« A Path Integral Approach to Interacting Economic systems with Multiple Heterogenous Agents », working paper (June 2017)

This paper presents an analytical treatment of a broad class of economic systems with an arbitrary number of agents, while keeping tracks of the system’s interactions and complexity at the individual level. In this respect, our approach is similar to the Agent-Based one, in that it does not seek to aggregate all agents, and considers the interaction system in itself. However, we depart from the Agent Based Model in that we do not aggregate the agents in several different types and aim at considering the system as a whole set of large number of interacting agents. This point of view is close to the Econophysics approach in which the agents are often considered as a statistical system. Nethertheless, our objective is to transcribe, at the level of these statistical systems, the usual characteristics of a system of optimizing agents. How to to include the forward looking behaviors, the constraints of an economic system, the heterogeneity of agents or of information, the strategic dominations, at least approimatively at the statistical level are the goals of this work.
In that, our approach is in between statistical models and economics ones. We keep from the statistical models the idea of dealing with a large number of degrees of freedom of a system without aggregating the quantities, but we keep from standard economic models the relevant concepts developed in the past decades to describe the behaviors of rational, or partly rational agents. A natural question arising in that context is the relevance of these concepts at the scale of the statistical system, i.e. the macro level. It is actually known that some microscopic feature may fade away at large scales, whereas some others may reinforce and become predominant at the macroscopic scale. This phenomenon, the relevance, or the irrelevance of some micro interactions could, indirectly, yield some hints, about the aggregation problem in economics, at least at the phenomenological level.
Our work is an attempt and a first step toward an answer, but already shows that translating standard economic models to statistical ones requires to introduce some statistical field models that differ from the usual ones used in dealing with physical systems. The models we introduce keep track of the individual behaviors and these behaviors influence the description in terms of fields and the results at the macro scale.
The advantage of a statistical field theories are threefolds. First, they allow, at least approximatively, to deal with a system with large degrees of freedom in an analytical way, without reducing it, at first, to an aggregate. Second, they make possible to transcribe micro relations into macro ones. Last but not least they display features that would not appear obviously in an aggregated context. Actually, they allow to switch from macro description to micro ones, and vice-versa, taht is to interpret one scale at the light of the other. Moreover, and this point is relevant for economic systems, these model present the possibility of phase transition. Depending on the parameters of the model, the system may experience some switches in behaviors, at the individual and collective scale. In that, they allow to approach the question of multiple equilibria.
The statistical approach to an economic system presented here is done in two steps. The first one modifies the usual model of optimizing agent and replaces it with a probabilistic point of view. We consider an interacting system, involving an arbitrary number of agents, in which each agent is still represented by an intertemporal utility function, or any quantity to optimize depending on an arbitrary number of variables, But we assume that each agent’s utility function is subject to unpredictable shocks. In such a setting, individual optimization problems need not be resolved. Each agent is described by a time-dependent probability distribution centered around its utility optimum. Unpredictable shocks will deviate each agent from its optimal action, depending on the shocks’ variances. When these variances are null, we recover the standard optimization result. It furthermore takes into account the statistic nature of a system of several agents by including uncertainty on the agents’ behavior. It nonetheless preserves the analytical treatment by slightly modifying the agents’ standard optimization problem.
Remark that this form of modelling is close to the usual optimization of an agent when some unpredictable schocks are introduced, and this is the reason why one can recover the standard optimization equations in some cases in the limit of no uncertainty. However, the uncertainty we introduce at the basis of our description is different from the one usually considered in economics model. Actually, the uncertainty we deal with is an internal uncertainty, about the agent behavior, about it’s goal, or some unobservable shocks. As such it is present at the beginning of the model, and not considered as a random and external perturbation.
The system composed by the set of all agents is consequently defined by a composite probability depending on time, agents’ interactions, relations of strategic dominations, agents’ information sets and expectations. This setting allows for heterogeneous agents with different utility functions, strategic domination relations, heterogeneity of information, etc.
This dynamic system is described by a stochastic process whose characteristics (mean, variance, etc.) determine the transition probabilities of the system and its mean values. For example, the process mean value at time t describes the mean state of each agent at time t. Besides, we can define transition probabilities that describe the evolution of the system from t to t+1.
This setup is actually a path integral formalism in an abstract space — the space of the agents’ actions — and is very similar to the statistical physics or quantum mechanics techniques. We show that this description, applied to the space composed of all the agents’ actions, reduces to the usual optimization results in simple cases, inasmuch as the unpredictable shocks’ variances are null. This description is a good approximation of standard descriptions and allows to solve otherwise intractable problems. Compared to the standard optimization, such a description markedly eases the treatment of a system with a small number of agents. As a consequence, this approach is closed and useful in itself and provides an alternative to the standard modelling in the case of a small number of interacting agents. It allows to recover an average dynamics, which is closed, or in some cases even identical to, the standard approach, and study the dynamics of the set of agents, as well as it’s fluctuations if we introduce some external shocks. Our main examples will be the models developpend in L, GL, GLW, describing systems of interacting agents, or structures in interactions,where some of them have information and strategic advantage. We show through this examples the possibilities of our approach in term of resolution.
However, this model becomes useless for a large number of agents, and has to be modified in an other formalism based on statistical fields, more efficient in that case. Nethertheless, this first step was necessary since the statistical fieds model is grounded on our preliminary probabilistic description. Actually, this one, by it’s form in terms of path integrals for a small number of interacting agents, can be transformed in a straightforward way in a description for large systems. As a consequence, the first step is also a preparatory one, needed for our initial goal, a model of large number of interacting agents.
The second step to reach this goal, therefore, consists in replacing the path integrals description by a model of field theory that replicates the properties of the system when n, the number of agents, is large. Actually, in that case, we can show that the previous description is equivalent to a more compact description in terms of field theory. It allows an analytical, although approximate, treatment of the system.
Hence, a double transformation, with respect to the usual optimization models has been performed. The usual optimization system is first described by a statistical system of n agents. It can then itself be replaced by a specific field theory with a large number of degrees of freedom. This field theory does not represent an aggregation of microeconomic systems in the usual sense, but rather describes an environment of an infinity of agents, from which various phases or equilibria may be retrieved, as well as the behavior of the agent(s), and the way they are influenced by, or interact with, their environment.
This double transformation allows first, for a small number of agents, to solve a system without recurring to aggregation, and second, for a large number of agents, to aggregate them so as to shape an environment whose characteristics will in turn induce and impact agents’ interactions. This environment, or “medium”, allows to reconstruct aggregate quantities and study their dynamics, without reducing the system to mere relations between aggregates. Indeed, the fundamental environment from which these quantities are drawn can witness fluctuations that may invalidate relations previously established. The environment is not macroscopic in itself, but rather describes a multitude of agents in interaction. It does not necessarily have a unique or stable equilibrium. Relations between macroeconomic quantities ultimately depend on the state or “phase” of this environment (“medium”), and can vary with the state of the environment. This phenomenon is the so-called “phase transition” in field theory: The configuration of the ground state represents an equilibrium for the whole set of agents, and shapes the characteristics of interactions and individual dynamics. Various forms for this ground state, depending on the parameters of the system, may change drastically the description at individual level.
For illustrative purposes, this paper presents several economic models of consumer/producer agents facing binding constraints in competitive markets, generalized to a large number of agents and presenting various phases or equilibria.

« A Path Integral Approach to Interacting Economic systems with Multiple Heterogenous Agents (June 2017),