmetic or the geometric mean of numbers or it might be an abstract mean M in the sense that min{v j |1 ≤ j ≤ n} ≤ M (v1 , , v n ) ≤ max{v j |1 ≤ j ≤ n} (2) To formulate such sandwich inequalities (see [4]) we want the values to be a lattice that is a set which is partially ordered by a relation ≤ and such that minima and maxima of finitely 2

many elements do exist. In the particular case, where the value domains V j consist of real numbers, the inequalities (2) define just the ordinary abstract means as considered in [1] und [4]. Beside the abstract mean as defined by (2) there is another general type of mean which also extends the one dimensional ordinary abstract mean and which seems more appropriate for higher dimensions. For value domains Vi which are convex subsets of some finite dimensional space V, a convex mean M is defined by the property that M (v1 , , v n ) ∈ conv {v1 , , vn } for all (v1 , , v n ) ∈ V1 × × Vn

(3)

n n where conv {v1 , , v n } denotes the set of all convex combinations ∑ α j v j |0 ≤ α j , ∑ α j = 1 j =1 j =1 (see[7] for the concept of a convex mean). Dealing with abstract means for real numbers one needs that the mean is strict. Similarly, the more abstract mean defined by (2) is called strict if an inequality becomes an equality only in the case that all v j are equal. A convex mean

defined by (3) is called strict if M (v1 , , v n ) is in the relative interior of conv {v1 , , v n } . (Instead of a finite dimensional space V one could consider also a general vector space which for the formulation of “strict” has to be assumed a topological vector space.) All these notations coincide with the ordinary ones for the particular case that value domains Vi are sets of real numbers. As for the ordinary abstract means, also for the means defined by (2) and (3), respectively, a partial version can be defined which fits to model bounded confidence within the corresponding context. In this paper, however, we will abstain from that and take up the issue of bounded confidence for value domains Vi in a forthcoming paper. Now, equation (1) induces a dynamics on the joint state space

x(t + 1) = f ( x(t ) ) for t = 0,1, 2,

(4)

where x(⋅) is an element of the joint state space S1 × × Sn and f is a selfmapping of this space with the i -th component f i given by xi∗ = f i ( x1 , , xn ) for xi ∈ Si . To define f in this way, however, requires some assumption. We will assume that all vi are bijections on their value domains. (Otherwise, we could treat f not as a mapping in the common sense but as a set-valued mapping.) In other words, the dynamics following (1) is given by

xi (t + 1) = vi−1 ( M i ( vi ( x1 (t ),l , xn (t ) ) )

(5)

(See [6] for a similar model.) In the next section we will present examples for these dynamics together with some general results.

4. The Gauss soup and its dynamics A simple example of the collective dynamics considered in the previous section which, however, is not trivial at all, is the following. Consider two agents, n = 2 , with state spaces S1 = S2 = + , the set of (strict) positive real numbers. Let the value domains be V1 = + , V2 = and let agent 1’s valuation be given by v1 (r ) = r and agents 2’s valuation by v2 (r ) = log r . Thus, we are considering two heterogeneous agents which differ in their valua-

3

tions which may be looked at as utility functions. Furthermore, as mean procedure choose simply for both agents the arithmetic mean. The dynamics given by (5) means that

( x1 (t ) + x2 (t ) ) x2 (t + 1) = log −1 ( 12 ( log x1 (t ) + log x2 (t ) ) ) x1 (t + 1) =

1 2

(

(6)

)

= log −1 log x1 (t ) x2 (t ) = x1 (t ) x2 (t )

That is, in the joint state space 2+ agent 1 acts by the arithmetic mean, whereas agent 2 acts by the geometric mean. The joint iteration of the arithmetic and the geometric mean has been explored already by Carl Friedrich Gauss, who found out the iteration converges to the socalled arithmetic-geometric mean which is given by an elliptic integral. In the light of the previous section there is a huge variety of possible generalizations of Gauss’ arithmetic-geometric mean. Instead of two agents we may consider an arbitrary number of n of agents. As with the arithmetic-geometric mean we may consider agents who all apply an arithmetic mean but differ in choosing arbitrarily between identity and logarithm as their valuations. A collection of n agents (or particles) interacting in this manner we call a Gauss soup. For n = 2 the Gauss soup reduces essentially to the setting of Gauss’ arithmetic-geometric mean. (The two other possible cases where both agents choose identity or logarithm, respectively, are trivial.) There are, however, much more ways in speaking of a Gauss soup. For example, instead of identity and logarithm we may consider other valuations admitting that all are different from each other. Or, we consider agents applying all the same valuations but different mean procedures. (Of course, the setting of the arithmetic-geometric mean can be viewed in this way.) In general, we can consider a Gauss soup of n interacting heterogeneous agents who differ in valuations as well as in mean procedures. Moreover, we can consider also a Gauss soup where valuations and/or means procedures change with time (see below). As interesting as these Gauss soups of heterogeneous agents may be, as difficult are they to explore. An important tool, of course, are computer simulations. (Some results in [3, 4] can be viewed as dealing with particular Gauss soups.) Much more difficult seem analytical investigations to be. The first question to answer is whether the dynamics converges to some limit pattern, in particular, to a consensus. In the following we present four results in this direction. All results state condition under which a collective dynamics as modelled by equation (5) converges to a consensus c, that is lim xi (t ) = c for all 1 ≤ i ≤ n where c ∈ S can be a real x →∞

number or a collection (vector) of real numbers. To determine the value of the consensus c, however, is in general a task much more formidable. The following result is a classical result for Markov chains where it is easy to determine the value of c. (See also [2, Theorem 3], [3, Theorem 1].)

Result 1 • • •

The state spaces as well as the value domains of all agents are equal to + and the valuations of all agents are given by identity. All agents apply weighted arithmetic means which may be different means. The means are such that any two agents put a positive weight jointly on some third agent.

4

The next result includes the previous one and may also be considered to be a classical result. It applies also to the arithmetic-geometric means of Gauss which is not covered by Result 1. A proof of Result 2 can be found in [1, Theorem 8.8]. In [4, Corollary] , Result 2 is obtained as a special case of a more general result on partial abstract means (PAM).

Result 2 • •

The state spaces as well as the value domains of all agents are equal to + and the valuations of all agents are given by identity. All agents apply arbitrary strict abstract means (which, of course, can be different).

The following result deals with valuations that are not given by identity. This result appears as a consequence of a more general theorem in [6, Corollary 1].

Result 3 •

The state space of all agents is equal to

•

equal to . The valuations v for all agents are given by logarithm or by v(r ) = r α for some α ≠ 0 . All agents apply weighted arithmetic means with weights which may change with time. There is a condition on the weights to decrease not “too fast” with time.

•

+

and the valuation domain of all agents is

The final result deals with multi-dimensional values and the convex mean as mean procedure in the sense of (3). For the special case of one dimension only see [6, Theorem 1]. For higher dimensions see [7, Theorem 1] and also [3, Theorem 4]. For bounded confidence in higher dimensions see [8].

Result 4 • • •

The state spaces as well as the value domains of all agents are given by a closed convex subset in a finite-dimensional real space. The valuations of all agents are given by identity. All agents apply strict convex means which may be different from each other and with coefficients changing in time. There is a condition on the weights to decrease not “too fast” (in particular, the coefficients are constant in time).

References [1]

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.

[2]

J.C. Dittmer, Consensus formation under bounded confidence, Nonlinear Analysis 47 (2001), pp. 4615 – 4621.

5

[3]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: models, analysis, and simulation. Journal of Artificial Societies and Social Simulation 5 (2002), 33 pages. http://jasss.soc.surrey.ac.uk/5/3/2.html.

[4]

R. Hegselmann and U. Krause, Collective dynamics of interacting agents when driven by PAM. Paper presented at the Conference Complexity 2003, Aix-en-Provence, 2003.

[5]

U. Krause, Hierarchical structures in multicriteria decision making. In J. Jahn, W. Krabs (Eds.), Recent Advances and Historical Development of Vector Optimization, Springer, Berlin etc. 1987, 183-193.

[6]

U. Krause, A discrete nonlinear and non-autonomous model of consensus formation. In S. Elaydi, G. Ladas, J. Popenda, and J. Rakowski (Eds.), Communications in Difference Equations, Gordon and Breach Publ., Amsterdam 2000, 227-236.

[7]

U. Krause, Positive particle interaction. To appear in the Proceedings of the Conference on Positive Systems, Theory and Applications, Rome, 2003.

[8]

J. Lorenz, Multidimensional opinion dynamics when confidence changes. Paper presented at the Conference Complexity 2003, Aix-en-Provence, 2003.

[9]

I. Steedman and U. Krause, Goethe’s Faust, Arrow’s Possibility Theorem and the individual decision taker. In J. Elster (Ed.), The Multiple Self, Cambridge University Press, Cambridge, 1986, 197-231.

[10]

D. Urbig, Attitude dynamics with limited verbalisation capacities, Journal of Artificial Societies and Social Simulation 6 (2003), 22 pages. http://jasss.soc.surrey.ac.uk./6/1/2.html.

6