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.

近年来，通过将设施的地理位置视为空间域中的一个点，在设施位置博弈和令人讨厌的设施博弈中出现了各种策略证明机制的设计。在本文中，我们将这一点扩展为连续间隔，并研究了一种新颖的活动调度游戏，根据所有代理的偏好/冲突时间报告在标准化时间域 [0, 1] 中安排活动。该活动从时间点y开始，持续固定时间段d，其中\(0\le d\le 1\)。每个代理\(i\in N = \{1, \cdots , n\}\)希望他的首选时间间隔\([t_i,t_i+l_i]\)接近或与活动间隔\([ y,y+d]\)。由于代理是异构的，因此当活动安排在其时间间隔之后或之前时，我们分别认为每个代理i具有权重\(\alpha_i\)或\(\beta_i\) 。因此，每个代理i的成本是他的权重（\(\alpha _i\)或\(\beta _i\)）乘以他的时间间隔\([t_i,t_i+l_i]\)和活动之间的时间差区间\([y,y+d].\)社会成本是所有代理人成本的总和。在这个游戏中，代理人的首选时间间隔\([t_i,t_i+l_i]\)是私人信息，他们可能会向社会规划者错误报告此类信息。我们的目标是选择活动开始时间y，以便机制具有策略证明性（即，所有代理都应该如实报告\(t_i\)和\(l_i\)），并在最小化社会成本。我们设计了一种输出最优解的机制，并证明它是群体策略证明的。为了最小化代理之间的最大成本，我们设计了另一种策略证明机制，当\(\alpha _i = \alpha , \beta _i = \beta\) for \(i\in N,\)并证明它是最好的策略证明机制。在令人讨厌的活动调度游戏中，每个代理更喜欢其冲突时间间隔\([t_i,t_i+l_i]\)远离活动间隔\([y,y+d]\)。我们设计了确定性和随机的群体策略证明机制，并将其可证明的近似率与下限进行比较。最后，我们将每个代理的成本/效用视为 0-1 指标函数，并找到最小化社会成本和最大化社会效用的群体策略证明机制。