Incentives in effort games

We consider Effort Games, a game theoretic model of co-operation in open environments, which is a variant of the principal-agent problem from economic theory. In our multiagent domain, a common project depends on various tasks; achieving certain subsets of the tasks completes the project successfully, while others do not. The probability of achieving a task is higher when the agent in charge of it exerts effort, at a certain cost for that agent. A central authority, called the principal, attempts to incentivize agents to exert effort, but can only reward agents based on the success of the entire project. We model this domain as a normal form game, where the payoffs for each strategy profile are defined based on the different probabilities of achieving each task and on the boolean function that defines which task subsets complete the project and which do not. We view this boolean function as a simple coalitional game, and call this game the under-lying coalitional game. We show that finding the minimal reward that induces an agent to exert effort is at least as hard computationally as finding the Banzhaf power index in the underlying coalitional game, so this problem is #P-hard in general. We also show that in a certain restricted domain, where the underlying coalitional game is a unanimity weighted voting game with certain properties, it is possible to solve all of the above problems in polynomial time.