Random searchers cope with cognitive errors and uncertainty better than path planners

There is a widespread belief in ecology that the capacity of animals to orchestrate systematic and planned paths should represent a significant benefit for efficient search and exploration. Within this view, stochasticity observed in real animal trajectories is mostly understood as undesirable noise caused by internal or external effects. Far less is known, however, about the case when cognitive errors and limitations inherent to living systems are explicitly put into play. Here we compare within this context the search efficiency of (i) walkers driven by Bayesian rules generating deterministic paths, (ii) standard random walkers, and (iii) human trajectories obtained from search experiments in a soccer field and on the computer screen. Our results clearly challenge the view that deterministic paths are generally better for exploration than random strategies, as the latter are more resilient to cognitive errors. Instead, we provide numerical and experimental evidence that stochasticity would provide living organisms with a sufficient and cognitively simple exploration solution to the problem of uncertainty.


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Search theory is a branch of applied mathematics aimed at identifying optimal search paths (paths 26 that promote encounters between 'searchers' and 'targets') and assessing the efficiency of search 27 strategies under different conditions. The general mathematical framework encompasses a wide 28 range of situations, including a continuum of strategies from information-guided schemes for 29 Search and Rescue (SAR) (Koopman, 1980;Stone, 1975; Kagan, 2015) to purely stochastic, non-  (a) How should a walker orchestrate its trajectory in order to optimize the probability of finding a target (marked with the cross) within the area ? (b) Following prescriptions both from Search And Rescue (SAR) manuals and from Bayesian inference rules, given a uniform prior expectation (HPE scenario) a parallel sweep or similar self-avoiding trajectory is recommended (case b1). If prior expectations include (PPE scenario) the existence of a most likely position for the target, , then an expanding (square or spiral) trajectory starting from that point should be used, as in b2. (c) When walkers face uncertainty (due to severe sensory/cognitive limitations) a random trajectory is expected. The new paradigm in the present work explores how such random trajectories perform against mathematically optimal solutions under the existence of uncertainties and errors.

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We developed algorithms to examine the performance of searchers following Bayesian inference 141 rules in the HPE and PPE scenarios. For this, the searcher's trajectory is continuously updated 142 based on the expected information gain (see the Methods section for further details). This gain is 143 driven by a detection function ( , ) which determines the expected probability that the target will 144 be found as a function of the searcher's speed and its distance to the target . Here we assume where and are assumed to be independent. The positive parameter represents the character-146 istic detection distance, so for values > the probability of detection starts to decay drastically. 147 Equivalently, the parameter can be interpreted as the characteristic speed above which detec-   Figure 2. Search efficiency as a function of searcher speed for systematic (full symbols) and random (open symbols) search strategies in the HPE and PPE scenarios. Different values of (for the systematic case, with = 10 for circles and = 2 for triangles) and (for the random case, with = 100 for circles, = 10 for triangles and = 1 for inverted triangles) are shown. The rest of the parameters used are = 400 × 400, = 2, = 2, = 25000. The initial conditions considered and further details about the algorithms are specified in the Methods Section. are rather small in practice.

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Errors in sweep width estimation 188 Both SAR manuals and works on optimal search theory (Koopman, 1980) agree that the success of conservative strategies based on a too low will easily lead to that situation.

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This effect becomes especially dramatic in the PPE scenario, where only a fine choice of makes 205 deterministic paths more efficient than random strategies (Fig. 3b). The performance of random 206 strategies within this scenario becomes almost independent of the specific value of their move-207 ment parameter , which makes clear that random walks are more resilient strategies. It is still 208 more remarkable that the relatively narrow window of values for which deterministic paths out-209 perform random walks is explicitly dependent on the rest of the search parameters (i.e. , and ).

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This makes the picture even more complicated for systematic searchers, which have to accurately 211 adapt their movement patterns to each particular case. 213 The view given in the preceding Section for the PPE scenario can be completed by taking into ac-214 count that initial prior expectations may have limited reliability, or may be simply wrong. If the  initial expectation about where to find the target is wrong, then the effects of choosing an erro-216 neous value of will presumably become more severe.

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For simplicity, we focus on the case where erroneous prior expectations result in erroneous 218 estimations of the most likely position of the target̄ (this will not affect the HPE scenario, so we 219 omit it in this Section). Then, the most likely position of the target in the spatial domain does not 220 coincide with̄ but is shifted a distance Δ from it. The impact of this expectation error on the search 221 efficiency is observed in Figure 4, which corresponds to the same situation as in Figure 3

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The results from the previous Sections suggest that the benefits of planned and systematic strate-  237 Here, two very different settings were used to assess search efficiency in human subjects under 238 relatively homogeneous media. In the first one, the subjects were presented on a 17" computer 239 screen several images consisting of a homogeneous cloud of numbers from '1' to '9' over a white 240 background, and were asked to look for the number '5', which was present only once in each image   In order to test whether more systematic strategies performed better than random ones, we clas- subjects, in contrast, relied more on a free-style strategy with a higher level of stochasticity (in the 279 Methods Section a representative sample of the experimental trajectories is shown for illustration). 280 We measured the amount of randomness of the trajectories based on the computation of the 281 turning angle distribution, ( ), carried out between consecutive segments of the trajectory. Sys-282 tematic strategies (both parallel sweeps and spirals) should exhibit a distribution of turning angles 283 peaked at one or two specific values, while random trajectories would be characterized by a more 284 uniform distribution, ( ) ≈ 1∕ , for ∈ (0, ). We used then as our measure of randomness. Note that there is nothing but the Kulblack-Leibler divergence 286 between the actual turning angle distribution ( ) and the uniform one, so it gives the statistical 287 distance between both. The prefactor −1∕ log and the rest of the definition for is chosen such 288 that randomness takes only values between 0 (totally systematic strategy) and 1 (totally random 289 strategy); the higher the value of , the more random is the strategy. 290 We first explored whether the POS obtained from the experimental trajectories showed any 291 correlation with the randomness parameter . Non-parametric (Spearman's) correlation tests 292 were used to assess the independence between both variables.

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As depicted in Fig. 6 (see also are slightly more efficient than systematic ones when = 5. In that case, testing the one-sided 299 null hypothesis of no-difference against the alternative that random search is more efficient, we 300 find a p-value of 0.0825 -a low probability that the observed difference from 0 is merely the result 301 of random variation. For higher values of , the coefficients are also positive but the p-values are 302 much higher making it impossible to distinguish this from random variation.

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Overall, we find no evidence that efficiency is reduced by random strategies, regardless of de-304 tection distance, and even some slight evidence that it can increase. Thus, the benefits of using 305 Figure 6. Relation between search efficiency (POS) and trajectory randomness ( ) for the three scenarios studied experimentally (from left to right, HPE on the screen, PPE on the screen and HPE on the soccer field). Different symbols correspond to = 15 (diamonds), = 10 (circles), = 5 (triangles), with = 1 used in all cases. Results from the statistical tests to discern the existence or not of significant correlations between the variables are given below in the Methods Section.

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For the HPE scenarios, there were no significant correlations at all between trajectory randomness 327 ( ) and coverage ( ), and the result is valid for any detection distance ( ) class (Fig. 7). However,

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in the PPE scenarios, homogeneous coverage decreased significantly with increasing randomness 329 ( Fig. 7) but the signal being much weaker than in the case of POS, given that it only gets significance 330 when pooling the data for all classes of detection distances ( ). Instead, when looking at detection 331 distance classes ( ) per separate classes (see Table 2) we did not find correlations between and 332 .

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In conclusion, in HPE scenarios we were not able to observe clear significant benefits from using 334 a more systematic strategy, provided that stochasticity is introduced by the subjects in a minimally  344 One could argue that our comparison above between experimental trajectories of naïve subjects 345 is not that meaningful, since all the subjects are prone to commit cognitive errors, as they were 346 not experts or trained individuals. In this Section we extend our analysis by comparing how the 347 empirical trajectories would theoretically perform against perfectly trained individuals aware of 348 SAR manuals' prescriptions.

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For this, we took the velocity series from the experimental trajectories (see the Appendix for 350 a characterization of these speeds) and we generated parallel sweeps (for the HPE scenarios) or 351 Archimedean spirals (PPE scenario) that follow that series. By doing so, we ensure that the effects 352 coming from the speed-perception tradeoff are the same for both the real trajectory and the sys-353 tematic one, so we can compare only the effects from the path shapes. 354 We parameterized systematic strategies (both parallel sweepings and Archimedean spirals)  parallel sweeps and small favoring experimental (more noisy) paths. On the soccer field, both 375 types of search perform equally well regardless the detection scale . These results may be related 376 to the fact that completely exploring a screen with the eyes is a relatively easy task compared to 377 completely exploring a soccer field for coins (in many cases subjects did not even cover the whole 378 field by the end of the experiment; see Methods Section below for additional details). There seems 379 to be some kind of negative relationship between the level of difficulty in the search task (i.e. prior   Fig. 10 parameter takes twice that value. So, in Fig. 9 we have 386 more conservative and locally thorough strategies than in Fig. 10. As we can observe, locally thor-387 ough strategies (Fig. 9) improve systematic search on screens but not necessarily on the soccer 388 field, whereas broader systematic searching (larger ) becomes less efficient compared to real tra-389 jectories. The latter seem be to either equally efficient (independently of ) or even outperform 390 systematic paths (HPE on screen, for which a clear decreasing trend with exists).

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According to some additional tests not shown here, the optimal will probably be situated 392 somewhere between the values chosen for Fig. 8 and 9; we stress again, though, that the exact 393 position of the optimal will be slightly different for each specific choice of parameter values.  The evidence gathered, leads us to conclude that some degree of stochasticity is not necessarily 402 detrimental for search efficiency, or at least, the damage that stochasticity may cause may not 403 be significant in most real situations. As a consequence, Random Search Theory does not neces- with ≡ |⃗ 1 − ⃗ 0 |. So, the corresponding information gain produced by the move would be given by 461 the entropy production The walker moves then to the position 1 (among all possible) which provides a maximum informa- For the case of random paths, the move from ⃗ 0 to ⃗ 1 is implemented by generating a flight whose 469 speed has a previously fixed value, and whose duration is chosen at random from an exponential  (Cooper et al., 2003). This requires finding the target within a given 487 time. In order to preserve the essence of that definition, we propose a generalized measure for where the time horizon probability distribution ℎ( ) is introduced. This function determines the 490 (prior) probability that the search will be given up, in case it is unsuccessful, after a given time .

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For instance, if we know for certain that a search must be given up after some fixed search time  between subjects was found due to the stochastic nature of the task.

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The visual search trajectories were monitored by a Tobii T60 fixed eye-tracker (which accuracy 514 around 0.5 • ), from which the raw data for binocular eye positions were extracted. All the experi-515 ments were conducted in the same room and with the same device, with soft light conditions and 516 avoiding any kind of noise or distraction for the subjects during the task. All participants wearing 517 glasses or contact glasses were allowed to decide freely if they wanted to use them during the 518 experiment.

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The second experiment was conducted in a soccer field of 100x68 metres of size located within 520 the Servei d'Activitat Física, in the Campus of the Universitat Autónoma de Barcelona. 31 subjects 521 (aged 18 to 35, 19 males and 12 females) were selected for this experiment, which was conducted 522 during three consecutive sunny mornings. The subjects were left (one at a time) on the soccer field 523 and they were asked to look during ten minutes for ten (20 cent euro) coins which had been previ-524 ously left on the ground (within the soccer field boundaries) according to a Poisson distribution. A 525 prize money was offered to the subject who were able to find more coins within the ten minutes, 526 in order to motivate the subjects for the task. Once a coin was found by the subject he/she marked 527 it (with some plastic red disks they wear in a pocket) to prevent counting the same coin twice as 528 'found'. The number of coins found by the subjects within the given time ranged from 0 to 8 (which 529 again illustrates the high variability that one can find in this kind of search experiments), with an 530 average of 4.2 ± 2.1 coins.

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The trajectories followed on the field were monitored through four small GPS data loggers 532 (GlobalSat DG-100) embedded on an adjustable helmet that the participants wore during the ex-

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The raw data obtained both from the eye-tracker and from the GPS devices was subsequently 546 filtered before generating the final movement trajectories. Points left in blank were filled by using 547 a linear interpolation between the immediately previous point and the immediately posterior point.

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However, data files where more than 10% of the points were in blank or gave incorrect values were 549 discarded due to its low significance (this only happened for 5 participants wearing glasses in the  Again, a sample of trajectories (corresponding to four different subjects) is presented in figures 561 12 and 13 for the HPE and PPE scenarios, respectively. This allows us to see that strategies followed 562 by the subjects differed considerably from completely systematic (sweeps, spirals) to those varying 563 with time or those using a free-style.

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Once the ( , ) series were obtained, the direction of motion at the -th step was determined

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Çompeting interests 578 The authors declare that no competing interests exist. For the sake of completeness, we show here the frequency histograms for the speeds used by the subjects during the three search tasks (HPE ad PPE scenarios in the screen, and HPE scenario on the soccer field). As described in the section about experimental results, we use experimental velocity series as an input for the model, and we generate parallel sweeps and spirals (for the HPE and PPE scenarios, respectively) following those experimental velocity series to compare their search efficiency to those obtained directly from the experimental trajectories. So that, it is interesting to check what is the general shape of the velocity distributions. According to the histograms shown in Fig.1 in all cases we find similar results, with distributions peaked at a typical speed (around 40 pixels/s in the screen, and 1 m/s in the soccer field), and relatively infrequent steps done are at large speeds (specially in the soccer field). For the PPE scenario there is a slightly higher tendency to use slower speeds more frequently, since the prior expectation that the target must be close to the center point made subject behave cautiously specially at the beginning of the task.