TIME INCONSISTENCY, PRECOMMITMENT, AND EQUILIBRIUM STRATEGIES FOR A STACKELBERG GAME

We consider a Stackelberg game in discrete time where a manager (leader) hires an agent (follower) for the completion of a project. The project has two states: either uncompleted or completed. The manager gets a fixed reward upon the completion of the project. At the beginning of each stage, given that the project has not been completed, the manager proposes an amount that will be paid to the agent if the project is completed in the next period. The agent controls the level of effort which is equivalent to the probability of the project being completed in the next period. Both the manager and agent aim to maximize their own expected payoffs subject to exponential discounting. It turns out that the manager's problem is time-inconsistent, albeit with exponential discounting. We provide the manager's optimal Markov, optimal precommitment, and equilibrium strategies. Next, we extend our results to the problem where the project has two milestones, i.e., three states. Moreover, we show that if the manager is weakly less patient than the agent, then under optimal Markov and precommitment strategies there is no need for the manager to pay the agent upon the completion of the intermediate milestone, while that may not be the case for the equilibrium strategy.