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Showing or telling? Local interaction and organization of behavior

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Abstract

We present a choice model based on agent interaction. Interaction is modeled as face-to-face communication that takes place on a regular periodic lattice with decision-makers exchanging information only with immediate neighbors. We investigate the long-run (equilibrium) behavior of the resulting system and show that for a large range of initial conditions clustering in economic behavior emerges and persists indefinitely. Unlike many models in the literature, our model allows for the analysis of multi-option environments. Therefore, we add to existing results by deriving the equilibrium distribution of option popularity and thus, implicitly, of market shares. Additionally, the model sheds new light on the emergence of the novel behavior in societies.

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Notes

  1. A good example might be found in the pricing options of telephone or internet services.

  2. This is a crucial requirement for localization of interaction.

  3. It is important to distinguish between these rumors and rumors as understood by Banerjee (1993). In the latter case rumors diffuse only though practices, while the essence of the former is the diffusion of information that has not been verified by the experience of the sender.

  4. The assumption of circularity of (physical) space has become a standard in economics in recent years. This assumption allows for the analytical solution to various problems ranging from human capital accumulation (Bala and Sorger 2001) to spatial agglomeration processes (Chincarini and Asherie 2008; Ikeda et al. 2012).

  5. The only complication that this approach introduces is that evolution of valuations now depends on expected, rather than actual choices. However, a large number of numerical exercises confirm that as long as the gradient of the function mapping valuation to behavior is steep enough near 1, expected choices overlap with the actual choices and expected habituation is sufficiently close to to its actual counterpart. While making this change will affect particular individuals at particular moments, in terms of the aggregate behavior of the model it is innocuous.

  6. Here and in what follows we drop the expectation sign, although it should be remembered that all the discussion in this section is about the expected values of the variables.

  7. Note that in (9) the valuation of option \(n\) does not depend on the valuations of other options. This is the technical characteristic of our approach that allows us to analyze the multi-option environment in Sect. 4.

  8. Thinking literally in terms of Euclidean space: If agents are arrayed on a circle of radius \(r\), then \(\delta =2\pi r/N\). Clearly \(\delta \) goes to zero as the ratio \(r/S\) goes to zero. Essentially the issue then is how densely the space is populated by the agents.

  9. Note that making higher order approximations in (13) and (14) will leave only the even number terms in the expression (15). Odd number terms will always cancel out. Thus, the third order term, the one with the order of significance from the omitted terms, can be safely ignored. Fourth, Sixth and higher order terms can be sacrificed for tractability without changing the results.

  10. Note that as agents are located on a periodic lattice, the identity of agent zero is arbitrary, and thus can be placed anywhere on the circle. To write Lemma 2 we have set label 0 such that \( s_0 = \mathop {\arg \max }\nolimits _{x \in \left[ {0,{l/k}} \right] } \cos \left( {2\pi kx/l} \right) \), which implies that we label agents such that the sinusoid identified in Lemma 2 reaches its maximum at agent zero.

  11. This effectively means that we fix \(\delta =1\). Moving back to decision-maker addresses is convenient for relating parameters in the solution to the parameters of the model. As demonstrated by Turing (1952, sections 6 and 7, pp. 46–50) and Ellis (1985, section V.10, pp. 190–198) this move does not undermine the results of Lemmas 1–3.

  12. Unless \(\mu =0\), which is not a very interesting case as it implies no interaction. In this case the existing choice pattern is reinforced indefinitely.

  13. From Eq. (17) one can easily see that negative \(\sigma \) is a result of higher rate of communication \(\mu \).

  14. For example, in the small economy that we have simulated (\(S=100\)), \(H=49\) implies that the speed of habituation, \(\alpha \), must be roughly 80 times as high as the influence of neighbors, \(\mu \), in order the system to be stable for the largest possible cluster (\(\underline{c}= S/2\)).

  15. With notable exceptions being Brock and Durlauf (2002) and Durlauf and Ioanides (2010).

  16. More precisely, one wave for each difference in valuation between an option and a numeraire option.

  17. We might reasonably expect that as the number of options increases the probability of finding one large cluster covering half the social space increases. This is simply because the more options the more likely the dominant wave will have a wave frequency \(k=1\). To simplify the development here, we assume this is the case.

  18. In the limit the sine waves become infinitely steep, and with peaks at infinity. Because different waves have different growth rates, it will remain the case that one will dominate others.

  19. “Ranking” is slightly tricky here, since all options are on average valued equally. To rank options in this sense we use the maximal (absolute) valuation over the population.

  20. This is similar to the case presented in Fig. 1.

  21. This argument suggests that one might need many options to guarantee this condition. In fact, however, the probability that a function decomposed into sinusoids has a low frequency wave of zero amplitude is vanishingly small. Thus a small number of options will typically be enough to produce this condition.

  22. If the number of decision-makers is finite, due to the integer problem (cluster size cannot be \(<\)1 agent), there will always be only a finite number of clusters in the economy.

  23. Even though emergence of novelty can also be observed in a similar model without rumors (agents transmitting information only about the products that they have consumed during the period), the richer communication channel including rumors substantially expands the conditions under which emergent novelty can be observed.

  24. This means that we are analyzing the evolution of valuations where habituation depends on expected choices. This does not necessarily coincide with the evolution of choices, as choices are only probabilistically determined by valuations. Because actual behavior is “noisy” in this sense, it is more difficult to observe the emergence of pure novelty in behavior, unless the function mapping valuation to behavior has a very steep gradient near 1. However, even with less severe gradients, it is not uncommon to observe behavior that was rare (and sometimes completely absent) in a neighborhood emerging and growing to become common in that neighborhood, and beyond.

  25. To see more easily why the second summand is zero, one can discuss the discrete case and thus use Eq. (11) instead of Eq. (15). In the discrete case the second summand is \(\sum \nolimits _s \left( (z^{s+1}-z^s)-(z^s - z^{s-1}) \right) \). As decision-makers are indexed by \(s\) around a circle, it is obvious that this sum is zero.

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Correspondence to Zakaria Babutsidze.

Appendix

Appendix

1.1 Appendix A: Proof of Lemma 1

Proof

In the continuous case the average over space can be defined as \(\bar{z} = (1/S) \int \nolimits _0^S z ds\). This implies that

$$\begin{aligned} \frac{\partial \bar{z}}{\partial t} = \frac{1}{S} \int \limits _0^S \frac{\partial z}{\partial t} ds. \end{aligned}$$

Then, using Eq. (15) we can write

$$\begin{aligned} \frac{\partial \bar{z}}{\partial t} = \alpha \frac{1}{S} \int \limits _0^S z ds + \tilde{\mu }\frac{1}{S} \int \limits _0^S \frac{\partial ^2 z}{\partial s^2} ds. \end{aligned}$$
(20)

As space in our system is a periodic lattice the second summand in Eq. (20) is zero.Footnote 25 Then, using the definition of average again we can write Eq. (20) as

$$\begin{aligned} \frac{\partial \bar{z}}{\partial t} = \alpha \bar{z}. \end{aligned}$$
(21)

This is an ordinary differential equation with the solution described in the lemma. \(\square \)

1.2 Appendix B: Proof of Lemma 3

Proof

From Propositions 1 and 2, we know that

$$\begin{aligned} z(s;t) = e^{\alpha t} \bar{z}(0) + e^{\sigma t} \cos \left( k \frac{2 \pi }{l} s \right) \tilde{z}(0;0). \end{aligned}$$

Substituting this into Eq. (15) and noticing that

$$\begin{aligned} \partial ^2 \cos (\beta x) / \partial x^2 = - \beta ^2 \cos (\beta x), \end{aligned}$$

allows us to solve for \(\sigma \). \(\square \)

1.3 Appendix C: Proof of Proposition 4

Proof

Consider the case of arbitrary neighborhood size of \(2H\). In this case after assuming that the distance between two neighboring agents is \(\delta \) and considering the two-option case, continuous version of Eq. (9) can be rewritten as

$$\begin{aligned} \frac{\partial z(s)}{\partial t} = \alpha z(s) + \frac{\mu }{2H} \left[ \int \limits _{ - H}^H {z(s + \delta h)dh} - 2H z(s) \right] . \end{aligned}$$
(22)

Using second order taylor approximation we can rewrite the part of (22) under the integral as

$$\begin{aligned} \int \limits _{ - H}^H {z(s)dh} + \int \limits _{ - H}^H {\delta h \frac{\partial z(s)}{\partial s}dh} + \int \limits _{ - H}^H {\frac{\delta ^2 h^2}{2}\frac{\partial ^2 z(s)}{\partial s^2}dh}. \end{aligned}$$

Which, after integration of first two summands, is equal to

$$\begin{aligned} 2Hz(s)+0+\frac{\delta ^2}{2}\frac{\partial ^2z(s)}{\partial s^2}\int \limits _{ - H}^H {h^2}dh. \end{aligned}$$

To obtain more accurate values for smaller neighborhood size, we go back to discrete space and replace the integral in expression above with the sum of squares of integer values.

Substituting this result back to (22) yields

$$\begin{aligned} \frac{\partial z(s)}{\partial t} = \alpha z(s) + \frac{\mu \delta ^2}{4H} \sum \limits _{h = -H}^H {h ^2 } \frac{\partial ^2z(s)}{\partial s^2}. \end{aligned}$$

Thus, it follows that the only modification that this generalization brings to the system can be captured by the definition of \(\tilde{\mu }\) in the text being changed to

$$\begin{aligned} \tilde{\mu }= \frac{\mu \delta ^2}{4H} \sum \limits _{h = -H}^H {h ^2}. \end{aligned}$$
(23)

Going back to agent addresses (\(\delta = 1\)), using new definition of \(\tilde{\mu }\), and the identity \(\sum \nolimits _{n = 1}^x {n ^2 } = \frac{x^3}{3} + \frac{x^2}{2} + \frac{x}{6}\) we can rewrite Eq. (17) as

$$\begin{aligned} \sigma _H = \alpha - 2\mu \left( k \frac{\pi }{l} \right) ^2 \left( \frac{H^2}{3} + \frac{H}{2} + \frac{1}{6} \right) , \end{aligned}$$
(24)

which results in

$$\begin{aligned} \bar{k}_{H} = \frac{S}{\pi }\sqrt{\alpha \left( 2 \mu \left( \frac{H^2}{3} + \frac{H}{2} + \frac{1}{6} \right) \right) ^{-1}}, \end{aligned}$$
(25)

and further in

$$\begin{aligned} \underline{c}_{H} = \frac{\pi }{2\sqrt{3}} \sqrt{2H^2+3H+1} \sqrt{\frac{\mu }{\alpha }}. \end{aligned}$$
(26)

\(\square \)

1.4 Appendix D: Proof of Proposition 5

Proof

In order to derive the distribution of popularity it is useful to split the popularity rankings in three parts: \(R_n=1, R_n=2\) and \(R_n \ge 3\). We consider each of these cases separately.

\(R_n=1\): The fact that \(F_1=1/2\) is demonstrated by Remark 3.

\(R_n=2\): Consider the effect of large number of options. We know that highest \(\sigma \) guarantees the championship of the wave. However, as each equation in system (19) has the same parameters, we know that there will be many waves with the same values of \(\sigma \). Consider the grouping the waves in subsets, where waves in each subset have the same value of \(\sigma \). Then we can rank these subsets starting from the highest to lowest. We also know that for winning the championship in case of equal \(\sigma \)s what matters is the initial amplitude. Then in each subset we can rank waves in decreasing order of their initial amplitude values. Now we have a unique ranking of all the waves. We call this a preliminary ranking as some of the waves might get dropped from the top places due to the subsequent refinement. We will demonstrate that not every option will appear in the long run frequency distribution. As higher waves in ranking have higher chances for ending up in the frequency distribution we assume that the set of options is so large that all the ultimate practices will be selected from the highest ranked subgroup. Therefore, we simply disregard lower ranked subgroups. Thus, large number of options ensures that every option present in the long run frequency distribution with a non-zero weight has the wave length of \(k=1\).

We know that the most popular option has half of the market size. As large number of options ensures that the champion wave has the wave length of \(k=1\), and thus the most popular option has one cluster (of size \(S/2\)) in the social space. As our social space is circular we can reindex the agents without loss of generality. Assume the champion sinusoid starts at agent \(s=0\). This would mean, that the cluster of the champion practice comprises the social space between \(s=0\) and \(s=S/2\). Now, what becomes important for identifying the size of the second largest cluster is the offset of the second ranked wave from the champion. Offset is the difference in social space between the sinusoid under discussion and the champion sinusoid. As we normalized the champion to start at \(s=0\), the offset of any wave will simply be equal to the location \(s\) where it starts. To identify which option is going to be the second most popular in the long run we go down the preliminary ranking. If the second ranked option in the preliminary ranking has offset exactly equal to zero this means that this sinusoid is positive in space \((0;S/2)\) and negative in \((S/2;1)\). But so is the champion wave. And we know that the champion dominates any other wave completely in the space \((0;S/2)\). Therefore, the wave with offset zero will never show up in the long run frequency distribution with the positive weight. Thus, we can discard the wave and remove it from the rankings.

Then we go down to the rankings until we find the wave with offset \(s_i>0\). Consider how the share of social space dominated by this option depends on \(s_i\). If \(s_i<S/2\) we know that this wave will be positive on \((s_i; S/2+s_i)\) and negative on \((0;s_i)\cup (S/2+s_i;1)\). However, on \((s_i;S/2)\) it will be dominated by the champion wave, therefore this option will only acquire \(S/2+s_i-S/2=s_i\). If \(s_i>S/2\) the wave is positive on \((s_i;1)\cup (0;S/2-1+s_i)\). But it is dominated by the champion on fraction \((0;S/2-1+s_i)\), and thus, it obtains the section \(1-s_i\). It can be easily seen that as \(s_i\) goes from \(s_i=0\) to \(s_i=S/2\), the part dominated by the second ranked wave also increases linearly from zero to \(S/2\). As \(s_i\) continues move to the right after passing \(S/2\), the part dominated by the wave decreases linearly from \(S/2\) to zero (when \(s_i=1\)).

Now, as initial conditions are random and agents are distributed uniformly over the social space, the probability of choice of any \(s_i\) is constant. Therefore, we can calculate that the average market share of the second ranked practice in the long run \(F_2=1/4\). In order to build the case for \(R_n>2\) notice that there are two actual waves corresponding to the market share of \(1/4\). These are \(s_i=S/4\) and and \(s_i=3S/4\). It does not matter for the further calculations which of them we choose to be present while considering \(R_n>2\) options. Because of the circularity of social space, there will always be two waves corresponding to each share distribution. Without loss of generality we always choose to consider that the wave with the smaller \(s_i\) is at place. Thus, for later options there will always be some space \((0;W>S/2)\) that we be occupied by stronger waves and the space \((W;1)\) left to be distributed among the weaker waves.

\(R_n=m>2\): As pointed out in case \(R_n=2\), by now the social space \((0;W)\) is already distributed. Then \(W=\sum \nolimits _{j=1}^{m-1}F_j\). Denote the size of the remaining social space \(w=1-W\). Then, \(w\) is the size of the not-yet-distributed portion. In this case, the we know that the weakest wave already assigned its long-run share is the wave with the positive part on \((S/2-w;1-w)\). Therefore, any wave to be placed next on the social space has to have the offset more than \(s_i>S/2-w\). This is because the waves with less offset will always be dominated by the already the most popular \(m-1\) waves. Therefore, while going down the preliminary ranking we through out all the waves with offset less then \(S/2-w\), and concentrate only on offsets with higher offsets.

Consider how long-run market share depends on \(s_i\) in this case. With \(s_i\) increasing from \(S/2-w\) till \(S/2\) the share increases linearly from zero to \(w\). In the section where \(s_i \in (S/2;1-w)\) the share is constant at \(w\). Once \(s_i\) passes \(1-w\) the share decreases linearly and reaches zero at \(s_i=1\). In this case taking the average long run market size and converting it to shares results in

$$\begin{aligned} F_m=\frac{w}{1+2w}. \end{aligned}$$

It is easy to check that \(m=2\) also obeys this formula (although the calculation of \(F_2\) was slightly different, it was in fact the specific case of these calculations). \(\square \)

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Babutsidze, Z., Cowan, R. Showing or telling? Local interaction and organization of behavior. J Econ Interact Coord 9, 151–181 (2014). https://doi.org/10.1007/s11403-013-0117-x

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