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.
A Proof of Theorem 1
Suppose that the inequality (4) holds. Then, it is clear that is a Nash equilibrium: it satisfies the first-order condition (3) during the interval , and any player has no incentive to select before , because any other player selects before .
Suppose that the inequality (4) does not holds, implying and . Then, it is clear that is not a Nash equilibrium: it does satisfy the first-order condition (3) during the interval , but any player prefers the selection of the initial time 0 to any later time because of .
Suppose that the strict inequality of (4) holds:
We prove that is the unique Nash equilibrium in the following manner. Consider an arbitrary symmetric Nash equilibrium . From the strict inequality, it is clear that .
We show that is continuous. Suppose that is not continuous; there exists such that . Then, by selecting any time slightly earlier than , any player can dramatically increase his winning probability. This implies that no player selects . This is a contradiction.
Note that . We show that is increasing in . Suppose that is not increasing in ; there exist and such that , , and the selection of is a best response. Since no player selects any in , it follows from the continuity of that by selecting instead of , a player can increase his winner’s payoff without decreasing his winning probability. This is a contradiction.
Since is increasing in , any selection must satisfy the first-order condition (3). This implies
Since any player has no incentive to select before , it follows that , that is, is the unique symmetric Nash equilibrium.
Next, we prove that is a unique Nash equilibrium, even if we consider all asymmetric strategy profiles. We set any Nash equilibrium arbitrarily.
We show that must be continuous. Suppose that is not continuous; there exists such that . Then, any other player can drastically increase his winning probability by selecting any time slightly earlier than . Hence, no other player selects any time that is either the same as or slightly later than ; player can postpone the time without decreasing his winning probability. This is a contradiction.
Note that . We show that must be increasing in . Suppose that is not increasing in ; there exist and such that , , and the selection of is a best response for some player. Since no player selects any time in , it follows from the continuity of that, by selecting instead of , any player can postpone the time from to without decreasing his winning probability. This is a contradiction.
We show that must be symmetric. Suppose that is asymmetric. From the strict inequality, the selection of time zero is a dominated strategy. Hence, we have , and
Since is asymmetric and continuous and is increasing in , there must exist , , and such that
Since is continuous and increasing in , any selection of time in must be a best response for any player satisfying
This implies . Hence, the first-order condition (2) must hold for this player during the interval :
However, from (13),
implying that the first-order condition (2) does not hold for player during the interval ; instead holds for all . This implies that player prefers to any time in ;
where is set to be close to zero. This is a contradiction, because the inequality in (14) implies . 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 ,
Hence, a necessary and sufficient condition for to be a Nash equilibrium is given by
This inequality is equivalent to
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