Discrete-Space Social Interaction Models: Stability and Continuous Limit

Akamatsu, Takashi and Fujishima, Shota and Takayama, Yuki (2014): Discrete-Space Social Interaction Models: Stability and Continuous Limit.

Abstract

We study the equilibrium properties, including stability, of discrete-space social interaction models with a single type of agents, and their continuous limit. We show that, even though the equilibrium in discrete space can be non-unique for all finite degree of discretization, any sequence of discrete-space models' equilibria converges to the continuous-space model's unique equilibrium as the discretization of space is refined. Showing the existence of multiple equilibria resorts to the stability analysis of equilibria. A general framework for studying equilibria and their stability is presented by characterizing the discrete-space social interaction model as a potential game.

• Item Type: MPRA Paper
• Language: English
• Keywords: Social interaction; Agglomeration; Discrete space; Potential game; Stability; Evolutionary game theory
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C62 - Existence and Stability Conditions of Equilibrium; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C73 - Stochastic and Dynamic Games ; Evolutionary Games ; Repeated Games; D - Microeconomics > D6 - Welfare Economics > D62 - Externalities; R - Urban, Rural, Regional, Real Estate, and Transportation Economics > R1 - General Regional Economics > R12 - Size and Spatial Distributions of Regional Economic Activity

References

Anas, A. and R. Xu. (1999), "Congestion, land use, and job dispersion: A general equilibrium model," Journal of Urban Economics, 45, 451-473.

Bapat, R. B. and T. E. S. Raghavan. (1997), Nonnegative Matrices and Applications, Cambridge University Press.

Beckmann, M. (1976), "Spatial equilibrium in the dispersed city," in: G. J. Papageorgiou. (eds.), Mathematical Land Use Theory, Lexington, Mass. : Lexington Books, 117-125.

Blanchet, A., P. Mossay., and F. Santambrogio. (2013), "Existence and uniqueness of equilibrium for a spatial model of social interactions," RIETI Discussion Paper Series 13-E-055.

Borukhov, E. and O. Hochman. (1977), "Optimum and market equilibrium in a model of a city without a predetermined center," Environment and Planning A, 9, 849-856.

Braid, R. M. (1988), "Heterogeneous preferences and non-central agglomeration of firms," Regional Science and Urban Economics, 18, 57-68.

Brown, G. W. and J. von Neumann. (1950), "Solutions of games by differential equations," in: H. W. Kuhn and A. W. Tucker. (eds.), Contributions to the Theory of Games I, Princeton University Press.

Caruso, C., D. Peeters., J. Cavailhes., and M. Rounsevell. (2009), "Space-time patterns of urban sprawl, a 1D cellular automata and microeconomic approach," Environment and Planning B, 36, 968-988.

Caruso, C., D. Peeters., J. Cavailhes., and M. Rounsevell. (2007), "Spatial configurations in a periurban city. A cellular automata-based microeconomic model," Regional Science and Urban Economics, 37, 542-567.

Dupuis, P. and A. Nagurney. (1993), "Dynamical systems and variational inequalities," Annals of Operations Research, 44, 9-42.

Fujishima, S. (2013), "Evolutionary implementation of optimal city size distributions," Regional Science and Urban Economics, 43, 404-410.

Fujita, M., P. Krugman., and A. J. Venables. (1999), The Spatial Economy: Cities, Regions, and International Trade, MIT Press.

Fujita, M. and J.-F. Thisse. (2013), Economics of Agglomeration, Cambridge University Press.

Gilboa, I. and A. Matsui. (1991), "Social stability and equilibrium," Econometrica, 59, 859-867.

Griva, I., S. G. Nash., and A. Sofer. (2009), Linear and Nonlinear Optimization, Society for Industrial and Applied Mathematics.

Hofbauer, J. and W. H. Sandholm. (2002), "On the global convergence of stochastic fictitious play," Econometrica, 70, 2265-2294.

Hofbauer, J. and K. Sigmund. (1988), The Theory of Evolution and Dynamical Systems, Cambridge University Press.

Horn, R. A., and C. R. Johnson. (2013), Matrix Analysis, Cambridge University Press.

LeVeque, R. J. (2007), Finite Difference Methods for Ordinary and Partial Differential Equations, The Society for Industrial and Applied Mathematics.

Matsuyama, K. (2013), "Endogenous ranking and equilibrium Lorenz curve across (ex ante) identical countries," Econometrica, 81, 2009-2031.

Monderer, D. and L. S. Shapley. (1996), "Potential games," Games and Economic Behavior, 14, 124-143.

Mossay, P. and P. M. Picard. (2011), "On spatial equilibria in a social interaction model," Journal of Economic Theory, 146, 2455-2477.

O'Hara, D. J. (1977), "Location of firms within a square central business district," Journal of Political Economy, 85, 1189-1207.

Oyama, D. (2009a), "History versus expectations in economic geography reconsidered," Journal of Economic Dynamics and Control, 33, 394-408.

Oyama, D. (2009b), "Agglomeration under forward-looking expectations: potentials and global stability," Regional Science and Urban Economics, 39, 696-713.

Sandholm, W. H. (2007), "Pigouvian pricing and stochastic evolutionary implementation," Journal of Economic Theory, 132, 367-382.

Sandholm, W. H. (2005), "Excess payoff dynamics and other well-behaved evolutionary dynamics," Journal of Economic Theory, 124, 149-170.

Sandholm, W. H. (2001), "Potential games with continuous player sets," Journal of Economic Theory, 97, 81-108.

Simsek, A., A. Ozdaglar., and D. Acemoglu. (2007), "Generalized Poincare-Hopf theorem for compact nonsmooth regions,” Mathematics of Operations Research, 32, 193-214.

Swinkels, J. M. (1993), "Adjustment dynamics and rational play in games," Games and Economic Behavior, 5, 455-484.

Tabuchi, T. (1986), "Urban agglomeration economies in a linear city," Regional Science and Urban Economics, 16, 421-436.

Tabuchi, T. (1982), "Optimal distribution of city sizes in a region," Environment and Planning A, 14, 21-32.

Taylor, P. D. and L. Jonker. (1978), "Evolutionary stable strategies and game dynamics," Mathematical Biosciences, 40, 145-156.

Turner, M. A. (2005), "Landscape preferences and patterns of residential development," Journal of Urban Economics, 57, 19-54.

Yueh, W.-C. and S. S. Cheng. (2008), "Explicit eigenvalues and inverses of tridiagonal Toeplitz matrices with four perturbed corners," ANZIAM Journal, 49, 361-387.