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• W4312592995 abstract "Article Figures and data Abstract Editor's evaluation Introduction Results Discussion Materials and methods Appendix 1 Appendix 2 Appendix 3 Appendix 4 Data availability References Decision letter Author response Article and author information Metrics Abstract Many everyday life decisions require allocating finite resources, such as attention or time, to examine multiple available options, like choosing a food supplier online. In cases like these, resources can be spread across many options (breadth) or focused on a few of them (depth). Whilst theoretical work has described how finite resources should be allocated to maximize utility in these problems, evidence about how humans balance breadth and depth is currently lacking. We introduce a novel experimental paradigm where humans make a many-alternative decision under finite resources. In an imaginary scenario, participants allocate a finite budget to sample amongst multiple apricot suppliers in order to estimate the quality of their fruits, and ultimately choose the best one. We found that at low budget capacity participants sample as many suppliers as possible, and thus prefer breadth, whereas at high capacities participants sample just a few chosen alternatives in depth, and intentionally ignore the rest. The number of alternatives sampled increases with capacity following a power law with an exponent close to 3/4. In richer environments, where good outcomes are more likely, humans further favour depth. Participants deviate from optimality and tend to allocate capacity amongst the selected alternatives more homogeneously than it would be optimal, but the impact on the outcome is small. Overall, our results undercover a rich phenomenology of close-to-optimal behaviour and biases in complex choices. Editor's evaluation The authors describe human behavior in a novel task to understand how humans seek information about uncertain options when having a limited sampling budget – the 'breadth-depth' trade-off. They show that human information search approximates the optimal allocation strategy, but deviates from it by favoring breadth in poor environments and depth in rich environments. This study will likely be of interest to a broad range of behavioral and cognitive neuroscientists. https://doi.org/10.7554/eLife.76985.sa0 Decision letter eLife's review process Introduction When choosing an online food supplier as we settle into a new area, we need to trade off the number of alternative shops that we check with the time or money that we want to invest in each of them to learn about the quality of their products. Distributing resources widely (breadth search) allows us to sample many suppliers but very superficially, thus limiting our ability to distinguish which one is best. Allocating our resources to check just a few suppliers (depth search) allow us to learn detailed information but only from a few, at the risk of neglecting potentially much better ones. Striking the right balance between breadth and depth is critical in countless other endeavours such as when selecting which courses to register in college (Schwartz et al., 2009) or developing marketing strategies (Turner et al., 1955). Its implications are far reaching when it comes to understand behaviours on the internet, where breadth and depth search has been related to navigation through lists of search results (Klöckner et al., 2004) or web site menus (Miller, 1991). Despite its relevance to understand how humans make decisions under finite resources, it is remarkable that the breadth-depth (BD) dilemma has mostly been investigated outside cognitive neuroscience (Moreno-Bote et al., 2020), in contrast to other well-studied trade-offs like speed-accuracy and exploration-exploitation (Cohen et al., 2007; Costa et al., 2019; Daw et al., 2006; Ebitz et al., 2018; Wilson et al., 2014). The BD dilemma underlies virtually all cognitive problems, from allocating attention amongst multiple alternatives in multi-choice decision making (Busemeyer et al., 2019; Hick, 1958; Proctor and Schneider, 2018), splitting encoding precision to items in working memory (Joseph et al., 2016; Ma et al., 2014), to dividing cognitive effort into several ongoing subtasks (Feng et al., 2014; Musslick and Cohen, 2021; Shenhav et al., 2013). In all these problems, finite resources, such as attention, memory precision, or amount of control, need to be allocated amongst many potential options simultaneously, making the efficient balancing between breadth and depth a fundamental computational conundrum. The dynamics of resource allocation can be very complex, and thus it has been studied in decision making in simplified cases with few alternatives (Callaway et al., 2021b; Jang et al., 2021; Krajbich et al., 2010) or in multi-tasking using a low number of simultaneously active tasks (Musslick and Cohen, 2021; Sigman and Dehaene, 2005). In these and other cases, resource allocation can be changed on the fly if feedback is immediate or is available within very short delays. In some real-life situations, however, feedback about the quality of the allocation is necessarily delayed. This happens for instance in problems such as investing (Blanchet-Scalliet et al., 2008; Reilly et al., 2016), choosing college (Schwartz et al., 2009) or, in ant colonies, sending scouts for exploration (Pratt et al., 2002); feedback can come after seconds, days, or even years. As resources should then be allocated beforehand, the dynamic aspect of the allocation is less relevant. One-shot resource allocation is important in cognition as well, as building-in stable attentional or control strategies that work in a plethora of situations could relieve the burden of solving a taxing BD dilemma for optimal resource allocation every time (Mastrogiuseppe and Moreno-Bote, 2022). Previous theoretical work has shown how to optimally trade off breadth and depth over multi-alternative problems in the situations described above, where resources are allocated all at once before feedback is received (Mastrogiuseppe and Moreno-Bote, 2022; Moreno-Bote et al., 2020; Ramírez-Ruiz and Moreno-Bote, 2022). A central result is that the optimal trade-off depends on the search capacity of the agent: while at low capacity resources should be split in as many alternatives as capacity permits (breadth), at high capacity resources should be focused on a relatively small number of selected alternatives so that available resources are more focused (depth) (Moreno-Bote et al., 2020). In rich environments, where finding options rendering good outcomes is more likely, depth should be further favoured. Despite of the existence of precise predictions describing ideal behaviours in these scenarios, how humans solve BD trade-offs in many-alternative decision making is largely unknown (Brown et al., 2008; Callaway et al., 2021a; Chau et al., 2014; Cohen et al., 2017; Hawkins et al., 2012; Moreno-Bote et al., 2020; Roe et al., 2001; Usher and McClelland, 2004; Vul et al., 2014). To fill this gap, we designed a novel, many-alternative task where search capacity was parametrically controlled on a trial-by-trial basis. Human participants are immersed in a context where they are asked to allocate finite search capacity over a large number (several dozens) of apricot suppliers with the goal to choose the best one. We compare human sampling behaviour to optimality (Moreno-Bote et al., 2020, see Materials and methods for more details) by using two different ranges of capacities to zoom in relevant regimes. Concerning the adaptation of the sampling strategy as a function of capacity, we expected a pure-breadth behaviour at low capacity followed by a sharp transition towards a trade-off between breadth and depth, characterized by an increase of the number of alternatives sampled with the square root of capacity. We also addressed the effect of environment richness (the overall probability of alternatives rendering good outcomes) on the sampling strategy. We used three different environments with either a majority of poor, neutral, or rich alternatives and expected sampling strategy to shift towards depth as the environment became richer. Finally, we looked at whether systematic deviations from optimality can be observed regarding how samples are distributed amongst each sampled alternative. We employed model comparison to adjudicate between optimal and other heuristic models of behaviour. Results Humans switch from breadth to a BD balance as capacity increases We developed a novel experimental approach, the ‘BD apricot task’, to study many-alternative decisions under uncertainty and limited resources by confronting participants with a BD dilemma. Participants played a gamified version of the task in which they were presented with several virtual apricot suppliers (10 or 32) having different, and unknown, probabilities of good-quality apricots (Materials and methods; see Video 1). These probabilities were independently generated for each supplier in each trial from a beta prior distribution. At the beginning of each trial, participants received a budget of a few coins, which defined their sampling capacity in that trial. Each coin was used to buy one apricot from a selected supplier (Figure 1a). Once all coins had been spent (Figure 1b), participants discovered which of the sampled apricots were of good quality, and which ones were bad (Figure 1c), whose outcomes followed a Bernoulli process with the unknown probabilities of good-quality apricots for each supplier. These probabilities were sampled independently for each supplier and trial from a beta distribution. Based on the observed outcomes, participants could estimate the probabilities from the sampled suppliers, and make a final purchase of 100 apricots from the one they considered to be the best (Figure 1d). Only one of the sampled alternatives could be selected for the final purchase. Their goal was to maximize the number of good-quality apricots collected throughout the experiment through the implementation of an informative sampling strategy, adapted both to the sampling capacity and to the environment richness. The task was intuitive and easily grasped by most participants. No instructions about the underlying probability generative model was provided as the context was informative enough to aid task understanding (Schustek et al., 2019). Figure 1 Download asset Open asset The breadth-depth (BD) apricot task. Human participants allocate a finite search capacity (coins) to learn about the quality of good apricots in different suppliers (sampling phase) and then make a final purchase of 100 apricots from one of the sampled suppliers (purchase phase). Each black section of the wheel represents a different supplier. The number of coins represents the search capacity of the participants on each trial and varies randomly from trial to trial within a finite range (see Materials and methods). The available coins (panel a; yellow green dots) at any time during the trial are displayed within the centre of the wheel. To allocate the coins to suppliers, participants have first to click on the designated active coin displayed at the centre (green dot) and then select the supplier to sample from (panel a) – both touch screen events are indicated by a large grey dot. One of the inactive (yellow) coins is then automatically activated and displayed, in green, at the centre. This sequence repeats until all coins are allocated. Then, each of the allocated samples turn either orange, representing a good-quality apricot, or purple, representing a bad-quality apricot (panel c). Finally, after this information is revealed, the participant selects one of the sampled suppliers for the final purchase of 100 apricots (with a touch screen, indicated by a large grey dot) and the choice outcome is immediately displayed (panel d). Video 1 Download asset This video cannot be played in place because your browser does support HTML5 video. You may still download the video for offline viewing. Download as MPEG-4 Download as WebM Download as Ogg Demonstration video of the task (design W10). Two different ranges of sampling capacity, narrow (2–10 samples per trial) and wide (2–32), were tested to zoom-in relevant behavioural regimes (see Materials and methods and Table 1). For each range, we used three environments differing in the parameters of the beta distribution used to generate probabilities of good-quality apricots for each supplier. By increasing the average probability of good alternatives, we can increase the richness of the environment to move from a ‘poor’, to a ‘neutral’, up to a ‘rich’ environment (beta prior means: poor, 0.25; neutral, 0.50; and rich, 0.75). While environments and capacity ranges were run in a block fashion, capacity in each was chosen randomly in each trial from the underlying range in that block. Environments were tested within and between subjects in two different sets of experiments, to address potential learning effects. Table 1 Summary of the experimental designs. DesignCapacityN suppliersN trialsN subjectW10Within-subject2–101021618B10Between-subject2–10107245W32Within-subject2,4,8,16,323212018B32Between-subject2,4,8,16,32324045 We observed in all environments that participants’ sampling behaviour of suppliers follows a pure-breadth strategy at low capacity (Figure 2A–B), whereby they sample close to as many suppliers as coins they have. Indeed, at low capacity (C∼2,3) the number M of sampled suppliers averaged over trials and participants is close to C (Figure 2A) and the ratio M/C reaches very close to 1 (Figure 2B). At higher capacities, sampling progressively evolves towards a trade-off between breadth and depth. Therefore, although participants could have sampled more suppliers to learn about, they preferred to focus sampling capacity on a rather small fraction of suppliers, as shown by the decline of the ratio M/C towards values of around 0.4 at the highest capacities tested (Figure 2B; right panels). Although these are predicted features and thus signs of optimal behaviour, we did not observe a fast transition between the low- and high-capacity regimes, as previously predicted (Moreno-Bote et al., 2020, see also Appendix 1—figure 1A). This could simply result from averaging over participants having different transition points, or by having allocation noise, which would smooth out fast transitions. As shown below, models with allocation noise account for the smoothness of this transitions and are better models in predicting the behaviour than the noise-free optimal model. Figure 2 Download asset Open asset Number of sampled suppliers increases with capacity and is strongly sensitive to the richness of the environment, consistent with theoretical predictions. (A) Number of alternatives sampled M as a function of capacity averaged across participants (points), for each of the three different environments (colours), for the low-capacity between-subject design (B10). Dashed lines indicate unit slope line. Optimal observer predictions are displayed in black and grey lines represent individual data. Error bars correspond to s.e.m. (B) Number of alternatives sampled M divided by capacity as a function of capacity. Colour code as in panel A. Samples sizes per environment condition: designs B10 and B32: n=15, designs W10 and W32: n=18. Richer environments promote depth The effect of environmental richness was also clearly visible in our data, with richer environments typically causing a stronger preference for depth sampling (Figure 2A–B; different colours), consistent with predictions. Therefore, as it becomes easier for the participant to find good options, they prefer to neglect an even larger number of suppliers and focus capacity to examine fewer ones. We also observed systematic deviations from optimality (Figure 2A; black lines). In particular, despite participants’ strategy was overall strongly dependent on environment richness, as predicted, this was mostly due to a strong effect of the poor environment (Figure 2A–B; red lines), whereas they were not sensitive to the difference between neutral and rich environments (green and blue lines). To test quantitatively the effect of environmental richness on the BD trade-offs, we fitted participant’s individual data in each environment using three models. These models were chosen based both on our predictions and on previous observations. First, a piece-wise power-law model (W), having a fast transition at some arbitrary capacity value, was selected as it is the one anticipated by the ideal observer. Second, as previously said, we visually noticed that for a majority of participants the transition between pure-breadth and BD trade-off was gradual (see Figure 2), so we decided to capture their sampling strategy using two simpler models: a linear model (L) and a power-law model (P). Indeed, it has been predicted that once the BD trade-off established, the number of alternatives sampled M approximately increases with a power-law behaviour (Moreno-Bote et al., 2020). We observed that the power-law model (Radj2=0.96±0.04 mean ± s.d.) showed significantly better fits than the linear model (Radj2=0.92±0.06 mean ± s.d.; V=404, p<2.2×10−16). Moreover, the linear piece-wise model was not significantly better than the power-law model for all participants in all environments individually (ANOVAs, with α=.05). We compared the exponent estimated from the power-law fits in each environment and experimental design (Figure 3) using within- (W10 and 32) or between-subjects (B10 and 32) designs. The results demonstrated a significant effect of the environment on the exponent in all designs (one-way ANOVAs, B10: F1=13.17, p=7.51×10-4 ; W10: F1=31.05, p=3.37×10-5 ; W32: F1=15.12, p=.0012), except for the between-subjects design and the larger range of capacities tested (B32: F1=0.82, p=.37). Overall, effect sizes were larger in designs with narrow (B10: ωG2=.213, W10: ωG2=.121) compared to wide ranges of capacities (B32: ωG2=-.004, W32: ωG2=.072) and the non-result observed may stem from an insufficient sample size (achieved power in B32 is 14.7%, against 97.8% in W32, 95.2% in B10 and 94.7% in W10). As suggested by visual inspection above, there is no significant difference in the exponents between the neutral (median values, W10: mneutral=0.72, B10: mneutral=0.68, W32: mneutral=0.70) and rich (W10: mrich=0.71, B10: mrich=0.66, W32: mrich=0.66) environments in any of the designs (t-tests with Bonferroni correction, W10: t17=0.57, padj=1, B10: t23.9=0.24, padj=1, W32: t17=2.04, padj=.17), although strong and significant differences were typically observed between rich and poor (W10: mpoor=0.82, B10: mpoor=0.84, W32: mpoor=0.75) environments (W10: t17=5.57, padj=1.01×10-4 , B10: t20.8=3.51, padj=.006, W32: t17=3.89, padj=.004). Significant differences were also observed between poor and neutral environments in the designs with smaller capacities (W10: t17=3.25, padj=.014 , B10: t26.7=4.45, padj=4.11×10-4). Overall, we observed in the three designs mentioned a decrease of the exponent as the environment gets richer. This suggests an adaptation of participants’ sampling behaviour to the environment, with sampling depth increasing with the richness of environment. Figure 3 Download asset Open asset Participants’ strategy is modulated by the richness of the environment. Distribution of power factors extracted from fitting a linear model to values M vs. capacity in a log-log scale. Colour dots represent subjects, black dots represent means across participants, and bars are 95% confidence intervals. Results of post hoc comparisons are displayed according to adjusted p-values (‘ns’: p>0.05, ‘**’: p<0.01, ‘***’: p<0.001). Samples sizes per environment condition: designs B10 and B32: n=15, designs W10 and W32: n=18. Deviations from optimality We have demonstrated that, as expected, participants’ sampling strategy is modulated by the environment richness. We now investigate whether the strategy used coincides with the optimal sampling behaviour or if some deviations are observable. Pooling the data of the four experimental designs together, we confirmed our previous results and found a significant effect overall of the environment on the exponent of the power law (ANOVA, F1=21.03, p=8.23×10-6). Importantly, there was no significant effect of the experimental design (ANOVA, F3=0.76, p=.52) nor a significant interaction between the environment and the experimental design (ANOVA, F3=1.40, p=.24), thus confirming that the effect of the environment can be studied on the whole dataset independently of the experimental design. In order to quantify deviations of participants’ sampling strategies from the optimal strategy, we fitted the optimal values of the number M of sampled suppliers for all capacities (C={2−10,16,32}) using the power-law model previously described (see Materials and methods) and extracted the power-law exponent. Comparing the observed values of the exponent to the optimal ones (see Table 2), we observed that participants’ sampling strategy significantly shifted towards excessive depth in the poor environment (t-test, t65=-5.66, padj=3.72×10-7), and a similar tendency occurred in the neutral environment (t-test, t65=-1.74, padj=.087). In contrast, participants’ sampling strategy deviated significantly from optimality in the direction of excessive breadth in the rich environment (t-test, t65=4.06, padj=1.36×10-4). Table 2 Participants’ sampling strategy significantly deviates from optimality. Values of the factor a predicted (first row) or observed (averaged across participants ± s.d., second row) depending on capacity using a power-law function with free exponent and fixed intercept (see Materials and methods for more details). Results of comparisons between factors a using one-sample t-tests with Bonferroni corrections (third row) show that participants’ sampling strategy extracted from fitting the number of alternatives sampled is significantly tilted towards depth in the poor and neutral (tendency) environments compared to optimality, while in the rich environment participants are sampling in a breather way than predicted. EnvironmentValue of power factor aPoorNeutralRichOptimal (predicted)0.8770.7320.612Data (observed)0.795±0.1180.705±0.1270.688±0.152Comparisont65=-5.66,padj=3.72×10-7t65=-1.74,padj=.087t65=4.06,padj=1.36×10-4 Looking in greater detail at which capacities these deviations occur, we computed the differences between the optimal number Mopt of sampled suppliers and the observed M for each capacity and environment and observed again a significant effect of the environment on the differences (Scheirer-Ray-Hare test, H2=159.67, p<2.2×10−16), but also a significant effect of capacity (H10=98.07, p=2.2×10−16) and a significant interaction between the environment and capacity (H10=84.08, p=7.89×10-10), showing that the participants' bias towards excessive depth or breadth varies with the environment and the capacity. In particular, we observed that at low capacity, the difference between Mopt and the observed M tended to be positive, which is especially visible in the poor and neutral environments (Appendix 1—figure 1 and exhaustive analyses presented in Supplementary file 1). In contrast, at high capacity this difference tends to be negative, which is especially noticeable in the neutral and rich environments. The results of these analyses indicate some deviations from optimality. Participants have a bias to sample more deeply than optimal at low capacities while this bias is reversed at high capacities. These overall biases could be accounted for by assuming that participants have a biased model of the richness of the environment, not estimating as much as they should the extreme nature of poor or rich environments (Schustek et al., 2019). We next studied whether the deviations from optimality weakened over time or were persistent. We fitted the power-law model to individual BD trade-off in each environment (block) for the first and second halves of each block separately (median split, Figure 4, BD trade-offs are presented in Figure 4—figure supplement 1). We observed that participants’ strategy significantly shifted towards the optimal regime from the first to the second half of the block, in the poor (V=615, padj=.0085) and the rich environment (V=1651, padj=.0015), but not in the neutral environment (V=1271, padj=.88). Therefore, through experience within a block, participants become closer to optimal. The magnitude of this improvement was not significantly different in the rich compared to the poor environment (W=2222, p=.84), suggesting that the amount of reward accumulated doesn’t influence performance. Figure 4 with 1 supplement see all Download asset Open asset Participants’ sampling strategy gets closer to the optimal breadth-depth (BD) trade-offs with experience. Distribution of the power factor a in the power-law model when fitting the number of alternatives sampled M as a function of the capacity in each environment, separately for each block’s half (median split on the number of trials). Each line connects a subject, black dots represent the power factor a when fitting the optimal BD trade-offs. Results of post hoc comparisons are displayed according to adjusted p-values (‘ns’: padj >0.1, ‘**’: padj <0.01). Lower and upper hinges correspond to the 1st and 3rd quartiles and vertical lines represent the interquartile range (IQR) multipled by 1.5. Sample sizes n=66. Finally, we wondered whether the above deviations from optimality were more pronounced in the within-subject designs (W10 and W32), where it is possible that experience on one environment carried over the next experienced environment. We observe that in several cases sampling behaviours seem to shift towards breadth in environments directly following the presentation of a poorer environment, or towards depth if a richer environment was presented immediately before (Figure 5—figure supplement 2). For instance, participants’ sampling strategy in the neutral environment (Figure 5—figure supplement 1 middle panels) seems to differ depending on whether it was presented first or not. If presented after the poor environment, we observe a clear deviation towards breadth, whereas a shift towards depth is observed if presented after the rich environment. To statistically test the presence of a sequential, or contamination effect, on participants’ sampling strategy, we compared the exponent estimated from fitting the number of observed and optimal alternatives sampled M depending on the capacity using the power-law model (P) previously described (Figure 5). We observed significant positive deviations from optimality in the power factor after the presentation of a poor environment (deviation mean ± s.d.: 0.08±0.12, one-sample t-test, t23=3.26, padj=.014), suggesting that participants sample with more breadth when previously presented with a poor environment. In contrast, we observed negative deviations after the presentation of a rich environment (–0.08±0.13, t23=−2.85, padj=.036), suggesting that participants sample more deeply when previously presented with a rich environment. Blocks presented first (in within-subject designs or independently in between-subject designs) or after a neutral environment do not significantly deviate from optimality (first: –0.01±0.14, t126=−0.81, padj=1, after neutral: –0.04±0.18, t23=−1.13, padj=1). Overall, we observe a significant effect of block history on participants’ sampling strategy (measured by the difference between the observed and the optimal exponent: aobserved- aoptimal) (ANOVA, F3=5.23, p=.002). To confirm this, post hoc analyses revealed that participants’ sampling strategies shifted towards depth in blocks presented after a poor environment compared to blocks presented after a neutral (t-test with Bonferroni correction: t24=2.79, padj=.048) and rich environments (t24=4.32, padj=5.1×10−4), and compared to blocks presented first (t=−3.27, padj=.014). No significant shift was found after being presented to a rich environment compared to blocks presented first (t=2.21, padj=.20). Although based on exploratory analyses, these observations provide some evidence that participants’ sampling strategy was affected by the previous environment presented and that adaptation to a new environment requires to overcome the behavioural pattern implemented at earlier blocks. Figure 5 with 2 supplements see all Download asset Open asset Participants’ sampling strategy is affected by the previous environment presented. Distribution of the individual deviation from optimality (difference between power factors a fitting the data and the optimal BD trade-off) for each block’s history (presented first or independently, or after another environment). ’*’: p<0.05, ‘**’: p<0.01, ‘***’: p<0.001. Lower and upper hinges correspond to the 1st and 3rd quartiles and vertical lines represent IQR*1.5. Sample sizes: 'first': n=126, 'after poor/neutral/rich': n=24. To follow up on this history effects we further investigated if this overcoming from the previous strategy happens totally, partially or at all within the timescale of a block. We divided each block in two halves as a function of trial number (median split) and submitted the data to an ANOVA with experience (first or second block half) and environment context (poor, neutral, or rich) inside both block history types (with an environment change or not). First, we found a significant interaction between block half and environment in both cases (ANOVAs, blocks no-change: F2=5.29, p=.0063, change: F2=3.95, p=.024) showing that parti" @default.
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• W4312592995 title "Decision letter: Balance between breadth and depth in human many-alternative decisions" @default.
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