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

Shared effort games model people investing resources in public projects,where the share of the generated values is predefined. In linear θ sharing(effort) games, a project’s value is linear in the total contribution, while thethreshold θ for effort defines which contributors win and receive their (equal)share. Thresholds between 0 and 1 model games such as paper co-authorshipand shared assignments, where a minimum positive contribution is requiredfor sharing in the value. We constructively characterise the conditions for theexistence of a pure equilibrium for θ ∈ {0, 1}, and for two-player games witha general threshold, and find the prices of anarchy and stability. For moreplayers we also provide existence and efficiency results, and use generalisedfictitious play simulations to show when a pure equilibrium exists and whatits efficiency is. We also prove mixed equilibria always exist and bound theirefficiency. This facilitates reaching socially efficient equilibria