Collective Dynamics of Faustian Agents ULRICH KRAUSE University of Bremen Department of Mathematics, Bibliothekstr. MZH, 28334 Bremen, Germany
[email protected] http://www.informatik.uni-bremen.de/~krause/ Paper dedicated to Ian Steedman on the occasion of his retirement
1 Introduction In the meantime things have changed a lot – the rational agent, representative or not, is no longer the main character on stage. Other characters came into play like bounded rational agents, heterogeneous agents, reciprocators, imitators, .... A recent survey on heterogeneous agent models in economics and finance starts with the following statement [7] “Economics and finance are witnessing an important paradigm shift, from a representative, rational agent approach towards a behavioral, agent-based approach in which markets are populated with boundedly rational, heterogeneous agents using rule of thumb strategies.” This was very different when Ian Steedman and I published about twenty years ago an article on multi-faceted individuals or Faustian decision-takers. (See [19]). Opening up the black box of the rational decision-taker in that article, we found a huge variety of diverse characters, ranging from a Pareto-type over hierarchical-types to the harmonic-type, of which the rational character is a special case. This paper discussed also the question of the identity of a person in spite of this variety. Interestingly, this became recently an issue in economics (see [2], [9], [16]). The present paper starts with the concept of a Faustian agent, and analyzes the dynamics of many interacting agents of this kind. Though the social and economic life is full of interactions among different kinds of people, the dynamics of interaction has been widely neglected in economic theory. The concept of a representative agent, for example, wipes out any interaction. One may think also of the difficulties in general equilibrium theory to prove stability for equilibria (see the critical assessments in [1], [8]). The dynamics of interaction, as modelled in the present paper, originates from the field of opinion dynamics (see for this [5], [6], [12]). One of the main issues in this field is whether the agents will reach a consensus. Thus, in this paper conditions will be examined under which heterogeneous agents approach a joint action by using thumb rules of compromising. In Section 2 the concept of a Faustian agent and of an agent's character will be outlined following [19]. In Section 3 we concentrate on a particular kind of character which seeks for a compromise by taking a mean of values. Since there are many kinds of means, there are many possible compromising characters. In Section 4 the interaction among compromising 1
agents is modeled, which then in Section 5 is considered in discrete time to analyze the collective or joint dynamics of many agents. To illustrate this general model, in Section 6 a first example is discussed. Being linear, the example is rather simple but it exhibits an important condition for the agents to approach a joint action. This condition is the principle of the third agent, which requires interaction and communication be structured in such a way that any two agents take a third one positively into account. In Section 7 the general model is illustrated by a seemingly simple nonlinear example of two agents. Whereas one agent compromises by using an arithmetic mean, the other compromises using a geometric mean. The result, already obtained by Gauß in 1799, is that both agents approach a “consensus”, the value of which is given by an elliptic integral. In Section 8 these two examples are taken in a broader perspective. The linear example lives within the framework of Markov chains. Here the famous Basic Limit Theorem supplies conditions for the agents to reach a consensus (cf. [15, p. 230/231]). The nonlinear example leads to the concept of a Gauß soup, where each agent uses either a weighted arithmetic mean or a weighted geometric mean. In sharp contrast to what is true for Markov chains, it is not possible until today to determine the value of consensus for Gauß soups in general. The paper concludes with Section 9 which presents some further results.
2 Faustian agents “Two souls, alas, do dwell within his breast; The one is ever parting from the other.” GOETHE, Faust, Part I “Faust complained that he had two souls in his breast. I have a whole squabbling crowd. It goes on as in a republic.” BISMARCK (quotations from [19, p. 197])
In everyday life one often faces conflicting aspects when taking a decision between alternatives. This does not necessarily mean that there are several souls in the agent's breast. It may simply mean that in a complex world different things are perceived differently by different people. What then, in spite of this heterogeneity does it mean that agents decide in a rational manner? How does an agent arrive at a consistent overall assessment which is based on the different aspects things have? Or, how do Johann Faust and Otto von Bismarck aggregate the many souls in a consistent way? Dismissing any analysis of how a decision can be taken in spite of conflicts, the model of the rational agent simply equips the agent with some unifying preference ordering or utility function. The crucial question behind this assumption is, how in a complex world an agent can arrive at a conclusive decision at all. Such a multi-faceted agent or Faustian decisiontaker is modeled in [19] by its character. The character of an agent comprises the number of aspects, the domain of conflict D admitted by the agent and as its core the particular formation rule F, applied by the agent to aggregate the various aspects into one overall picture. This rule maps a profile ( R1 , R2 ,..., Rn ) within the domain of conflict D to an overall assessment R. Thereby, Ri , the assessment according to aspect i and R are relations on the given set of alternatives. The crucial point then is that the relations Ri may be assumed to
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be complete orderings (e.g., because aspects are simple), but that R will not be a complete ordering in general. The standard rational agent is now just a special case within a manifold of many characters. Examples of characters are, among others, the Pareto-type (R equals the intersection of the Ri ), the hierarchical-type (R is built up from the Ri by a lexicographical principle) or the counting-type (R counts the number of aspects in favour of one alternative against another). By representing the Ri by functions and applying numerical aggregation rules as forming sums or maxima, a whole family of cardinal characters is obtained. A particular kind of cardinal character will be dealt with in the next section, where the numerical aggregation is done by taking a mean, e.g., a weighted arithmetic mean. Furthermore, in Section 4 we will specialize the general framework of Faustian agents to a situation, where the aspects are given by other agents including individuals, groups or institutions. There are many more interesting aspects concerning the concept of Faustian agents which we will, however, only mention in passing. As brought up, a Faustian agent does simply take care of the different aspects within a complex world. The several assessments Ri may be viewed also as several selves and the Faustian agent himself as a “multiple self” [4, Section 3] or as a “partitioned mind” [16, Section 3]. This raises the question of the unity of an agent and of the “social identity and personal identity” [3, Section 2], [2], [9],[17], [18]. Also, for a Faustian agent the notions of “rationality” and “egoism” are questionable, there is simply no ego which could act rational. To pursue selfish goals, a Faustian agent has to know his ego. This raises the question of self-consciousness (cf. [11]).
3 Compromising characters Imagine a Faustian agent who aggregates the various aspects into one overall picture by making a compromise. Compromising is a ubiquitous feature in everyday decision-making. But how can we describe this in more precise terms? Consider the following example discussed by I. Levi [14, p. 11]: “Suppose that Jones, the manager of an office, is looking for a new typist-stenographer. He requires all applicants to take standardized typing and stenography tests. Three persons apply: Jane, Dolly, and Lilly. Their typing scores are 100, 99, and 90, respectively, and their stenography scores are 90, 99, and 100, respectively.” Manager Jones can be described as a Faustian agent faced with a set of alternatives (Jane, Dolly, Lilly) and aspects “performance in typing”, “performance in stenography” with ordering relations R1 : Lilly < Dolly < Jane for typing and R2 : Jane < Dolly < Lilly for stenography. Obviously, there are conflicts, since for no pairwise comparism of the three women both relations point in the same direction. Manager Jones could make a compromise by taking the average score as formation rule. This gives for Jane, Dolly, and Lilly the overall scores 95, 99, 95. Thus, by this rule he would select Dolly. But instead of using the arithmetic mean for averaging, he could also apply a geometric mean or a harmonic mean. All these different means can be considered to model a compromise. (The reader will verify that in the case of Manager Jones all three means will result in the same overall ordering.) The general feature of this example is that an agent attributes a value to each alternative and then applies a compromise. Let V denote the set of values and c the compromise rule, that is a mapping c : V n → V ,where V n is the Cartesian product V × L × V (n-times). If V is a set of positive real numbers, one could as compromise rule c use one of the following means
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arithmetic mean
c ( v1 ,..., vn ) =
1 n
( v1 + L + vn )
geometric mean
c ( v1 L vn ) =
n
v1 L vn
harmonic mean
c ( v1 , K , vn ) =
n 1 1 v1 + L + vn
power mean
c ( v1 ,K, vn ) =
p 1 n
(v
p 1
+ L + vnp )
(for p ≠ 0 ).
(For the various kinds of means see [6].) These are common means but there are many more. For example one could take weighted versions of these means, by assigning weights to the values v1 , K , vn . In the case of Manager Jones, he could take a weighted arithmetic mean by valuing “typing” ten times more important than “stenography” or the latter twenty times moreimportant than the former. (The reader will verify that in the first case Jones would select Jane and in the second one Lilly.) Also, one could take means of means, for example a harmonic mean of an arithmetic mean and a geometric mean, that is c ( v1 , K , vn ) = 2
n 1 + v1 + L + vn n v L v 1 n
.
An agent using values from a set V of positive numbers corresponds to an agent with cardinal character as in the previous section. For an aspect i the relation Ri is given for two alternatives a and b by according to aspect i.
( a, b ) ∈ Ri
iff
vi ( a ) ≤ vi ( b ) , where vi denotes the valuation
Intuitively, to take a mean value suggests taking a value between the smallest and biggest possible value. Indeed, the above examples as well as the extensions mentioned satisfy the following sandwich inequalities
min {v j |1 ≤ j ≤ n} ≤ c ( v1 , K , vn ) ≤ max {v j |1 ≤ j ≤ n} . This inequalities could be used to define a mean in an abstract sense. Such a mean models the essential feature of a compromise to be within the extremes. For this it is not required that the values in the set V can be added or multiplied. The only assumption needed to formulate the sandwich inequalities, is that the values in V can be partially ordered by a relation “ ≤ ”, in such a way that minima and maxima of finitely many values exist. An example would be a subset V of a higher dimensional space R m . For u , v ∈ V let u ≤ v mean that ui ≤ vi for all components. If V is a box { x | u ≤ x ≤ v} , then minima and maxima of finitely many elements in V are in V again. An abstract mean would be a mapping which assigns to finitely many points in V some point in V. Thus, making a compromise can be made precise also in case the values attributed to alternatives are multi-dimensional, which is particularly interesting when dealing with Faustian agents. A special case would be the componentwise composition of one dimensional means. Concerning compromise for multi-dimensional values there are other processes possible. Beside the above possibility based on the concept of mean in an abstract sense, it is also possible to make the notion of a compromise precise by taking convex combinations of
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finitely many values [13]. In the next section we will consider several agents with compromising characters – compromising agents, for short – who interact with each other.
4 Interaction of compromising agents The general framework developed, we will now apply to the particularly interesting case of a group of agents, whose actions depend on each other. The set of alternatives is given by all the actions which can be taken by one of the agents. For a fixed agent the possible aspects are given by what the other agents are doing. The assessment of the fixed agent of how agent j is acting induces a relation R j on the set of actions. Depending on his formation rule, the agent considered will make his own course of action based on his assessments of the actions of the other agents. For example, an agent with Pareto-type character will prefer one action to another, if all other agents do. A hierarchical-type character bases his course of action by sorting the other agents according to their importance to him. A counting-type character will act according to what the majority is doing. In the following we consider the interaction of n agents of cardinal character. Each agent values the actions of all agents (including himself) and applies a specific compromise rule to get the value of his future action. To be precise, let for 1 ≤ j ≤ n x j denote the action of agent j within a certain region S j of the action space S. v j the valuation of agent j, v j : S → V j , where V j is a certain region of the value space V. Furthermore, let S1 × L × S n be the joint state space, V1 × L × Vn be the joint value space and ci : Vi × L × Vi → Vi be the compromise rule applied by agent i. Though it would be possible to consider general compromise rules as discussed in the previous section, in the following we let Vi be a set of positive numbers and ci a concrete mean. The interaction of these n agents can be described as follows. For a fixed agent the value of his future action is a compromise of the values he attaches to the actions of the other agents. That is, if xi* denotes the future action of agent i we have that vi ( xi* ) = ci ( vi ( x1 ) , K , vi ( xn ) )
(*)
for each agent 1 ≤ i ≤ n .
5 Collective dynamics In general, a future action xi* is not uniquely determined by equation (*) but is described by a set valued mapping. To simplify, we shall assume that the valuations v j can be inverted. If we denote the action of agent j in period t by x j ( t ) , we obtain from (*) for the action
xi ( t + 1) = xi* of agent i in the next period
( (
xi ( t + 1) = vi−1 ci vi ( x1 ( t ) ) ,K , vi ( xn ( t ) )
))
(**)
for 1 ≤ i ≤ n and t = 0,1, 2, K .
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x ( t ) = ( x1 ( t ) ,K , xn ( t ) ) ∈ S1 × L× Sn we obtain the following joint state space dynamics x ( t + 1) = f ( x ( t ) ) , where f : S1 × L× Sn → S1 × L× Sn (***)
For
has i-th component function
(
)
f i ( x ) = vi−1 ci ( vi ( x1 ) , K , vi ( xn ) ) . The system (***) is a
discrete dynamical system on the state space S1 × L × S n .Depending on the functions vi and ci the system will be nonlinear in general and, hence, not easy to analyze. Important questions addressed to this system will be its asymptotic behaviour, the possibility of a consensus and when a consensus can be reached. Before presenting general results in Sections 8 and 9, we will discuss in detail two instructive examples. In the next section we consider a linear example and in Section 7 we consider a simple but nontrivial example which is not linear.
6 Linear example: The principle of the third agent Consider three interacting agents and let us simplify issues of valuation drastically by assuming that the action spaces, as well as the value spaces for all agents, are all equal and given by the set of positive numbers. Furthermore, assume all valuations are simply the identity map. Consider the following compromise rules for the agents c1 ( x1 , x2 , x3 ) = vx1 + (1 − v ) x2
c2 ( x1 , x2 , x3 ) = x2 c3 ( x1 , x2 , x3 ) = (1 − w ) x2 + wx3 All three rules are given by a weighted arithmetic mean. Whereas agents 1 and 3 put beside a positive weight, v and w, respectively, on themselves, also put a positive weight on another agent, agent 2 puts weight only on himself. Though agent 2 does not interact with the other agents, the collective dynamics in this example approaches a consensus for all agents, that is a joint state with equal components. This can be seen by iterating the process of compromise formation again and again. The dynamical system (***) in this example is x ( t + 1) = f ( x ( t ) ) , where f : S 3 → S 3 , S the set of positive numbers, is given by
f i ( x ) = ci ( x1 , x2 , x3 ) for 1 ≤ i ≤ 3 . The easiest way to do the iteration is perhaps by using matrix notation, that is v (1 − v ) 0 f ( x ) = A ⋅ x with A = 0 1 0 0 (1 − w ) w Then x ( t ) = At x ( 0 ) , x ( 0 ) an initial joint state. The matrix product At is not
0 1 0 difficult to evaluate, and one finds that A converges to 0 1 0 irrespective of v and w 0 1 0 as long as their value is strictly less than 1. This implies that for every initial joint state all agents approach the position initially held by agent 2. The reason that the collective dynamics approaches a consensus is the principle of the third agent: Any two agents give a strictly positive weight to some third agent. In the example given, agent 2 is a third agent for all t
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agents, as long as 0 ≤ v, w < 1 . In the case of v = 1 or w = 1 a consensus will not be reached. In these cases there exists a third agent only for agents 2 and 3 or 2 and 1. There are many more examples to illustrate the principle of the third agent. The general formulation of this principle is as follows: For n agents using weighted arithmetic means as compromise rules, the collective dynamics approaches a consensus if and only if for the matrix A with rows given by the weights, some power has the property that any two rows have jointly a strictly positive entry in some column. This general principle is strongly related to Markov chains as will be explained in Section 8. Before that, however, we will examine a simple but intriguing example of nonlinear interaction.
7 Nonlinear example: The Arithmetic-Geometric Mean The next example is almost like the last one with a seemingly small modification, making it more difficult. Simpler than in the last example there are only two agents. Both agents apply an arithmetic mean, but whereas agent 1 uses, as the agents in the previous example, the trivial valuation given by identity, agent 2 uses a nonlinear valuation, given by the (natural) logarithm. (This valuation has the characteristic features of the utility functions used in economics, it is strictly increasing with strictly decreasing marginal utility.) Thus the general model (**) becomes x1 ( t + 1) = 12 ( x1 ( t ) + x2 ( t ) ) , x2 ( t + 1) = log −1
( ( log x ( t ) + log x ( t ) ) ) , 1 2
1
2
where we take the state spaces of both agents as well as the value space of agent 2 to be the positive numbers, and for the value space of agent 2 all real numbers. Using the fact that the logarithm converts multiplication into addition, the second equation is equivalent to
(
1
)
x2 ( t + 1) = log −1 log ( x1 ( t ) x2 ( t ) ) 2 = x1 ( t ) x2 ( t ) . Thus, we arrive at the system x1 ( t + 1) =
1 2
( x (t ) + x (t )) 1
2
(****)
x2 ( t + 1) = x1 ( t ) x2 ( t ) .
In other words, agent 1 applies an arithmetic mean and agent 2 applies a geometric mean. What could be said about the dynamical behaviour of this system? Since the geometric mean is less than or equal to the arithmetic mean it follows that x1 ( t ) and x2 ( t ) are decreasing and increasing functions, respectively. Therefore, x1 ( t ) and x2 ( t ) will converge for t tending to infinity to some
y1 ( t ) and y2 ( t ) , respectively. Furthermore, y1 = lim x1 ( t ) = lim x1 ( t + 1) = lim 12 ( x1 ( t ) + x2 ( t ) ) t →∞
t →∞
t →∞
= lim x1 ( t ) + lim x2 ( t ) = 12 y1 + 12 y2 . 1 2
t →∞
1 2
t →∞
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This implies that y1 = y2 .Thus, we obtain that both agents approach the same value, that is a consensus. The value y of this consensus is determined solely by the initial conditions a = x1 ( 0 ) , b = x2 ( 0 ) and an obvious and interesting question is, how the value y depends on a and b. Surprisingly, the answer is that y is given by an elliptic integral with parameters a and b. This was discovered by Carl Friedrich Gauß on May 31 in 1799.(Of course not thinking of interacting agents.) Experimentally he detected that the iterations of the system (****) converge in both components to an elliptic integral which he called the ArithmeticGeometric Mean. The proof for this is not difficult but not obvious at all. This discovery by Gauß is considered to be a crucial moment in the history of mathematics in that it led to the new field of elliptic functions. The example (****) we may vary a little bit by replacing the geometric mean by the quadratic mean
1 2
(x
2 1
+ x22 ) .
In this case the quadratic mean dominates the arithmetic mean and, by the same argument as before, both agents tend to a consensus. Again, one may ask: How does the consensus depend on the initial conditions? I do not know!
8 Markov chains and Gauß soups Markov chains were already mentioned in connection with the principle of the third agent in Section 6. Let A be a matrix with n rows, n columns, and nonnegative entries only and such that each row sums up to 1. In other words, A collects n weighted arithmetic means as discussed at the end of Section 6. A Markov chain is a discrete dynamical system x ( t + 1) = x ( t ) ' A , with given initial conditions x ( 0 ) . ( x ' denotes the row vector for a column vector x.) The famous Basic Limit Theorem for Markov chains [15, p. 230/231] states that for a regular Markov chain the powers At converge to a matrix with all rows equal to a vector q which can be computed from the equation q ' A = q ' . There, a Markov chain is called regular if some power of A has all its entries strictly positive. This Basic Theorem can be used to make sure and to compute the consensus for collective dynamics given by a linear mapping f ( x ) = Ax ; the n
value of the consensus is given by
∑ q x ( 0 ) . Concerning our linear example in Section 6, i i
i =1
we find, however, that the Basic Theorem is not applicable since no power of A is strictly positive. (The reader will verify that in this example the north-east corner of each power of A is zero.) Nevertheless, the conclusion of the Basic Theorem holds true also for this example. Actually, it is true in general that the conclusion of the Basic Theorem applies if and only if some power of A satisfies the principle of the third agent. Thus, this principle is not only relevant for the example discussed in Section 6 but for the general linear case, viz. for Markov chains. What can be said in the nonlinear case about the collective dynamics of Faustian agents as modelled in Section 5? The example in the previous section may indicate that this is very difficult to answer. Yes and no. Yes – it is very difficult, even impossible up to now, to determine the dependence on initial conditions. This is in sharp contrast to the linear case of Markov chains. No – it can be shown under rather mild conditions that all agents will approach a consensus. This will be explained in the following for the so called “Gauß soups” which generalize the example discussed in the previous section. In a Gauß soup of n agents each agent can take a compromise procedure which is either a weighted power mean or a weighted geometric mean. More precisely, a Gauß soup is the particular case of collective
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dynamics as defined in Section 5 where
Si is the set of positive numbers and vi is the
identity for all agents i and where ci ( x1 ,K, xn ) is either 1
r n r a weighted power mean ∑ pij x j j =1
n
or a weighted geometric mean
∏x
pij j
.
j =1
n
Thereby, the
pij are nonnegative weights with
∑p
ij
= 1 for all i and r ≠ 0
is a
j =1
parameter; the special case r = 1 gives a weighted arithmetic mean. It can be proven that for strictly positive weights pij the actions of all agents converge to the same action, that is to a consensus. For this, it does not matter which agents or how many apply one or the other mean out of the set described above. (Therefore, the term “soup”.) For the proof one observes first that each of the means specified is an abstract mean, as discussed in Section 3 – for this one does not need the positivity of the pij . Second, each of the means has the property that equality on both sides of the sandwich inequality implies the v j must all be equal. For this, one needs an assumption on the positivity of the weights. The assumption of strict positivity is sufficient but could be weakened. (See [13]. The proof given there applies also to the case of multi--dimensional values relevant for Faustian agents.) Obviously, a Markov chain is a special case of a Gauß soup, where each agent uses a power mean with r = 1. As mentioned, in this special case the consensus can easily be computed from the initial conditions with the help of the eigenvector q. This is completely different for proper Gauß soups where agents use arithmetic means as well as geometric means. Whereas for the example of Section 7 it is known that the consensus can be computed from initial conditions via an elliptic integral, almost nothing is known about other examples – not to speak of a Gauß soup in general.
9 Further results It was remarked already in Section 3 that all common means are means in an abstract sense, as defined by the sandwich inequalities. The convergence to consensus, discussed in the previous section for power means and geometric means, can be extended to means in the abstract sense. Furthermore, means in an abstract sense can be studied also for multidimensional values, if one replaces the sandwich inequalities by the condition that the compromise c ( v1 ,K, vn ) is always contained in the convex hull of v1 , K , vn . (See [13]). In the one dimensional case, the convex hull reduces to an interval with endpoints, given by minimum and maximum.) The interaction of compromising agents, as modelled in Section 4, links a fixed agent with potentially all other agents. If an agent places strictly positive weights to all agents, as in the previous section, this agent indeed interacts with all other agents. It would be more realistic to admit that each agent can interact only locally with agents in “some neighbourhood”. An example would be agents with bounded confidence, who place positive weights only on agents they trust in. Approaching a consensus is then less likely and more difficult to analyze. 9
It can be shown, however, that from a certain confidence level onwards and depending on the initial conditions, a consensus will be reached in finite time. (See [5] for analytical results as well as computer simulations in that respect and for a historical account of consensus formation.) Combining the models, one can analyze local interaction for agents applying means in an abstract sense. This yields a partial abstract mean, where the sandwich inequalities are modified in such a way that minimum and maximum are not taken over all agents, but for a fixed agent only over his neighbourhood, which will depend on this agent as well as on the values. (See [6] for analytical results as well as computer simulations in that respect.) Considering the interaction of agents in reality, one should admit, finally, also that the structure and strength of interaction may change in the course of time. For example, for agents with bounded confidence it may be that confidence decreases (or increases) by interaction, a phenomenon known as “hardening of positions”. Under certain conditions on the structure and strength, it is still possible that for this time-dependent or non-autonomous interaction the agents approach a consensus. (Cf. [12].)
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