Online games: a novel approach to explore how partial information influences human random searches

Many natural processes rely on optimizing the success ratio of a search process. We use an experimental setup consisting of a simple online game in which players have to find a target hidden on a board, to investigate how the rounds are influenced by the detection of cues. We focus on the search duration and the statistics of the trajectories traced on the board. The experimental data are explained by a family of random-walk-based models and probabilistic analytical approximations. If no initial information is given to the players, the search is optimized for cues that cover an intermediate spatial scale. In addition, initial information about the extension of the cues results, in general, in faster searches. Finally, strategies used by informed players turn into non-stationary processes in which the length of e ach displacement evolves to show a well-defined characteristic scale that is not found in non-informed searches.

The problem of searching for targets whose location is unknown arises in many fields and at different scales [1][2][3].Numerous examples appear in the natural sciences including in ecology [4][5][6], biochemistry [7][8][9] and chemistry [10].In addition, many human activities involve situations where a target has to be found.Some instances are the location of a lost object, rescue operations, or fugitive prosecutions [11].More recently, the development of eye-tracking technology has allowed the study of visual searches on screens [12][13][14].Having empirical evidence of the performance of different search strategies and how they are affected by environmental cues, regardless of whether they are employed by humans, other animals or bacteria, is essential to understand the biological and physical mechanisms behind these processes.Such data are also required to verify the mathematical models that have been proposed [15][16][17][18][19][20][21][22], and to develop improved protocols.
Situations in which a target has to be located appear in a large variety of scenarios, which allows the design of multiple strategies to find a successful solution.Such strategies can be classified in many different ways, according to one or more of their properties [1].
For instance, stochastic or systematic processes are distinguished depending on the type of search rule [2].The amount of directional information available determines the existence of bias towards preferred regions [23,24].Finally, differences may also be attributable to the movement pattern, such as cruising versus ambush [25] and to the frequency of the reorientation events, such as intensive (frequent) versus extensive (infrequent) [26,27].The accuracy of a particular choice within each category is determined by the properties and the state of the searcher, the target and the environment where the task has to be accomplished.
For instance, searchers with memory that navigate relatively predictable environments do not employ purely random strategies but combine a stochastic component with knowledge acquired through previous experience.There is therefore a learning process that plays an important role in the emergence of new rules [28,29].In other scenarios, individuals who live in groups may incorporate information gathered by conspecifics with their own in order to improve foraging efficiency.It has been recently showed that intermediate combinations between both types of cues result in more efficient searches regardless of the nature of the mobility pattern [30] and the spatial distribution of the targets [31,32].However, the precise optimal balance between social and individual information is determined by each specific setup.
In all these examples, interactions among the three components of the searching problem (searcher, target and environment) establish fluxes of information in the system (cues) that may alter the suitability of a given strategy over the course of the process.Therefore, in the most general case, search methodologies must be understood as dynamical processes consisting of several components rather than as fixed procedures.For instance, when the amount and the quality of the information varies over time, individuals may alternate between stochastic and deterministic strategies [33].Changes between habitats may also trigger switches in the movement behavior [34,35].
In this work we propose the use of computer games as a new experimental approach in which it is possible to address these and related questions in humans.This is particularly intriguing since, due to their cognitive abilities, individuals might show a large diversity of complex responses to the same stimulus.Despite substantial efforts aimed at understanding the theoretical concepts behind many searching processes, a reliable and unifying empirical framework in which these ideas may be tested is still lacking.The family of games presented here is a good candidate to fill this gap, as they can be accessed online by a large number of players.This results in the generation of large and clean datasets.In addition, the rules and setup of the game can be experimentally manipulated so that different mechanisms or strategies can be rigorously tested.We address several questions related to search efficiency and investigate how the strategies change due to the amount and the quality of the information acquired by the player at different stages of the game.In a second step, the main features of these patterns are extracted from the data and used to develop a family of random walk models that can be applied to predict human search behavior in other configurations of the game.The variety of experiments shown in this work reinforces the flexibility of our approach and aims to open a new route for the study of searching problems.
In the following section, after presenting the characteristics of the game, we show the empirical results obtained from two different setups.In the first case, players have no information about the configuration of the board, whereas in the second study they are provided with partial information about the target location.Then, we formulate a family of models that capture the main mechanisms behind the experimental results and derive analytical approximations to show the robustness of the results.Finally, all the previous steps are combined to develop a comprehensive framework in which it is possible to predict the optimal configuration of the landscape that yields faster searches.The paper finishes with a discussion of the results and opportunities for new lines of research.

A. Experimental setup
We consider a simple game in which a single target has to be found.It slightly resembles the classic minesweeper, although the objective is to find a unique target (mine) instead of avoiding a collection of them.The interface consists of N × N squares that can be explored by the player through successive clicks with the mouse.There are three classes of cells depending on their color after being clicked (unclicked cells are always blue): (i) black cells are typically far from the target, (ii) yellow cells indicate that the target may be one of the neighboring cells and (iii) the single red cell is the target.The target is randomly located within a patch of yellow cells that provides partial information about the configuration of the board through the fact that it contains the target.Two different geometries for this set of yellow cells are explored here.First, they form a N y × N y neighborhood square region (Fig. 1A) and second, they will outline a random patch whose size will be measured in terms of the number of yellow cells.The discovery of a yellow cell indicates that the player is in the neighborhood of the target and thus reduces the area that needs to be searched.For simplicity we fixed N = 20 in all the experiments and then manipulated N y .
To generate the dataset players access the game online and are asked to find the target using as few clicks (jumps on the board) as possible.The rounds are all independent and each one is represented by the trajectory traced by the player on the board.Finally, the experimental setup also includes a timer.Players are requested to find the target as quickly as possible.In the following sections we investigate i) how the length of the search, represented by the number of mouse clicks, changes with the size of the target's neighborhood (also called yellow region); and ii) the statistical properties of the searching patterns as defined by the distance between clicks d i (jump length) and the turn angles θ i .By definition, we consider turns to the left to be between 0 • and 180 • and turns to the right to be between 180 • and 360 • (see Fig. 1B for a definition of both quantities).
We consider two classes of experiments: a) blind searches, where the player is given no a priori knowledge of the size of the neighborhood, and b) searches with initial information, where the value of N y is given to the player at the beginning of the round.The objective of performing both classes of experiments is twofold: on the one hand to investigate whether players adapt their searching strategies when they have better information about the landscape and, on the other hand, to examine how search efficiency changes when the reliability of the information provided by the yellow cells increases.

B. Experiments with blind searchers
For this first series of experiments neither the exact size, the position of the yellow region, nor a range of possible dimensions was given to the searchers.Before starting the round, each player only knew that a target (red square) was hidden in the board and it might be randomly placed inside a square vicinity of yellow cells of unknown size.The uncertainty in the size of the neighborhood reduces the reliability of the information acquired by the player when a yellow cell is open and favors the efficiency of random strategies [1].Our dataset consists on 500 rounds with N y ranging from N y = 1, which means that the target does not have a neighborhood, to N y = 13.We first measure the mean number of clicks needed to find the target as a function of the lateral length of its yellow neighborhood (black squares in Fig. 2).
Because of the way the experiments are designed, there is a tradeoff between finding the yellow region and finding the target inside it.Larger neighborhoods are easier to locate but make the final detection of the target inside them harder.Smaller neighborhoods, however, need on average more steps to be found but make the target within them easier to locate.
According to our results, this tradeoff is balanced at intermediate sizes of the neighborhood, N opt y = 5.This resembles the foraging dynamics of animals that exchange information about food location with their conspecifics, so that both spreading information over distances that are either too large or too short may slow down the search [32].Following this analogy, we refer to the the size of the yellow area that minimizes the number of clicks needed to find the target as the optimal interaction range.The standard deviation of the number of jumps is also minimal at the optimal range, which means a narrowing in the distribution of clicks used to detect the target and therefore a reduction in the stochasticity of the search.In the limit of zero information (i.e.no yellow cells or N y = 1, or the whole board is yellow, N y = N ), the probability of finding the target on the first click is given by the inverse of the number of available cells, 1/N 2 .In any subsequent movement, m, this probability is given by where the first term yields the probability of not having found the target in the previous m−1 clicks and the second term yields the probability of hitting the target once m−1 squares have been visited.Equation (1) reduces to 1/N 2 regardless of the value of m.Therefore, the probability of detection in the limit N y = 1 (and N y = N ) follows a uniform distribution of mean N 2 /2 = 200 and standard deviation N 2 / √ 12, which is in good agreement with data (black squares in Fig. 2).
Next, we analyze all the trajectories traced by the players in every round.To this aim the experimental setup saves the sequence of clicks in each round, from where it is possible to calculate the length of each displacement and every reorientation turn.We identify an extensive and an intensive searching mode that depend on whether the player has detected a yellow cell or not respectively (Fig. 3a).In both situations the jump lengths can be fitted using exponential distributions, but with a lower mean value in the proximity of the target; 1/λ in = 2.04 and 1/λ out = 3.70.Therefore, the typical size of the jumps is reduced once the player finds the yellow area as the detection of the cue (represented by a yellow cell) changes the searcher's internal state [34,36,37].Although the player does not know how big the neighborhood is and therefore how reliable the information is, the trajectories recorded after the discovery of the yellow region still show shorter distances between turns, suggesting that players switch to an intensive search mode once they find the yellow region [1].It is important to remark that, although alternation between extensive (motion phase) an intensive modes (scanning phase) is also characteristic of intermittent searches, the player is not performing an intermittent search as the switch between reorientation modes is triggered by the external cue instead of taking place at random [2].
Regarding the turn angles, both the extensive (before the first encounter with a yellow cell) and the intensive phases (after detecting the first yellow cell) show correlations between subsequent turn angles (Fig. 3b,c, respectively).This indicates that the strategies are not purely random and players try to design systematic ways of scanning the board.One of the most abundant strategies consists of tracing a series of short jumps in the same direction.In order to reduce searching times players show a tendency to scan a direction doing several consecutive clicks.This behavior is also shown by the distributions of jump lengths, which show a large deviation from the exponential for one-cell length jumps, which are overrepresented in the dataset (Figure 3a).As an exception, movements done starting from a black cell reached from a yellow cell show a strong tendency to reverse the direction, as this sequence in the colors of the cells indicates that the player is moving away from the target (Fig. 3d).

C. Experiments with initial information
In this second series of experiments the players know the size of the yellow region, which is fixed at N y = 5.This increases the quality of the information obtained when one of its cells is found as the player can limit the search area.The position of this area is random and changes from round to round, as well as the place of the target inside it.
Data from 230 rounds was collected.As a general result, a priori information accelerates the search and reduces its stochasticity.Whereas blind searchers need on average 31.30clicks to find the target when N y = 5 (subset of 65 rounds from the 500 trajectories analyzed in Section I B), advised players use 25.5 clicks.The two-tailed P value of these mean values obtained using an unpaired t-test, 3 × 10 −4 , is considered to be extremely statistically significant.The standard deviation also decreases, indicating a narrowing in the distribution of the number of displacements and therefore in the randomness of the process: σ b = 14.10 for blind searchers and σ a = 10.50 for the informed (advised) ones.
In addition, as we showed in Section I B, the strategies are composed of an extensive mode before the detection of the first cue, and an intensive mode after that encounter.During the extensive phase players adapt their displacements, concentrating them around the size of the yellow neighborhood (Fig. 4a).If we analyse the whole set of advised rounds, we observe a strong dominance of movements of length 5 (green squares in Figure 4a).This is due to the presence of approximately 50 rounds in the dataset where players performed optimally designed systematic strategies that consist of moving in jumps of fixed length N y during the extensive phase.We will come back to this in Section I E. For the purposes of this section we will remove these systematic rounds and focus on the subset of stochastic strategies formed by the other 180 rounds.The distribution of the displacements lengths is still dominated by jumps covering a distance of the order of N y (red circles in Figure 4a).This arises from the existence of a feedback between the searcher and the environment that allows a narrowing in the distribution of displacement lengths as the extensive phase evolves.As the player has perfect memory about his trajectory (visited cells remain open) searches that start with large displacements tend to create landscapes that are fragmented in patches of length N y where long movements are inefficient.To show the existence of this feedback we split the data of the extensive phase in four subdivisions: (i) from jump 1 to 5, (ii) 6 to 10, (iii) 11 to 15, and (iv) 16 to the end.The distributions for each of these pieces are shown in Figure 4c, d, e and f respectively, and they can be fitted by a family of gamma distributions (dashed lines in each panel) of decreasing mean, mode and variance (See Table I for numerical values of these parameters and details of the distributions).Then the total distribution of Figure 4a can be approximated by a gamma distribution defined in terms of the parameters of the distributions of the pieces (dashed line in Fig. 4a).This approach shows an excellent agreement with a direct fitting of the whole extensive phase (full line in Fig. 4a).The subset of blind searchers with N y = 5 however do not have a well defined typical scale and instead explore several scales as they look for the yellow region (Fig. 4 b).Finally, the intensive phase of the advised searches also shows a higher abundance of one-cell displacements than the distribution of the blind searches (inset of Figure 4a and 4b respectively).This result is independent of whether or not the systematic deterministic strategies are included within the analyzed dataset and is due to the fact that knowing the neighborhood size reduces exploration during this phase.
Despite the change in the jump-length distributions from exponential to gamma, informed players are not able to improve the efficiency of the extensive phase as they need on average  Jump length 11.65 movements to find a cue (10.90 if systematic optimized strategies are included), which is the same result obtained for blind searches, 11.30.The question is then which part of the advised strategies changes and yields the improvement in the efficiency.Having information about the size of the neighborhood of the target allows a faster detection of its limits and therefore reduces the number of movements spent in order to find the target.A quantity that explains this feature of the search is the distribution of turns made by the searcher immediately after moving from inside the yellow region to outside of it.This is shown in Figure 3d for blind searches and in Figure 4g for informed strategies.Although in both cases the movement shows a strong bias backwards, informed players are represented by narrower distributions.They tend to make turns of 180 • since they do not have to find out the size of the neighborhood of the target.
At this point we have shown the existence of an optimal size for the neighborhood of the target, as well as an improvement in the search efficiency when the size of the neighborhood is revealed before each round.In addition, these informed strategies evolve through information gathering during the extensive phase towards a dominant jumping distance equal to the lateral length of the neighborhood of the target.In the following sections we develop a theoretical framework and a family of models based on random walks to study the basic principles behind these results and how they can be transfered to more general scenarios, with irregular shapes for the information region.Assuming an exponential distribution for the jump length we neglect the higher frequency of one-cell length displacements shown in the data (see Fig. 3a).These movements introduce correlations in the turn angles as we already discussed in Section I B (see Fig. 3b,c).Using a uniform distribution for the turn angles is thus consistent with drawing the length of the displacements from exponential distributions neglecting the peak for one-cell jump lengths.
The results of the simulations (green curve in Fig. 2) show an excellent agreement with the experimental data (black curve) both in the mean average number of jumps and in its standard deviation.Simulations reproduce at least the two first moments of the number of clicks distribution.
Except in the limits N y = 1 (no yellow cells) and N y = N (yellow cells occupy the whole board), it is hard to obtain exact analytical expressions for the average total number of jumps needed to find the target.However, it is possible to obtain the distribution for the length of the extensive phase where P i (N y ) is the probability of having an extensive phase of i jumps when the neighborhood of the target has a lateral length N y and p i (N y ) = Ny N −i+1 is the probability of finding a yellow cell in the i − th mouse click.In words, the probability of having an extensive phase with i jumps is given by the probability of not finding a yellow cell in all the previous moments multiplied by the probability of finding one in the i − th movement.Given the equation ( 2), the mean length of the extensive phase is For the length of the intensive phase however we can only give and upper and a lower limit, assuming that after the detection of the first yellow cell all the movements are to neighboring cells.Therefore, the target is found on average after N 2 y /2 jumps in the intensive phase when the neighborhood of the target is large and after (N y + 2) 2 /2 movements when the neighborhood is small.These two limits account for the decreasing probability of visiting cells outside the neighborhood when increasing its size.For small values of N y it is very likely to reach the border of the neighborhood before detecting the target and thus to return to the black region.Combining these two results for the intensive phase with the length of the extensive phase obtained in equation 3, we obtain two theoretical approximations to the total number of clicks The combination of these two expressions gives an approximated range for the length of the search (magenta region in Figure 2) that shows an excellent agreement with empirical data and numerical simulations of the model.
E. Model for searches with initial information.The design of optimal strategies.
Knowing the size of the yellow region at the beginning of the game changes the nature of the search as the information gathered by the player with each movement may be used to design the next displacement.This reinforces the non-Markovian nature of the informed search process as the player uses all the previous steps to discard cells that have not been visited yet and results in self-adaptive strategies that evolve towards displacements of length N y .Also, as the value of N y is known, the number of exits from the neighborhood of the target diminishes.In a first approach to model this effect, we modify the model used for blind searches using the new experimental distribution of the length of the displacements in both the extensive and the intensive modes (Figure 4a).Therefore, instead of using the exponential distributions of Fig. 3 we sample the histograms of Fig. 4a (red circles) and its inset, that are obtained from experimental searches with initial information.This approach overestimates both the length of the extensive and the intensive phases, which results in a clearly higher average number of movements; 33.50 jumps, σ = 19.00 for the model and 25.50 jumps, σ = 10.50 in the data (MB green bars and DI gray bars in Figure 5 respectively).This is due to the fact that the model does not integrate the information about the size of the target to a priori discard some of the cells during the intensive and the extensive phase.
In a first approach to remove this discrepancy, we hypothesize that the most important differences arise in the modeling of the intensive phase.During this stage, given a certain number of yellow cells and some of their neighboring black squares, our experimental results suggest that human players are able to discriminate the real border of the neighborhood of the target and thus reduce the number of erroneous displacements.The model that we developed for blind searches lack this ingredient, which increases the duration of the intensive phase.To correct this, we modify the model to include the effect that previous movements, together with knowing the size of the neighborhood of the target, has on the intensive phase (See Methods for a detailed description).In this new approach, once the first yellow cell has been detected and based on all the previous movements, only those cells that can possibly be part of a 5 × 5 yellow square have a non-zero probability of being visited by the searcher.This mechanism reduces the number of times that black cells are visited once a yellow cell has been found as the model is able to discriminate all the possible borders of the neighborhood of the target.With this new ingredient the efficiency of the model increases (MI blue bar in Figure 5a) and the number of jumps in the intensive phase shows excellent agreement with the experimental data (DI gray and MI blue bars in Figure 5c).However, despite this substantial improvement as compared to the blind model, significant differences still remain between empirical data and numerical results.The source of this disagreement arises from the extensive phase (DI gray and MI blue bars in Figure 5b).To correct this, we next modify the extensive phase of the model.
During the extensive phase, players are able to discriminate regions where the target cannot be placed as a 5 × 5 square would not fit.To incorporate this in the model, we first compute the probability of jumping to each of the non-visited cells of the board according to the histogram in Figure 4a.Then, for each cell we obtain all the possible squares of lateral length 5 to which it could belong and set the probability of jumping to that cell to zero if all these squares contain at least one open black cell (See Methods for more details).With this mechanism the extensive phase becomes more efficient and the agreement of the model with the experimental data is excellent.This comes from a precise fitting of both the intensive and the extensive phase individually (DI gray and MII black bars in 5a,b,c).
Optimal strategy.-However,both actual player strategies and random walk models are much less efficient than entirely systematic protocols.Knowing a typical size of the target in advance allows the design of optimized strategies that minimize the number of incorrect steps.Particularly important is to shorten the extensive phase, as within the neighborhood of the target all the cells are equivalent and it is equally likely to find the target in any position.In fact, during the experimental rounds with initial information, one of the players developed one of these searching methods by repeatedly playing in the same landscape.This strategy optimizes the extensive phase and only allows for two movements into black cells during the intensive phase (Fig. 6a).Given a value for N y , the search rule is given the following steps: 3. Once a yellow cell is found, visit consecutive squares in a given direction (horizontal in Fig. 6a for N y = 5) until finding a black position.Then, if the number of yellow cells in the row is lower than N y , complete it.
4. Repeat the same operation in the other direction starting from one already known yellow cells.

5.
Once the the neighborhood of the target has been delimited, move inside it until finding the target.
In the particular case of N y = 5, the average number of movements before target detection is 19.03 (10 4 realizations) and it is always lower than 42.In addition, 5.90 movements are needed to find the yellow region, which is about 50% lower than the experimental result.
Applying this strategy to many sizes of the yellow region (Fig. 6b) we observe that the tradeoff between finding the neighborhood of the target (yellow diamonds in Fig. 6b) and finding the target inside it (blue circles in Fig. 6b) balances at intermediate values of N y .
Following theoretical results in Section I D, analytical expressions can be obtained for the mean number of movements during both phases and therefore for the optimal interaction range.The mean number of clicks during the intensive phase is N 2 y /2 as the target can be in any cell with the same probability (green dashed line in Fig. 6b) (we only consider the lower bound from Section I D since this optimal protocol minimizes the number of erroneous movements).To obtain the mean number of movements in the extensive stage, we assume that the upper right corner of each subdivision of the board (Fig. 6a) is equally likely to have a yellow cell.Therefore, the number of steps is given by N 2 /2N 2 y .This is not completely true, as cells close to the border have a lower probability of being yellow, but it is a good approximation (black dashed line in Fig. 6b fitting yellow diamonds).
At the optimal interaction range both functions intersect, which gives N opt y = √ N = 4.47 for our experimental setup with N = 20.This result is in excellent agreement with the value obtained from the experiments (Fig. 6c) and suggests, together with the theoretical approximation derived in Section I D that the optimal interaction range is independent of the searching strategy.This result suggests the possibility of using this theoretical framework to predict the optimal size of the neighborhood of the target in more general scenarios.
F. Anticipating the optimal range of interaction for random neighborhoods.
In this section we allow the target to adopt different sizes and random shapes across rounds.In order to facilitate the formulation of theoretical predictions, the neighborhood is built starting from a triangle of varying base b y (see Methods for a detailed description and Figure 7) where the target is embedded.Then, the region is randomized by turning black a 30% of the cells.In this way, we implement random neighborhoods that vary in form and size from round to round but with an underlying fixed pattern.Before starting the game players know that the neighborhood has now a varying form and size (Fig. 7), but they are given no information about the way it is constructed.
The optimal interaction range can be evaluated from an independent estimation of the number of movements needed in the extensive and the intensive phases.The length of the extensive phase is obtained following the same steps used for square neighborhoods; the probability of finding a yellow cell in the i − th movement is given by Eq. ( 2) from where the mean length of the extensive phase is obtained using Eq.3.This quantity is shown by the magenta circles in Figure 8b.To approximate the number of movements used in the intensive phase, which will give us the optimal interaction range we used the underlying triangle shape of the neighborhood of the target.This calculation provides lower and upper bounds for the average duration of the intensive phase.The lower bound is obtained assuming that all the cells from the original target have the same probability of being visited but all the cells that do not belong to it will never be clicked.The total number of cells that form this original triangle is (b y /2 + 0.5) 2 and since all the cells can be visited with the same probability, the lower limit for the length of the intensive phase is given by (by/2+0.5) 2

2
. The upper limit is obtained assuming that the first cells that do not belong to the triangle in each direction has also a non-zero probability of being visited.This results in an upper bound for the length of the intensive phase given by [(by+4)/2+0.5] 2

2
. Both limits are shown by the magenta circles in Figure 8c.Finally, the total number of movements, i.e., the sum of the extensive and the intensive phase, is shown by the magenta circles in Figure 8a, with an estimated optimal neighborhood size in between 18 and 25 yellow cells.It is important to remark the difference between the optimal interaction range for random and square neighborhoods, which shows the nontriviality of predicting optimal interaction ranges for different geometries.
We tested these a priori predictions with a series of experiments using an experimental setup with neighborhoods that consist of 5, 16, 33, 55 and 69 cells (plus the target red cell).
301 rounds were analyzed and the observed mean number of clicks is shown in Figure 8a.
We also split each round into the extensive and intensive phases and the results are shown in panels b and c of Figure 8.The good agreement between the predicted values and the results obtained with the experiments shows the robustness of the theoretical approach developed in simpler scenarios.

II. DISCUSSION
We have developed a novel approach to study human searching problems by building a simple game that can be accessed online.This approach facilitates the collection of large and clean experimental datasets.By combining data analysis with probabilistic calculations and numerical simulations of existing and new models, it is possible to obtain a deeper understanding of how humans approach simple searching tasks and how their strategies differ from optimal patterns.
A comprehensive analysis of the trajectories on the board of the game (length of the displacements and turn angles) shows that players follow strategies that consist of two modes.The detection of cues about the possible location of the target triggers a response in the searcher, which leads to shorter movements on average.In the context of existing works these strategies can be modeled by composite random walks, which are frequently used to model area-restricted search processes (e.g., foraging) in heterogeneous landscapes [37][38][39].
These two-behavior models typically consist of an extensive phase and an intensive phase that is triggered by encountering a food item and and is characterized by shorter steps and larger turning angles (relative to the extensive mode) [1,[39][40][41] In the simplest scenario studied here, in which no information is given about the size of the neighborhood of the target, developing a systematic searching rule as opposed to following a stochastic trajectory does not provide a significant advantage.A systematic scan of the environment usually provides higher efficiencies by minimizing the probability of revisiting a certain region.In this setup, however, cells remain open once they are visited, providing players with a perfect memory about the history of their movements.As a consequence, neither random nor systematic players click more than once on a cell, regions are not revisited, and both protocols offer equivalent results.This scenario however changes when some information about the nature of the target is provided to the players.In that case an optimal systematic strategy can be constructed based on this information.Interestingly, our data show that one of these optimal strategies was developed by a particular player who repeatedly played several rounds in the same landscape.This result opens the door to explore a broad range of questions at the interface between landscape variability, the searcher's memory, and learning abilities, which has been recently become an important topic in movement ecology [29].Most animals do not follow completely random strategies, but combine this stochastic component with spatial memory and learning [28,42,43].To investigate the importance of cognitive skills such as learning or memory in the development of optimal strategies, our approach could easily be extended to allow landscapes where the position of the target exhibits a certain degree of persistence across rounds of the game.
In fact, we have shown that, when they have some knowledge about the landscape (size of the neighborhood of the target), players use the additional information obtained in each movement step to increase search efficiency.In this scenario, the effect of the information gathered during the whole process has to be included in theoretical models to reproduce experimental results.Introducing a more realistic finite memory by allowing clicked cells to revert back to the unclicked state after some time arises as a future line of research.
More importantly, however, the excellent agreement between our experimental data and simple theoretical models suggest that this online-game based methodology could be applicable to address more complex scenarios.Energy budget related questions can be addressed by introducing a metabolic cost that penalizes longer movements and evolutionary aspects of searching problems may be addressed by allowing pairs of players to compete and selecting those using more efficient strategies.This would mimic environments where different individuals compete for limited resources and could shed some light on the driving forces behind the evolution of optimal searching.Finally, the effect of cooperative interactions among players on search efficiency could also be addressed.Many species forage in groups as opposed to individually.The methodology that has been presented here would allow, given a certain landscape, exploration of the level of confidence that players place on movements performed by previous participants.Before every movement of the new player, the choice of previous searchers at that same moment can be shown to the new player to investigate the level of trust the current player places on previous participants.In addition, if the neighborhood of the target is changed, or multiple targets are included, it would be possible to explore the relationships between use of social information versus personal experience for tasks of increasing difficulty.
In summary, and in view of the large and exciting range of possibilities for future exploration, we expect that this general framework will complement purely theoretical efforts to unveil the fundamental mechanisms that drive a wide variety of search scenarios.

III. METHODS
A. Fitting of the partial distributions of displacement lengths to gamma distributions in advised searches In Section I C we showed that the lengths of the displacements when players are given a priori information about the landscape follow a series of gamma distributions whose proba-  bility density function is given by where α and β are real positive parameters.For known values of α and β, the mean value of the distribution can be obtained as αβ, the variance as αβ 2 and the mode (the value that appears most often in the distribution) as β(α − 1).All the parameters shown in Table I were obtained using the maximum likelihood estimation.Results shown in the last row of Table I correspond to a distribution that is a mixture of all four component distributions.
Given the mean value and variance of these distributions, we can assume that they all have the same weight in the composition since all the subsets of the trajectory have the same length.The mixed distribution can be obtained as: B. Implementation of the random walk model for blind searches We have developed a minimalistic model based on composite random walks to understand the basic features of the searching strategies used by the players.We initialize the model from a random configuration of the board in which the target is placed in a random position inside a smaller square of lateral length N y .To mimic the experimental setup, we fix the the target is randomly placed inside a square region of lateral length N y = 5.The position of this region is also random in the board and changes across realizations.The searcher is placed at an initial random position and the dynamics starts.The algorithm has two well differentiated parts for the intensive and the extensive phase: • Extensive phase: 1. Obtain the distance from the current position of the searcher, i, to every other cell in the board, j, and assign a jumping probability, P ij , by taking a random sample from the histogram in Figure 4a.
2. As in the game the player has perfect memory of previous moves, so the probability of jumping to already visited cells is set to zero. 4. Renormalize all the jumping probabilities so they sum one.
5. Sort a uniform random number u between 0 and 1 and move to a cell k when k j=1 P ij ≥ u.
• Intensive phase, after the first yellow cell is hit: 1. Obtain all the possible neighborhoods of the target to which the first detected yellow cell can belong.5. From the histogram in the inset of Figure 4a, obtain the probability P ij of jumping to the cells that belong to the chosen 5 × 5 squares.
6. Renormalize all the jumping probabilities so they sum one.
7. Sort a uniform random number u between 0 and 1 and move to a cell k when k j=1 P ij ≥ u.

FIG. 1 :
FIG.1: A) Single realization as shown in the game interface.Blue cells have not been visited, black and yellow cells represent the two types of cues and the red square is the target.B) Reconstruction of the round in A from the saved data.Small circles correspond to black cells, bigger circles to the yellow ones and the biggest circle is the target.Circles are labeled with blue numbers, d i is the distance jumped starting from node i and θ i is the turn angle relative to the direction at node i.

FIG. 2 :
FIG.2: Number of jumps needed to detect the target as a function of the lateral length of the yellow neighborhood N y .N y = 1 means that there are no yellow cells around the target.Black squares are averages taken from experimental data, light green squares are obtained from numerical simulations (averages over 10 4 realizations) and the magenta region is the theoretical approximation.Dashed lines are interpolations.

FIG. 3
FIG.3: a) (Linear-log plot) Jump length distribution during the extensive (blue squares) and intensive (green circles) phase.Magenta lines are exponential fits to the data with mean value given by 1/λ.b, c) Turn angle distributions during the extensive and the intensive mode respectively.d) Distribution of the turns made outside the neighborhood region when the previous move was inside it.

FIG. 4 :
FIG. 4: Jump length and turning angle distributions for advised searchers.a) Displacements done during the extensive phase for searches with initial information, green squares correspond to the whole set of rounds and red circles to the subset of random strategies.The dashed and full lines show two analytical approximations.Inset: distribution for the intensive phase.b) Equivalent to a) but for the subset of blind searchers with constant N y = 5. c,d,e,f) Jump length distributions when the extensive phase is divided in four parts: steps 1-5 (c), 6-10 (d), 11-15 (e) and 16-end (f).g) Turning angle distribution for movements in the intensive phase made immediately after leaving the yellow region.

D.
Model for blind searchers: numerical simulations and analytical approximationWe develop a minimalistic searching model based on random walks to explain previous experimental results on the basis of simple dynamical rules.The model has the three main ingredients obtained from the data analysis: (i) two modes of movement defined by the mean length of the displacements: λ in and λ out ; (ii) in the absence of any information (no yellow cell clicked) the direction is completely random (uniform distribution in the turning angles); and (iii) when cues are obtained (a yellow cell has been detected), the searcher has a bias towards unvisited cells surrounding a yellow one.The last assumption aims to capture the influence of the information provided to the searcher when a yellow cell is found, as well as the strong directionality exhibited by the distribution of turning angles in Fig.3d.More details of the model as well as the simulation setup are provided in Section III.

FIG. 5 :
FIG. 5: Mean number of movements comparing the informed searches experimental data and a series of random walk based models.a) Total number of movements before detecting the target.b) Number of movements during the extensive phase.c) Number of movements during the intensive phase.Labels of the x-axis: DI data informed, MB model blind, MI model informed, MII model informed 2 and OS optimal strategy.

FIG. 6
FIG. 6: a) Typical realization of an optimized systematic search strategy.The color of the squares indicates the temporal sequence of the jumps and its size the location outside (smaller squares) or inside the neighborhood (intermediate squares).The biggest square represents the target.b)Mean number of jumps to detect the target using an optimized search rule.This quantity is divided between the extensive (outside) and the intensive (inside) stages to represent the tradeoff that appears in the game.Analytical approximations are also shown using dashed lines.c) Mean number of jumps to detect the target using an optimized search rule (red squares) and in the experimental data for blind searchers (black squares).Error bars represent the standard deviation.

FIG. 7 :
FIG. 7: Construction of random information neighborhoods starting from triangles of different size.Black cells highlighted in green are cells that belonged to the original triangle and have been removed in the randomization process.

FIG. 8 :
FIG. 8: Search efficiency when the information is provided by an irregular neighborhood.Black squares correspond to experimental data and magenta circles to theoretical predictions.a) Total number of movements, b) extensive phase, c) intensive stage.Error bars represent the standard error, when not shown they are smaller than the size of the square.

3 .
For every cell j in the board obtain all the possible 5 × 5 squares to which it can belong.If all of them have any open black cell, then set the probability of jumping to j to zero.This step is skipped in the intermediate model where only the intensive phase is improved.

2 . 3 . 4 .
Count the number of open cells of both classes (black and yellow) in each of those possible neighborhoods of the target.Pick those 5 × 5 squares that include all the open yellow cells and none of the black ones.Set the probability of jumping to all other of the rest of the cells of the board to zero.

TABLE I :
Numerical values of the parameters that are obtained fitting the jump length distributions to gamma distributions.Advised searches.