Strategic Exit With Information and Payoff Externalities

I consider a stopping game between two players, where observations related to an unknown state of the nature arrive at random. Players not only learn from observing each other, but their payoffs also depend on the presence of the counterpart. I derive a general characterization of an equilibrium in this game. As applications, I consider two stopping time games which can be viewed as models of sponsored research - one is the model where researchers get funded until (if ever) a research project experiences the first failure, the other one is the model where researchers get rewarded if a success is achieved. In either case, the researchers start working on a project of unknown quality. The quality of the project is identified with its ability to generate failures or successes, in the first and second models, respectively. The rate of arrival of success conditioned on the quality of the project is an increasing function of the total time spent on the sponsored research. Observations of failures or successes are public information.I find subgame perfect equilibria in both models and show that in case of two competing researchers, neither equilibrium outcomes, nor cooperative solutions are efficient unless research creates no payoff externalities. In either model, at least one of the researchers experiments inefficiently long, so that a designer of a grant competition would like to stop sponsoring one of the players earlier than in equilibrium. Surprisingly, this result holds in the model where the first success is rewarded no matter whether the laggards are rewarded with a smaller prize or punished.