Running head: PATH PLANNING UNDER SPATIAL UNCERTAINTY

We present experiments studying path planning under spatial uncertainties. In the main experiment, participants’ task was to navigate the shortest possible path to find an object, hidden in one of 4 places, and to bring it to the final destination. The probability of finding the object was different for the 4 places (probability matrix) and varied between conditions. Given such uncertainties about the object’s location, planning a single path is not sufficient. Participants had to generate multiple consecutive plans (meta-plans): if the object is found in A, proceed to the destination; if the object is not found, proceed to B, etc. The optimal solution depends on the specific probability matrix. In each condition, participants learned a different probability matrix and were then asked to report the optimal meta-plan. Results demonstrate effective integration of the probabilistic information about the object’s location during planning. We present a hierarchical planning scheme that could account for participants’ behavior as well as for systematic errors and differences between conditions. C337 Planning under uncertainties 3 Introduction Path planning under spatial uncertainty The selection between path alternatives and the planning of novel paths are essential and frequent tasks for human navigators and several studies addressed the underlying cognitive strategies and mechanisms (e.g., Christenfeld, 1995; Gärling & Gärling, 1988, Golledge, 1995; Wiener& Mallot, 2003; Wiener, Schnee & Mallot, 2004; Hölscher, Meilinger, Vrachliotis, Brösamle, & Knauff, 2006). All prior studies on path selection and path planning behavior assumed or assured that all required spatial information was available. In real life, however, navigators have to deal with incomplete or imprecise spatial knowledge resulting in spatial uncertainties. They might, for example, be confronted with situations in which the exact location of a goal or goal place is unknown or can be judged with a certain probability only. This work aimed at investigating the strategies and heuristics navigators employ during path planning under such spatial uncertainties. The main experimental task was analogous to the following scenario: “On your way home from work you realize that you are missing your keys. You know, however, that your roommate, who has another set of keys, has an appointment for dinner in a nearby restaurant after work. Obviously, the probability to find your roommate at work or in one of the restaurants depends on the time of day. Given different timings, what is the best path (i.e., in which order should you visit your roommate’s workplace and the nearby restaurants) to find him/her, get the keys, and get C337 Planning under uncertainties 4 home as quickly as possible”. What is being described here is a path planning task with a destination and with an intermediate target whose exact whereabouts is uncertain, but which can be described by a probability matrix over multiple locations. This probability matrix might change with different timings: if you leave work early, the probability to find your roommate still at work is rather high. With advancing time, however, the probability to find your roommate at work decreases, while the probability to find him/her in one of the nearby restaurant increases. Decision making and uncertainties. In the scenario described above, decisions about the order in which different locations are visited have to be made, although the exact whereabouts of the target can be described with a certain probability only. Probabilistic decision making has been extensively studied in non-spatial contexts in probability learning experiments in which participants are faced with response alternatives that differ in terms of payoff (e.g., Shanks, Tunney, & McCarthy, 2002; West & Stanovich, 2003). Alternative A, for example, is rewarded in 70% of the choices, while alternative B is rewarded in 30% of the choices. Rational choice theory requires that after learning these probabilities, participants should eventually allocate all their responses to the alternative with the higher payoff (alternative A). However, many studies demonstrated that participants match their response probabilities to the payoff probabilities: that is, participants choose alternative A in ~70% of the trials and alternative B in ~30% of the trials (for an overview, see Vulkan, 2000). In contrast to rational choice, such probability matching behavior C337 Planning under uncertainties 5 results in sub-optimal overall payoff. In context of the spatial planning scenario introduced above, in which one faces a single trial only, both rational choice and probability matching predict that the different locations are visited in an order resembling their probability to hold the desired target. Note however, that the spatial layout is not considered. In spatial terms, trajectories resulting from such non-spatial decision making theories could therefore be rather long and inefficient. Strategies and heuristics involved in path planning. Even without spatial uncertainties, planning reasonably short paths for visiting several locations is a complex and computationally expensive task. Several planning strategies and heuristics have been described that allow for a reduction of computational effort: A particularly simple strategy to solve path planning tasks with multiple target locations is the nearest neighbor (NN) strategy (e.g., Gärling, Säisä, Böök & Lindberg, 1986; Hirtle & Gärling, 1992; Bures, Buresova & Nerad, 1992; Gärling & Gärling, 1988). It states that repeatedly the closest target location is chosen until all targets locations have been visited. The NN strategy is a purely local strategy that is universally applicable if distances between locations are available and that is known to result in reasonably short paths in many situations (Golden, Bodin, Doyle & Stewart, 1980). A number of further planning strategies can be subsumed under the term hierarchical planning schemes. Essentially, hierarchical planning schemes such as the cluster strategy C337 Planning under uncertainties 6 (Gallistel & Cramer, 1996; Wiener et al., 2004), region-based planning strategies (Wiener & Mallot, 2003), or the hierarchical Nearest Neighbor strategy (Vickers, Bovet, Lee & Hughes, 2003a), reduce mental effort during planning by first generating approximate solutions using abstractions from the actual planning task. These approximate or coarse solutions (path plans) are then refined in later planning steps. The cluster strategy, for example, assumes that neighboring target locations collapse into clusters and that paths are planned such that the largest cluster is visited first (c.f. Gallistel & Cramer, 1996; Wiener et al., 2004). By collapsing multiple targets into one cluster that becomes a target in the first planning step, the problem size is considerably reduced, many suboptimal solutions are eliminated, and a simple yet coarse path plan is generated that is easily remembered and that can be refined during navigation by planning a detailed path within each cluster. Similarly, region-based-planning strategies assume that target locations are assigned to different environmental regions and that paths are planned towards that region(s) rather than towards the single target places (e.g., Wiener & Mallot, 2003, Wiener et al. 2004, Hölscher et al., 2006). During planning, detailed spatial information is made available for the close surrounding only, while distant locations are represented at super-ordinate region levels. Such region-based planning will result in a detailed path plan for the close surrounding only, allowing for immediate movement decisions. During navigation the plan then has to be refined as the navigator enters new regions. Path planning and optimization behavior has also been investigated by means of visual versions of the Traveling Salesman Problem (e.g., MacGregor, Ormerod, & Chronicle, 1999; C337 Planning under uncertainties 7 Graham, Joshi, & Pizlo, 2000; Vickers, Lee, Dry & Hughes, 2003b; Vickers, Lee, Dry, Hughes & McMahon, 2006), in which participants are confronted with a number of locations (usually dots) on a computer monitor. Their task is to connect these locations such that the resulting path (tour) is optimal with respect to overall length. There is an ongoing debate on the strategies participants apply in these experiments. MacGregor and Ormerod (1996) proposed the convex hull method, assuming that participants use the convex hull as part of their strategy (see also MacGregor, Ormerod, & Chronicle, 2000; MacGregor, Chronicle, & Ormerod, 2004). They argue that the fact that a tour that follows the convex hull method is by definition free of crossings and that humans tend to avoid crossings supports the convex hull strategy. Van Rooij, Stege, and Schactman (2003), however, argue that participants know that crossings will result in sub-optimal solutions. They therefore proposed the crossing avoidance hypothesis, stating that, rather than following the convex hull method, humans avoid crossings when solving TSPs. While the abovementioned strategies are specific for the traveling salesman problem, requiring to plan closed loops, the hierarchical nearest neighbor (hNN) methods is more general (Vickers et al., 2003a). The hNN method assumes that participants first establish clusters of several dots based on NN distances, which they then sequentially link into a tour, using some variant of the nearest neighbor algorithm. To the authors’ knowledge, path planning has not yet been investigated in situations in which participants were confronted with spatial uncertainties. All planning strategies described above are based solely on spatial information; hence they do not allow predicting how navigators C337 Planning under uncertainties 8 cope with spatial uncertainties. However, if not stated otherwise, the reported strategies can be applied to the spatial aspects