## 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.

**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

Suppose that the inequality (4) does not holds, implying

Suppose that the strict inequality of (4) holds:

We prove that

We show that

Let

Note that

Since

Since any player has no incentive to select before

Next, we prove that

We show that

Let

Note that

We show that

Since

and

Since

This implies

However, from (13),

implying that the first-order condition (2) does not hold for player

where

From the above observations, we have proved Theorem 1.

### B Proof of Theorem 2

For every

and

Hence, a necessary and sufficient condition for

This inequality is equivalent to

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**Published Online:**2019-08-23

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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