Structure and emergence in a nested logit model with social and spatial interactions

Abstract

Suppose you have the possibility to choose to adopt one of a number of different behaviors or to choose to buy one of a number of different products, and suppose your choice is influenced by your individual perception of the average choices made by others. Economists Brock and Durlauf (in Am. Econ. Rev. 92(2):298, 2002; The Economy as an Evolving Complex System III. Oxford University Press, New York, 2006) have derived seminal theoretical results for the equilibrium behavior of the multinomial discrete choice model with social interactions, assuming homogeneous decision-makers, global interactions and laws of large of numbers. The research presented in this paper extends Brock and Durlauf’s model to allow for unobserved preference heterogeneity between choice alternatives by studying the nested logit model. Next, by drawing on the computational possibilities permitted through social simulation of multi-agent systems (MAS), this paper relaxes the assumption of global interactions and considers instead local interactions within several hypothesized social and spatial network structures. Additional heterogeneity is thus hereby induced by the influence on a given decision-maker’s choice by the particular network connections he or she has and the particular perceived percentages, for example, of the agent’s neighbors or socio-economic peers making each choice. Discrete choice estimation results controlling these heterogeneous individual preferences are embedded in a multi-agent based simulation model in order to observe the evolution of choice behavior over time with socio-dynamic feedback due to the network effects. The MAS approach also gives us an additional advantage in the possibility to test size effects, and thus relax the assumption of large numbers, as well as test the effect of different initial conditions. Finally an extra benefit is gained via the MAS approach in that we are not confined to study only the equilibrium behavior, and have the possibility here to observe the time-varying trajectories of the choice behavior. This is important since smaller network sizes are revealed to be associated with higher volatility of the choice behavior in this model, and consequently stochastic cycling between equilibria. Averaged over time, the emergent behavior in such case yields a quite different picture than the theoretical results predicted by Brock and Durlauf. Furthermore being able to observe the emergent behavior allows us to see the subtle role of the unobserved heterogeneity in the nested logit model in breaking the symmetry of the multinomial logit model. We can see the temporal patterns by which theoretically predicted dominant equilibria emerge or not according to different social and spatial network scenarios. With an eye towards application in the context of transportation mode choice, we conclude highlighting limitations of our present study and recommendations for future work.

Appendix

1.1 A.1 Individual choice probabilities for trinary nested logit

Recall the definition of the nested logit model presented in Sect. 3.1 and consider a scenario as in Fig. 1 where elemental alternative 0 is an isolate in its own “nest”, and elemental alternatives 1 and 2, assumed to be correlated, are nested together. The probability of choosing alternative 0 within nest0, conditional on having chosen nest0 is:

$$ P(i = 0|\mathrm{nest}0) = \frac{e^{\mu _{m}V_{in}}}{\sum_{\forall j \in \mathrm{nest}0} e^{\mu _{m}V_{jn}}} = \frac{e^{\mu _{L}\beta p_{0}}}{e^{\mu _{L}\beta p_{0}}} =1 $$

(21)

where the systematic utility for choosing elemental alternative 0 is a linear-in-parameter β function of the proportion p ₀ of decision-making agents that have chosen alternative 0, and μ _L is the scale parameter that characterizes the distribution of the Gumbel error terms within the nest. The inclusive value for nest0 is then:

$$ I_{\mathrm{nest}0} = \ln\sum_{\forall j \in \mathrm{nest}0} e^{\mu_{m}V_{jn}} = \ln e^{\mu_{L}\beta p_{0}} = \mu_{L}\beta p_{0} $$

(22)

The probabilities of choosing respectively alternative 1 and 2 within nest1, conditional on having chosen nest1, are likewise the binary choice probabilities:

$$ \everymath{\displaystyle }\begin{array}{rcl} P(i = 1|\mathrm{nest1}) &=& \frac{e^{\mu_{m}V_{in}}}{\sum_{\forall j \in \mathrm{nest1}} e^{\mu_{m}V_{jn}}} = \frac{e^{\mu _{L}\beta p_{1}}}{e^{\mu _{L}\beta p_{1}} + e^{\mu _{L}\beta p_{2}}} \\ \noalign {\vspace {5pt}} P(i = 2|\mathrm{nest1}) &=& \frac{e^{\mu _{m}V_{in}}}{\sum_{\forall j \in \mathrm{nest1}} e^{\mu _{m}V_{jn}}} = \frac{e^{\mu _{L}\beta p_{2}}}{e^{\mu_{L}\beta p_{1}} + e^{\mu _{L}\beta p_{2}}} \end{array} $$

(23)

where the systematic utilities for choosing elemental alternative 1 or 2 are respectively a linear-in-parameter β function of the proportions p ₁ and p ₂ of decision-making agents that have chosen alternatives 1 and 2, and μ _L is again the scale parameter that characterizes the distribution of the Gumbel error terms within the nest. The inclusive value for nest1 is then:

$$ I_{\mathrm{nest1}} = \ln\sum_{\forall j \in \mathrm{nest1}} e^{\mu_{m}V_{jn}} = \ln\bigl(e^{\mu_{L}\beta p_{1}} + e^{\mu_{L}\beta p_{2}}\bigr) $$

(24)

The probabilities of choosing respectively nest0 and nest1 among the set of nests are:

(25)

where we have normalized the upper level scale parameter μ to unity. Thus the probabilities of choosing respectively alternatives 0, 1, 2 among all possible elemental alternatives in the choice set are:

$$ \begin{array}{rcl} P(i = 0|C) &=& P(i = 0|\mathrm{nest}0) \cdot P(\mathrm{nest}0|C) \\ \noalign {\vspace {5pt}} &=& \frac{e^{\beta p_{0}}}{e^{\beta p_{0}} + (e^{\mu _{L}\beta p_{1}} + e^{\mu _{L}\beta p_{2}})^{\frac{1}{\mu _{L}}}} \\ \noalign {\vspace {5pt}} P(i = 1|C) &=& P(i = 1|\mathrm{nest1}) \cdot P(\mathrm{nest1}|C) \\ \noalign {\vspace {5pt}} &=& \frac{e^{\mu _{L}\beta p_{1}}}{e^{\mu_{L}\beta p_{1}} + e^{\mu _{L}\beta p_{2}}} \cdot\frac{(e^{\mu_{L}\beta p_{1}} + e^{\mu _{L}\beta p_{2}})^{\frac{1}{\mu _{L}}}}{e^{\beta p_{0}} + (e^{\mu _{L}\beta p_{1}} + e^{\mu_{L}\beta p_{2}})^{\frac{1}{\mu _{L}}}} \\ \noalign {\vspace {5pt}} P(i = 2|C) &=& P(i = 2|\mathrm{nest1}) \cdot P(\mathrm{nest1}|C) \\ \noalign {\vspace {5pt}} &=& \frac{e^{\mu _{L}\beta p_{2}}}{e^{\mu _{L}\beta p_{1}} + e^{\mu _{L}\beta p_{2}}} \cdot \frac{(e^{\mu _{L}\beta p_{1}} + e^{\mu _{L}\beta p_{2}})^{\frac{1}{\mu _{L}}}}{e^{\beta p_{0}} + (e^{\mu _{L}\beta p_{1}} + e^{\mu _{L}\beta p_{2}})^{\frac{1}{\mu _{L}}}} \end{array} $$

(26)

1.2 A.2 Evaluation of eigenvalues of the Jacobian matrix

Let parameters β and μ _L be real and finite with mode shares p ₀,p ₁ defined on [0,1]. Recall the converted system of equations from Sect. 3.1 which defines the steady-state equilibrium of the sociodynamic process for the trinary nested logit with social interactions:

$$ \everymath{\displaystyle }\begin{array}{rcl} g_{0} &\equiv & e^{\beta p_{0}} - p_{0}e^{\beta p_{0}} - p_{0}\bigl(e^{\mu_{L}\beta p_{1}} + e^{\mu_{L}\beta (1 - p_{0} -p_{1})}\bigr)^{\frac{1}{\mu_{L}}} =0 \\ \noalign {\vspace {5pt}} g_{1} &\equiv & e^{\mu_{L}\beta p_{1}}\bigl(e^{\mu_{L}\beta p_{1}} + e^{\mu_{L}\beta (1 - p_{0} -p_{1})}\bigr)^{\frac{1 - \mu_{L}}{\mu_{L}}} - p_{1}e^{\beta p_{0}} \\ \noalign {\vspace {5pt}} &&{} - p_{1}\bigl(e^{\mu_{L}\beta p_{1}} + e^{\mu_{L}\beta (1 - p_{0} -p_{1})}\bigr)^{\frac{1}{\mu_{L}}} =0 \end{array} $$

(27)

This planar system of equations can be solved numerically for given empirical values of β and μ _L . Depending on these values of β and μ _L , the system may have more than one solution.

By evaluating the Jacobian J at a given solution p=(p ₀,p ₁), we can determine the stability of that solution.

$$ \mathbf{J} \equiv D\mathbf{g}(\mathbf{p}) = \left[ \begin{array}{c@{\quad }c} \frac{\partial g_{0}}{\partial p_{0}}( \mathbf{p} ) & \frac{\partial g_{0}}{\partial p_{1}}( \mathbf{p} ) \\ \noalign {\vspace {3pt}} \frac{\partial g_{1}}{\partial p_{0}}( \mathbf{p} ) & \frac{\partial g_{1}}{\partial p_{1}}( \mathbf{p}) \end{array} \right] $$

(28)

Namely, if all the eigenvalues of the Jacobian matrix J have negative real parts, then the equilibrium point is asymptotically stable; if at least one of the eigenvalues of the Jacobian matrix has positive real part, then the equilibrium point is unstable. For the system given by (27), the four terms in the Jacobian matrix of g=(g ₀,g ₁) can be computed using the sum rule, the product rule and the chain rule as follows:

(29)

(30)

(31)

(32)

The characteristic equation to determine the eigenvalues of the Jacobian matrix is:

(33)

Solving (33) for eigenvalues using the quadratic formula we have:

$$ \lambda_{1,2} = \frac{1}{2}\operatorname {trace}\mathbf{J} \pm\frac{1}{2}\sqrt{(\operatorname {trace}\mathbf{J})^{2} - 4\det \mathbf{J}} $$

(34)

For example, for the case study in Sect. 4.4 we can determine the stability of the solutions as in Table 9.

Table 9 Stability analysis of the sociodynamic trinary nested logit model with estimated coefficients in Sect. 4.4 (β=2.76, μ _L =1.03)

1.3 A.3 Analytical benchmark: a counter example

In this section we provide a hypothetical counter example to the results in Table 3 and Table 4 in Sect. 4.4 to demonstrate that: (1) the estimated parameter for the nest scale parameter need necessarily be statistically weak; and (2) the initial modal split in the data need not necessarily be a saddle point equilibrium of the minimal sociodynamic nested logit model. For convenience of comparison, suppose we again have 2913 decision-making agents, but instead the number of respondents in the data sample that chose to commute to work by bicycle or moped/motorcycle is 100, the number of respondents that chose to commute by external and/or internal system public transit is 500, and the number of respondents that chose to commute by car as driver or passenger is 2313. In such case, the modal split is roughly: 3,4 % mode share bicycle/moped/motorcycle; 17,2 % mode share public transit; and 79,4 % mode share car driver/passenger. Such a commuter modal split would be roughly similar to that in 2011 in the greater metropolitan areas of two of the top five most bicycle-friendly cities in the United States, Seattle and Minneapolis.

As in Sect. 4.4, we again estimate a minimal nested logit model, here on the basis of the hypothetical data. The only observed explanatory variable in the model is again the network interaction variable. Unobserved heterogeneity across the transportation mode choice alternatives is again captured by nesting the alternatives that are assumed to be correlated. Since we again have only three elemental choices, there are only three possible nesting structures, namely public transit nested with bicycle, public transit nested with car, and bicycle nested with car. Estimation of the three successive nested logit models shows both the first and the third nesting structure to be indicated, namely public transit nested with bicycle, and bicycle nested with car. The unobserved heterogeneity might represent here for example individual preference for freedom from parking requirements, and freedom from schedules, respectively. The estimation results for both of these models are given Table 10; in both cases the nest scale parameter is highly significant as evidenced by the t-statistics against unity, 10.1 and 9.14 respectively. The nesting of public transit with car is not indicated in this example.

Table 10 Estimation results for sociodynamic nested logit models with hypothetical modal split^a

Using the approach outlined in Sect. 3.1 and the previous section of this Appendix, we determine the equilibrium solutions for the long-run behavior of the minimal nested logit model with sociodynamic feedback with global interactions and characterize their stability, for the particular estimated parameter values in Table 10. We find that there are seven equilibrium solutions for the hypothetical model with the transit-bicycle nest and five equilibrium solutions for the hypothetical model with the bicycle-car nest. See Tables 11 and 12, respectively. These estimated parameters show the feedback in the system without any other variables is again “dominant enough” to cause runaway flocking as evidenced by the multiple stable steady-state equilibria. However, here contrary to the case in Table 4, the initial modal split is indeed one of the stable equilibria (solution nr. 3 in both Tables 11 and 12).

Table 11 Benchmark equilibria for hypothetical sociodynamic model with transit-bicycle nest

Table 12 Benchmark equilibria for hypothetical sociodynamic model with bicycle-car nest