A Study of Hamilton-Jacobi-Bellman Equation Systems Arising in Differential Game Models of Changing Society
This paper is concerned with a system of
Hamilton-Jacobi-Bellman equations coupled with an autonomous
dynamical system. The mathematical system arises in the differential
game formulation of political economy models as an infinite-horizon
continuous-time differential game with discounted instantaneous
payoff rates and continuously and discretely varying state variables.
The existence of a weak solution of the PDE system is proven and
a computational scheme of approximate solution is developed for a
class of such systems. A model of democratization is mathematically
analyzed as an illustration of application.
[1] D. Acemoglu and J. Robinson. Economic Origins of Dictatorship and
Democracy. Cambridge University Press. New York, 2006.
[2] D. Acemoglu, G. Egorov, and K. Sonin. Political model of social
evolution. Proceedings of the National Academy of Sciences of the
United States of American, 108 Suppl 4:21292–21296, 2011.
[3] D. Acemoglu, G. Egorov, and K. Sonin. Dynamics and stability of
constitutions, coalitions and clubs. Am. Econ. Rev., 102(4), 2012.
[4] Jinhui H. Bai and Roger Lagunoff. On the Faustian dynamics of policy
and political power. Review of Economic Studies, 78:17–48, 2011.
[5] S. Barber, M. Maschler, and J. Shalev. Voting for voters: A model of
the electoral evolution. Games Econ. Behav., 37:40–78, 2001.
[6] L. D. Berkovitz. Two person zero sum differential games: an overview.
In J. D. Grote, editor, The theory and application of differential games.
D. Reidel Publishing Company, 1974.
[7] Bruce Bueno de Mesquita. Game theory, political economy, and the
evolving study of war and peace. American Political Science Review,
4:638–642, 2006.
[8] R. J. Elliott. Introduction to differential games ii. stochastic games and
parabolic equations. In J. D. Grote, editor, The theory and application
of differential games. D. Reidel Publishing Company, 1974.
[9] A. Friedman. Differential Games. Wiley, 1971.
[10] Arieh Gavious and Shlomo Mizrahi. A signaling model of peaceful
political change. Soc. Choice Welfare, 20:119–136, 2003.
[11] A. Gomes and P. Jehiel. Dynamic processes of social and economic
interactions: On the persistence of inefficiencies. J. Polit. Econ., 113:
626–667, 2005.
[12] F. Huang. The coevolution of economic and political development
from monarchy to democracy. International Economic Review, 53(4):
1341–1368, 2012.
[13] F. Huang. Why did universities precede primary schools? a political
economy model of educational change. Economic Inquiry, 50:418–434,
2012.
[14] Paul E. Johnson. Formal theories of politics: The scope of mathematical
modelling in political science. Mathl Comput. Modelling, 12(4/5):
397–404, 1989.
[15] Roger Lagunoff. Dynamic stability and reform of political institutions.
Games and Economic Behavior, 67:569–583, 2009.
[16] Leonardo Martinez. A theory of political cycles. Journal of Economic
Theory, 144:1166–1186, 2009.
[17] Nolan McCarty and Adam Meirowitz. Political Game Theory, An
Introduction. Cambridge University Press, 2007.
[18] Akira Okada, Kenichi Sakakibara, and Koichi Suga. The dynamic
transformation of political systems through social contract: a game
theoretic approach. Soc. Choice Welf., 14:1–21, 1997.
[19] Lawrence Perko. Differential Equations and Dynamical Systems. 3
edition.
[20] A. W. Starr and Y. C. Ho. Further properties of nonzero-sum differential
games. J. Optimization Theory and Applications, 3:207–219, 1969.
[21] K. Uchida. On existence of a nash equilibrium point in n-person nonzero
sum stochastic differential games. SIAM J. Control Optim., 16:142–149,
1978.
[22] P. P. Varaiya. N-player stochastic differential games. SIAM J. Control
Optim., 4:538–545, 1976.
[23] Eelco Zandberg, Jakob de Haan, and J. Paul Elhorst. The political
economy of financial reform: How robust are Huang’s findings. J. Appl.
Econ., 27:695–699, 2012.
[1] D. Acemoglu and J. Robinson. Economic Origins of Dictatorship and
Democracy. Cambridge University Press. New York, 2006.
[2] D. Acemoglu, G. Egorov, and K. Sonin. Political model of social
evolution. Proceedings of the National Academy of Sciences of the
United States of American, 108 Suppl 4:21292–21296, 2011.
[3] D. Acemoglu, G. Egorov, and K. Sonin. Dynamics and stability of
constitutions, coalitions and clubs. Am. Econ. Rev., 102(4), 2012.
[4] Jinhui H. Bai and Roger Lagunoff. On the Faustian dynamics of policy
and political power. Review of Economic Studies, 78:17–48, 2011.
[5] S. Barber, M. Maschler, and J. Shalev. Voting for voters: A model of
the electoral evolution. Games Econ. Behav., 37:40–78, 2001.
[6] L. D. Berkovitz. Two person zero sum differential games: an overview.
In J. D. Grote, editor, The theory and application of differential games.
D. Reidel Publishing Company, 1974.
[7] Bruce Bueno de Mesquita. Game theory, political economy, and the
evolving study of war and peace. American Political Science Review,
4:638–642, 2006.
[8] R. J. Elliott. Introduction to differential games ii. stochastic games and
parabolic equations. In J. D. Grote, editor, The theory and application
of differential games. D. Reidel Publishing Company, 1974.
[9] A. Friedman. Differential Games. Wiley, 1971.
[10] Arieh Gavious and Shlomo Mizrahi. A signaling model of peaceful
political change. Soc. Choice Welfare, 20:119–136, 2003.
[11] A. Gomes and P. Jehiel. Dynamic processes of social and economic
interactions: On the persistence of inefficiencies. J. Polit. Econ., 113:
626–667, 2005.
[12] F. Huang. The coevolution of economic and political development
from monarchy to democracy. International Economic Review, 53(4):
1341–1368, 2012.
[13] F. Huang. Why did universities precede primary schools? a political
economy model of educational change. Economic Inquiry, 50:418–434,
2012.
[14] Paul E. Johnson. Formal theories of politics: The scope of mathematical
modelling in political science. Mathl Comput. Modelling, 12(4/5):
397–404, 1989.
[15] Roger Lagunoff. Dynamic stability and reform of political institutions.
Games and Economic Behavior, 67:569–583, 2009.
[16] Leonardo Martinez. A theory of political cycles. Journal of Economic
Theory, 144:1166–1186, 2009.
[17] Nolan McCarty and Adam Meirowitz. Political Game Theory, An
Introduction. Cambridge University Press, 2007.
[18] Akira Okada, Kenichi Sakakibara, and Koichi Suga. The dynamic
transformation of political systems through social contract: a game
theoretic approach. Soc. Choice Welf., 14:1–21, 1997.
[19] Lawrence Perko. Differential Equations and Dynamical Systems. 3
edition.
[20] A. W. Starr and Y. C. Ho. Further properties of nonzero-sum differential
games. J. Optimization Theory and Applications, 3:207–219, 1969.
[21] K. Uchida. On existence of a nash equilibrium point in n-person nonzero
sum stochastic differential games. SIAM J. Control Optim., 16:142–149,
1978.
[22] P. P. Varaiya. N-player stochastic differential games. SIAM J. Control
Optim., 4:538–545, 1976.
[23] Eelco Zandberg, Jakob de Haan, and J. Paul Elhorst. The political
economy of financial reform: How robust are Huang’s findings. J. Appl.
Econ., 27:695–699, 2012.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:76566", author = "Weihua Ruan and Kuan-Chou Chen", title = "A Study of Hamilton-Jacobi-Bellman Equation Systems Arising in Differential Game Models of Changing Society", abstract = "This paper is concerned with a system of
Hamilton-Jacobi-Bellman equations coupled with an autonomous
dynamical system. The mathematical system arises in the differential
game formulation of political economy models as an infinite-horizon
continuous-time differential game with discounted instantaneous
payoff rates and continuously and discretely varying state variables.
The existence of a weak solution of the PDE system is proven and
a computational scheme of approximate solution is developed for a
class of such systems. A model of democratization is mathematically
analyzed as an illustration of application.", keywords = "Differential games, Hamilton-Jacobi-Bellman
equations, infinite horizon, political-economy models.", volume = "11", number = "7", pages = "320-14", }
