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The B.E. Journal of Theoretical Economics

Editor-in-Chief: Schipper, Burkhard

Ed. by Fong, Yuk-fai / Peeters, Ronald / Puzzello , Daniela / Rivas, Javier / Wenzelburger, Jan

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Source Normalized Impact per Paper (SNIP) 2018: 0.186

Mathematical Citation Quotient (MCQ) 2018: 0.08

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Timing Games with Irrational Types: Leverage-Driven Bubbles and Crash-Contingent Claims

Hitoshi Matsushima
Published Online: 2019-08-23 | DOI: https://doi.org/10.1515/bejte-2018-0088


This study investigates strategic aspect of leverage-driven bubbles from the viewpoint of game theory and behavioral finance. Even if a company is unproductive, its stock price grows up according to an exogenous reinforcement pattern. During the bubble, this company raises huge funds by issuing new shares. Multiple arbitrageurs strategically decide whether to ride the bubble by continuing to purchase shares through leveraged finance.

We demonstrate two models that are distinguished by whether crash-contingent claim, i. e. contractual agreement such that the purchaser of this claim receives a promised monetary amount if and only if the bubble crashes, is available. We show that the availability of this claim deters the bubble; without crash-contingent claim, the bubble emerges and persists long even if the degree of reinforcement is insufficient. Without crash-contingent claim, high leverage ratio fosters the bubble, while with crash-contingent claim, it rather deters the bubble.

We formulate these models as specifications of timing game with irrational types; each player selects a time in a fixed time interval, and the player who selects the earliest time wins the game. We assume that each player is irrational with a small but positive probability. We then prove that there exists the unique Nash equilibrium; according to it, every player never selects the initial time. By regarding arbitrageurs as players, we give careful conceptualizations that are necessary to interpret timing games as models of leverage-driven bubbles.

Keywords: timing games with irrational types; uniqueness; awareness heterogeneity; leverage-driven bubbles; crash-contingent claim

JEL Classification: C720; C730; D820; G140


  • Abreu, D., and M. Brunnermeier. 2003. “Bubbles and Crashes.” Econometrica 71: 173–204.CrossrefGoogle Scholar

  • Allen, F., and G. Gorton. 1993. “Churning Bubbles.” Review of Economic Studies 60: 813–36.CrossrefGoogle Scholar

  • Awaya, Y., K. Iwasaki, and M. Watanabe. 2018. “Rational Bubbles and Middleman.” mimeo.Google Scholar

  • Brunnermeier, M., and M. Oehmke. 2013. “Bubbles, Financial Crises, and Systemic Risk.” In Handbook of the Economics of Finance, vol. 2, edited by G. Constantinides, M. Harris, and R. Stulz, 1221–88. North Holland: Elsevier.Google Scholar

  • Brunnermeier, M., and L. Pedersen. 2009. “Market Liquidity and Funding Liquidity.” Review of Financial Studies 22: 2201–38.CrossrefWeb of ScienceGoogle Scholar

  • Che, Y.-K., and R. Sethi. 2010. “Credit Derivatives and the Cost of Capital.” mimeo.Google Scholar

  • De Long, J., A. Shleifer, L. Summers, and R. Waldmann. 1990. “Noise Trader Risk in Financial Markets.” Journal of Political Economy 98 (4): 703–38.CrossrefGoogle Scholar

  • Fostel, A., and J. Geanakoplos. 2012. “Tranching, CDS and Asset Prices: Bubbles and Crashes.” AEJ: Macroeconomics 4 (1): 190–225.Google Scholar

  • Friedman, M., and A. Schwarz. 1963. A Monetary History of the United States 1867–1960. Princeton, USA: Princeton University Press.Google Scholar

  • Geanakoplos, J. 2010. “The Leverage Cycle.” In NBER Macroeconomics Annual, vol. 24, 1–65. Chicago, USA: University of Chicago Press.Google Scholar

  • Harrison, J., and D. Kreps. 1978. “Speculative Investor Behavior in a Stock Market with Heterogeneous Expectations.” Quarterly Journal of Economics 89: 323–36.Google Scholar

  • Kindleberger, C. 1978. Manias, Panics, and Crashes: A History of Financial Crises. USA: Basic Books.Google Scholar

  • Kreps, D., P. Milgrom, J. Roberts, and R. Wilson. 1982. “Rational Cooperation in the Finitely Repeated Prisoners’ Dilemma.” Journal of Economic Theory 27: 245–52.CrossrefGoogle Scholar

  • Kreps, D., and R. Wilson. 1982. “Reputation and Imperfect Information.” Journal of Economic Theory 27: 253–79.CrossrefGoogle Scholar

  • Martin, A., and J. Ventura. 2012. “Economic Growth with Bubbles.” American Economic Review 147 (2): 738–58.Web of ScienceGoogle Scholar

  • Matsushima, H. 2013. “Behavioral Aspects of Arbitrageurs in Timing Games of Bubbles and Crashes.” Journal of Economic Theory 148: 858–70.CrossrefWeb of ScienceGoogle Scholar

  • Shleifer, A., and R. Vishny. 1992. “The Limits of Arbitrage.” Journal of Finance 52 (1): 35–55.Web of ScienceGoogle Scholar

  • Simsek, A. 2013. “Belief Disagreements and Collateral Constraints.” Econometrica 81: 1–53.Web of ScienceCrossrefGoogle Scholar

  • Tirole, J. 1985. “Asset Bubbles and Overlapping Generations.” Econometrica 53 (5): 1071–100.CrossrefGoogle Scholar

About the article

Published Online: 2019-08-23

This research was supported by a grant-in-aid for scientific research (KAKENHI 21330043, 25285059) from the Japan Society for the Promotion of Science (JSPS) and the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of the Japanese government. I am grateful to the editor-in-chief, an anonymous referee, and Professor Takashi Ui for their valuable comments. All errors are mine.

Citation Information: The B.E. Journal of Theoretical Economics, Volume 20, Issue 1, 20180088, ISSN (Online) 1935-1704, DOI: https://doi.org/10.1515/bejte-2018-0088.

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