A family of strategyproof mechanisms for activity scheduling

Recent years have seen various designs of strategyproof mechanisms in the facility location game and the obnoxious facility game, by considering the facility’s geo-location as a point in the spatial domain. In this paper, we extend this point to be a continuous interval, and study a novel activity scheduling game to schedule an activity in the normalized time domain [0, 1] based on all agents’ time reports for preferences/conflicts. The activity starts at time point y and lasts for a fixed time period of d with \(0\le d\le 1\). Each agent \(i\in N = \{1, \cdots , n\}\) wants his preferred time interval \([t_i,t_i+l_i]\) to be close to or overlap with the activity interval \([y,y+d]\). Since agents are heterogeneous, we consider each agent i has weight \(\alpha _i\) or \(\beta _i\) when the activity is scheduled after or before his time interval, respectively. Thus each agent i’s cost is his weight (\(\alpha _i\) or \(\beta _i\)) multiplied by the time difference between his time interval \([t_i,t_i+l_i]\) and the activity interval \([y,y+d].\) The social cost is the summation of all agents’ costs. In this game, agents’ preferred time intervals \([t_i,t_i+l_i]\)’s are private information and they may misreport such information to the social planner. Our objective is to choose the activity starting time y so that the mechanisms are strategyproof (i.e., all agents should be truthful to report \(t_i\)’s and \(l_i\)’s) and perform well with respect to minimizing the social cost. We design a mechanism outputting an optimal solution and prove that it is group strategyproof. For the objective of minimizing the maximum cost among agents, we design another strategyproof mechanism with the approximation ratio \(1+\min \{\alpha /\beta ,\beta /\alpha \}\) when \(\alpha _i=\alpha , \beta _i = \beta\) for \(i\in N,\) and prove it is the best strategyproof mechanism. In the obnoxious activity scheduling game, each agent prefers his conflicting time interval \([t_i,t_i+l_i]\) to be far away from the activity interval \([y,y+d]\). We design deterministic and randomized group strategyproof mechanisms, and compare their provable approximation ratios to the lower bounds. Finally, we consider the cost/utility of each agent as a 0-1 indicator function and find group strategyproof mechanisms for minimizing the social cost and maximizing the social utility.