servables
3.2.1 Micro and Macro in ABMs
What do we look at when we analyze an ABM? Typically, we try to cap- ture the dynamical behavior of a model by studying the time evolution of parameters or indicators that inform us about the global state of the system. Although, in some cases, we might understand the most important dynami- cal features of a model by looking at repeated visualizations of all details of the agent system through time, basic requirements of the scientific method will eventually enforce a more systematic analysis of the model behavior in the form of systematic computational experiments and »extensive sensitivity analysis« (Epstein, 2006, 28). In this, there is no other choice than to leave the micro level of all details and to project the system behavior or state onto global structural indicators representing the system as a whole. In many cases, a description like that will even be desired, because the focus of atten- tion in ABMs, the facts to be explained, are usually at a higher macroscopic level beyond the microscopic description. In fact, the search for microscopic foundations for macroscopic regularities has been an integral motivation in the development of ABMs (see Macy and Willer, 2002; Squazzoni, 2008).
It is characteristic of any such macroscopic system property that it is invariant with respect to certain details of the agent configuration. In other words, any observation defines, in effect, a many-to-one relation by which sets of micro configurations with the same observable value are subsumed into the same macro state. Consider the population dynamics in the sug- arscape model by Epstein and Axtell (1996) as an example. The macroscopic indicator is, in this case, the number of agents N . This aggregate value is not sensitive with respect to the exact positions (the sites) at which the agents are placed, but only to how many sites are occupied. Consequently, there are many possible configurations of agent occupations in the sugarspace with an equal number of agents N and all of them correspond to the same macro state. Another slightly more complicated example is the skewed wealth dis- tribution in the sugarscape model. It is not important which agents con- tribute to each specific wealth (sugar) level, but only how many there are in each level. This describes how macro descriptions of ABMs are related to observations, system properties, order parameters and structural indicators, and it also brings into the discussion to the concepts of aggregation and decomposition.
Namely, aggregation is one way (in fact, a very common one) of realiz- ing such a many-to-one mapping from micro-configurations to macroscopic system properties and observables. For simple models of opinion dynamics inspired by spin physics, for instance, it is very common to use the average opinion – due to the spin analogy often called »system magnetization« – as
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an order parameter and to study the system behavior in this way. Magne- tization, computed by summation over the spins and division by the total number of spins, is a true aggregative measure. Magnetization levels or val- ues are then used to classify spin or opinion configurations, such that those configurations with the same magnetization value correspond to the same macro state. This many-to-one mapping of sets of micro configurations onto macro states automatically introduces a decomposition of the state space at the micro level Σ.
3.2.2 Observables, Partitions and Projected Systems
The formulation of an ABM as a Markov chain developed in the previous section allows a formalization of this micro-macro link in terms of projections. Namely, a projection of a Markov chain with state space Σ is defined by a new state space X and a projection map Π from Σ to X. The meaning of the projection Π is to lump sets of micro configurations in Σ according to some macro property in such a way that, for each X ∈ X, all the configurations of Σ in Π−1(X) share the same property.
Therefore, such projections are important when catching the macroscopic properties of the corresponding ABM because they are in complete cor- respondence with a classification based on an observable property of the system. To see how this correspondence works let us suppose that we are interested in some factual property of our agent-based system. This means that we are able to assign to each configuration the specific value of its cor- responding property. Regardless of the kind of value used to specify the property (qualitative or quantitative), the set X needed to describe the con- figurations with respect to the given property is a finite set, because the set of all configurations is also finite. Let then φ : Σ → X be the function that assigns to any configuration x ∈ Σ the corresponding value of the considered property. It is natural to call such φ an observable of the system. Now, any observable of the system naturally defines a projection Π by lumping the set of all the configurations with the same φ value. Conversely any (projection) map Π from Σ to X defines an observable φ with values in the image set X. Therefore these two ways of describing the construction of a macro-dynamics are equivalent and the choice of one or the other point of view is just a matter of taste.
The price to pay in passing from the micro to the macro-dynamics in this sense (Kemeny and Snell, 1976; Chazottes and Ugalde, 2003) is that the projected system is, in general, no longer a Markov chain: long memory (even infinite) may appear in the projected system. This »complexification« of the macro dynamics with respect to the micro dynamics is a fingerprint of dynamical emergence in agent-based and other computational models (cf. Humphreys, 2008).
40 Agent–Based Models as Markov Chains 3.2.3 Lumpability and Symmetry
Under certain conditions, the projection of a Markov chain (Σ, ˆP ) onto a coarse-grained partition X, obtained by aggregation of states, is still a Markov chain. In Markov chain theory this is known as lumpability (or strong lumpability), and necessary and sufficient conditions for this to hap- pen are known. Let us restate the respective Thm. 6.3.2 of Kemeny and Snell (1976) using our notations, where Σ denotes the configuration space of the micro chain and ˆP the respective transition matrix, and X = (X1, . . . , Xr)
is a partition of Σ. Let ˆpxY = P y∈Y
ˆ
P (x, y) denote the conjoint probability for x ∈ Σ to go to the set of elements y ∈ Y where Y ⊆ Σ is a subset of the configuration space.
Theorem 3.2.1 (Kemeny and Snell, 1976, 124) A necessary and sufficient condition for a Markov chain to be lumpable with respect to a partition X = (X1, . . . , Xr) is that for every pair of sets Xi and Xj, ˆpxXj have the same
value for every x in Xi. These common values {ˆpij} form the transition
matrix for the lumped chain.
In general it may happen that, for a given Markov chain, some projections are Markov and others not. Therefore a judicious choice of the macro properties to be studied may help the analysis.
In order to establish the lumpability in the cases of interest we shall use symmetries of the model. For further convenience, we state a result for which the proof is easily given Thm. 6.3.2 of Kemeny and Snell (1976):
Theorem 3.2.2 Let (Σ, ˆP) be a Markov chain and X = (X1, . . . , Xn) a
partition of Σ. Suppose that there exists a group G of bijections on Σ that preserve the partition (∀x ∈ Xi and ∀ˆσ ∈ G we have ˆσ(x) ∈ Xi). If the
Markov transition probability ˆP is symmetric with respect to G, ˆ
P (x, y) = ˆP (ˆσ(x), ˆσ(y)) : ∀ˆσ ∈ G, (3.8) the partition (X1, . . . , Xn) is (strongly) lumpable.
Proof. For the proof it is sufficient to show that any two configurations x and x′ with x′ = ˆσ(x) satisfy
ˆ pxY = X y∈Y ˆ P (x, y) = X y∈Y ˆ P (x′, y) = ˆpx′Y (3.9)
for all Y ∈ X. Consider any two subsets X, Y ∈ X and take x ∈ X. Because G preserves the partition it is true that x′ ∈ X. Now we have to show that
Eq. (3.9) holds. First the probability for x′ = ˆσ(x) to go to an element y∈ Y is ˆ pσ(x)Yˆ = X y∈Y ˆ P (ˆσ(x), y). (3.10)
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Because the ˆσ are bijections that preserve the partition X we have ˆσ(Y ) = Y and there is for every y ∈ Y exactly one ˆσ(y) ∈ Y . Therefore we can substitute ˆ pˆσ(x)Y = X y∈Y ˆ P (ˆσ(x), ˆσ(y)) = X y∈Y ˆ P (x, y) = ˆpxY, (3.11)
where the second equality comes by the symmetry condition (3.8) that ˆ
P (x, y) = ˆP (ˆσ(x), ˆσ(y)).
The usefulness of the conditions for lumpability stated in Thm. 3.2.2 becomes apparent recalling that ABMs can be seen as random walks on regular graphs defined by the functional graph or »grammar« of the model Γ = (Σ, FZ). The full specification of the random walk (Σ, ˆP ) is obtained
by assigning transition probabilities to the connections in Γ and we can interpret this as a weighted graph. The regularities of (Σ, ˆP ) are captured by a number of non-trivial automorphisms which, in the case of ABMs, reflect the symmetries of the models.
In fact, Thm. 3.2.2 allows to systematically exploit the symmetries of an agent model in the construction of partitions with respect to which the micro chain is lumpable. Namely, the symmetry requirement in Thm. 3.2.2, that is, Eq. (3.8), corresponds precisely to the usual definition of automorphisms of (Σ, ˆP ). The set of all permutations ˆσ that satisfy (3.8) corresponds then to the automorphism group of (Σ, ˆP ).
Lemma 3.2.3 Let G be the automorphism group of the micro chain (Σ, ˆP ). The orbits of G define a lumpable partition X such that every pair of micro configurations x, x′ ∈ Σ for which ∃ˆσ ∈ G such that x′ = ˆσ(x) belong to the same subset Xi ∈ X.
Remark 3.2.1 Lemma 3.2.3 actually applies to any G that is a proper sub- group of the automorphism group of (Σ, ˆP ). The basic requirement for such a subset G to be a group is that be closed under the group operation which establishes that ˆσ(Xi) = Xi. With the closure property, it is easy that any
such subgroup G defines a lumpable partition in the sense of Thm. 3.2.2.