Interacting Agents

Ph.D Thesis, Doctoral Programme in European Economic Studies,_x000d_ _x000d_ 2004 (discussed)_x000d_ _x000d_ This work deals with the broad topic of modelling interaction, and explores the usefulness of agent-based simulations as complements or substitutes of more traditional analytical approaches._x000d_ _x000d_ The main argument of the thesis is that agent-based computational economics (ACE), described as “the computational study of economies modelled as evolving systems of autonomous interacting agents”, is a powerful tool for economic analysis, allowing for a more flexible model specification and the inclusion of more tailored assumptions._x000d_ _x000d_ As all new methodologies, the use of ACE must be justified, from a methodological point of view. The burden of the proof rests on ACE practitioners. However, instead of definitely establishing the technique the stream of methodological work has ironically confirmed many mathematical economists in their belief that agent-based modelling is inconclusive. In Chapter 1 (“The promises and perils of ACE”) I have rationalised the main theoretical critiques that can be moved to agent-based computational models. The general claim, by sceptical economists, that “simulations do not prove anything”, boils down to the belief that agent-based simulations are (i) difficult to interpret, (ii) difficult to estimate and (iii) difficult to generalise. I carefully examine these three statements, and highlight the existence of appropriate solutions. I hope that the arguments provided will help in clarifying the strengths and weaknesses of agent-based computational economics, and convince the more sceptical readers that the methodology is sound and potentially useful. _x000d_ _x000d_ Non-ACE practitioners sometimes feel that ACE has prompted more methodological discussion than applications. I therefore turn to the empirical relevance of ACE model as a tool for expanding knowledge about economic systems._x000d_ _x000d_ I build on Robert Axtell[1] (Axtell, 2000) identification of three distinct uses of agent-based computation in the social sciences, and rank them according to their auxiliary nature, with respect to analytical modelling._x000d_ _x000d_ The first use is numerical computation of analytical models. Note with Axtell that «[t]here are a variety of ways in which formal models resist full analysis. Indeed, it is seemingly only in very restrictive circumstances that one ever has a model that is completely soluble, in the sense that everything of importance about it can be obtained solely from analytical manipulations». Situations in which resort to numerical computation may prove useful include (a) when a model is not analytically soluble for some relevant variable, (b) when a model is stochastic, and the empirical distribution of some relevant variable needs to be compared with the theoretical one, of which often few moments are known, (c) when a model is solved for the equilibrium, but the out-of-equilibrium dynamics are not known. In particular, with reference to the last point, it may happen that multiple equilibria exist, that the equilibrium or (at least some of) the equilibria are unstable, that they are realized only in the very long run. Conversely, it may happen that equilibria exist but are not computable. Axtell (2000) provides references and examples for each case. Finally, it may be the case that the equilibrium is less important than the out-of-equilibrium fluctuations or extreme events._x000d_ _x000d_ Clearly, agent-based simulations are not the only way to perform numerical computations of a given analytical model. However, they may prove effective and simple to implement, especially for models with micro-foundations._x000d_ _x000d_ The second use is testing the robustness of analytical models with respect to departures from some of the assumptions. Assumptions may relate to the behaviour of the agents, or to the structure of the model. ACE models can easily include bounded rationality (Sargent, 1993; Leijonhufvud, 1993; Conlisk, 1996) and heterogeneity at an individual level, and investigate variations in the way agents interact with each other or with the institutional setting. One important feature of ACE is that in considering departures from the assumptions of the reference model, a number of different alternatives can be investigated, thus offering intuition towards a generalization of the model itself._x000d_ _x000d_ The first two uses of ACE models are complementary to mathematical analysis. The third use is a substitute, going beyond the existence of an analytical reference model. It provides stand-alone simulation models for (a) problems that are analytically intractable, or (b) problems for which an analytical solution bears no advantage. The latter may happen when negative results are involved, for instance. A simulation may be enough to show that some institution or norm is wrong, or does not work in the intended way. Analytical intractability may arise when more complicated assumptions are needed, or when the researcher wants to investigate the overall effect of a number of mechanisms (each possibly already analytically understood in simpler models), at work at the same time._x000d_ _x000d_ The three applications that form the core of this dissertation are examples of the three different purposes in writing an ACE model presented above[2]. The first application (chapter 2) makes use of agent-based simulation techniques in accordance with the first purpose described above, i.e. to investigate the dynamical behaviour of a system, beyond the few statistical indicators for which analytical results are available. Then, simulation is used for considering important extensions of the basic model, in accordance with the second purpose described above. The second application (chapter 3) develops an analytical search model that is used as a benchmark for analysing more interesting variations by means of an agent-based simulation, thus providing again an example of how ACE models and analytical ones can successfully complement each other. The third model (chapter 4) is a stand-alone simulation model, without analytical counterparts._x000d_ _x000d_ Chapter 2 (“Generalizing Gibrat: Reasonable multiplicative stochastic models of firm dynamics”) deals with the literature on stochastic models of firm dynamics, originated by the well-known work by Gibrat in the ‘30s. I show that most multiplicative stochastic models developed in the literature, while being characterised by nice analytical properties such as a Power Law distribution of firm size, produce very unreasonable dynamics, with the number and the size of the firms either collapsing to zero or increasing without limits. I show that this is due to an intrinsic feature of multiplicative models, and relate it with recent works on stochastic multiplicative models, mainly developed by physicists. Then, I develop a general simulation model in order to encompass and extend the results produced in the existing literature. My question is: “which modifications to the standard models should be made in order to produce more ‘reasonable’ dynamics?” In particular, I show that stochastic multiplicative models of firm size that abstract completely from considering firm interaction fail to do it. I then test a number of different entry and exit mechanisms, and characterize a class that is able to produce the desired results._x000d_ _x000d_ Chapter 3 (“A search model of unemployment and industrial dynamics”) tries to build a bridge between industrial dynamics models – such as those explored in chapter 2 – and search models of the labour market. Such a connection is still largely missing in the literature. Here, the simulation is complementary to an analytical approach that is itself different from those normally developed in the literature. The micro-foundations of the labour market are analysed by means of an equilibrium probabilistic model, in the framework of optimising behaviour. This allows the joint investigation of employment and firm dynamics. An agent-based simulation is then developed in order to test whether the model is able to reach the equilibrium, and to investigate its implications regarding firm demography. The simulation is finally used to extend the model in two directions: first, relaxing some assumptions regarding the stochastic nature of some structural parameters, and – second – allowing some departures from individual optimisation towards more realistic rule-of-thumb decision processes. The model can also be considered as an example of how well the bounded rationality paradigm fits with the simulation approach._x000d_ _x000d_ The model in chapter 4 (“The new Italian road code and the virtues of the ‘shame lane’”) studies the effects of institutional change on the performance of a system, and is an example of how non-linear systems can be successfully investigated by means of simulation models alone. It deals with a problem for which «writing down equations is not a useful activity. In such circumstances, resort to agent-based computational models may be the only way available to explore such processes systematically» (Axtell, 2000)._x000d_ _x000d_ During the spring of 2003 I had to spend a lot of time driving on Italian motorways. My mind continuously waved between attention to the road and the models I was developing for this thesis. Then, I started to realize that at an entire world of individual agents where interacting around me and with me. We were coordinating but at the same time competing for scarce resources, space and time. All of us were following very simple rules of behaviour, yet the emerging dynamics were very interesting, and bore a strong resemblance with many economic phenomena: «Why are vehicles sometimes stopped by so-called “phantom traffic jams”, although they all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction of the traffic volume cause a lasting traffic jam?» (Helbing, 2003a). Speed limits can speed up traffic under certain conditions. Traffic forecasts are often invalidated by the actual behaviour of individual drivers, but this is exactly what they are produced for._x000d_ _x000d_ As in July 2003 a new Road Code was approved by the Italian parliament, I got the intuition that some of the changes brought in could have an effect opposed to the one claimed by the legislators. In particular, the new law abolished the so-called ‘shame lane’, i.e. the reservation of the right lane on three-lane motorways for slow vehicles alone. As in two-lane roads, all vehicles must now drive on the right lane, as long as it is not occupied by other vehicles. I started to think that it was not so obvious that this rule should perform better than the old one. So, I decided to exploit one distinct feature of agent-based simulations, namely their ability to study how different rules of behaviour lead, through the repeated interaction of many, and possibly heterogeneous, self-motivated agents, to different aggregate outcomes, and build a simulation for comparing the new and the old rule. In the end, my intuition proved right._x000d_ _x000d_ References_x000d_ _x000d_ Axtell R. (2000), " Why Agents? On the Varied Motivations for Agent Computing in the Social Sciences”, in Proceedings of the Workshop on Agent Simulation: Applications, Models and Tools, Argonne National Laboratory, IL._x000d_ _x000d_ Conlisk J. (1996), “Why Bounded Rationality?'', Journal of Economic Literature, Vol. 34, No. 2_x000d_ _x000d_ Helbing D. (2003a), “Agent-Based Simulation of Traffic Jams, Crowds, and Supply Networks”, in Proceeding of the IMA ‘Hot Topics’ Workshop, IMA, Minneapolis, MN_x000d_ _x000d_ Leijonhufvud A. (1993), “Towards a not-too-rational macroeconomics”, Southern Economic Journal, Vol. 50, No. 1_x000d_ _x000d_ Sargent T.J. (1993), Bounded Rationality in Macroeconomics, Clarendon Press,Oxford_x000d_ _x000d_ [1] The three categories identified below correspond only partially to Axtell’s._x000d_ _x000d_ [2] All agent-based simulations developed in this thesis are written in Java code, and make use of JAS, a powerful software which provides a set of simulation libraries, developed by Michele Sonnessa at the University of Torino (http://jaslibrary.sourceforge.net/index.html).

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