When Effort May Fail: Equilibria of Shared Effort with a Threshold

People, robots, and companies mostly divide time and effort between projects, and \defined{shared effort games} model people investing resources in public endeavors and sharing the generated values. In linear $\theta$ sharing (effort) games, a project's value is linear in the total contribution, thus modelling predictable, uniform, and scalable activities. The threshold $\theta$ for effort defines which contributors win and receive their share, equal share modelling standard salaries, equity-minded projects, etc. Thresholds between 0 and 1 model games such as paper co-authorship and shared assignments, where a minimum positive contribution is required for sharing in the value. We constructively characterise the conditions for the existence of a pure equilibrium for $\theta\in\{0,1\}$, and for two-player games with a general threshold, and find the prices of anarchy and stability. We also provide existence and efficiency results for more than two players, and use generalised fictitious play simulations to show when a pure equilibrium exists and what its efficiency is. We propose a novel method for studying solution concepts by defining a new concept and proving its equivalence to a previously known on a large subclass of games. This means that the original concept narrows down to a more demanding concepts on certain games, providing new insights and opening a path to study both concepts conveniently. We also prove mixed equilibria always exist and bound their efficiency.