Starting small in project choice: A discrete-time setting with a continuum of types

We add to the literature on long-term relationships with variable stakes and incomplete information by analyzing a discrete-time trust game between a principal and agent, with a continuum of types. In each period the principal selects the level of a project and the agent then decides whether to cooperate or betray; payoffs in the period scale with the level. The agent's benefit of betraying is privately known. The discrete-time framework allows for analysis of renegotiation in terms of an internal consistency condition that compares actual equilibria in the continuation of the game from any period, improving on the prior literature. Our condition assumes the principal has full power to alter the equilibrium selection. Our main result shows that the resulting perfect Bayesian equilibria converge as the period length shrinks to zero, and we provide a closed-form solution. In equilibrium, the relationship starts small and the level gradually rises until it reaches its maximum; cooperation is viable regardless of the type distribution.