Abstract
The exploreexploit dilemma describes the trade off that occurs any time we must choose between exploring unknown options and exploiting options we know well. Implicit in this trade off is how we value future rewards — exploiting is usually better in the short term, but in the longer term the benefits of exploration can be huge. Thus, in theory there should be a tight connection between how much people value future rewards, i.e. how much they discount future rewards relative to immediate rewards, and how likely they are to explore, with less ‘temporal discounting’ associated with more exploration. By measuring individual differences in temporal discounting and correlating them with exploreexploit behavior, we tested whether this theoretical prediction holds in practice. We used the 27item DelayDiscounting Questionnaire to estimate temporal discounting and the Horizon Task to quantify two strategies of exploreexploit behavior: directed exploration, where information drives exploration by choice, and random exploration, where behavioral variability drives exploration by chance. We find a clear correlation between temporal discounting and directed exploration, with more temporal discounting leading to less directed exploration. Conversely, we find no relationship between temporal discounting and random exploration. Unexpectedly, we find that the relationship with directed exploration appears to be driven by a correlation between temporal discounting and uncertainty seeking at short time horizons, rather than information seeking at long horizons. Taken together our results suggest a nuanced relationship between temporal discounting and exploreexploit behavior that may be mediated by multiple factors.
Introduction
The exploreexploit dilemma refers to a ubiquitous problem in reinforcement learning in which an agent has to decide between exploiting options it knows to be good and exploring options whose rewards are unknown^{1}. For example, when ordering sushi at a favorite restaurant, should we exploit our usual favorite (the Rainbow Roll), which is guaranteed to be good, or explore the Burrito Roll, which could be delicious, disgusting or somewhere in between. As anyone who has agonized over a dining decision will know, making exploreexploit choices can be hard, and there is considerable interest in how these decisions are made by humans and other animals^{2}.
Recently, a number of studies have shown that people make exploreexploit decisions using a mixture of two strategies: directed exploration and random exploration^{3,4,5,6,7,8}. In directed exploration, choices are biased towards more informative options by an ‘information bonus,’ that increases the relative value of unknown options^{9}. In random exploration, behavioral variability, perhaps driven by random noise processes in the brain, causes exploratory options to be chosen by chance^{1,10}. Further work suggests these two types of exploration have different computational properties^{4}, age dependence^{11}, and may be controlled by different systems in the brain^{12,13,14,15}.
Regardless of the type of exploration, the benefits of exploring over exploiting lie in the possibility of earning larger rewards in the future. For example, in our restaurant example, if the Rainbow Roll is an aboveaverage item on the menu, then, in the short term, exploiting it will usually be best. In the longer term, however, if the Burrito Roll turns out to be sublime, then we could order this roll again and again for years to come. Thus, how much we care about future rewards, that is how we discount them relative to immediate rewards, should play a critical role in how we make our exploreexploit choice.
Optimal models of exploreexploit decision making formalize this relationship between temporal discounting and exploration, at least for directed exploration^{9}. In these models, the exploreexploit choice is made by choosing the option that maximizes the expected discounted future reward. Because this maximizing behavior is deterministic (apart from rare cases in which options are tied), optimal models do not exhibit random exploration. Thus, while they predict a negative relationship between temporal discounting and directed exploration, they say nothing about the relationship with random exploration. Suboptimal models of exploreexploit decision making do include random exploration, but most of them predict no relationship with temporal discounting^{1,10,16}.
Thus, in theory, one might predict a negative relationship between temporal discounting and directed exploration, and no relationship between temporal discounting and random exploration. In practice, however, previous experimental work suggests a more nuanced picture because of how temporal discounting covaries with our attitudes toward risk. In particular, high temporal discounting is associated with greater impulsivity^{17}, and higher impulsivity is associated with greater risk taking^{18}. This suggests that more temporal discounting is associated with more risk seeking^{19,20} (However, by defining risk seeking in terms of probability discounting, some studies on the relationship between temporal and probability discounting have yielded ambiguous results on this suggestion^{21,22,23,24,25}). In most exploreexploit paradigms, such increased risk taking would look a lot like increased directed exploration, because the more informative option is usually more uncertain, i.e. risky, too. Thus, while theory might predict a negative correlation between temporal discounting and directed exploration, this effect could be countered by a positive correlation between temporal discounting and risk taking.
In the current study, we investigated the correlation between temporal discounting and the two kinds of exploration using an individual differences approach. That is, we asked whether people with higher temporal discounting show less directed and/or random exploration. We used the 27item Delay Discounting Questionnaire^{26} to measure temporal discounting. In this questionnaire, participants choose between between small but immediate amounts of money and a larger but delayed amounts of money (e.g. $11 now or $30 in two weeks). Based on participants’ pattern of choosing between immediate and delayed options, a parameter k^{27} is calculated for each participant which estimates their average discounting rate for delayed rewards.
We used the Horizon Task^{3} to measure directed and random exploration. In this task participants make a series of choices between two slot machines (onearmed bandits). When played, each machine pays out a reward from a Gaussian distribution. The average payout is different for each machine such that one option is always better on average. Thus, to maximize their rewards, participants need to exploit the option with the highest average payout, but can only find out which option is best by exploring both options first. By manipulating key parameters in this task (distribution of rewards, time horizon, and the amount of uncertainty for each bandit), the Horizon Task allows us to quantify directed and random exploration, and, crucially, to dissociate them from baseline risk seeking and behavioral variability.
Thus, by comparing individual differences in behavior on the Horizon Task with individual differences in temporal discounting, we aimed to quantify the relationship between the two types of exploration and temporal discounting.
Methods
Participants and sample size
We collected data from a total of 82 participants (ages 18–25, average = 19.10; Females = 47, Males = 35). To estimate the sample size, we chose the conventional level of significance at α = 0.05, and the typical power at P = 0.8. A priori power analysis provided by Cohen^{28} and implemented in G*Power 3 software^{29}, estimated n = 82 as the appropriate sample size for a desired medium effect size of r = 0.3 at α = 0.05 and P = 0.8. We aimed to recruit around 100 participants but ended up with 82 which is sufficient for our desired level of significance and power. Participants were recruited through the Psychology subject pool at the University of Arizona and received course credit for their participation. All participants gave informed consent and the study was approved by the Institutional Review Board at the University of Arizona and all experiments were performed in accordance with relevant guidelines and regulations.
Temporal discounting measure
To measure temporal discounting we used the Delay Discounting Questionnaire developed by^{30}. In this instrument there are 27 questions asking participants’ preferences between two hypothetical monetary rewards: one of which pays immediately but is smaller, and the other pays more but is delayed. For example, one item asks: Do you prefer $11 today or $30 in 7 days? The amount of smallerimmediate reward (“today” option), largerdelayed reward (“later” option) and the delay (in terms of days) vary in those 27 questions (“today” reward between $11–$80; “later” reward between $25–$85; Delay between 7–186 days). The exact values are reported in^{30}Table 3.
One out of four participants were selected by chance (by drawing a card at the end of experiment) to receive the actual money according to their responses. If a participant drew a winning card (%25 chance), they then would proceed to draw a numbered chip from a bag (out of 27 chips numbered from 1 to 27 according to the number of items in the monetary choice questionnaire). The number on the chip corresponds to the number of the question we would look at for the actual payout. For example, if the winning participant picked the number 19 and they answered “later” on the question #19: “Do you prefer $33 now or $80 in 14 days?”, they need to come back to lab in 14 days and receive $80 in cash after signing a receipt form.
To quantify temporal discounting we used a number of different measures. The simplest was just the number of today options chosen, with greater temporal discounting associated with larger number of “today” choices.
More sophisticated measures of temporal discounting were obtained by fitting a hyperbolic discount factor to the data. In particular, we assume that future reward, A, arriving after a delay D, is discounted according to a hyperbolic discount factor^{31}:
where k is the subjectspecific discount factor. Fitting k was done using the spreadsheet provided by^{32} based on the method described in^{27}. In addition to computing an overall k using all 27 items, this approach also computes separate discount factors for small, medium and large reward items, based on the idea that delay discounting may be different for different range of rewards, and also the geometric mean of the small, medium and large ks. Based on the range of monetary values, the 27 choices are divided into three 9item categories: small, medium and large ranges. Then, based on the hyperbolic discounting equation (Eq. 1), it finds a k value for each item as a point in which there is no difference between choosing “today” and “later” options for that item. Then for each participant based on his/her answers and the patterns of switches from “today” to “later” options and the reverse, it gives us a kvalue for each 9item category: Small k, Medium k, Large k.
For example, in question 2 it asks: Would you prefer $55 today, or $75 in 61 days? The indifference point is when the $75 in 61 days worth as $55 today. We can calculate the k for the indifference point, in which the "today” and “later” choices look the same, by plugging V = 55, A = 75, D = 61 in Eq. (1):
If a participant choose “today” for this question, they have a k > 0.00596125.
Similarly, if the same participant answer “later” in question 7: Would you prefer $15 today, or $35 in 13 days?, the indifference point would be k = ((35/15) − 1)/13 = 0.102564103 so our participant would have a k < 0.102564103. So for this participant given these two questions, we can estimate their k to be between 0.00596125 < k < 0.102564103. By adding more questions, we can obtain better estimates for k.
Thus we have six measures of temporal discounting for each subject: the fraction of “today” choices, overall k, small k, medium k, large k, and the geometric mean of small, medium and large ks.
Horizon task
The Horizon Task^{3} is a recently developed task that allows for the measurement of directed and random exploration. The key manipulation in the Horizon Task is the time horizon, the number of trials participants will make in the future. The idea being, that in a long time horizon, people should explore, while in a short time horizon, people should exploit. Thus the change in behavior between short and long horizons can be used to quantify directed and random exploration.
More specifically, in the Horizon Task participants choose between two onearmed bandits. When chosen, the bandits pay out rewards sampled from a Gaussian distribution whose standard deviation is always fixed at 8 points, but whose mean is different for each bandit and can change from game to game. Each game lasts for 5 or 10 trials and participants’ job is to make multiple choices between the two bandits to try to maximize their reward. Because they know nothing about the mean of each bandit at the start of each game, they can only find out which option is best by exploring.
To control the amount of information, the first four trials of each game are predetermined (Fig. 1B). Participants are instructed to pick either the left or right bandit during these four “forced trials”. By changing the number of forced choices for each bandit, we manipulate the amount of “uncertainty” or information participants have about the payoffs from each bandit. In the unequal uncertainty (or [1 3] condition) participants are forced to choose one option once and the other three times; whereas in the equal uncertainty (or [2 2] condition) participants play both options twice. After the forcedchoice trials, the rest of trials are “free trials” in which participants make their own choice. The number of free trials varies between horizon conditions with 1 free choice in the horizon 1 condition and 6 free choices in the horizon 6 condition.
These two information conditions allow us to quantify directed and random exploration by looking at the first free choice in each game, immediately after the four forced choices (Fig. 1A). Because directed exploration involves information seeking, it can be quantified as the probability of choosing the more informative option in the [1 3] condition, p(high info). Conversely, because random exploration involves decision noise, it correlates with choosing the low mean option in the [2 2] condition, p(low mean). Computing these measures separately for each horizon condition allows us to quantify four key properties of exploreexploit behavior:
uncertainty preference as p(high info) in horizon 1
baseline behavioral variability as p(low mean) in horizon 1
directed exploration as Δp(high info), the change in information seeking with horizon
random exploration as Δp(low mean), the change in variability with horizon
In Supplementary Materials4 you can find the onscreen instructions used to instruct participants at the beginning of the Horizon Task.
Modelbased analysis
In addition to the abovementioned modelfree parameters (p(high info) and p(low mean)) we also fit a logistic model that was previously shown to be adequate in capturing the basics of the Horizon Task^{3}. With this model we estimate two main parameters: “information bonus” and “decision noise” which corresponds to the modelfree measures of directed and random exploration, respectively. The description of the model is provided in the Supplementary Materials1. The modeling will help us to disentangle directed and random exploration more clearly. However, since there was a high correlation between modelfree and modelbased parameters (Supplementary Materials2 Fig. S1) and both the modelbased and modelfree parameters yielded the same relationships with the temporal discounting (Supplementary Materials2 Fig. S2), and since the modelfree approach requires less assumptions than the modelbased approach, we chose to include the modelfree analysis in the main article and move the modeling part to the Supplementary Materials.
Statistical analysis
To evaluate the basic behavior on the Horizon Task, we used the paired (dependent) sample ttest. For directed exploration we looked to see whether there was a significant increase in the mean of p(high info) from horizon 1 to horizon 6 condition using the paired sample ttest. Similarly, for the random exploration we used paired sample ttest to see whether there was a significant increase in the p(low mean) between horizon 1 and horizon 6 condition.
To evaluate the relationship between measures of temporal discounting and the Horizon Task parameters, we simply calculated the Pearson correlation coefficients between the 6 measures of temporal discounting (the 5 k’s: overall k, small k, medium k, large k, geometric k and the total number of today items chosen) on one hand and the Horizon Task parameters (directed exploration, random exploration, p(high info) in horizon 1 and 6, p(low mean) in horizon 1 and 6, reaction time on the first free trial in horizon 1 and 6, and accuracy in horizon 1 and 6) on the other hand. Accuracy is defined as choosing the high mean option.
Compliance with ethical standards
All procedures performed in experiments were in accordance with the ethical standards of the institutional research committee and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards.
Informed consent
Informed consent was obtained from all individual adult participants included in the study.
Results
Behavior on the horizon task (Modelfree)
Table 1 shows the range, mean and standard deviation (SD) for the basic task parameters in the sample. Figure 2 shows the distribution of basic task parameters in the sample (N = 82). Behavior on the Horizon Task was consistent with that previously reported in our studies^{3}. Specifically we see a significant increase in p(low mean) with horizon (p(low mean)h1_average = 0.2883; p(low mean)h6_average = 0.3554; t(81) = 3.87; p < 0.001; and the effect size of d = 0.40^{33}) (Fig. 3B) and we see a clear trend (but not significant) in p(high info) with horizon (p(high info)h1_average = 0.5146; p(high info)h6_average = 0.5486; t(81) = 1.75; p = 0.084; d = 0.24) (Fig. 3A), consistent with participants using both types of exploration in this paradigm. Figure 4 shows the scatter plots comparing p(high info) and p(low mean) for individual participants in horizon 1 and horizon 6 conditions. Out of 82 participants, 57 individuals showed random exploration (p(low mean) h6 > p(low mean) h1) and 47 individuals showed directed exploration (p(high info) h6 > p(high info) h1) on average.
Behavior on the temporal discounting task
For the temporal discounting measure we obtained 5 different k values for each participant as a measure of how much they discount future reward. We also can simply estimate that measure just by counting the number of times participants chose the immediate versus delayed reward (Supplementary Materials3). Table 2 shows the range, mean and standard deviations of temporal discounting indices (k’s and # today items) in 82 participants of our study which is similar to previous studies using the same measure^{26,30}. Figure 5 shows the histogram of distribution of temporal discounting indices in the sample (N = 82).
In our research, it turned out that all of these indices are highly correlated with each other (Supplementary Materials3) and all have very similar relationship with directed and random exploration. The more simple measure of # today items has a Pearson’s correlation coefficient between 0.89–1 with the more complicated k measures (Supplementary Materials3 Fig. S3).
Correlation between temporal discounting and exploreexploit behavior
Table 3 shows the correlations between a measures of temporal discounting (log k overall) and the horizon task parameters: directed and random exploration, p(high info) & p(low mean) at horizons 1 & 6, reaction times and accuracy (the percentage of times the “accurate” option (the higher mean option) was chosen for each horizon (1 & 6) conditions. We found a significant negative correlation between between temporal discounting and directed exploration, with more temporal discounting associated with less directed exploration. Closer inspection revealed that this negative correlation was driven by a positive correlation between temporal discounting and p(high info) at horizon 1 and a zero correlation between temporal discounting and p(high info) at horizon 6.
In contrast to directed exploration, temporal discounting did not correlate with random exploration. There was, however, a positive correlation between temporal discounting and overall behavioral variability, p(low mean) in both horizon conditions. This suggests that people with higher temporal discounting perform worse on the task overall.
Finally, to demonstrate that the significant correlations were not driven by outliers, we plot the correlations between measures of directed and random exploration and the number of today items chosen in Fig. 6.
Modelbased analysis
We also utilized a logistic model (further explained in the Supplementary Materials1) to estimate two main parameters, “information bonus” and “decision noise”, which are assumed to correspond to p(high info) and p(low mean) in the modelfree analysis, respectively. Figure S1 in the Supplementary Materials2 shows that in fact there are high correlations between modelfree and modelbased parameters. Additionally, Fig. S2 shows that the correlations between temporal discounting and modelbased parameters are similar to the correlations between temporal discounting and modelfree parameters (Fig. 6).
Discussion
In this study we investigated the correlation between temporal discounting measured by a monetary choice questionnaire^{30} and two types of exploration (directed and random) measured by the Horizon Task^{3}. We found a negative correlation between temporal discounting and directed exploration that was driven by a positive correlation between temporal discounting and uncertainty seeking in horizon 1. Conversely, we found no correlation between temporal discounting and random exploration, although we did see a positive correlation between temporal discounting and overall behavioral variability.
While the negative correlation between temporal discounting and directed exploration (i.e. Δp(high info) is consistent with the theory, the correlation with p(high info) in each horizon condition is not. In particular, normative models predict a negative correlation between temporal discounting and p(high info) in horizon 6 and no correlation in horizon 1. Conversely, we found no correlation with horizon 6 behavior and a positive correlation with horizon 1 behavior.
One reason for this discrepancy could be the possible positive association between temporal discounting and risk taking^{19,20} (See^{21,22,23,24,25} for suggesting otherwise). In both horizon conditions in the Horizon Task, the more informative option is also the more uncertain, riskier option. Thus, by this account, people who discount more would show greater p(high info) in both horizon conditions, but this would be counteracted by a negative relationship between temporal discounting and directed exploration in horizon 6. That is, in horizon 1, directed exploration is not present, and so the positive association with temporal discounting is revealed. In horizon 6, directed exploration is present, and this negative relationship with temporal discounting counteracts the positive relationship with risk taking leaving no correlation overall. Testing this hypothesis requires a future study that includes appropriate measures of risk taking.
The fact that random exploration does not correlate with temporal discounting is also consistent with theories of random exploration^{1,10}. Moreover, this apparent dissociation between directed and random exploration is consistent with other findings showing that directed and random exploration have different computational properties^{4}, different age dependence^{11}, and may rely on dissociable neural systems^{12,14,15}. In this regard it is notable that directed exploration appears to rely on the same frontal systems thought to underlie temporal discounting^{5,12,14,34,35,36}, while random exploration does not. Thus, an intriguing prediction is that the relationship between directed exploration and temporal discounting may be mediated by the integrity of frontal circuits, something that future neuroimaging studies could address.
There are several limitations in the current study. First, the chosen measures for both temporal discounting and exploratory behavior are very specific. This questions the generalizibility of our results. Although a strong correlation between different measures of temporal discounting has been demonstrated in several studies^{37,38}, most of these measures are monetary which may have weak relationships with delay discounting in other domains^{39}. Exploratory behavior also has been studied in different settings including foraging, repeated choice and sequential choice paradigms and it seems there is no shared factor underlying exploratory behavior in all of these tasks^{40}. Replicating the current study using other measures of exploration and temporal discounting, will provide us with more evidence to better assess the generlizablilty of the current results.
Another important limitation of our study is recruiting university students as participants. Between all possible biases that such a selective sample may introduce in our study, age seems the most obvious one. It has been shown that temporal discounting^{41}, exploratory behavior^{42} and risktaking behavior^{43}, all varies significantly through the lifespan. So it is unclear how the results of the current study would look like in different age groups. This would be an interesting topic for a future study.
Lastly, we hypothesised the mediating role of risk taking to explain the results while we haven’t included appropriate scales to measure it in the current study. A future study can shed more light on this hypothesis by adding measures of risk taking.
Data availability
All the raw data and MATLAB codes for the analysis and plots are available at https://github.com/hashem20/temporaldiscountingexploreexploit.
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Acknowledgements
The authors thank Shlishaa Savita and Kathryn Lui Kellohen for their help in collecting and organizing data.
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H.S., S.W., A.S. and R.C.W. designed the experiment. H.S., M.R.A. and H.M.K. ran the experiment. H.S. and S.W. analyzed the data with supervision from R.C.W. H.S. and R.C.W. wrote the manuscript with input from all other authors.
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Sadeghiyeh, H., Wang, S., Alberhasky, M.R. et al. Temporal discounting correlates with directed exploration but not with random exploration. Sci Rep 10, 4020 (2020). https://doi.org/10.1038/s41598020605764
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DOI: https://doi.org/10.1038/s41598020605764
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