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

Hitoshi Matsushima

doi:10.1515/bejte-2018-0088

40% Rabatt

auf Fachbücher bei De Gruyter Brill *

Artikel

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

Hitoshi Matsushima

Veröffentlicht/Copyright: 23. August 2019

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

Aus der Zeitschrift The B.E. Journal of Theoretical Economics Band 20 Heft 1

Abstract

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

Funding statement: 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.

Appendix

A Proof of Theorem 1

Suppose that the inequality (4) holds. Then, it is clear that q~ is a Nash equilibrium: it satisfies the first-order condition (3) during the interval [τ~,1], and any player has no incentive to select before τ~, because any other player selects before τ~.

Suppose that the inequality (4) does not holds, implying τ~=0 and q~1(0)>0. Then, it is clear that q~ is not a Nash equilibrium: it does satisfy the first-order condition (3) during the interval (0,1], but any player prefers the selection of the initial time 0 to any later time because of q~1(0)>0.

Suppose that the strict inequality of (4) holds:

ε < exp ⁡ [ − 1 n − 1 ∫ 0 1 R ( τ ) d τ ] .

We prove that q~ is the unique Nash equilibrium in the following manner. Consider an arbitrary symmetric Nash equilibrium q. From the strict inequality, it is clear that q1(0)<1.

We show that q1 is continuous. Suppose that q1 is not continuous; there exists τ′>0 such that limτ↑τ′⁡q1(τ)<q1(τ′). Then, by selecting any time slightly earlier than τ′, any player can dramatically increase his winning probability. This implies that no player selects τ′. This is a contradiction.

Let

τ ^ = max { τ ∈ [ 0 , 1 ] : q 1 ( τ ) = q 1 ( 0 ) } .

Note that τ^<1. We show that q1 is increasing in [τ^,1]. Suppose that q1 is not increasing in [τ^,1]; there exist τ′∈[τ^,1] and τ″∈[τ^,1] such that τ′<τ″, q1(τ′)=q1(τ″), and the selection of τ′ is a best response. Since no player selects any τ in (τ′,τ″), it follows from the continuity of q1 that by selecting τ″ instead of τ′, a player can increase his winner’s payoff without decreasing his winning probability. This is a contradiction.

Since q1 is increasing in [τ^,1], any selection τ∈[τ^,1] must satisfy the first-order condition (3). This implies

q 1 ( t ) = q ~ 1 ( t ) for all t ∈ [ τ ^ , 1 ] .

Since any player has no incentive to select before τ^, it follows that τ^=τ~, that is, q~ is the unique symmetric Nash equilibrium.

Next, we prove that q~ is a unique Nash equilibrium, even if we consider all asymmetric strategy profiles. We set any Nash equilibrium q∈Q arbitrarily.

We show that qi must be continuous. Suppose that qi is not continuous; there exists τ′>0 such that limτ↑τ′⁡qi(τ)<qi(τ′). Then, any other player can drastically increase his winning probability by selecting any time slightly earlier than τ′. Hence, no other player selects any time that is either the same as or slightly later than τ′; player i can postpone the time without decreasing his winning probability. This is a contradiction.

Let

τ 1 = max { τ ∈ [ 0 , 1 ] : q i ( τ ) = q i ( 0 ) f o r a l l i ∈ N } .

Note that τ1<1. We show that D(τ;q) must be increasing in [τ1,1]. Suppose that D(τ;q) is not increasing in [τ1,1]; there exist τ′∈(τ1,1] and τ″∈(τ1,1] such that τ′<τ″, D(τ′;q)=D(τ″;q), and the selection of τ′ is a best response for some player. Since no player selects any time τ in (τ′,τ″), it follows from the continuity of q that, by selecting τ″ instead of τ′, any player can postpone the time from τ′ to τ″ without decreasing his winning probability. This is a contradiction.

We show that q must be symmetric. Suppose that q is asymmetric. From the strict inequality, the selection of time zero is a dominated strategy. Hence, we have τ1>0, and

q i ( τ ) = 0 for all i ∈ N and τ ∈ [ 0 , τ 1 ] .

Since q is asymmetric and continuous and D(τ;q) is increasing in [τ1,1], there must exist τ′>0, τ″>τ′, and i∈N such that

q 1 ( t ) = q j ( t ) for all j ∈ N and t ∈ [ 0 , τ ′ ] ,

(13) D ′ i ( τ ; q − i ) 1 − D i ( τ ; q − i ) > min h ≠ i ⁡ D ′ h ( τ ; q − h ) 1 − D h ( τ ; q − h ) for all t ∈ ( τ ′ , τ ″ )

and

(14) D ′ i ( τ ; q − i ) 1 − D i ( τ ; q − i ) = min h ≠ i ⁡ D ′ h ( τ ; q − h ) 1 − D h ( τ ; q − h ) > 0.

Since D(τ;q) is continuous and increasing in [τ1,1], any selection of time t in (τ′,τ″) must be a best response for any player j∈N satisfying

D ′ j ( τ ; q − j ) 1 − D j ( τ ; q − j ) = min h ≠ i ⁡ D ′ h ( τ ; q − h ) 1 − D h ( τ ; q − h ) .

This implies ∂qj(t)∂t>0. Hence, the first-order condition (2) must hold for this player during the interval (τ′,τ″):

D ′ j ( t ; q − j ) 1 − D j ( t ; q − j ) = min h ≠ i ⁡ D ′ h ( t ; q − h ) 1 − D h ( t ; q − h ) = v ¯ ′ ( t ) v ¯ ( t ) − v _ ( t ) for all t ∈ ( τ ′ , τ ″ ) .

However, from (13),

D ′ i ( τ ; q − i ) 1 − D i ( τ ; q − i ) > v ¯ ′ ( t ) v ¯ ( t ) − v _ ( t ) ,

implying that the first-order condition (2) does not hold for player i during the interval (τ′,τ″); instead ∂ui(τ,q−i)∂τ<0 holds for all τ∈(τ′,τ″). This implies that player i prefers τ′ to any time in (τ′,τ″+η];

D i ′ ( τ ; q − i ) = 0 for all τ ∈ ( τ ′ , τ ″ + η ]

where η>0 is set to be close to zero. This is a contradiction, because the inequality in (14) implies Di′(τ″;q−i)>0. Hence, we have proved that any Nash equilibrium must be symmetric.

From the above observations, we have proved Theorem 1.

B Proof of Theorem 2

For every t∈(0,1],

u 1 ( t , q ^ − 1 ) = ε n − 1 v ¯ ( t ) + ( 1 − ε n − 1 ) v _ ( 0 )

≤ ε n − 1 v ¯ ( 1 ) + ( 1 − ε n − 1 ) v _ ( 0 ) = u 1 ( 1 , q ^ − 1 ) ,

and

u 1 ( 0 , q ^ − 1 ) = { ∑ 1 ≤ l ≤ n − 1 ( n − 1 ) ! l ! ( n − 1 − l ) ! ( 1 − ε ) l ε n − 1 − l 1 l + 1 } v ¯ ( 0 )

+ { 1 − ∑ 1 ≤ l ≤ n − 1 ( n − 1 ) ! l ! ( n − 1 − l ) ! ( 1 − ε ) l ε n − 1 − l 1 l + 1 } v _ ( 0 ) .

Hence, a necessary and sufficient condition for q^ to be a Nash equilibrium is given by

u 1 ( 0 , q ^ − 1 ) = { ∑ 1 ≤ l ≤ n − 1 ( n − 1 ) ! l ! ( n − 1 − l ) ! ( 1 − ε ) l ε n − 1 − l 1 l + 1 } v ¯ ( 0 ) + { 1 − ∑ 1 ≤ l ≤ n − 1 ( n − 1 ) ! l ! ( n − 1 − l ) ! ( 1 − ε ) l ε n − 1 − l 1 l + 1 } v _ ( 0 ) ≥ ε n − 1 v ¯ 1 ( 1 ) + ( 1 − ε n − 1 ) v _ 1 ( 0 ) ≥ u 1 ( 1 , q ^ − 1 ) .

This inequality is equivalent to

∑ 1 ≤ l ≤ n − 1 ( n − 1 ) ! l ! ( n − 1 − l ) ! ( 1 − ε ε ) l 1 l + 1 ≥ v ¯ 1 ( 1 ) − v ¯ 1 ( 0 ) v ¯ 1 ( 0 ) − v _ 1 ( 0 ) = R .

References

Abreu, D., and M. Brunnermeier. 2003. “Bubbles and Crashes.” Econometrica 71: 173–204.10.1111/1468-0262.00393Suche in Google Scholar

Allen, F., and G. Gorton. 1993. “Churning Bubbles.” Review of Economic Studies 60: 813–36.10.1093/acprof:osobl/9780199844333.003.0002Suche in Google Scholar

Awaya, Y., K. Iwasaki, and M. Watanabe. 2018. “Rational Bubbles and Middleman.” mimeo.10.2139/ssrn.3380383Suche in 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.10.1016/B978-0-44-459406-8.00018-4Suche in Google Scholar

Brunnermeier, M., and L. Pedersen. 2009. “Market Liquidity and Funding Liquidity.” Review of Financial Studies 22: 2201–38.10.3386/w12939Suche in Google Scholar

Che, Y.-K., and R. Sethi. 2010. “Credit Derivatives and the Cost of Capital.” mimeo.10.2139/ssrn.1654222Suche in 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): 703–38.10.1086/261703Suche in Google Scholar

Fostel, A., and J. Geanakoplos. 2012. “Tranching, CDS and Asset Prices: Bubbles and Crashes.” AEJ: Macroeconomics 4 (1): 190–225.10.1257/mac.4.1.190Suche in Google Scholar

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

Geanakoplos, J. 2010. “The Leverage Cycle.” In NBER Macroeconomics Annual, vol. 24, 1–65. Chicago, USA: University of Chicago Press.10.1086/648285Suche in Google Scholar

Harrison, J., and D. Kreps. 1978. “Speculative Investor Behavior in a Stock Market with Heterogeneous Expectations.” Quarterly Journal of Economics 89: 323–36.10.2307/1884166Suche in Google Scholar

Kindleberger, C. 1978. Manias, Panics, and Crashes: A History of Financial Crises. USA: Basic Books.10.1007/978-1-349-04338-5Suche in 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: 245–52.10.1016/0022-0531(82)90029-1Suche in Google Scholar

Kreps, D., and R. Wilson. 1982. “Reputation and Imperfect Information.” Journal of Economic Theory 27: 253–79.10.1016/0022-0531(82)90030-8Suche in Google Scholar

Martin, A., and J. Ventura. 2012. “Economic Growth with Bubbles.” American Economic Review 147 (2): 738–58.10.3386/w15870Suche in Google Scholar

Matsushima, H. 2013. “Behavioral Aspects of Arbitrageurs in Timing Games of Bubbles and Crashes.” Journal of Economic Theory 148: 858–70.10.1016/j.jet.2012.08.002Suche in Google Scholar

Shleifer, A., and R. Vishny. 1992. “The Limits of Arbitrage.” Journal of Finance 52 (1): 35–55.10.3386/w5167Suche in Google Scholar

Simsek, A. 2013. “Belief Disagreements and Collateral Constraints.” Econometrica 81: 1–53.10.3982/ECTA9956Suche in Google Scholar

Tirole, J. 1985. “Asset Bubbles and Overlapping Generations.” Econometrica 53 (5): 1071–100.10.2307/1911012Suche in Google Scholar

Published Online: 2019-08-23

Sie haben derzeit keinen Zugang zu diesem Inhalt.

Artikel in diesem Heft

https://doi.org/10.1515/bejte-2018-0088

Schlagwörter für diesen Artikel

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