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

This study investigates a 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 the possibility of irrational types in that each player is irrational with a positive probability, thus selecting the terminal time. We show that there exists the unique Nash equilibrium; according to it, every player never selects the initial time. As an application, we analyze a strategic aspect of leverage-driven bubbles; even if a company is unproductive, its stock price grows up according to an exogenous reinforcement pattern. During the bubble, this company is willing to raise huge funds by issuing new shares. We regard players as arbitrageurs, who decide whether to ride the bubble or burst it. We demonstrate two models, which are distinguished by whether crash-contingent claim, i.e., contractual agreement such that the purchaser of this claim receives a promised monetary amount from its seller if and only if the bubble crashes, is available. The availability of this claim deters the bubble; without crash-contingent claim, the bubble emerges and persists 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.