diff --git "a/txt/2105.06948.txt" "b/txt/2105.06948.txt" new file mode 100644--- /dev/null +++ "b/txt/2105.06948.txt" @@ -0,0 +1,1985 @@ +People construct simplified mental representations to plan +Mark K. Ho1,2,*, David Abel3,+, Carlos G. Correa4, Michael L. Littman3, Jonathan D. Cohen1,4, +and Thomas L. Griffiths1,2 +1Princeton University, Department of Psychology, Princeton, NJ, USA;2Princeton University, Department +of Computer Science, Princeton, NJ, USA;3Brown University, Department of Computer Science, +Providence, RI, USA;+Now at DeepMind, London, United Kingdom;4Princeton University, Princeton +Neuroscience Institute, Princeton, NJ, USA;*Corresponding author: Mark K Ho, +mho@princeton.edu +One of the most striking features of human cognition is the capacity to plan. Two aspects +of human planning stand out: its efficiency and flexibility. Efficiency is especially impres- +sive because plans must often be made in complex environments, and yet people successfully +plan solutions to myriad everyday problems despite having limited cognitive resources1–3. +Standard accounts in psychology, economics, and artificial intelligence have suggested hu- +man planning succeeds because people have a complete representation of a task and then use +heuristics to plan future actions in that representation4–11. However, this approach gener- +ally assumes that task representations are fixed . Here, we propose that task representations +can be controlled and that such control provides opportunities to quickly simplify problems +and more easily reason about them. We propose a computational account of this simplifica- +tion process and, in a series of pre-registered behavioral experiments, show that it is subject +to online cognitive control12–14and that people optimally balance the complexity of a task +representation and its utility for planning and acting. These results demonstrate how strate- +gically perceiving and conceiving problems facilitates the effective use of limited cognitive +resources. +1arXiv:2105.06948v2 [cs.AI] 26 Nov 2022In the short story “On Exactitude in Science,” Jorge Luis Borges describes cartographers who +seek to create the perfect map, one that includes every possible detail of the country it represents. +However, this innocent premise leads to an absurd conclusion: The fully detailed map of the +country must be the size of the country itself, which makes it impractical for anyone to use. Borges’ +allegory illustrates an important computational principle. Namely, useful representations do not +simply mirror every aspect of the world, but rather pick out a manageable subset of details that +are relevant to some purpose (Figure 1a). Here, we examine the consequences of this principle for +how humans flexibly construct simplified task representations to plan. +Classic theories of problem solving distinguish between representing a task andcomputing a +plan4,15,16. For instance, Newell and Simon17introduced heuristic search , in which a decision- +maker has a full representation of a task (e.g., a chess board, chess pieces, and the rules of chess), +and then computes a plan by simulating and evaluating possible action sequences (e.g., sequences +of chess moves) to find one that is likely to achieve a goal (e.g., checkmate the king). In artificial +intelligence, the main approach to making heuristic search tractable involves limiting the com- +putation of action sequences (e.g., only thinking a few moves into the future, or only examining +moves that seem promising)5. Similarly, psychological research on planning largely focuses on +how limiting, prioritizing, pruning, or chunking action sequences can reduce computation6–11,18–20. +However, people are not necessarily restricted to a single, full, or fixed representation for a +task. This matters since simpler representations can make better use of limited cognitive resources +when they are tailored to specific parts or versions of a task. For example, in chess, considering the +interaction of a few pieces, or focusing on part of the board, is easier than reasoning about every +piece and part of the board. Furthermore, it affords the opportunity to adapt the representation, +tailoring it to the specific needs of the circumstance—a process that we refer to as controlling a +task construal . Although studies show that people can flexibly form representations to guide action +(e.g., forming the ad hoc category of “things to buy for a party” when organizing a social gath- +ering21), a long-standing challenge for cognitive science and artificial intelligence is explaining, +predicting, and deriving such representations from general computational principles22,23. +2a +Task Action PlanDecision-Maker +Decision-Maker +Plan ConstrualTask +Task Actionb +cFigure 1. Construal and planning. a, A satellite photo of Princeton, NJ (top) and maps of +Princeton for bicycling versus automotive use cases (bottom). Like maps and unlike photographs, +a decision-maker’s construal picks out a manageable subset of details from the world relevant to +their current goals. Imagery ©2022 Google, Map data ©2022. b,Standard models assume that a +decision-maker computes a plan, , with respect to a fixed task representation, T, and then uses +it to guide their actions, a.c,According to our model of value-guided construal , the decision- +maker forms a simplified task construal, Tc, that is used to compute a plan, c. This process can +be understood as two nested optimizations: an “outer loop” of construal and an “inner loop” of +planning. +Our approach to studying how people control task construals starts with the premise that ef- +fective decision-making depends on making rational use of limited cognitive resources1–3. Specif- +ically, we derive how an ideal, cognitively-limited decision-maker should form value-guided con- +3struals that balance the complexity of a representation and its utility for planning and acting. We +then show that pre-registered predictions of this account explain how people attend to task elements +in several planning experiments (see Data Availability Statement). Our analysis and findings sug- +gest that controlled, moment-to-moment task construals play a key role in efficient and flexible +planning. +Task construals from first principles +We build on models of sequential decision-making expressed as Markov Decision Processes24. +Formally, a taskTconsists of a state space, S; an initial state, s02S; an action space, A; a +transition function P:SAS ! [0;1]; and a utility function U:S ! R. In standard +formulations of planning, the value of a plan:SA! [0;1]from a state sis determined +by the expected, cumulative utility of using that plan25:V(s) =U(s) +P +a(ajs)P +s0P(s0j +s;a)V(s0). Standard planning algorithms5(e.g., heuristic search methods) attempt to efficiently +compute plans that optimize value by directly planning over a fixed task representation, T, that +is not subject to the decision-maker’s control (Figure 1b). Our aim is to relax this constraint and +consider the process of adaptively selecting simplified task representations for planning, which we +call the construal process (Figure 1c). +Intuitively, a construal “picks out” details in a task to consider. Here, we examine construals +that pick out cause-effect relationships in a task. This focus is motivated by the intuition that a key +source of task complexity is the interaction of different causes and their effects with one another. +For instance, consider interacting with various objects in someone’s living room. Walking towards +the couch and hitting it is a cause-effect relationship, while pulling on the coffee table and moving +itmight be another such relationship. These individual effects can interact and may or may not be +integrated into a single representation of moving around the living room. For example, imagine +pulling on the coffee table and causing it to move, but in doing so, backing into the couch and +hitting it. Whether or not a decision-maker anticipates and represents the interaction of multiple +4effects depends on what causes and effects are incorporated into their construal; this, in turn, can +impact the outcome of behavior. +Related work has studied how attention guides learning about how different state features pre- +dict rewards26. By contrast, to model construals, we require a way to express how attention flexibly +combines different causes and their effects into an integrated model to use for planning. For this, +we use a product of experts27, a technique from the machine learning literature for combining dis- +tributions that is similar to factored approximations used in models of perception28. Specifically, +we assume that the agent has Nprimitive cause-effect relationships that each assign probabili- +ties to state, action, and next-state transitions, i:SAS ! [0;1],i= 1;:::;N . Each +i(s0js;a)is a potential function representing, say, the local effect of colliding with the couch or +pulling on the coffee table. Then a construal is a subset of these primitive cause-effect relation- +ships,cf1;:::;Ng, that produces a task construal, Tc, with the following construed transition +function: +Pc(s0js;a)/Y +i2ci(s0js;a): (1) +Here, we assume that task construals ( Tc) and the original task ( T) share the same state space, +action space, and utility function. But, crucially, the construed transition function can be simpler +than that of the actual task. +What task construal should a decision-maker select? Ideally, it would be one that only includes +those elements (cause-effect relationships) that lead to successful planning, excluding any others +so as to make the planning problem as simple as possible. To make this intuition precise, it is +essential to first distinguish between computing a plan with a construal and using the plan induced +by a construal. In our example, suppose the decision-maker forms a construal of their living +room that includes the effect of pulling on the coffee table but ignores the effect of colliding with +the couch. They might then compute a plan in which they pull on the coffee table without any +complications, but when they usethat plan in the actual living room, they inadvertently stumble +over their couch. This particular construal is less than optimal. +Thus, we formalize the distinction between the computed plan associated with a construal and +5its resulting behavioral utility : If the decision-maker has a task construal Tc, denote the plan that +optimizes it as c. Then, the utility of the computed plan when starting at state s0is given by its +performance when interacting with the actual transition dynamics, P: +U(c) =U(s0) +X +ac(ajs0)X +s0P(s0js0;a)Vc(s0): (2) +Put simply, the behavioral utility of a construal is determined by the consequences of using it to +plan and act in the actual task. +Having established the relationship between a construal and its utility, we can define the value +of representation (VOR) associated with a construal. Our formulation resembles previous models +of resource-rationality2and the expected value of control13by discounting utilities with a cognitive +cost,C. This cost could be further enriched by specifying algorithm-specific costs29or hard con- +straints30. However, our aim is to understand value-guided construal with respect to the complexity +of the construal itself and with minimal algorithmic assumptions. To this end, we use a cost that +penalizes the number of effects considered: C(c) =jcj, wherejcjis the cardinality of c. Intuitively, +this cost reflects the description length of a program that expresses the construed transition func- +tion in terms of primitive effects31. It also generalizes recent economic models of sparsity-based +behavioral inattention32. The value of representation for construal cis then its behavioral utility +minus its cognitive cost: +VOR (c) =U(c)C(c): (3) +In short, we introduce the notion of a task construal (Equation 1) that relaxes the assumption +of planning over a fixed task representation. We then define an optimality criterion for a construal +based on its complexity and its utility for planning and acting (Equations 2-3). This optimality +criterion provides a normative standard we can use to ask whether people form optimal value- +guided construals33,34. We note that the question of precisely how people identify or learn optimal +construals is beyond the scope of our current aims. Rather, here our goal is to simply determine +whether their planning is consistent with optimal construal. If so, then understanding how people +6achieve (or approximate) this ability will be a key direction for future research (see Supplementary +Discussion of Construal Optimization Algorithms). +A paradigm for examining construals +Do people form construals that optimally balance complexity and utility? To answer this question, +we designed a paradigm analogous to the example in Figure 1a, in which participants were shown +a two-dimensional map of a maze and had to move a blue dot to reach a goal location. On each +trial, participants were shown a new maze composed of a starting location, a goal location, center +black walls in the shape of a +, and an arrangement of blue obstacles. The goal, starting state, +and the blue obstacles (but not the center black walls) changed on every trial, which required +participants to examine the layout of the maze and plan an efficient route to the goal (Figure 2a). +In our framework, each obstacle corresponds to a cause-effect relationship, i—i.e., attempting to +move into the space occupied by the obstacle and then being blocked. This is analogous to the +effect of being blocked by a piece of furniture in our earlier example. +Two key features make our maze-navigation paradigm useful for isolating and studying the +construal process. First, the mazes are fully observable : Complete information about the task +is immediately accessible from the visual stimulus. Second, each instance of a maze emerges +from a particular composition of individual elements (e.g., the obstacles). This means that while +all the components of a particular maze are immediately accessible, participants need to choose +which ones to integrate into an effective representation for planning (i.e., select a construal). Fully +observable but compositionally-structured problems occur routinely in everyday life—e.g., using +a map to navigate through exhibits in a museum—as well as in popular games—e.g., in chess, +figuring out how to move one’s knight across a board occupied by an opponent’s pieces. By +providing people with immediate access to all the components of a task while planning, we can +examine which ones they attend to versus ignore and whether these patterns of awareness reflect +a process of value-guided construal (Methods, Model Implementations, Value-guided Construal +7Implementation; Code Availability Statement). Furthermore, this general paradigm can be used in +concert with several different experimental measures to assess attention (Extended Data Figures +1-3; Supplementary Experimental Materials; Data Availability Statement). +b +An obstacle was either in the yellow or +green location (not both), which one was it? +How confident are you?Goal, agent, and +obstacles appearObstacles are invisible +during navigationRecall probe +Confidence probea +Trial BeginsGoal, agent, and +obstacles appearParticipant navigatesAwareness probe +How aware of the highlighted +obstacle were you at any point? +Figure 2. Maze-navigation paradigm and design of memory probes, Value-guided con- +strual predicts how people will form representations that are simple but useful for planning +and acting. These predictions were tested in a new paradigm in which participants controlled +a blue circle and navigated mazes composed of center black walls in the shape of a cross, blue +tetronimo-shaped obstacles, and a yellow goal state with a shrinking green square. We assume +that attention to obstacles as a result of construal is reflected in memory of obstacles and used +two types of probes to assess memory. a,In our initial experiment, participants were shown the +maze and navigated to the goal (dashed line indicates an example path). After navigating, partic- +ipants were given awareness probes in which they were asked to report their awareness of each +obstacle on an 8-point scale (for analyses, responses were scaled to range from 0 to 1). b,In a +subsequent experiment, obstacles were only visible prior to moving in order to encourage plan- +ning up-front, and participants were given recall probes in which they were shown a pair of ob- +stacles in green and yellow, only one of which had been present in the maze they had just com- +pleted. They were then asked which one had been in the maze as well as their confidence. +8Traces of construals in people’s memory +We assume that the obstacles included in a construal will be associated with greater awareness +and thereby memory; accordingly, we began by probing memory for obstacles after participants +completed each maze to test whether they formed value-guided construals of the mazes. In our ini- +tial experiment, participants received awareness probes in which, following navigation, they were +shown a picture of the maze they had just completed with one of the obstacles highlighted. Then, +they were asked, “How aware of the highlighted obstacle were you at any point?” and responded +on an 8-point scale that was later scaled to range from 0 to 1 for analyses (Figure 2a). If participants +formed representations of the mazes that balance utility and complexity, their responses should be +positively predicted by value-guided construal. This is precisely what we found: Value-guided con- +strual predicted awareness judgments (likelihood ratio test comparing hierarchical linear models +with and without z-score normalized value-guided construal probabilities: 2(1) = 2297:21;p < +1:01016; = 0:133, S.E. = 0:003; Methods, Experiment Analyses; Figure 3). Furthermore, +we also observed the same results when participants could not see the obstacles while moving and +so needed to plan their route entirely up front ( 2(1) = 726:95;p < 1:01016; = 0:115, +S.E.= 0:004). This was also the case when we probed awareness judgments immediately after +planning but before execution (2(1) = 679:20;p< 1:01016; = 0:106, S.E. = 0:004; Meth- +ods, Experimental Design, Up-front Planning Experiment; Supplementary Memory Experiment +Analyses). +9Value-Guided Construal +Expected Obstacle Probability +≤ 0.5 > 0.5ab +c +0.00.51.0 +Participant mean awareness response (experiment) +Value-guided construal probability (predicted) +0.00 0.25 0.50 0.75 1.00 +Initial Experiment Mean Awareness051015Count +Figure 3. Initial experiment results, In our initial planning experiment (out of four), each +person (n= 161 independent participants) navigated twelve 2D mazes, each of which had seven +blue tetronimo-shaped obstacles. To assess whether attention to obstacles reflects a process of +value-guided construal, participants were given an awareness probe (see Figure 2a) for each ob- +stacle in each maze. a,For our first analysis, we split the set of 84 obstacles across mazes based +on whether value-guided construal assigned a probability less than or equal to 0:5or greater than +0:5. Here, we plot two histograms of participants’ mean awareness responses corresponding to +the two sets of obstacles ( 0:5in grey,>0:5in blue; individual by-obstacle mean awareness un- +derlying the histograms are represented underneath). We then similarly split the obstacles based +on whether mean awareness responses were less than or equal to 0:5or greater than 0:5and, us- +ing a chi-squared test for independence, found that this split was predicted by value-guided con- +strual (2(1;N= 84) = 23:03,p= 1:6106, effect size w= 0:52).b,Value-guided construal +predictions for three of the twelve mazes used in the experiment (blue circle indicates the starting +location, green and yellow square indicates the goal; obstacle colors represent model probabilities +according to the colorbar). c,Participant mean awareness judgments for the same three mazes +(obstacle colors represent mean judgments according to the colorbar). Responses in this initial +experiment generally reflect value-guided construal of mazes. Participants were recruited through +the Prolific online experiment platform. +While the awareness probes provide useful insight into people’s task construals, it is a step +removed from their memory (which is already a step removed from the construal process itself) +since it requires participants to reflect on their earlier awareness during planning. To address this +limitation, we developed a second set of critical mazes with two properties. First, the mazes were +10designed to test the distinctive predictions of value-guided construal (e.g., Figure 4a). Second, +these new mazes allowed us to use a more stringent measure of memory for task elements. Specif- +ically, we used obstacle recall probes , in which, following navigation, participants were shown a +grid with the black center walls, a green obstacle, a yellow obstacle, and no other obstacles. Either +the green or yellow obstacle had actually been present in the maze, whereas the other obstacle did +not overlap with any of those that had been present. Participants were then asked, “An obstacle +was either in the yellow or green location (not both), which one was it?” and could select either op- +tion, followed by a confidence judgment on an 8-point scale (Figure 2b; Extended Data Figure 4a). +The recall probes thus provided two measures, accuracy and confidence, and using hierarchical +generalized linear models (HGLMs) we found that value-guided construal predicted both types of +responses (likelihood ratio tests comparing models on accuracy: 2(1) = 249:34;p< 1:01016; + = 0:648, S.E. = 0:042; and confidence: 2(1) = 432:76;p < 1:01016; = 0:104, +S.E.= 0:005. Methods, Experiment Analyses). Additionally, when we gave a separate group +of participants the awareness probes on these mazes, value-guided construal was again predictive +(Awareness: 2(1) = 837:47;p < 1:01016; = 0:175, S.E. = 0:006). Thus, using three +different measures of memory (recall accuracy, recall confidence, and awareness judgments), we +found further evidence that when planning, people form task representations that optimally balance +complexity and utility. +11a b +c +0.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 +Accuracy0.00.30.40.50.60.70.80.9ConfidencePlanning +0.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 +Accuracy0.00.30.40.50.60.70.80.9Perception Control +Obstacle Type +Relevant/Near +Relevant/Far (Critical) +Irrelevant +0.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 +Accuracy0.00.30.40.50.60.70.80.9Execution Control +Optimal PathIrrelevantIrrelevant +Relevant/Far +(Critical)Relevant/Near +0 20 40 60 80 100 120 +Change in AICVGC +Traj HS +Graph HS +Bottleneck +SR Overlap +Nav Dist +Nav Dist Step +Goal Dist +Start Dist +Wall Dist +Center DistLesioned +Predictor +Figure 4. Critical mazes recall experiment, model comparisons, and control studies. a, +The critical mazes recall experiment ( n= 78 independent participants; one version of one of the +four planning experiments) used critical mazes that included critical obstacles that were highly +relevant to planning but far from an optimal path (dashed line). Value-guided construal predicts +critical obstacles will be included in a construal while irrelevant obstacles will not, indepen- +dent of distance to the optimal path. b,We fit a global model to recall responses that included +the fixed parameter value-guided construal modification model (VGC) along with ten alternative +predictors based on heuristic search models, successor representation-based predictors, and low- +level perceptual cues (see Methods, Experiment Analyses). Then, each predictor was removed +from this global model, and we calculated the resulting change in fit (in AIC). Removing value- +guided construal led to the largest degradation of fit (greatest increase in AIC), underscoring its +unique explanatory value. c,In a pair of non-planning control experiments, new participants ei- +ther viewed patterns that looked exactly like the mazes (perceptual control; n= 88 independent +participants) or followed “breadcrumbs” through the maze along a path taken by a participant +from the original experiment (execution control; n= 80 independent participants). They then an- +swered the exact same recall questions. Value-guided construal remains a significant factor when +explaining recall in the original critical mazes experiment (planning) while including mean re- +call from the perceptual and execution controls as covariates (likelihood ratio test for accuracy: +2(1) = 106:36;p= 6:21025; confidence: 2(1) = 18:56;p= 1:6105;p-values +are unmodified). This confirms that responses consistent with value-guided construal are not a +simple function of perception and execution. Participants were recruited through the Prolific on- +line experiment platform. Plotted are the mean values for each obstacle, with relevant/near, rele- +vant/far (critical), and irrelevant obstacle types distinguished. Error bars are standard errors about +the mean. +12Controlling for perception and execution +The memory studies provide preliminary confirmation of our hypothesis, but they have several +limitations. One is that, although participants were engaged in planning , they were also necessarily +engaged in other forms of cognitive processing, and these unrelated processes may have influenced +memory of the obstacles. In particular, participants’ perception of a maze or their execution of a +particular plan through a maze may have influenced their responses to the memory probes. This +potentially confounds the interpretation of our results, since a key part of our hypothesis is that +task construals arise from planning , rather than simply perceiving or executing. +Thus, to test that responses to the memory probes cannot be fully explained by perception +and/or execution, we administered two sets of yoked controls that did not require planning (Meth- +ods, Experimental Design, Control Experiments). In the perceptual controls , new participants were +shown patterns that looked exactly like the mazes, but they performed an unrelated, non-planning +task. Each pattern was presented to a new participant for the same amount of time that a partic- +ipant in the original experiments had examined the corresponding maze before moving—i.e., the +amount of time the original participant spent examining the maze to plan. The new participant then +responded to the same probes, in the same order, as the original participant. For the execution con- +trols, we recruited another group of participants and gave them instructions similar to those in the +planning experiments. However, unlike the original experiments, the task did not require planning. +Rather, these mazes included “breadcrumbs” that needed to be collected and that appeared every +two steps. Breadcrumbs appeared along the exact path taken by one of the original participants, +meaning that the new participant executed the same actions but without having planned . After +completing each maze, the participant then received the same probes, in the same order, as the +original participant. +We assessed whether responses in the planning experiments can be explained by a simple +combination of perception and/or execution by testing whether value-guided construal remained +a significant factor after accounting for control responses. Specifically, we used the mean by- +obstacle responses from the perceptual and execution controls as predictors in HGLMs fit to +13the corresponding planning responses. We then tested whether adding value-guided construal +as a predictor improved fits. For the awareness, accuracy, and confidence responses in the re- +call experiment, we found that including value-guided construal significantly improved fits (like- +lihood ratio tests comparing models on accuracy: 2(1) = 106:36;p= 6:21025; confi- +dence:2(1) = 18:56;p= 1:6105; and awareness: 2(1) = 55:34;p= 1:01013) +and that value-guided construal predictions were positively associated with responses (coefficients +for accuracy: = 0:58;S.E. = 0:058; confidence: = 0:039;S.E. = 0:009; and awareness: + = 0:054;S.E. = 0:007). Thus, responses following planning are not reducible to a simple +combination of perception andexecution , and they can be further explained by the formation of +value-guided construals (Figure 4c; Supplementary Control Experiment Analyses). +Externalizing the planning process +Another limitation of the previous planning experiments is that they assess construal after planning +is complete (i.e., by probing memory). To obtain a measure of the planning process as it unfolds , +we developed a novel process-tracing paradigm . In this version of the task, participants never +saw all of the obstacles at once. Instead, at the beginning of the trial, after being shown the start +and goal locations, they could use their mouse to reveal individual obstacles by hovering over them +(Methods, Experimental Design, Process-tracing Experiments; Extended Data Figure 4b). This led +participants to externalize the planning process, and so their behavior on this task provides insight +into how planning computations unfolded internally. We tested whether value-guided construal +accounted for behavior by analyzing two measures: whether an obstacle was hovered over and, if +it was hovered over, the duration of hovering. Value-guided construal was a significant predictor +for both these measures on both the initial mazes (likelihood ratio tests comparing HGLMs for +hovering:2(1) = 1221:76;p < 1:01016; = 0:704, S.E. = 0:021; and hover duration [log +milliseconds]: 2(1) = 169:90;p < 1:01016; = 0:161, S.E. = 0:012) and on the critical +mazes (hovering: 2(1) = 1361:92;p < 1:01016; = 0:802, S.E. = 0:023; hover duration +14[log milliseconds]: 2(1) = 540:63;p < 1:01016; = 0:369, S.E. = 0:016). These results +thus complement our original memory-based measurements of people’s task representations and +strengthen the interpretation of them in terms of value-guided construal during planning. +Value-guided construal modification +Thus far, our account of value-guided construal has assumed that an obstacle is either always +or never included in a construal. This simplification is useful since it enables us to derive clear +qualitative predictions based on whether a plan is influenced by an obstacle, but it overlooks graded +factors such as how much of a plan is influenced by an obstacle. For example, an obstacle may only +be relevant for planning a few movements around a participant’s initial location in a maze and, as +a result, could receive less total attention than one that is relevant for deciding how to act across +a larger area of the maze. To characterize these more fine-grained attentional processes, we first +generalized the original construal selection problem to a one in which the decision-maker revisits +and potentially modifies their construal during planning. Then, we derived obstacle awareness +predictions based on a theoretically optimal construal modification policy that balances complexity +and utility (Methods, Model Implementation, Value-Guided Construal). +To assess value-guided construal modification, we re-analyzed our data using three versions of +the model with increasing ability to capture variability in responses. First, we used an idealized +fixed parameter model to derive a single set of obstacle attention predictions and confirmed that +they also predict participant responses on the planning tasks (Supplementary Construal Modifica- +tion Analyses). Second, for each planning measure and experiment, we calculated fitted parameter +models in which noise parameters for the computed plan and construal modification policy were +fit (Methods, Model Implementation, Value-Guided Construal). Scatter plots comparing mean by- +obstacle responses and model outputs for parameters with the highest R2are shown in Figure 5. +Finally, we fit a set of models that allowed for biases in computed plans (e.g., a bias to stay along +the edge of a maze or an explicit penalty for bumping into walls) and found that this additional ex- +15pressiveness led to obstacle attention predictions with an improved correspondence to participant +responses (Supplementary Construal Modification Analyses). Together, these analyses provide +additional insight into the fine-grained dynamic structure of value-guided construal modification. +0.00 0.25 0.50 0.75 1.00 +Fitted Value-Guided +Construal Modification Prob0.20.40.60.8Initial Exp +Awareness Judgment +R2=0.53 +0.00 0.25 0.50 0.75 1.00 +Fitted Value-Guided +Construal Modification Prob0.20.40.60.8Up-Front Planning Exp +Awareness Judgment +R2=0.44 +0.00 0.25 0.50 0.75 1.00 +Fitted Value-Guided +Construal Modification Prob0.40.50.60.70.80.91.0Critical Maze Exp +Recall Accuracy +R2=0.87 +0.00 0.25 0.50 0.75 1.00 +Fitted Value-Guided +Construal Modification Prob0.40.50.60.70.80.9Critical Maze Exp +Recall Confidence +R2=0.81 +0.00 0.25 0.50 0.75 1.00 +Fitted Value-Guided +Construal Modification Prob0.20.40.60.8Critical Maze Exp +Awareness Judgment +R2=0.74 +0.00 0.25 0.50 0.75 1.00 +Fitted Value-Guided +Construal Modification Prob0.00.20.40.60.81.0Process-Tracing +(Initial Mazes) +Hovering +R2=0.42 +0.0 0.5 1.0 +Fitted Value-Guided +Construal Modification Prob5.05.56.06.57.07.5Process-Tracing +(Initial Mazes) +Log-Hover Duration +R2=0.30 +0.00 0.25 0.50 0.75 1.00 +Fitted Value-Guided +Construal Modification Prob0.00.20.40.60.81.0Process-Tracing +(Critical Mazes) +Hovering +R2=0.61 +0.00 0.25 0.50 0.75 1.00 +Fitted Value-Guided +Construal Modification Prob5.56.06.57.0Process-Tracing +(Critical Mazes) +Log-Hover Duration +R2=0.48 +Figure 5. Fitted value-guided construal modification. Our initial model of value-guided +construal focuses on whether an obstacle should or should not be included in a construal. We de- +veloped a generalization that additionally accounts for how much an obstacle influences a plan if +a decision-maker is optimally modifying their construal during planning (Methods, Model Im- +plementations, Value-Guided Construal). We used an "-softmax noise model [35] for computed +action plans and construal modification policies and, for each experiment and measure, searched +for parameters that maximize the R2between model predictions and mean by-obstacle responses. +Shown here are plots comparing scores that the fitted construal modification model assigns to +each obstacle with participants’ mean by-obstacle responses for the nine measures. +Accounting for alternative mechanisms +While the analyses so far confirm the predictive power of value-guided construal, it is also im- +portant to consider alternative planning processes. For instance, differential awareness could have +been a passive side-effect of planning computations , rather than an active facilitator of planning +computations as posited by value-guided construal. In particular, participants could have been +planning by performing heuristic search over action sequences without actively construing the task, +which would have led to differential awareness of obstacles as a byproduct of planning. Differ- +ential awareness could also have arisen from alternative representational processes, such as those +16based on the successor representation36or related subgoaling mechanisms37. Similarly, perceptual +factors, such as the distance to the start, goal, walls, center, optimal path, or path taken, could have +influenced responses. +Based on these considerations, we identified ten alternative predictors (Methods, Model Imple- +mentations; Extended Data Figures 5, 6, and 7; Code Availability Statement). All ten predictors +plus the fixed value-guided construal modification predictions were included in global models that +were fit to each of the nine planning experiment measures, and, in all cases, value-guided construal +was a significant predictor (Extended Data Table 1; see Supplementary Alternative Mechanisms +Analyses for the same analyses with the single-construal model). +Furthermore, to assess the relative importance of each predictor, we calculated the change in +fit (in terms of AIC) that resulted from removing each predictor from a global model (Methods, +Experiment Analyses). Across all planning experiment measures, removing value-guided con- +strual led to the first or second largest reduction in fit (Figure 4b; Extended Data Table 1). These +“knock-out” analyses demonstrate the explanatory necessity of value-guided construal. To assess +explanatory sufficiency , we fit a new set of single-predictor and two-predictor models using all pre- +dictors and then calculated their AICs (Methods, Experiment Analyses; Extended Data Figure 8). +For all nine experimental measures, value-guided construal was one of the top two single-predictor +models and was one of the two factors included in the best two-predictor model. Together, these +analyses confirm the explanatory necessity and sufficiency of value-guided construal. +Discussion +We tested the idea that when people plan, they do so by constructing a simplified mental representa- +tion of a problem that is sufficient to solve it—a process that we refer to as value-guided construal. +We began by formally articulating how an ideal, cognitively-limited decision-maker should con- +strue a task so as to balance complexity and utility. Then, we showed that pre-registered predictions +of this model explain people’s awareness, ability to recall problem elements (obstacles in a maze), +17confidence in recall ability, and behavior in a process-tracing paradigm, even after controlling for +the baseline influence of perception and execution as well as ten alternative mechanisms. These +findings support the hypothesis that people make use of a controlled process of value-guided con- +strual, and that it can help explain the efficiency of human planning. More generally, our account +provides a framework for further investigating the cognitive mechanisms involved in construal. For +instance, how are construal strategies acquired? How is construal selection shaped by computation +costs, time, or constraints? 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To derive empirical predictions for the maze +tasks, we assume a set of primitive cause-effect relationships, each of which is analogous to the +example of interacting with furniture in a living room (see main text). For each maze, we modeled +the following: The default effect of movement (i.e., pressing an arrow key causes the circle to +move in that direction with probability 1"and stay in place with probability ","= 105),Move; +the effect of being blocked by the center, plus-shaped ( +) walls (i.e., the wall causes the circle to +notmove when the arrow key is pressed), Walls; and effects of being blocked by each of the N +obstacles,Obstacle i;i= 1;:::;N . Since every maze includes the same movements and walls, the +model only selected which obstacle effects to include. The utility function for all mazes was given +by a step cost of1until the goal state was reached. +Value-guided construal posits a bilevel optimization procedure involving an “outer loop” of +construal and an “inner loop” of planning. Here, we exhaustively calculate potential solutions to +this nested optimization problem by enumerating and planning with all possible construals (i.e., +subsets of obstacle effects). We exactly solved the inner loop of planning for each construal us- +ing dynamic programming40and then evaluated the optimal stochastic computed plan under the +actual task dynamics (i.e., Equation 2). For planning and evaluation, transition probabilities were +multiplied by a discount rate of :99was used to ensure values were finite. The general procedure +for calculating the value of construals is outlined in the algorithm in Extended Data Table 2. To +be clear, our current research strategy is to derive theoretically optimal predictions for the inner +loop of planning and outer loop of construal in the spirit of resource-rational analysis2. Thus, +this specific procedure should not be interpreted as a process model of human construal. In the +Supplemental Discussion of Algorithms for Construal Optimization, we discuss the feasibility of +22optimizing construals and how an important direction for future research will involve investigating +tractable algorithms for finding good construals. +Given a value of representation function, VOR, that assigns a value to each construal, we model +participants as selecting a construal according to a softmax decision-rule: +P(c)/exp + 1VOR (c) +; (4) +where > 0is a temperature parameter (for our pre-registered predictions = 0:1). We then +calculated a marginalized probability for each obstacle being included in the construal, from the +initial state, s0, corresponding to the expected awareness of that obstacle: +P(Obstacle i) =X +c1[Obstacle i2c]P(c); (5) +where, for a statement X, 1[X]evaluates to 1ifXis true and 0ifXis false. We implemented this +model in Python 3.7 using the msdm library (see Code Availability Statement). +The basic value-guided construal model makes the simplifying assumption that the decision- +maker plans with a single static construal. We can extend this idea to consider a decision-maker +who revisits and potentially modifies their construal at each stage of planning. In particular, we +can conceptualize this process in terms of a sequential decision-making problem induced by the +interaction between task dynamics (e.g., a maze) and the internal state of an agent (e.g., a con- +strual) [41]. The solution to this problem is then a sequence of modified construals associated with +planning over different parts of the task (e.g., planning movements for different areas of the maze). +Formally, we denote the set of possible construals as C=P(f1;:::;Ng), the powerset of +cause-effect relationships, and define a construal modification Markov Decision Process , which +has a state space corresponding to the Cartesian product of task states and construals, (s;c)2SC , +and an action space corresponding to possible next construals, c02 C. Having chosen a new +construalc0, the probability of transitioning from task state stos0comes from first calculating +a joint distribution using the actual transition function P(s0js;a)and planc0(ajs)and then +23marginalizing over task actions a: +P(s0js;c0) =X +ac0(ajs)P(s0js;a): (6) +In this construal modification setting, the analogue to the value of representation (VOR; Equa- +tion 3) is the optimal construal modification value function , defined over all s;c: +V(s;c) =U(s) + max +c0"X +s0P(s0js;c0)V(s0;c0)C(c0;c)# +; (7) +whereC(c0;c) =jc0cjis the number of additional1cause-effect relationships in the new construal +c0compared to c. Importantly, this cost on modifying the construal encourages consistency—i.e., +withoutC(c0;c), a decision-maker would have no disincentive to completely change their construal +for each state. Note that in the special case where c=?, we recover the original static construal +cost for a single step. Finally, using the construal modification value function, we define a softmax +policy over the task/construal state space, (c0js;c)/expf 1 +c[P +s0P(s0js;c0)V(s0;c0)C(c0;c)]g. +For the fixed parameter model we set c= 0:1(as with the single-construal model). +The construal modification formulation allows us to consider not just whether an obstacle ap- +pears in a construal, but also how long it appears in a construal. In particular, we would like to +compute a quantity that is analogous to Equation 5 that assigns model values for each obstacle. +To do this, we use the normalized task/construal state occupancy induced by a construal policy  +from the initial task/construal state, (s;cjs0;c0)/M(s0;c0;s;c), wherec0=?andMis +the successor representation under (for a self-contained review of M, see the section on Succes- +sor Representation-based Predictors below). Given a policy and starting task state s0, for each +obstacle, we calculate the probability of having a construal that includes that obstacle: +P(Obstacle i) =X +s;c1[Obstacle i2c](s;cjs0;c0): (8) +1For sets AandB, the set difference AB=fa:a2Aanda =2Bg. +24To calculate the optimal construal modification value function, V(s;c), for each maze, we con- +structed construal modification Markov Decision Processes in Python (3.7) using scipy (1.5.2) +sparse matrices [42]. We then exactly solved for V(s;c)using a custom implementation of policy +iteration [43] designed to take advantage of the sparse matrix data structure (see Code Availability +Statement). For the fitted parameter models, we used separate "-softmax noise models [35] for the +computed plans, c(ajs), and construal modification policy, (c0js;c), and performed a grid +search over the four parameters for each of the nine planning measures ( 1 +a2f1;3;5;7g;"a2 +f0:0;0:1;0:2g; 1 +c2f1;3;5;7;9g;"c2f0;0:05;0:1;0:2;0:3g). Additionally, for parameter fit- +ting, we limited the construals c02C to be of size three. This improves the speed of parameter +evaluation and yields results comparable to the fixed parameter model, which uses the full con- +strual set. Finally, to obtain obstacle value-guided construal probabilities we simulate 1000 rollouts +of the construal modification policy to estimate (js0;c0). As with the initial model, we empha- +size that these procedures are not intended as an algorithmic account of construal modification, but +rather allow us to derive theoretically optimal predictions of the fine-grained dynamics of value- +guided construals during planning. +Heuristic Search Over Action Sequences +Value-guided construal posits that people control their task representations to actively facilitate +planning , which, in the maze navigation paradigm, leads to differential attention to obstacles. How- +ever, differential attention could also occur as a passive side-effect of planning , even in the absence +of active construal. In particular, heuristic search over action sequences is another mechanism for +reducing the cost of planning, but it accomplishes this in a different way: by examining possible +action sequences in order of how promising they seem, not by simplifying the task representation. +If people are simulating candidate action sequences via heuristic search (and not engaged in an ac- +tive construal process), differential attention to task elements could have simply been a side-effect +of how those simulations unfolded. +Thus, we wanted to derive predictions of differential awareness as a byproduct of search over +25action sequences. To do so, we considered two general classes of heuristic search algorithms. +The first, a variant of Real-Time Dynamic Programming (RTDP)44,45, is a trajectory-based search +algorithm that simulates physically realizable trajectories (i.e., sequences of states and actions that +could be generated by repeatedly calling a fixed transition function). The algorithm works by +first initializing a heuristic value function (e.g., based on domain knowledge). Then, it simulates +trajectories that greedily maximize the heuristic value function while also performing Bellman +updates at simulated states44. This scheme then leads RTDP to simulate states in order of how +promising they are (according to the continuously updated heuristic value function) until the value +function converges. Importantly, RTDP can end up visiting a fraction of the total state space, +depending on the heuristic. Our implementation was based on the Labeled RTDP algorithm of +Bonet & Geffner45, which additionally includes a labeling scheme that marks states where the +estimate of the value function has converged, leading to faster overall convergence. +To derive obstacle awareness predictions, we ran RTDP (implemented in msdm ; see Code +Availability Statement) on each maze and initialized it with a heuristic corresponding to the optimal +value function assuming there are plus-shaped walls but no obstacles . This models the background +knowledge participants have about distances, while also providing a fair comparison to the initial +information provided to the value-guided construal implementation. Additionally, if at any point +the algorithm had to choose actions based on estimated value, ties were resolved randomly, making +the algorithm stochastic. For each maze, we ran 200 simulations of the algorithm to convergence +and examined which states were visited by the algorithm over all simulations. We calculated the +mean number of times each obstacle was hitby the algorithm, where a hit was defined as a visit +to a state adjacent to an obstacle such that the obstacle was in between the state and the goal. +Because the distribution of hit counts has a long tail, we used the natural log of hit counts +1as +the obstacle hit scores. The reason why the raw hit counts have a long tail is due to the particular +way in which RTDP calculates the value of regions where the heuristic value is much higher than +the actual value (e.g., dead ends in a maze). Specifically, RTDP explores such regions until it has +confirmed that it is no better than an alternative path, which can take many steps. More generally, +26trajectory-based algorithms are limited in that they can only update states by simulating physically +realizable trajectories starting from the initial state. +The limitations of trajectory-based planning algorithms motivated our use of a second class +ofgraph-based planning algorithms. We used LAO46, a version of the classic Aalgorithm47 +generalized to be used on Markov Decision Processes (implemented in msdm ; see Code Availabil- +ity Statement). Unlike trajectory-based algorithms, graph-based algorithms like LAOmaintain a +graph of previously simulated states. LAOin particular builds a graph of the task rooted at the +initial state and then continuously plans over the graph. If it computes a plan that leads it to a state +at the edge of the graph, the graph is expanded according to the transition model to include that +state and then the planning cycle is restarted. Otherwise, if it computes an optimal plan that only +visits states in the simulated graph, the algorithm terminates. By continuously expanding the task +graph and performing planning updates, the algorithm can intelligently explore the most promising +(according to the heuristic) regions of the state space being constrained to physically realizable se- +quences. In particular, graph-based algorithms can quickly “backtrack” when they encounter dead +ends. +Obstacle awareness predictions based on LAOwere derived by using the same initial heuristic +as was used for RTDP and a similar scheme for handling ties. We then calculated the total number +of times an obstacle was hit during graph expansion phases only, using the same definition of a hit +as above. For each maze, we generated 200 planning simulations and used the raw hit counts as +the hit score. +Algorithms like RTDP and LAOplan by simulating realizable action sequences that begin at +the start state. As a result, these models tend to predict greater awareness to obstacles that are near +the start state and are consistent with the initial heuristic, regardless of whether those obstacles +strongly affect or lie along the final optimal path. For instance, obstacles down initially promising +dead ends have a high hit score. This contrasts with value-guided construal, which predicts greater +attention to relevant obstacles, even if they are distant, and lower attention to irrelevant ones, even +if they are nearby. For an example of these distinct model predictions, see maze #14 in Extended +27Data Figure 6. +To be clear, our goal was to obtain predictions for search over action sequences in the absence +of an active construal process for comparison with value-guided construal. However, in general, +heuristic search and value-guided construal are complementary mechanisms, since the former is a +way to plan given a representation and the latter is a way to choose a representation for planning. +For instance, one could perform heuristic search over a construed planning model, or a construal +could help with selecting a heuristic to guide search over actions. These kinds of interactions +between action-sequence search and construal are important directions for future research that can +be built on the ideas developed here. +Successor Representation-based Predictors +We also considered two measures based on the successor representation , which has been proposed +as a component in several computational theories of efficient sequential decision-making36,48. Im- +portantly, the successor representation is not a specific model; rather it is a predictive coding of a +task in which states are represented in terms of the future states likely to be visited from that state, +given the decision-maker follows a certain policy. Formally, the value function of a policy (ajs) +can be expressed in the following two equivalent ways: +V(s) =U(s) +X +a(ajs)X +s0P(s0js;a)V(s0) (9) +=X +s+M(s;s+)U(s+); (10) +whereM(s;s+)is expected occupancy of s+starting from s, when acting according to . The +successor representation of a state sunderis then the vector M(s;). Algorithmically, Mcan +be calculated by solving a set of recursive equations (implemented in Python with numpy49; see +Code Availability Statement): +M(s;s+) = 1[s=s+] +X +a;s0(ajs)P(s0js;a)M(s0;s+): (11) +28Again, the successor representation is not itself an algorithm, but rather a policy-conditioned re- +coding of states that can be a component of a larger computational process (e.g, different kinds +of learning or planning). Here, we focus on its use in the context of transfer learning48,50and +bottleneck states37,51. +Research on transfer learning posits that the successor representation supports transfer that is +more flexible than pure model-free mechanisms but less flexible than model-based planning. For +example, Russek et al.50model agents that learned a successor representation for the optimal pol- +icy in an initial maze and then examined transfer when the maze was changed (e.g., adding in a +new barrier). While their work focuses on learning, rather than planning, we can borrow the ba- +sic insight that the successor representation induced by the optimal policy for a source task can +influence the encoding of a target task, which constitutes a form of construal. In our experiments, +participants were not trained on any particular source task, but we can use the maze with all obsta- +cles removed as a proxy (i.e., representing what all mazes had in common). Thus, we calculated +the optimal policy for the maze without any obstacles (but with the start and goal), computed the +successor representation M, and then calculated, for each obstacle iin the actual maze with the +obstacles, a successor representation overlap (SR-Overlap) score: +SR-Overlap (i) =X +s2ObsiM(s0;s); (12) +wheres0is the starting state and Obs iis the set of states occupied by the obstacle i. This quantity +can be interpreted as the amount of overlap between an obstacle and the successor representation of +the starting state. If the successor representation shapes how people represent tasks, this quantity +would be associated with greater awareness of certain obstacles. +The second predictor is related to the idea of bottleneck states . These emerge from how the +successor representation encodes multi-scale task structure37, and they have been proposed as a +basis for subgoal selection51. If bottlenecks guide subgoal selection, then distance to bottleneck +states could give rise to differential awareness of obstacles via subgoaling processes. Thus, we +29wanted to test that responses consistent with value-guided construal were not entirely attributable to +the effect of bottleneck states calculated in the absence of an active construal process. Importantly, +we note that as with alternative planning mechanisms like heuristic search, the identification of +bottleneck states for subgoaling is compatible with value-guided construal (e.g., one could identify +subgoals for a construed version of a task). +When viewing the transition function of a task (e.g., a maze) as a graph over states, bottleneck +states lie on either side of a partitioning of the state space into two regions such that there is high +intra-region connectivity and low inter-region connectivity. This can be computed for any transition +function using the normalized min-cuts algorithm52or derived from the second eigenvector of the +successor representation under a random policy37. Here, we use a variant of the second approach +as described in the appendix of37. Formally, given a transition function, P(s0js;a), we define an +adjacency matrix, A(s;s0) = 1[9as.t.P(s0js;a)>0], and a diagonal degree matrix, D(s;s) = +P +s0A(s;s0). Then, the graph Laplacian, a representation often used to derive low-dimensional +embeddings of graphs in spectral graph theory, is L=DA. We take the eigenvector with +the second largest eigenvalue, which assigns a positive or negative value to each state in the task. +This vector can be interpreted as projecting the state space onto a single dimension in a way that +best preserves connectivity information, with a zero point that represents the mid-point of the +projected graph. Bottleneck states correspond to those states nearest to 0. For each maze, we +used this method to identify bottleneck states and further reduced these to the optimal bottleneck +states , defined as bottleneck states with a non-zero probability of being visited under the optimal +stochastic policy for the maze. Finally, for each obstacle, we calculated a bottleneck distance score, +the minimum Manhattan distance from an obstacle to any of these bottleneck states. +Notably, value-guided construal also predicts greater attention to obstacles that form bottle- +necks because one often needs to carefully navigate through them to reach the goal. However, +the predictions of our model differ for obstacles that are distant from the bottleneck. Specifically, +value-guided construal predicts greater attention to relevant obstacles that affect the optimal plan, +even if they are far from the bottleneck (e.g., see model predictions for maze #2 in Extended Data +30Figure 5). +Perceptual Landmarks +Finally, we considered several predictors based on low-level perceptual landmarks and partici- +pants’ behavior. These included the minimum Manhattan distance from an obstacle to the start +location, the goal location, the center black walls, the center of the grid, and any of the locations +visited by the participant in a trial (navigation distance). We also considered the timestep at which +participants were closest to an object as a measure of how recently they were near an object. In +cases where navigation distance was not an appropriate measure (e.g., if participants never nav- +igated to the goal), we used the minimum Manhattan distance to trajectories sampled from the +optimal policy averaged over 100 samples. +Experimental Design +All experiments were pre-registered (see Data Availability Statement) and approved by the Prince- +ton Institutional Review Board (IRB). All participants were recruited from the Prolific online plat- +form and provided informed consent. At the end of each experiment, participants provided free- +response demographic information (age and gender, coded as male/female/neither). Experiments +were implemented with psiTurk53and jsPsych54frameworks (see Code Availability Statement). +Instructions and example trials are shown in the Supplementary Experimental Materials. +Initial experiment +Our initial experiment used a maze-navigation task in which participants moved a circle from +a starting location on a grid to a goal location using the arrow keys. The set of initial mazes +consisted of twelve 11 11 mazes with seven blue tetronimo-shaped obstacles and center walls +arranged in a cross that blocked movement. On each trial, participants were first shown a screen +displaying only the center walls. When they pressed the spacebar, the circle they controlled, the +goal, and the obstacles appeared, and they could begin moving immediately. In addition, to ensure +31that participants remained focused on moving, we placed a green square on the goal that shrank +and would disappear after 1000ms but reset whenever an arrow key was pressed, except at the +beginning of the trial when the green square took longer to shrink (5000ms). Participants received +$0.10 for reaching the goal without the green square disappearing (in addition to the base pay +of $0.98). The mazes were pseudo-randomly rotated or flipped, so the start and end state was +constantly changing, and the order of mazes were pseudo-randomized. After completing each +trial, participants received awareness probes, which showed a static image of the maze they had +just navigated, with one of the obstacles shown in light blue. They were asked “How aware of the +highlighted obstacle were you at any point?” and could respond using an 8-point scale (rescaled +from 0 to 1 for analyses). Probes were presented for the seven obstacles in a maze. None of the +probes were associated with a bonus. +We requested 200 participants on Prolific and received 194 complete submissions. Following +pre-registered exclusion criteria, a trial was excluded if, during navigation, >5000ms was spent at +the initial state, >2000ms was spent at any non-initial state, >20000ms was spent on the entire +trial, or>1500ms was spent in the last three steps in total. Participants with <80% of trials after +exclusions or who failed 2 of 3 comprehension questions were excluded, which resulted in n= 161 +participants’ data being analyzed (median age of 28;81male, 75female, 5neither). +Up-front planning experiment +The up-front planning version of the memory experiment was designed to dissociate planning +and execution. The main change was that after participants took their first step, all of the blue +obstacles (but not the walls or goal) were removed from the display (though they still blocked +movement). This strongly encouraged planning prior to execution. To provide sufficient time to +plan, the green square took 60000ms to shrink on the first step. Additionally, on a random half +of the trials, after taking two steps, participants were immediately presented with the awareness +probes ( early termination trials). The other half were fulltrials. We reasoned that responses +following early termination trials would better reflect awareness after planning but before execution +32(see Supplementary Memory Experiment Analyses for analyses comparing early versus full trials). +We requested 200 participants on Prolific and received 188 complete submissions. The exclu- +sion criteria were the same as in the initial experiment, except that the initial state and total trial +time criteria were raised to 30000ms and 60000ms, respectively. After exclusions, we analyzed +data fromn= 162 participants (median age of 28; 85 male, 72 female, 5 neither). +Critical mazes experiment +In the critical mazes experiment , participants again could not see the obstacles while executing +and so needed to plan up front, but no trials ended early. There were two main differences with +the previous experiments. First, we used a set of four critical mazes that included critical obsta- +cles chosen to test predictions specific to value-guided construal. These were obstacles relevant +to decision-making, but distant from the optimal path (see Supplementary Memory Experiment +Analyses for analyses focusing on these critical obstacles). Second, half of the participants re- +ceived recall probes in which they were shown a static image of the grid with only the walls, a +green obstacle, and a yellow obstacle. They were then asked “An obstacle was either in the yellow +or green location (not both), which one was it?” and could select either option, followed by a +confidence judgment on an 8-point scale (rescaled from 0 to 1 for analyses). Pairs of obstacles +and their contrasts in the critical mazes are shown in Extended Data Figure 4a. Participants each +received two blocks of the four critical mazes, pseudo-randomly oriented and/or flipped. +We requested 200 participants on Prolific and received 199 complete submissions. The trial +and participant exclusion criteria were the same as in the up-front planning experiment. After +exclusions, we analyzed data from n= 156 participants (median age of 26; 78 male, 75 female, 3 +neither). +Control Experiments +The aim of the control experiments was to obtain yoked baselines for perception and execution for +comparison with probe responses in the memory studies. The perceptual control used a variant of +33the task in which participants were shown patterns that were perceptually identical to the mazes. +Instead of solving a maze, they were told to “catch the red dot”: On each trial, a small red dot could +appear anywhere on the grid, and participants were rewarded based on whether they pressed the +spacebar after it appeared. Each participant was yoked to the responses of a participant from either +theup-front planning orcritical mazes experiments. On yoked trials , participants were shown +the exact same maze/pattern as their counterpart. Additionally, they were shown the pattern for +the amount of time that their counterpart took before making their first move—since the obstacles +were not visible during execution for the counterpart, this is roughly the time the counterpart spent +looking at the maze to plan. A red dot never appeared on these trials, and they were followed by +the exact same probes that the counterpart received. References to “obstacles” were changed to +“tiles” (e.g., “highlighted tiles” as opposed to “highlighted obstacle” for the awareness probes). +We also included dummy trials , which showed mazes in orientations not appearing in the yoked +trials, for durations sampled from the yoked durations. Half of the dummy trials had red dots. We +recruited enough participants such that at least one participant was matched to each participant +from the original experiments and excluded people who said that they had participated in a similar +experiment. This resulted in data from n= 164 participants being analyzed for the initial mazes +perceptual control (median age of 30:5; 84 male, 79 female, 1 neither) and n= 172 for the critical +mazes perceptual control (median age of 36.5; 86 male, 85 female, 1 neither). +The execution control used a variant of the task in which participants followed a series of +“breadcrumbs” through the maze to the goal and so did not need to plan a path to the goal. Each +participant was yoked to a counterpart in either the initial experiment or the critical mazes experi- +ment so that the breadcrumbs were generated based on the exact path taken by the counterpart. The +ordering of the mazes and obstacle probes (i.e., awareness or location recall) were also the same. +We recruited participants until at least one participant was matched to each participant from the +original experiments. Additionally, we used the same exclusion criteria as in the initial experiment +with the additional requirement that all black dots be collected on a trial. This resulted in data from +n= 163 participants being analyzed for the initial mazes execution control (median age of 29; 86 +34male, 77 female) and n= 161 for the critical mazes execution control (median age of 30; 94 male, +63 female; 4 neither). +Process-Tracing Experiments +We ran process-tracing experiments using the initial mazes and the critical mazes. These experi- +ments were similar to the memory experiments, except they used a novel process-tracing paradigm +designed to externalize the planning process. Specifically, participants never saw all the obstacles +in the maze at once. Rather, at the beginning of a trial, after clicking on a red X in the center +of the maze, the goal and agent appeared, and participants could use their mouse to hover over +the maze and reveal individual obstacles. An obstacle would become completely visible if the +mouse hovered over any tile that was part of it for at least 25ms, until the mouse was moved to a +tile that was not part of that obstacle. Once the participant started to move using the arrow keys, +the cursor became temporarily invisible (to prevent using the cursor as a cue to guide execution), +and the obstacles could no longer be revealed. We examined two dependent measures for each +obstacle: whether participants hovered over an obstacle, and if so, the duration of hovering in log +milliseconds. +For each experiment with each set of mazes, we requested 200 participants on Prolific. Partic- +ipants who completed the task had their data excluded if they did not hover over any obstacles on +more than half of the trials. For the experiment with the initial set, we received completed submis- +sions from 174 people and, after exclusions, analyzed data from n= 167 participants (median age +of 30; 82 male, 82 female, 3 neither). For the experiment with the critical set, we received com- +pleted submissions from 188 people and, after exclusions, analyzed data from n= 179 participants +(median age of 32; 89 male, 86 female, 4 neither). +Experiment Analyses +Hierarchical generalized linear models (HGLMs) were implemented in Python and R using the +lme455andrpy256packages (see Code Availability Statement). For all models, we included by- +35participant and by-maze random intercepts, unless the resulting model was singular, in which case +we removed by-maze random intercepts. For the memory experiment analyses testing whether +value-guided construal predicted responses, we fit models with and without z-score normalized +value-guided construal probabilities as a fixed effect and performed likelihood ratio tests to assess +significance. For the control experiment analyses reported in the main text, we calculated mean +by-obstacle responses from the perceptual and execution controls, and then included these values +as fixed effects in models fit to the responses in the planning experiments. We then contrasted +models with and without value-guided construal and performed likelihood ratio tests (additional +analyses are reported in the Supplementary Memory Experiment Analyses and Supplementary +Control Experiment Analyses). +For our comparison with alternative models, we considered 11 different predictors that assign +scores to obstacles in each maze: fixed-parameter value-guided construal modification probabil- +ity (VGC), trajectory-based heuristic search score (Traj HS), graph-based heuristic search score +(Graph HS), bottleneck state distance (Bottleneck), successor representation overlap (SR Over- +lap), minimum navigation distance (Nav Dist), timestep of minimum navigation distance (Nav +Dist Step), minimum optimal policy distance (Opt Dist), distance to goal (Goal Dist), distance to +start (Start Dist), distance to center walls (Wall Dist), and distance to the center of the maze (Cen- +ter Dist). We included predictors in the analysis of each experiment’s data where appropriate. For +example, in the up-front planning experiment, participants did not navigate on early termination +trials, and so we used the optimal policy distance rather than navigation distance. All predictors +were z-score normalized before being included as fixed effects in HGLMs in order to facilitate +comparison of estimated coefficients. +We performed three types of analyses using the 11 predictors. First, we wanted determine +whether value-guided construal captured variability in responses from the planning experiments +even when accounting for the other predictors. For these analyses, we compared HGLMs that +included all predictors to HGLMs with all predictors except value-guided construal and tested +whether there was a significant difference in fit using likelihood ratio tests (Extended Data Table 1). +36Second, we wanted to evaluate the relative necessity of each mechanism for explaining attention to +obstacles when planning. For these analyses, we compared global HGLMs to HGLMs with each +of the predictors removed and calculated the resulting change in AIC (see Extended Data Table +1 for estimated coefficients and resulting AIC values). Finally, we wanted to assess the relative +sufficiency of predictors in accounting for responses on the planning tasks. For these analyses, we +fit HGLMs to each set of responses that included only individual predictors or pairs of predictors, +and for each model we calculated the AIC relative to the best-fitting model (Extended Data +Figure 8). Note that for all of these models, AIC values are summed over participants. +Methods References +35. Nassar, M. R. & Frank, M. J. Taming the beast: extracting generalizable knowledge from +computational models of cognition. Current opinion in behavioral sciences 11,49–54 (2016). +40. Sutton, R. S. & Barto, A. G. Reinforcement learning: An introduction (MIT Press, 2018). +41. Parr, R. & Russell, S. Reinforcement learning with hierarchies of machines. Advances in +neural information processing systems 10(1997). +42. Virtanen, P. et al. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. +Nature Methods 17,261–272 (2020). +43. Howard, R. A. Dynamic programming and markov processes. (1960). +44. Barto, A. G., Bradtke, S. J. & Singh, S. P. Learning to act using real-time dynamic program- +ming. Artificial intelligence 72,81–138 (1995). +45. Bonet, B. & Geffner, H. 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Optimal behavioral hierarchy. PLoS Computational Biology 10,e1003779 +(2014). +52. Shi, J. & Malik, J. Normalized cuts and image segmentation. IEEE Transactions on pattern +analysis and machine intelligence 22,888–905 (2000). +53. Gureckis, T. M. et al. psiTurk: An open-source framework for conducting replicable behav- +ioral experiments online. Behavior research methods 48,829–842 (2016). +54. De Leeuw, J. R. jsPsych: A JavaScript library for creating behavioral experiments in a Web +browser. Behavior research methods 47,1–12 (2015). +55. Bates, D., M ¨achler, M., Bolker, B. & Walker, S. Fitting Linear Mixed-Effects Models Using +lme4. Journal of Statistical Software 67,1–48 (2015). +56. The rpy2 contributors. rpy2 version 3.3.6. Sept. 26, 2020. https://rpy2.github.io/ . +38Acknowledgements : The authors would like to thank Jessica Hamrick, Louis Gularte, Ceyda +Sayalı, Qiong Zhang, Rachit Dubey, and William Thompson for valuable feedback on this work. +This work was funded by NSF grant #1545126, John Templeton Foundation grant #61454, and +AFOSR grant # FA 9550-18-1-0077. +Author Contributions : All authors contributed to conceptualizing the project and editing the +manuscript. MKH, DA, MLL, and TLG developed the value-guided construal model. MKH im- +plemented it. MKH and CGC implemented the heuristic search models and msdm library. MKH, +JDC, and TLG designed the experiments. MKH implemented the experiments, analyzed the re- +sults, and drafted the manuscript. +Competing Interest Declaration : The authors declare no competing interests. +Supplementary Information is available for this paper. +Data Availability Statement : Data for the current study are available through the Open Science +Foundation repository http://doi.org/10.17605/OSF.IO/ZPQ69. +Code Availability Statement : Code for the current study are available through the Open Science +Foundation repository http://doi.org/10.17605/OSF.IO/ZPQ69, which links to a GitHub repository +and contains an archived version of the repository. The value-guided construal model and alterna- +tive models were implemented in Python (3.7) using the msdm (0.6) library, numpy (1.19.2), and +scipy (1.5.2). Experiments were implemented using psiTurk (3.2.0) and jsPsych (6.0.1). +Hierarchical generalized linear regressions were implemented using rpy2 (3.3.6), lme4 (1.1.21), +and R (3.6.1). +39Maze 0.68 +.42 .74.26 +.83 +.60.19Initial Exp +Awareness +.53 +.38 .69.25 +.73 +.57.20Up-front planning +Awareness +.72 +.69 .75.31 +.93 +.90.35Process-tracing +Hovering +6.6 +6.1 6.85.8 +7.2 +7.05.9Process-tracing +DurationMaze 1 +.66 .63.29.76.28 +.26.22 +.57 .69.28.70.23 +.27.21 +.81 .85.26.75.31 +.56.24 +6.6 7.05.76.76.1 +6.06.0Maze 2.70 +.50 +.59.50.79 +.31 +.44.56 +.41 +.65.50.71 +.35 +.47.94 +.67 +.75.64.88 +.27 +.796.6 +5.7 +6.56.36.6 +6.0 +6.6Maze 3.36.28 +.33 +.75.81 +.66.51.36.25 +.29 +.71.75 +.64.44.63.57 +.45 +.93.77 +.87.686.66.5 +6.4 +7.06.5 +6.86.0Maze 4.47 +.72.76 +.56.73 +.72.33 +.36 +.71.69 +.52.59 +.59.32 +.93 +.86.83 +.79.60 +.81.56 +6.5 +6.96.8 +5.96.0 +6.66.5Maze 5.71 +.37 +.41 +.76.54.83.24 +.61 +.29 +.31 +.71.53.73.23 +.77 +.52 +.71 +.84.85.89.52 +6.7 +6.1 +6.4 +7.47.17.36.4Extended Data Fig. 1 jExperimental measures on mazes 0 to 5, Average responses as- +sociated with each obstacle in mazes 0 to 5 in the initial experiment (awareness judgment), the +40up-front planning experiment (awareness judgment), and the process-tracing experiment (whether +an obstacle was hovered over and, if so, the duration of hovering in log milliseconds). Obstacle +colors are normalized by the minimum and maximum values for each measure/maze, except for +awareness judgments, which are scaled from 0 to 1. +41Maze 6.39 +.48.40 +.44.71.51 +.38Initial Exp +Awareness +.33 +.46.33 +.61.74.49 +.41Up-front planning +Awareness +.90 +.68.39 +.72.50.55 +.60Process-tracing +Hovering +6.5 +5.66.4 +6.76.46.4 +5.8Process-tracing +DurationMaze 7.36.19 +.74 +.20.43 .18 +.69 +.31.27 +.76 +.25.47 .26 +.69 +.71.16 +.67 +.25.27 .13 +.67 +6.25.4 +6.3 +6.15.8 5.8 +6.6Maze 8.18 +.61.29 +.41.25.35 +.70.20 +.72.28 +.35.24.40 +.79.09 +.51.49 +.70.18.14 +.885.0 +6.36.0 +6.25.75.5 +6.5Maze 9 +.30 +.20.79.81.39 +.34 +.78.27 +.23.73.80.42 +.30 +.78.50 +.66.82.88.79 +.82 +.915.8 +6.86.97.56.9 +6.6 +6.9Maze 10.66 +.77 .27 +.23 +.39.56 .43.74 +.73 .24 +.30 +.36.51 .39.92 +.65 .41 +.29 +.50.72 .416.8 +6.2 6.4 +5.9 +6.26.3 6.0Maze 11.71 +.47.23.80.83 +.19.41 +.65 +.32.22.77.79 +.23.38 +.59 +.57.21.77.93 +.15.63 +6.3 +6.45.97.17.5 +6.06.5Extended Data Fig. 2 jExperimental measures on mazes 6 to 11, Average responses as- +sociated with each obstacle in mazes 6 to 11 in the initial experiment (awareness judgment), the +42up-front planning experiment (awareness judgment), and the process-tracing experiment (whether +an obstacle was hovered over and, if so, the duration of hovering in log milliseconds). Obstacle +colors are normalized by the minimum and maximum values for each measure/maze, except for +awareness judgments, which are scaled from 0 to 1. +43Maze 12 +.67.51.57.92 +.78Critical Mazes Exp +Accuracy +.59.43.44.84 +.61Critical Mazes Exp +Confidence +.35.23.32.81 +.71Critical Mazes Exp +Awareness +.62.18.39.90 +.62Process-tracing +Hovering +6.45.35.36.8 +6.5Process-tracing +DurationMaze 13 +.70.60.51.76.92 +.58.46.47.66.78 +.34.35.26.69.84 +.65.44.18.52.89 +6.25.45.46.26.9Maze 14.65.49 +.54.79.94 +.54.53 +.40.68.81 +.41.35 +.23.81.79 +.78.69 +.28.88.85 +6.36.5 +5.37.26.8Maze 15.66 +.58.45 +.85.87 +.56 +.42.62 +.74.79 +.39 +.23.32 +.81.76 +.75 +.56.67 +.85.88 +6.5 +6.56.6 +6.96.8Extended Data Fig. 3 jExperimental measures on mazes 12 to 15, Average responses as- +sociated with each obstacle in mazes 12 to 15 in the critical mazes experiment (recall accuracy, +recall confidence, and awareness judgment) and the process-tracing experiment (whether an ob- +stacle was hovered over and, if so, the duration of hovering in log milliseconds). Obstacle colors +are scaled to range from 0.5 to 1.0 for accuracy, 0 to 1 for hovering, confidence, and awareness +judgments, and the minimum to maximum values across obstacles in a maze for hovering duration +in log milliseconds. +44Extended Data Fig. 4 jAdditional Experimental Details, a, Items from critical mazes exper- +iment. Blue obstacles are the location of obstacles during the navigation part of the trial. Orange +obstacles with corresponding number are copies that were shown during location recall probes. +During recall probes, participants only saw an obstacle paired with its copy. b,Example trial from +process-tracing experiment. Participants could never see all the obstacles at once, but, before nav- +igating, could use their mouse to reveal obstacles. We analyzed whether value-guided construal +predicted which obstacles people tended to hover over and, if so, the duration of hovering. +45Maze 0.15 +.05 .430.0 +.73 +0.00.0VGC +2.7 +3.6 3.82.9 +3.1 +2.0.04Traj HS +1.4 +3.0 4.0.82 +5.0 +2.00.0Graph HS +7.0 +5.0 1.011.0 +7.0 +1.09.0Bottleneck +.76 +1.3 1.4.32 +.62 +.01.12SR Overlap +1.0 +1.1 1.04.0 +1.0 +1.03.6Opt DistMaze 1 +.25 .040.0.400.0 +0.00.0 +1.7 .360.01.20.0 +0.00.0 +4.0 .530.01.30.0 +0.00.0 +11.0 3.07.01.07.0 +11.012.0 +1.6 .051.5.421.4 +.81.01 +1.0 1.64.01.03.0 +2.08.0Maze 2.42 +0.0 +.170.0.85 +0.0 +0.03.1 +4.5 +1.83.13.1 +0.0 +.311.0 +5.0 +1.25.05.0 +0.0 +.096.0 +5.0 +1.05.09.0 +3.0 +5.01.0 +0.0 +0.00.00.0 +0.0 +0.01.0 +1.2 +1.01.41.0 +2.0 +1.8Maze 3.160.0 +.02 +.70.80 +.23.363.33.6 +3.0 +1.71.1 +1.72.40.01.1 +2.4 +2.52.0 +1.72.38.07.0 +5.0 +8.09.0 +5.01.0.61.46 +.62 +.84.20 +.25.833.07.0 +4.1 +1.01.0 +1.21.0Maze 4.45 +.03.47 +.17.20 +.31.01 +4.3 +1.93.2 +4.5.86 +1.6.13 +3.0 +1.04.5 +5.01.0 +2.1.14 +9.0 +5.01.0 +7.010.0 +3.05.0 +1.0 +0.00.0 +0.00.0 +0.00.0 +4.0 +1.01.0 +1.01.0 +1.02.3Maze 5.14 +.02 +0.0 +.450.0.730.0 +3.8 +3.4 +2.6 +3.53.53.14.1 +0.0 +2.4 +2.0 +5.04.05.00.0 +6.0 +6.0 +5.0 +1.03.08.09.0 +.78 +.75 +.47 +.57.64.67.37 +1.0 +4.0 +1.6 +1.01.01.03.0Maze 6.38 +.230.0 +0.00.0.01 +.32 +3.9 +3.80.0 +0.00.0.10 +.52 +4.0 +5.00.0 +0.00.0.15 +.63 +6.0 +7.08.0 +5.01.04.0 +4.0 +2.0 +1.1.84 +0.00.00.0 +1.7 +4.1 +1.01.0 +3.01.02.1 +1.2Maze 7.170.0 +.29 +0.00.0 0.0 +.63 +2.60.0 +.86 +0.00.0 0.0 +.69 +.750.0 +.57 +0.00.0 0.0 +1.0 +6.08.0 +2.0 +11.04.0 10.0 +1.0 +.50.13 +.38 +0.0.03 .03 +.25 +5.53.6 +1.0 +4.02.6 3.0 +1.0Extended Data Fig. 5 jModel predictions on mazes 0 through 7, Shown are the predictions +for six of the eleven predictors we tested: fixed parameter value-guided construal modification +46obstacle probability (VGC, our model); trajectory-based heuristic search obstacle hit score (Traj +HS); graph-based heuristic search obstacle hit score (Graph HS); distance to optimal bottleneck +(Bottleneck); successor representation overlap score (SR Overlap); and distance to optimal paths +(Opt Dist) (see Methods, Model Implementations). Mazes 0 to 7 were all in the initial set of mazes. +Darker obstacles correspond to greater predicted attention according to the model. Obstacle colors +normalized by the minimum and maximum values for each model/maze. +47Maze 80.0 +0.0.05 +.120.00.0 +.33VGC +0.0 +0.01.5 +.970.00.0 +.98Traj HS +0.0 +0.0.26 +1.60.00.0 +1.0Graph HS +10.0 +2.06.0 +8.07.05.0 +1.0Bottleneck +0.0 +0.0.97 +.810.00.0 +.26SR Overlap +7.0 +1.03.5 +1.14.02.0 +1.0Opt DistMaze 9 +0.0 +0.0.46.58.45 +.28 +.143.3 +2.80.02.91.9 +4.0 +3.13.0 +2.0.675.01.0 +4.0 +4.07.0 +13.01.01.02.0 +8.0 +5.01.1 +.01.19.63.58 +.90 +1.34.0 +6.01.01.02.0 +5.0 +1.0Maze 10.19 +.43 .20 +0.0 +0.0.04 0.0.84 +2.6 4.1 +3.9 +1.9.28 .011.0 +4.4 1.1 +1.1 +1.3.36 .015.0 +1.0 8.0 +8.0 +6.08.0 11.0.06 +1.4 .90 +.58 +.59.29 1.51.2 +1.0 5.0 +6.9 +2.01.5 1.0Maze 11.32 +.220.0.54.68 +0.0.01 +3.7 +3.10.03.22.8 +0.0.22 +5.0 +3.00.05.05.0 +0.0.29 +4.0 +8.09.01.01.0 +12.07.0 +1.2 +.581.2.86.60 +.071.5 +1.0 +5.04.01.01.0 +7.91.0Maze 12 +.210.00.0.79 +.36 +3.50.03.73.4 +4.6 +3.00.05.05.0 +5.0 +8.08.06.04.0 +5.0 +.54.68.31.54 +.57 +6.04.02.01.0 +1.0Maze 13 +.190.00.0.38.84 +3.43.50.03.93.1 +4.05.00.04.05.0 +9.05.09.01.05.0 +.56.31.74.75.56 +6.02.05.01.01.0Maze 14.280.0 +0.0.29.82 +4.31.2 +3.83.63.1 +4.01.9 +4.03.05.0 +8.06.0 +6.01.010.0 +.93.02 +.73.64.58 +3.03.8 +4.01.01.0Maze 15.34 +0.00.0 +.30.87 +4.5 +3.51.5 +4.13.1 +5.0 +3.01.9 +4.05.0 +6.0 +10.07.0 +1.011.0 +1.0 +.02.02 +.61.56 +4.0 +6.53.7 +1.01.0Extended Data Fig. 6 jModel predictions on mazes 8 through 15, Shown are the predictions +for six of the eleven predictors we tested (see Methods, Model Implementations). Mazes 8 to 11 +48were part of the initial set of mazes, while mazes 12 to 15 constituted the set of critical mazes. +Darker obstacles correspond to greater predicted attention according to the model. Obstacle colors +normalized by the minimum and maximum values for each model/maze. +49R² = 0.50 +0.0 0.50.00.51.0Initial Exp. +Awareness JudgmentVGC +R² = 0.05 +0.0 2.5Traj HS +R² = 0.28 +0 5Graph HS +R² = 0.32 +5 10Bottleneck +R² = 0.00 +0 2SR Overlap +R² = 0.55 +2.5 5.0 7.5Opt Dist +R² = 0.00 +5 10 15Goal Dist +R² = 0.00 +5 10 15Start Dist +R² = 0.06 +2.5 5.0Wall Dist +R² = 0.05 +2.5 5.0 7.5Center Dist +R² = 0.40 +0.0 0.50.00.51.0Up-front Planning Exp. +Awareness JudgmentR² = 0.01 +0.0 2.5R² = 0.18 +0 5R² = 0.39 +5 10R² = 0.01 +0 2R² = 0.53 +2.5 5.0 7.5R² = 0.00 +5 10 15R² = 0.00 +5 10 15R² = 0.01 +2.5 5.0R² = 0.01 +2.5 5.0 7.5 +R² = 0.83 +0.0 0.50.00.51.0Critical Maze Exp. +Recall Accuracy +R² = 0.25 +0.0 2.5R² = 0.42 +0 5R² = 0.06 +5 10R² = 0.11 +0 1R² = 0.44 +2.5 5.0R² = 0.15 +5 10 15R² = 0.12 +10 20R² = 0.04 +2.5 5.0R² = 0.01 +5 10 +R² = 0.80 +0.0 0.50.00.51.0Critical Mazes Exp. +Recall ConfidenceR² = 0.05 +0.0 2.5R² = 0.19 +0 5R² = 0.05 +5 10R² = 0.02 +0 1R² = 0.42 +2.5 5.0R² = 0.27 +5 10 15R² = 0.21 +10 20R² = 0.02 +2.5 5.0R² = 0.07 +5 10 +R² = 0.71 +0.0 0.50.00.51.0Cricical Mazes Exp. +Awareness JudgmentR² = 0.15 +0.0 2.5R² = 0.27 +0 5R² = 0.20 +5 10R² = 0.05 +0 1R² = 0.69 +2.5 5.0R² = 0.11 +5 10 15R² = 0.07 +10 20R² = 0.00 +2.5 5.0R² = 0.01 +5 10 +R² = 0.38 +0.0 0.50.00.51.0Process-Tracing +(Initial Mazes 0-11) +Hovering +R² = 0.23 +0.0 2.5R² = 0.33 +0 5R² = 0.17 +5 10R² = 0.02 +0 2R² = 0.32 +2.5 5.0 7.5R² = 0.01 +5 10 15R² = 0.03 +5 10 15R² = 0.07 +2.5 5.0R² = 0.07 +2.5 5.0 7.5 +R² = 0.29 +0.0 0.5567Process-Tracing +(Initial Mazes 0-11) +Log-Hover Duration R² = 0.09 +0.0 2.5R² = 0.17 +0 5R² = 0.17 +5 10R² = 0.00 +0 2R² = 0.16 +2.5 5.0 7.5R² = 0.00 +5 10 15R² = 0.01 +5 10 15R² = 0.00 +2.5 5.0R² = 0.00 +2.5 5.0 7.5 +R² = 0.52 +0.0 0.50.00.51.0Process-Tracing +(Critical Mazes 12-15) +Hovering +R² = 0.22 +0.0 2.5R² = 0.30 +0 5R² = 0.03 +5 10R² = 0.00 +0 1R² = 0.21 +2.5 5.0R² = 0.18 +5 10 15R² = 0.13 +10 20R² = 0.07 +2.5 5.0R² = 0.17 +5 10 +R² = 0.42 +0.0 0.55.56.06.57.0Process-Tracing +(Critical Mazes 12-15) +Log-Hover Duration R² = 0.12 +0.0 2.5R² = 0.11 +0 5R² = 0.04 +5 10R² = 0.00 +0 1R² = 0.13 +2.5 5.0R² = 0.11 +5 10 15R² = 0.08 +10 20R² = 0.12 +2.5 5.0R² = 0.24 +5 10Extended Data Fig. 7 jSummaries of candidate models and data from planning experi- +ments, Each row corresponds to a measurement of attention to obstacles from a planning exper- +iment: Awareness judgments from the initial memory experiment, the up-front planning experi- +ment, and the critical mazes experiment; recall accuracy and confidence from the critical mazes +50experiment; and the binary hovering measure and hovering duration measure (in log milliseconds) +from the two process-tracing experiments. Each column corresponds to candidate processes that +could predict attention to obstacles: fixed parameter value-guided construal modification obsta- +cle probability (VGC, our model), trajectory-based heuristic search hit score (Traj HS), graph- +based heuristic search hit score (Graph HS), distance to bottleneck states (Bottleneck), successor- +representation overlap (SR Overlap), expected distance to optimal paths (Opt Dist), distance to the +goal location (Goal Dist), distance to the start location (Start Dist), distance to the invariant black +walls (Wall Dist), and distance to the center of the maze (Center Dist). Note that for distance-based +predictors, the x-axis is flipped. For each predictor, we quartile-binned the predictions across ob- +stacles, and for each bin we plot (bright red lines) the mean and standard deviation of the predictor +and mean by-obstacle response (overlapping bins were collapsed into a single bin). Black circles +correspond to the mean response and prediction for each obstacle in each maze. Dashed dark red +lines are simple linear regressions on the black circles, with R2values shown in the lower right +of each plot. Across the nine measures, value-guided construal tracks attention to obstacles, while +other candidate processes are less consistently associated with obstacle attention (data are based +onn= 84215 observations taken from 825independent participants). +51a +b +cExtended Data Table 1 jNecessity of different mechanisms for explaining attention to ob- +stacles when planning, For each measure in each planning experiment, we fit hierarchical gener- +alized linear models (HGLMs) that included the following predictors as fixed-effects: fixed param- +eter value-guided construal modification obstacle probability (VGC, our model); trajectory-based +heuristic search obstacle hit score (Traj HS); graph-based heuristic search obstacle hit score (Graph +HS); distance to optimal bottleneck (Bottleneck); successor representation overlap score (SR Over- +52lap); distance to path taken (Nav Dist); timestep of point closest along path taken (Nav Dist Step); +distance to optimal paths (Opt Dist); distance to the goal state (Goal Dist); distance to the start +state (Start Dist); distance to any part of the center walls (Wall Dist); and distance to the center of +the maze (Center Dist) (Methods, Model Implementations). If the measure was taken before par- +ticipants navigated, distance to the optimal paths was used, otherwise, distance to the path taken +and its timestep were used. a, b, Estimated coefficients and standard errors for z-score normalized +predictors in HGLMs fit to responses from the initial experiment, up-front planning experiment (F += full trials, E = early termination trials), the critical mazes experiment, and the process-tracing ex- +periments. We found that value-guided construal was a significant predictor even when accounting +for alternatives (likelihood ratio tests between full global models and models without value-guided +construal: Initial Exp, Awareness: 2(1) = 501:11;p< 1:01016; Up-front Exp, Awareness (F): +2(1) = 282:17;p< 1:01016; Up-front Exp, Awareness (E): 2(1) = 206:14;p< 1:01016; +Critical Mazes Exp, Accuracy: 2(1) = 114:87;p < 1:01016; Critical Mazes Exp, Confi- +dence:2(1) = 181:28;p < 1:01016; Critical Mazes Exp, Awareness: 2(1) = 486:99;p < +1:01016; Process-Tracing Exp (Initial Mazes), Hovering: 2(1) = 294:40;p < 1:01016; +Process-Tracing Exp (Initial Mazes), Duration: 2(1) = 177:58;p< 1:01016; Process-Tracing +Exp (Critical Mazes), Hovering: 2(1) = 183:52;p< 1:01016; Process-Tracing Exp (Critical +Mazes), Duration: 2(1) = 251:16;p < 1:01016).c,To assess the relative necessity of each +predictor for the fit of a HGLM, we conducted lesioning analyses in which, for each predictor in a +given global HGLM, we fit a new lesioned HGLM with only that predictor removed. Each entry of +the table shows the change in AIC when comparing global and lesioned HGLMs, where larger pos- +itive values indicate a greater reduction in fit as a result of removing a predictor. According to this +criterion, across all experiments and measures, value-guided construal is either the first or second +most important predictor.Largest increase in AIC after lesioning;ySecond-largest increase. +53VGCTraj HSGraph HS Bottleneck SR OverlapNav Dist +Nav Dist StepGoal Dist Start Dist Wall DistCenter DistVGC +Traj HS +Graph HS +Bottleneck +SR Overlap +Nav Dist +Nav Dist Step +Goal Dist +Start Dist +Wall Dist +Center Dist2110 +20704937 +204732463750 +1096309223573127 +20064938367231204993 +0990684113213041306 +2089486735363036494312744945 +20244926370929814986129249374988 +211149183602312949901305492849894995 +2105481537343122478511584761480647954815 +20994843374231274822119347924838482946954847Initial Exp. +Awareness +VGCTraj HSGraph HS Bottleneck SR OverlapGoal Dist Start Dist Opt Dist Wall DistCenter DistVGC +Traj HS +Graph HS +Bottleneck +SR Overlap +Goal Dist +Start Dist +Opt Dist +Wall Dist +Center Dist1405 +13003312 +138720112557 +279144610721444 +11493267235014203287 +130833142551137332893317 +1403329124131441328332883303 +0673496326500731725731 +13113298254713923214329832747323296 +129833062541137332313306328573032123304Up-front Planning Exp. +Awareness +VGCTraj HSGraph HS Bottleneck SR OverlapNav Dist +Nav Dist StepGoal Dist Start Dist Wall DistCenter DistVGC +Traj HS +Graph HS +Bottleneck +SR Overlap +Nav Dist +Nav Dist Step +Goal Dist +Start Dist +Wall Dist +Center Dist28 +24285 +26221225 +19285223350 +30271207321332 +22203180243223243 +16272220340334244364 +0209197252289222309313 +2219201261301224325276326 +27286227344330227352278297358 +28284227351308239364300317246368Critical Mazes Exp. +Accuracy +VGCTraj HSGraph HS Bottleneck SR OverlapNav Dist +Nav Dist StepGoal Dist Start Dist Wall DistCenter DistVGC +Traj HS +Graph HS +Bottleneck +SR Overlap +Nav Dist +Nav Dist Step +Goal Dist +Start Dist +Wall Dist +Center Dist39 +38638 +32481536 +0622522639 +21636535634662 +26438408440440440 +33591497584637429638 +17414394325475328471475 +8459424366523348524320523 +16577477614538430628477524657 +8543450579429406598468510413624Critical Mazes Exp. +Confidence +VGCTraj HSGraph HS Bottleneck SR OverlapNav Dist +Nav Dist StepGoal Dist Start Dist Wall DistCenter DistVGC +Traj HS +Graph HS +Bottleneck +SR Overlap +Nav Dist +Nav Dist Step +Goal Dist +Start Dist +Wall Dist +Center Dist394 +3671314 +39211091129 +011489321234 +3921297110212031453 +151687652715720740 +28013011126120414557331511 +13010971049782132070113241356 +99115210758351373712139710451411 +39512571096122613467421513134414061514 +393122510721196120673714981358141211221498Critical Mazes Exp. +Awareness +VGCTraj HSGraph HS Bottleneck SR OverlapGoal Dist Start Dist Opt Dist Wall DistCenter DistVGC +Traj HS +Graph HS +Bottleneck +SR Overlap +Goal Dist +Start Dist +Opt Dist +Wall Dist +Center Dist274 +2271153 +197743743 +124674443902 +27611497448241360 +209115374490213531427 +1761136740782129812831337 +0127204493551550441563 +26111547438961337135312935271357 +256115574490013461366130453713051370Process-Tracing (Initial Mazes) +Hovering +VGCTraj HSGraph HS Bottleneck SR OverlapGoal Dist Start Dist Opt Dist Wall DistCenter DistVGC +Traj HS +Graph HS +Bottleneck +SR Overlap +Goal Dist +Start Dist +Opt Dist +Wall Dist +Center Dist171 +142455 +167341414 +35246223246 +77421343229419 +169446402201409447 +144401323197383390399 +125298281206255286243297 +23406335150399394364247407 +0388315124386378350229306390Process-Tracing (Initial Mazes) +Duration +VGCTraj HSGraph HS Bottleneck SR OverlapGoal Dist Start Dist Opt Dist Wall DistCenter DistVGC +Traj HS +Graph HS +Bottleneck +SR Overlap +Goal Dist +Start Dist +Opt Dist +Wall Dist +Center Dist157 +1141176 +119875897 +10811568611452 +26117089914521588 +3173172995512941292 +0824771103013887911386 +158857775103510409069301038 +95748699141114241294138510181566 +315605471253936121212849132401409Process-Tracing (Critical Mazes) +Hovering +VGCTraj HSGraph HS Bottleneck SR OverlapGoal Dist Start Dist Opt Dist Wall DistCenter DistVGC +Traj HS +Graph HS +Bottleneck +SR Overlap +Goal Dist +Start Dist +Opt Dist +Wall Dist +Center Dist139 +140580 +72554552 +7524476530 +114582554529597 +0484511337526526 +5500521340540459541 +141414419412419393393417 +103367464481362513526384569 +682643934211724734843357513Process-Tracing (Critical Mazes) +Duration +01000200030004000 +050010001500200025003000 +050100150200250300350 +0100200300400500600 +0200400600800100012001400 +0200400600800100012001400 +0100200300400 +0200400600800100012001400 +0100200300400500Extended Data Figure 8 jSufficiency of individual and pairs of mechanisms for explaining +attention to obstacles when planning, To assess the individual and pairwise sufficiency of each +54predictor for explaining responses in the planning experiments, we fit hierarchical generalized +linear models (HGLMs) that included pairs of predictors as fixed effects. Each lower-triangle plot +corresponds to one of the experimental measures, where pairs of predictors included in a HGLM +as fixed-effects are indicated on the x- and y-axes. Values are the AIC for each model relative +to the best fitting model associated with an experimental measure (lower values indicate better +fit). Values along the diagonals correspond to models fit with a single predictor. According to this +criterion, across all experimental measures, value-guided construal is the first, second, or third best +single-predictor HGLM, and is always in the best two-predictor HGLM. +55Extended Data Table 2 jAlgorithm for Computing the Value of Representation Function +To obtain predictions for our our ideal model of value-guided construal, we calculated the value of +representation of all construals in a maze. This was done by enumerating all construals (subsets of +obstacle effects) and then, for each construal, calculating its behavioral utility and cognitive cost. +This allows us to obtain theoretically optimal value-guided construals. For a discussion of alterna- +tive ways of calculating construals, see the Supplementary Discussion of Construal Optimization +Algorithms. +56 \ No newline at end of file