New Tools of Economic Dynamics pp 3-24 | Cite as

# Modeling a Large Number of Agents by Types: Models as Large Random Decomposable Structures

## Summary

This paper introduces methods, based on decomposable random combinatorial analysis, to model a large number of interacting agents. This paper also discusses a largely ignored possibility in the mainstream economic literature that hitherto unknown types of agents may enter the models at some future time. We apply the notion of holding times, and introduce the results of the one- and two-parameter inductive methods of Ewens, Pitman and Zabell to economic literature. More specifically, we use the notion of exchangeable random partitions of a finite set to produce a simple rule of sucession, that is, the expressions for the probabilties for entries by new or known types, conditional on the observed data. Then Ewens equilibrium distriution for the sizes of clusters is introduced, and its use to examine market behavior is sketched, especially when a few types of agents are dominant. We suggest that the approaches of this paper and the notion of holding times are relevant to agent-based simulations because holding times can be used to randomly select agents that “act” first.

## Preview

Unable to display preview. Download preview PDF.

### References

- 1.Antoniak, C, (1969). “Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems”, Ph.D. dissertation, Univ. California, Los Angeles. Arratia, R., and A.D. Barbour and S. Tavar e, “ Poisson process approximations for the Ewens sampling formula”, Ann. Appl. Probab. 2 519–35, 1992.Google Scholar
- 2.Arratia, R., and S. Tavaré, “Independent process approximations for random combinatorial structures”, Adv. Math. 104, 90–154, 1994.CrossRefMathSciNetGoogle Scholar
- 3.Aoki, M.,(1996) New Approaches to Macroeconomic Modeling: Evolutionary Stochastic Dynamics, Multiple Equilibriua, and Externalities as Field effects, Cambridge University Press, New York.Google Scholar
- 4.—. (1998). “A simple model of asymmetrical business cycles: Interactive dynamics of a large number of agents with discrete choices”, Macroeconomic Dynamics, 2, 427–442Google Scholar
- 5.—, “Open Models of Share Markets with Several Types of Participants”, presented at 1999 Wheia Conference, Univ. Genova, Genova, Italy, June, 1999.Google Scholar
- 6.—. (2000a). “Cluster size distributions of economic agents of many types in a market,” J. Math. Analy. Appl. 249, 32–52.Google Scholar
- 7.—. (2000b). “Herd behavior and return dynamics in a share market with many types of agents,” in M. Tokuyama, and H. Stanley (eds). Statistical Physics, Amer. Inst. Phys., Melville, New York.Google Scholar
- 8.—. (2002a) Aggregate Behavior and Fluctuations in Economics. Cambridge Univ. Press, New York.Google Scholar
- 9.—. (2002b). “Open models of share markets with two dominant types of participants,” J. Econ. Behav. Org., 49 199–216.Google Scholar
- 10.—, (2002c), “A simple model of volatility fluctuations in asset markets”, pp. 180–85, in H. Takayasu (ed.) Empirical science of financial fluctuations: The advent of econophysics, Springer, Tokyo, and New York.Google Scholar
- 11.—, and M. Shirai. (2000). “Stochastic business cycles and equilibrium selection in search equilibrium.” Macroeconomic Dynamics 4, 487–505.MATHGoogle Scholar
- 12.—, and H. Yoshikawa. (2001). “A simple quantity adjustment model of economic fluctuatuions and growth,” presented at the 2001 Wehia annual meeting, Maastrict, June, 2001.Google Scholar
- 13.—, and H. Yoshikawa (2002). “Demand saturation-creation and economic growth”, J. Econ. Behav. Org. 48Google Scholar
- 14.Carlton, M. A., (1999). “Applications of the two-parameter Poisson-Dirichlet distribution”, Ph.D. dissertation, Univ. California. Los Angeles.Google Scholar
- 15.Costantini, D, and U. Garibaldi (1979) “A probabilistic foundation of elementary particle statistics” Stud. Hist. 28, 483–506MathSciNetGoogle Scholar
- 16.Costantini, D. (2000) “A probability theory for macroeconomic modelling”, Mimeo, Dep. Stat. University of Bologna.Google Scholar
- 17.Dixit, A., (1989). “Entry and exit decisions of firms under fluctuating real exchange rates,” J. Pol. Econ. 97, 620–637.Google Scholar
- 18.Ewens, W. J., (1972). “The sampling theory of selectively neutral alleles”, Theor. Pop. Biol. 3 87–112.MATHMathSciNetGoogle Scholar
- 19.—, (1990) “Population genetic theory-The past and the future’” in Mathematical and Statistical Developments of Evolutionary Theory, edited by S. Lessard, Kluwer Academic Publishers, LondonGoogle Scholar
- 20.—, (1996) “Remarks on the law of sucession”, in Athens Conf. on Applied Probab. on Time Series AnalysisGoogle Scholar
- 21.Feller, W. 1968 Introduction to Probability Theory, Vol.I, Wiley, New York.Google Scholar
- 22.Feng, S., and F. M. Hoppe (1998). “Large deviation principles for some random combinatorial structures in population genetics”, Ann. Appl. Probab. 8, 975–94.MathSciNetGoogle Scholar
- 23.Hoppe, F., (1984) “Polya-like urns and the Ewens sampling formula” J. Math. Biol. 20 91–94CrossRefMATHMathSciNetGoogle Scholar
- 24.— (1987) “The sampling theory of neutral alleles and an urn model in population genetics,” J. Math. Biol. 25, 123–59MathSciNetGoogle Scholar
- 25.Kelly, F., (1979) Reversibility and Stochastic Networks, J. Wiley, New York.Google Scholar
- 26.Kingman, J.F.C., (1992) Poisson Processes, Oxford Clarendon Press.Google Scholar
- 27.— (1980), Mathematics of Genetic Diversity, SIAM, PhiladelphiaGoogle Scholar
- 28.Pitman, J., (1992) “The two-parameter generalization of Ewens’ random partition structure”, Tech. Report 345, Department of Statistics, UC Berkeley, reissued with addendum in (2002).Google Scholar
- 29.—, (2002). “Combinatorial Stochastic Processes,” Lecture notes, St. Flour lecture course.Google Scholar
- 30.— (1995) “Partially exchangeable random partitions” Probab. Theory and Related Fields 102 145–58. van Lint, J. H., and R.M. Wilson, (1992) A Course in Combinatorics, Cambridge Univ. Press, New YorkGoogle Scholar
- 31.Sachkov, V. N., Probabilistic Methods in Combinatorial Anaysis, Cambridge Univ. Press, Cambridge, 1996.Google Scholar
- 32.Watterson, G.A., (1976) “The stationary distribution of the infinitely-many neutral alleles diffsusion models”, Journal of Applied Prob. 13, 639–651MATHMathSciNetGoogle Scholar
- 33.—, and Guess, H. A., (1977) “Is the most frequent allele the oldest?’ Theor. Popul. Biol. II, 141–60Google Scholar
- 34.Van Lint, J., and R. M. Wilson (1992) A Course in Combinatorics Cambridge Univ. Press, New York.Google Scholar
- 35.Whittle, P (1986) “Systems in Stochastic Equilibrium” Wiley, New YorkGoogle Scholar
- 36.Zabell, S., “Predicting the Unpredictable”, Synthese 90 205–32, 1992.CrossRefMATHMathSciNetGoogle Scholar
- 37.— (1982). “W. E. Johnson’s sufficientness postulate,” Ann. Stat. 10, 1090–1099.MathSciNetGoogle Scholar
- 38.— (1997) “The continuum of inductive methods revisited” in Earman, J., and J.D. Norton (eds) Cosmos of Science, Univ. Pittsburg Press.Google Scholar