Multistage Games

  • Jacek B. KrawczykEmail author
  • Vladimir Petkov
Reference work entry


In this chapter, we build on the concept of a repeated game and introduce the notion of a multistage game . In both types of games, several antagonistic agents interact with each other over time. The difference is that, in a multistage game, there is a dynamic system whose state keeps changing: the controls chosen by the agents in the current period affect the system’s future. In contrast with repeated games, the agents’ payoffs in multistage games depend directly on the state of this system. Examples of such settings range from a microeconomic dynamic model of a fish biomass exploited by several agents to a macroeconomic interaction between the government and the business sector. In some multistage games, physically different decision-makers engage in simultaneous-move competition. In others, agents execute their actions sequentially rather than simultaneously. We also study hierarchical games, where a leader moves ahead of a follower. The chapter concludes with an example of memory-based strategies that can support Pareto-efficient outcomes.


Collusive Equilibrium Discrete-Time Games Feedback (Markovian) Equilibrium Information Patterns Open-Loop Nash Equilibrium Sequential Games Stackelberg Solutions 


  1. Barro RJ (1999) Ramsey meets Laibson in the neoclassical growth model. Q J Econ 114: 1125–1152Google Scholar
  2. Başar T (1989) Time consistency and robustness of equilibria in noncooperative dynamic games. In: van der Ploeg F, de Zeeuw AJ (eds) Dynamic policy games in economics. North Holland, AmsterdamGoogle Scholar
  3. Başar T, Olsder GJ (1982) Dynamic noncooperative game theory. Academic Press, New YorkGoogle Scholar
  4. Başar T, Olsder GJ (1999) Dynamic noncooperative game theory. SIAM series in Classics in Applied Mathematics. SIAM, PhiladelphiaGoogle Scholar
  5. Bellman R (1957) Dynamic programming. Princeton University Press, PrincetonGoogle Scholar
  6. Clark CW (1976) Mathematical bioeconomics. Wiley-Interscience, New YorkGoogle Scholar
  7. DeFreitas G, Marshall A (1998) Labour surplus, worker rights and productivity growth: a comparative analysis of Asia and Latin America. Labour 12(3):515–539Google Scholar
  8. Diamond P, Köszegi B (2003) Quasi-hyperbolic discounting and retirement. J Public Econ 87(9):1839–1872Google Scholar
  9. Fahrig L (2002) Effect of habitat fragmentation on the extinction threshold: a synthesis*. Ecol Appl 12(2):346–353Google Scholar
  10. Fan LT, Wang CS (1964) The discrete maximum principle. John Wiley and Sons, New YorkGoogle Scholar
  11. Fudenberg D, Tirole J (1991) Game theory. The MIT Press, Cambridge/LondonGoogle Scholar
  12. Gruber J, Koszegi B (2001) Is addiction rational? Theory and evidence. Q J Econ 116(4):1261Google Scholar
  13. Halkin H (1974) Necessary conditions for optimal control problems with infinite horizons. Econom J Econom Soc 42:267–272Google Scholar
  14. Haurie A, Krawczyk JB, Zaccour G (2012) Games and dynamic games. Business series, vol 1. World Scientific/Now Publishers, Singapore/HackensackGoogle Scholar
  15. Kaldor N (1961) Capital accumulation and economic growth. In: Lutz F, Hague DC (eds) Proceedings of a conference held by the international economics association. McMillan, London, pp 177–222Google Scholar
  16. Kocherlakota NR (2001) Looking for evidence of time-inconsistent preferences in asset market data. Fed Reserve Bank Minneap Q Rev-Fed Reserve Bank Minneap 25(3):13Google Scholar
  17. Krawczyk JB, Shimomura K (2003) Why countries with the same technology and preferences can have different growth rates. J Econ Dyn Control 27(10):1899–1916Google Scholar
  18. Krawczyk JB, Tidball M (2006) A discrete-time dynamic game of seasonal water allocation. J Optim Theory Appl 128(2):411–429Google Scholar
  19. Laibson DI (1996) Hyperbolic discount functions, undersaving, and savings policy. Technical report, National Bureau of Economic ResearchGoogle Scholar
  20. Laibson D (1997) Golden eggs and hyperbolic discounting. Q J Econ 112(2):443–477Google Scholar
  21. Levhari D, Mirman LJ (1980) The great fish war: an example using a dynamic Cournot-Nash solution. Bell J Econ 11(1):322–334Google Scholar
  22. Long NV (1977) Optimal exploitation and replenishment of a natural resource. In: Pitchford J, Turnovsky S (eds) Applications of control theory in economic analysis. North-Holland, Amsterdam, pp 81–106Google Scholar
  23. Luenberger DG (1969) Optimization by vector space methods. John Wiley & Sons, New YorkGoogle Scholar
  24. Luenberger DG (1979) Introduction to dynamic systems: theory, models & applications. John Wiley & Sons, New YorkGoogle Scholar
  25. Maskin E, Tirole J (1987) A theory of dynamic oligopoly, iii: Cournot competition. Eur Econ Rev 31(4):947–968Google Scholar
  26. Michel P (1982) On the transversality condition in infinite horizon optimal problems. Econometrica 50:975–985Google Scholar
  27. O’Donoghue T, Rabin M (1999) Doing it now or later. Am Econ Rev 89(1):103–124Google Scholar
  28. Paserman MD (2008) Job search and hyperbolic discounting: structural estimation and policy evaluation. Econ J 118(531):1418–1452Google Scholar
  29. Phelps ES, Pollak RA (1968) On second-best national saving and game-equilibrium growth. Rev Econ Stud 35(2):185–199Google Scholar
  30. Pulliam HR (1988) Sources, sinks, and population regulation. Am Nat 652–661Google Scholar
  31. Romer PM (1986) Increasing returns and long-run growth. J Polit Econ 94(5):1002–1037MathSciNetCrossRefGoogle Scholar
  32. Samuelson PA (1937) A note on measurement of utility. Rev Econ Stud 4(2):155–161MathSciNetCrossRefGoogle Scholar
  33. Selten R (1975) Rexaminition of the perfectness concept for equilibrium points in extensive games. Int J Game Theory 4(1):25–55CrossRefGoogle Scholar
  34. Simaan M, Cruz JB (1973) Additional aspects of the Stackelberg strategy in nonzero-sum games. J Optim Theory Appl 11:613–626MathSciNetCrossRefGoogle Scholar
  35. Spencer BJ, Brander JA (1983a) International R&D rivalry and industrial strategy. Working Paper 1192, National Bureau of Economic Research,
  36. Spencer BJ, Brander JA (1983b) Strategic commitment with R&D: the symmetric case. Bell J Econ 14(1):225–235CrossRefGoogle Scholar
  37. Strotz RH (1955) Myopia and inconsistency in dynamic utility maximization. Rev Econ Stud 23(3):165–180CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Victoria University of WellingtonWellingtonNew Zealand

Personalised recommendations