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

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## 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.

## Keywords

Discrete choice Multi-agent based social simulation Social networks Spatial interaction Transportation demand## Notes

### Acknowledgements

The authors would like to gratefully acknowledge discussion with Harry Timmermans, Theo Arentze, Cars Hommes, Frank le Clercq, Loek Kapoen, George Kampis, József Váncza and András Márkus, as well as the valuable and insightful comments from three anonymous reviewers which greatly improved the exposition in this paper. Very special thanks are also due to Guus Brohm and Nelly Kalfs at the Agency for Infrastructure, Traffic and Transport of the Municipality of Amsterdam, to Willem Vermin and the High Performance Computing support team at SARA Computing and Networking Services, Amsterdam and to David Sallach, Michael North, Charles Macal and the RePast development team. In addition we would like to express our appreciation more generally to a number of other committed scholars and out-of-the box thinkers that influenced our own thinking during formative years: Nigel Gilbert, keynote speaker at the first Lake Arrowhead Conference where the authors first met, and one of the team of visionary signatories of the European Social Simulation Association (ESSA); Axel Leijonhufvud, Robert Axtell and Masanao Aoki for eye-opening introduction to the world of adaptive economic processes; Kathleen Carley, a beacon for inspiration on the realm of possibilities of network analysis coupled with population scale social simulation; and Scott Page and John Miller, organizers of the Santa Fe Institute Graduate Workshop on Computational Economics, for pointing us to William Brock and Steven Durlauf’s work on multinomial choice with social interactions during an intensive two weeks of learning. Finally we would like to thank Clara Smith, Fred Amblard, Paul Chapron, Matthias Maillard and Samuel Thiriot for their wonderful initiative to bring together researchers in social network analysis and multi-agent systems at the lively SNAMAS workshop at ESSA 2011. The authors claim full responsibility for any errors.

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