Generating good candidate actions is critical to the success of sample-based planners, particularly in continuous or large action spaces.

We study the mechanism design problem of a social planner for locating two facilities on a line interval [0, 1], where a set of n strategic agents report their locations and a mechanism determines the locations of the two facilities. We consider the requirement of a minimum distance 0 ≤ d ≤ 1 between the two facilities. Given the two facilities are heterogeneous, we model the cost/utility of an agent as the sum of his distances to both facilities. In the heterogeneous two-facility location game to minimize the social cost, we show that the optimal solution can be computed in polynomial time and prove that carefully choosing one optimal solution as output is strategyproof. We also design a strategyproof mechanism minimizing the maximum cost. Given the two facilities are homogeneous, we model the cost/utility of an agent as his distance to the closer facility. In the homogeneous two-facility location game for minimizing the social cost, we show that any deterministic strategyproof mechanism has unbounded approximation ratio. Moreover, in the obnoxious heterogeneous two-facility location game for maximizing the social utility, we propose new deterministic group strategyproof mechanisms with provable approximation ratios and establish a lower bound (7 − d)/6 for any deterministic strategyproof mechanism. We also design a strategyproof mechanism maximizing the minimum utility. In the obnoxious homogeneous two-facility location game for maximizing the social utility, we propose deterministic group strategyproof mechanisms with provable approximation ratios and establish a lower bound 4/3. Besides, in the two-facility location game with triple-preference, where each facility may be favorable, obnoxious, indifferent for any agent, we further motivate agents to report both their locations and preferences towards the two facilities truthfully, and design a deterministic group strategyproof mechanism with an approximation ratio 4.

In this paper we consider an online learning version of the Facility location problem where users need to be served one at a time in a sequence of trials. The goal is to select, at each trial, a subset of a given set of sites, and then pay a loss equal to their total "opening cost" plus the minimum "connection cost" for connecting the user to one of the sites in the subset. More precisely, we are given a set of N sites. At the beginning of each trial, an opening cost and a connection cost for the arriving user are associated with each site and are unknown. At each trial, the learner has to select a subset of sites and incurs a loss given by the minimum connection cost over the selected sites plus the sum of the opening costs of all selected sites. After each subset selection, the opening and connection costs of all sites are revealed. To solve this problem, we design and rigorously analyse an algorithm which belongs to the class of online learning algorithms that make use of the Exponentiated gradient method [15]. We measure, and rigorously analyse, the performance of our method by comparing its cumulative loss with that of any fixed subset of sites.

Sample-based planning is a powerful family of algorithms for generating intelligent behavior from a model of the environment. Generating good candidate actions is critical to the success of sample-based planners, particularly in continuous or large action spaces. Typically, candidate action generation exhausts the action space, uses domain knowledge, or more recently, involves learning a stochastic policy to provide such search guidance. In this paper we explore explicitly learning a candidate action generator by optimizing a novel objective, marginal utility. The marginal utility of an action generator measures the increase in value of an action over previously generated actions. We validate our approach in both curling, a challenging stochastic domain with continuous state and action spaces, and a location game with a discrete but large action space. We show that a generator trained with the marginal utility objective outperforms hand-coded schemes built on substantial domain knowledge, trained stochastic policies, and other natural objectives for generating actions for sampled-based planners.