Randomized Social Choice Functions Under Metric Preferences

Truthful Facility Location with Additive Errors

“…[19], [18], [14], [15]); Other papers look at heterogeneous facilities, i.e., facilities serving different purposes [26]; Several papers consider a setting in which every agent possesses multiple locations and pays the sum of the distances to her locations ( [23], [19]); Additional objective functions were considered besides the maximum cost and the social cost, for instance the L 2 norm [11] or the minimax envy [6] (in the latter, the approximation was done with an additive error); Additional papers consider different preferences of the agents, for instance doubly-peaked preferences [12], "obnoxious facility location" in which agents want to be as far away as possible from the facility [7], settings which combine agents with ordinary preferences and agents who wish to be far from the facility ([9], [28]); Another direction that was researched is the tradeoff between the approximation ratio and the variance ( [24]); Some papers consider different methods of voting, for instance by restricting the outcome to a discrete set of candidates ( [8], [27], [10]) or by using mediators [5]. When restricting the location of the facility to given candidates, minimizing the social cost of facility location problems has been associated to the notion of "distortion" ( [22], [3], [10], [4], [16]).…”

“…x At the end of the process, the highest additive error is reached in x (4) in the following configuration (see Figure 4) when there are n/5 agents at point 0 (location of OP T 1 ), one agent at F 1 , 3n/5 agents at OP T 2 and n/5 agents at point 1 (location of F 2 ). We denote the distances x = F 2 − OP T 2 , y = OP T 2 − F 1 and therefore F 1 = 1 − x − y. Denote the cost of the agent at F 1 by α.…”