Abstract
Humans and other animals are capable of inferring neverexperienced relations (for example, A > C) from other relational observations (for example, A > B and B > C). The processes behind such transitive inference are subject to intense research. Here we demonstrate a new aspect of relational learning, building on previous evidence that transitive inference can be accomplished through simple reinforcement learning mechanisms. We show in simulations that inference of novel relations benefits from an asymmetric learning policy, where observers update only their belief about the winner (or loser) in a pair. Across four experiments (n = 145), we find substantial empirical support for such asymmetries in inferential learning. The learning policy favoured by our simulations and experiments gives rise to a compression of values that is routinely observed in psychophysics and behavioural economics. In other words, a seemingly biased learning strategy that yields wellknown cognitive distortions can be beneficial for transitive inferential judgements.
Main
Humans routinely infer relational structure from local comparisons. For instance, learning that boxer Muhammad Ali defeated George Foreman can let us infer that Ali would probably win against other boxers that Foreman had defeated. More formally, generalizing from relational observations to new, unobserved relations (for example, knowing that A > B and B > C leads to A > C) is commonly referred to as transitive inference^{1,2,3,4}. Transitive inference is not a uniquely human capacity^{5} but can also be observed in nonhuman primates^{6,7,8}, rats^{9} and birds^{10,11,12}.
In the laboratory, transitive inference can be observed after teaching participants the relations between neighbouring elements from an ordered set of arbitrary stimuli (Fig. 1a). The neighbour relations are typically taught through pairwise choice feedback (Fig. 1b) where the relational information is deterministic (that is, if A > B, in our sporting analogy, A would never lose a match against B). Various theories have been proposed to describe how observers accomplish transitive inferences of nonneighbour relations (for example, A > D) in such settings. One class of models posits that observers learn implicit value representations for each individual element (A, B, C and so on), which then enables judgements of arbitrary pairings^{3,13,14}. Alternatively, transitive inference could be accomplished through more explicit, hippocampusbased memory processes^{15,16,17,18}, which we will return to below.
Before turning to transitive inference, we consider relational learning in a fullfeedback scenario (Fig. 1b) where choice feedback is provided for every possible pairing of items, such that no transitive inference is required. We model implicit value learning in this setting through a simple reinforcement learning (RL) mechanism (Qlearning; Methods) by which relational feedback (for example, ‘correct’ when selecting A over B) may increase the perceived value (Q) of item A and decrease that of item B (Model Q1, Fig. 2a). In this simple RL model, relational feedback symmetrically updates (with opposite signs) the value estimates for both items in a pair. For instance, if Muhammad Ali beat George Foreman, it seems rational to attribute this outcome to Ali’s greater skill as much as to Foreman’s deficit. We show in simulations that symmetric value updating is in fact optimal in the fullfeedback setting. An alternative model with asymmetric learning rates (α^{+} ≠ α^{−}) applied to the winner and loser in a pair (Model Q2; ‘2’ denotes dual learning rates) learns worse than the symmetric model (Q1) where α^{+} = α^{−} (Fig. 2b,c). Implicit value learning generally gives rise to a ‘symbolic distance effect’^{1,19,20}, where nearby elements are less discriminable (due to more similar value estimates) than elements with greater ordinal distance^{14,21}.
Next, we turn to a partialfeedback setting, which is the typical transitive inference scenario, with feedback being provided only for pairs of items with neighbouring values (Fig. 1b). Here, the simple RL models (Q1 and Q2) effectively learn about stimuli only at the extremes of the ordered set (for example, A and H; Extended Data Fig. 1a), since these are statistically more likely to be winners or losers than their neighbours (under uniform sampling). No value learning occurs for intermediate items (stimuli B to G), since these are equally likely to be paired with lower and highervalued stimuli^{3}. However, the model can easily be adapted to performing transitive inference when extending it with a simple assumption: value updates should scale with the difference between the estimated item values, Q(A) − Q(B) (for similar approaches, see refs. ^{14,21,22}). More specifically, to the extent that A is already higher valued than B, observing the expected outcome A > B should induce weaker value updates, whereas the unexpected outcome A < B should induce stronger updates. To illustrate, observing an unknown amateur boxer win against a world champion should induce stronger changes in belief than the opposite, less surprising result (champion > amateur). When we incorporate this simple assumption into our model (Model Q1*), it learns orderly structured values, Q(A) > Q(B) > … > Q(H), and thus accomplishes transitive inferences for all pairs of items (Fig. 2d; see also Supplementary Video for an illustration of how our Qlearning models accomplish transitive learning). We also observe a symbolic distance effect with this type of learning under partial feedback, similar to what we observed with simple RL under full feedback (Fig. 2a,d).
Notably, the effect of asymmetric learning rates (α^{+} ≠ α^{−}, Model Q2*) under partial feedback is strikingly different from what we observed with full feedback. Under partial feedback, optimal performance is achieved with a strongly asymmetric learning policy (α^{+} ≫ α^{−} or α^{+} ≪ α^{−}), where only the winner (or loser) in a pair is updated (Fig. 2e,f and Supplementary Video). In other words, in a setting where hidden relational structure is inferred from only local comparisons, it is surprisingly beneficial to ignore losers (or winners) in outcome attribution. Of note, the winner/loser asymmetry outlined here differs from, and is orthogonal to, previously described asymmetries in learning from positive/negative^{23,24,25,26} or (dis)confirmatory outcomes^{27,28}. A noteworthy aspect of our Model Q2* is that the superior, asymmetric learning policy results in a compression of the observer’s latent value structure (Fig. 2f). Selective updating therefore naturally gives rise to diminishing sensitivity towards larger values, as is universally observed in psychophysics^{29}, numerical cognition^{30,31} and behavioural economics^{32}.
Going beyond typical studies of transitive inference with deterministic outcomes, we examined whether our simulation results generalize to scenarios where relational outcomes can be variable, as is the case in many realworld domains such as sports, stock markets and social hierarchies. To this end, we added random variance to the comparison outcomes such that, for example, an item won over its lowervalued neighbour in approximately 80% of cases but lost in the other 20% (see Methods for the details). Intuitively, we allowed for the possibility that competitor A may sometimes lose against B, even if A is generally stronger. We found that our simulation results held for such probabilistic environments, just as they did for deterministic scenarios (Extended Data Fig. 2).
In the models discussed so far, the observer associates each individual item (A, B, C and so on) with an implicit value (Q(A), Q(B), Q(C), … ; itemlevel learning). An alternative strategy is to more directly learn response preferences for each individual item pairing (p_{A>B}, p_{B>C}, … ; pairlevel learning; Methods). For instance, in our partialfeedback setting (Fig. 1), observers might learn to choose A when comparing A and B, to choose B when comparing B and C, and so forth, even without relying on value estimates for the individual items. In its simplest form, such memory for pairwise preferences (Model P) only allows learning of pair relations that have been directly experienced (that is, only neighbouring pairs in our partialfeedback setting; Extended Data Fig. 1b, left). However, the pairlevel memory can also be extended to allow for transitive inference of more distant, never experienced relations^{8,33,34}: when asked to judge, for example, A versus C, observers might ‘chain together’ memories of the linking neighbour preferences (p_{A>B} and p_{B>C}) through associative recall^{17,35} or spreading activation^{36} to infer a transitive preference (p_{A>C}; Model Pi; Extended Data Fig. 1b, right). Transitive inference based on such pairlevel learning gives rise to an inverse symbolic distance effect (Extended Data Fig. 1b, right), where nearby pairs are more discriminable than more distant pairs, reflecting the high dimensionality of the underlying associative memory structure. In modelling our empirical data, we allow for itemlevel value learning (models denoted by a Q), pairlevel learning (models denoted by a P) and a combination of both, in explaining human transitive inference.
Results
We report the results of four experiments (n = 145) where we varied whether feedback was full or partial and whether it was probabilistic or deterministic (Methods). In all experiments, the participants were shown a pair of items (drawn from a set of eight) on each trial and were asked to make a relational choice (Fig. 1). The participants were given no prior knowledge about item values and could learn only through trialanderror feedback.
Full feedback
In Experiment 1 (Exp. 1; n = 17), probabilistic choice feedback (Methods) was provided after each of 448 sequential pair comparisons (‘full feedback’). Figure 3a shows the mean proportions of correctly choosing the highervalued item, averaged over all trials in Exp. 1. Descriptively, the choice matrix is dominated by a symbolic distance effect, as predicted by implicit value learning. Fitting our itemlevel learning models (Q1, Q2, Q1* and Q2*), the best fit to the data is provided by the simplest model (Q1), with a single learning rate for winners and losers (Fig. 3c,e; protected exceedance probability, pxp(Q1) > 0.99; mean Bayesian information criterion (BIC), 361.79 ± 24.68 s.e.m.). In other words, participant behaviour was consistent with a symmetrical updating policy, which our simulations showed to be optimal in the fullfeedback setting.
Partial feedback
In Exps. 2–4, choice feedback was provided only on neighbour pairs (‘partial feedback’) to study transitive inference. In these experiments, we increased the frequency at which the participants were shown neighbouring pairs relative to nonneighbouring pairs to provide more learning opportunities, since the task is inherently harder. We verified that our simulation results were invariant to this modification (Extended Data Fig. 3). Otherwise, the design of Exp. 2 (n = 31) was identical to that of Exp. 1. Experiment 3 (n = 48) was an online replication of Exp. 2, where the pair items on each trial were shown side by side instead of sequentially. Experiment 4 (n = 49) was similar to Exp. 3, but the feedback was made deterministic (100% truthful), as in previous studies of transitive inference (see Methods for details on the individual experiments).
The choice data from each of the partialfeedback experiments (Exps. 2–4, Fig. 3b) showed clear evidence for transitive inference, with abovechance performance for nonneighbouring pairs that never received feedback (mean accuracy averaged over nonneighbour trials: Exp. 2, 0.714 ± 0.028; Exp. 3, 0.698 ± 0.018; Exp 4, 0.709 ± 0.019; Wilcoxon signedrank tests against chance level (0.5): all P < 0.001, all r > 0.84; see Supplementary Table 1 for the details). Furthermore, the grand mean choice matrices showed the following descriptive characteristics: (1) a symbolic distance effect similar to that observed with full feedback, (2) an asymmetry with greater discriminability of lowervalued items and (3) relatively increased discriminability of neighbour pairs.
The modelling results for the partialfeedback experiments are summarized in Fig. 3d,f (see also Extended Data Figs. 4–8). We highlight two main findings. First, the partialfeedback data were better described by asymmetric models with different learning rates for winners and losers. This held true at every level of model complexity, with our asymmetric models (Q2, Q2*, Q2* + P and Q2* + Pi) always performing better than their symmetric counterparts (Q1, Q1*, Q1* + P and Q1* + Pi; Wilcoxon signedrank tests comparing BICs, Exps. 2–4 combined: all P < 0.001, all r > 0.35; see Supplementary Table 2 for details), and regardless of whether the partial feedback was probabilistic (Exps. 2 and 3) or deterministic (Exp. 4; comparison of mean BICs between asymmetric and symmetric models: all P < 0.001, all r > 0.67; see Supplementary Table 3 for the details). In other words, the participants adopted an asymmetric learning policy, which proved superior in our model simulations (Fig. 2e).
Second, behaviour in the partialfeedback scenario was not fully described by itemlevel value learning alone. The winning model in Exps. 2 and 4 (Q2* + P; pxp, 0.81 and 0.39; mean BIC, 609.15 ± 12.77 and 434.27 ± 6.29) incorporated pairlevel learning in addition to the value estimates of the individual items. This pairlevel memory (+P; ‘Models’) accounts for the increased performance for neighbouring pairs (Fig. 3b, first offdiagonals; see also Extended Data Fig. 1b). In Exp. 3, the model comparison was less clear, with model Q2* showing the highest pxp (0.41) but model Q2* + P providing a better average fit in terms of BIC (676.86 ± 19.40 versus 692.09 ± 8.61; Wilcoxon signedrank test: P < 0.001; z = −4.53; n = 48; r = 0.48; 95% confidence interval (CI), 0.29 to 0.63). However, we found no evidence that pairlevel memory contributed to transitive inference in our experiments. Incorporating associative recall of ‘linking’ neighbour pairs (+Pi) worsened the model fits, in terms of both pxp (all pxp < 0.07) and BIC (Exps. 2–4 combined; Q2* + Pi, 570.42 ± 15.72; compared with Q2* + P, 567.60 ± 15.38; Wilcoxon signedrank test: P < 0.001; z = −6.02; n = 128; r = 0.53; 95% CI, 0.37 to 0.69), which is in line with the absence of an ‘inverse’ symbolic distance effect (Extended Data Fig. 1b, right) in the empirical choice data (Fig. 3b).
Figure 4a illustrates how learning of nonneighbour comparisons in our experiments evolved over time. The value compression implied by asymmetric learning of winners (Fig. 2e) predicts relatively better performance for lowervalued pairs (for example, F–H) than for highervalued pairs (for example, A–C; Fig. 4b, right). We observed no such pattern in Exp. 1 with full feedback (Fig. 4a–c, left). In contrast, participants in Exps. 2–4 with partial feedback showed the critical pattern early on (Fig. 4a–c, right), as predicted by our asymmetric learning models (Fig. 4b, right). Turning to neighbouring pairs (Fig. 4d), which could additionally benefit from pairlevel learning (+P; see above), our asymmetric model (Q2* + P) predicts only a modest decline in accuracy for highervalued pairs (see also Fig. 3d), which also matched the empirical data (Fig. 4d, right).
The winner/loser asymmetries described so far might be explained by alternative learning biases, such as asymmetric learning weights for chosen versus unchosen items^{37}. Given abovechance performance, the chosen item will statistically be more likely to be the winning item. To test this alternative explanation, we repeated our modelling analyses using separate learning rates (α^{+}/α^{−}) for chosen/unchosen items instead of for the winning/losing item (Methods and equation (4)). This alternative model fit our partialfeedback data significantly worse (mean BIC collapsed across Exps. 2–4, 587.21 ± 15.91 versus 567.60 ± 15.38; Wilcoxon signedrank test: P < 0.001; z = −6.09; n = 128; r = 0.54; 95% CI, 0.40 to 0.65), corroborating our interpretation that transitive inference learning was better characterized by asymmetries between winners and losers. Previous RL studies have also highlighted potential differences in learning from positive (confirmatory) as opposed to negative (disconfirmatory) feedback^{24,25,28}. Extending our winning model to incorporate such confirmation bias (Extended Data Fig. 9) improved the overall model fit (mean BIC collapsed across Exps. 2–4, 537.91 ± 14.45 versus 567.60 ± 15.38; Wilcoxon signedrank test: P < 0.001; z = −8.32; n = 128; r = 0.72; 95% CI, 0.67 to 0.81), which is consistent with previous findings in other learning contexts^{24,25,28}. However, the addition of confirmation bias left the finding of winner/loser asymmetries unchanged (Extended Data Fig. 9 and Fig. 3f), thus illustrating the robustness of our results.
We also compared our model family against two previous models of transitive inference (Supplementary Methods): a classic valuetransfer model (VAT)^{21} and a more recent model based on ranking algorithms used in competitive sports such as chess (RLELO)^{22}. Both VAT and RLELO were outperformed by our winning model Q2* + P when fitted to our partialfeedback data (Exps. 2–4 combined; mean BIC VAT, 606.59 ± 14.04; RLELO, 617.96 ± 15.07; Q2* + P, 567.60 ± 15.38; Wilcoxon signedrank tests versus Q2* + P: both P < 0.001, both r > 0.65). This held true even when we modified VAT and RLELO to include pairlevel learning (+P) and separate learning rates for winners and losers (mean BIC, 572.95 ± 15.17 and 576.34 ± 15.94, respectively; both P < 0.02, both r > 0.21; see Supplementary Table 4 for detailed model comparison results). Our asymmetric Qlearning process thus explains the experimental data better than these earlier models of transitive inference.
Our model simulations (Fig. 2e) indicate two aspects of asymmetric learning that are not directly evident from the grouplevel results shown in Figs. 3 and 4. First, performance benefits under partial feedback emerged not only for selective updating of winners but likewise for selective updating of losers. Second, performance was highest for extreme asymmetries where the loser (or winner) in a pair was not updated at all. We examined these aspects more closely on the individual participant level (Fig. 5). Half of the participants in Exps. 2–4 (n = 64) were indeed characterized by extreme asymmetry towards winners (with α^{−} near zero). However, another subgroup (n = 15) showed the opposite, an extreme asymmetry towards losers (with α^{+} near zero). In other words, in the partialfeedback setting, most individuals showed an extreme bias towards winners or losers, either of which proved to be an optimal policy in our model simulations (Fig. 2e and Extended Data Fig. 2, right). In contrast, we found less substantial asymmetries under full feedback (Exp. 1) when allowing the learning rates for winners and losers to vary freely (that is, using model Q2 instead of the winning model Q1). Statistical analysis confirmed that the asymmetries under full feedback (Exp. 1) were significantly lower than under partial feedback (Mann–Whitney Utest of absolute asymmetry indices collapsed over Exps. 2–4 (n = 128) compared with Exp. 1 (n = 17): P = 0.007; z = −2.69; r = 0.22; 95% CI, 0.04 to 0.39; Methods).
A potentially surprising observation in the subgroup of participants in Exps. 2–4 who selectively updated winners (Fig. 5, right) is a tendency for belowchance performance for relatively highvalued nonneighbours (for example, A–D) despite each individual performing robustly above chance overall (‘Participants’). A potential explanation is that participants may sometimes have confused the two pair items in working memory at the time of feedback (Fig. 1b, right). Such memory confusions would result in the items occasionally being updated with the incorrect learning rate and the incorrect sign (‘Models’, equation (4)). Under an updating policy that ignores losers, the losing items would then be updated only in error (and always incorrectly), resulting in a negative net learning rate for losers. Indeed, repeating our analysis while allowing for negative values of α^{+} and/or α^{−} yielded a small but significant improvement in model fit (mean BIC, 562.52 ± 15.23 compared with 567.60 ± 15.38; Wilcoxon signedrank test: P < 0.001; z = −5.58; n = 128; r = 0.50; 95% CI, 0.36 to 0.62). More specifically, in the n = 64 participants who selectively updated winners (that is, with a positive α^{+}; mean = 0.063 ± 0.007), α^{−} estimates were weakly negative (mean = −0.009 ± 0.0016; Wilcoxon signedrank test against zero: P < 0.001; z = 4.95; n = 64; r = 0.64; 95% CI, 0.47 to 0.78). Memory confusions may thus explain the systematically false inferences about certain item pairs (Fig. 5, right). Together, these findings are consistent with a strongly asymmetric learning mechanism that is also prone to occasional memory errors.
To summarize our empirical findings, when transitive relations could be inferred only from local comparisons (Exps. 2–4), human learning was characterized by an asymmetric outcome attribution to either winners or losers, which proved to be surprisingly optimal in model simulations. In contrast, a symmetric attribution of relational outcomes emerged in a setting where all pair relations could be directly experienced (Exp. 1), and for which our simulations identified symmetric updating to be the most efficient.
Discussion
Reasoning about the relationships between arbitrary pairings of items is a key component of human intelligence. Through simulations, we showed how different learning regimes perform better in full and partialfeedback contexts. Under full feedback, the best learning model used symmetric learning to update the value estimates for the winning and losing items in opposite directions, with the same magnitude. However, under partial feedback (only for neighbouring items), the best learning model used asymmetric learning to only update the value representations for either the winner or the loser. Across four experiments, we found robust evidence that human learners used the best learning rule to match their feedback context. The participants used symmetric learning under full feedback (Exp. 1) and asymmetric learning under partial feedback (Exps. 2–4). While our asymmetric models allowed for a wide range of possible learning rate combinations, a majority of the participants showed onesided learning, where value representations were only updated for either winners or losers.
An important feature found both in our model simulations and participant behaviour is a compression of the emerging implicit value structure, which results in a systematic decrease in discriminability of highervalued items (Fig. 2f and Fig. 4). This resembles the Weber–Fechner law in psychophysics^{29}, where sensitivity to stimulus differences diminishes with increasing magnitude (see also refs. ^{32,38}). While there exist alternative theoretical accounts for this ubiquitous phenomenon^{39}, our findings add a new perspective: compressed representations of magnitude emerge naturally from a learning policy that is optimized for inferring global relationships from local comparisons. From this perspective, subjective compression might not only reflect an efficient adaptation to the distribution of stimuli in the environment^{40,41,42} but could also result from learning policies that enhance transfer to novel relationships.
In other contexts, previous RL studies have discovered different types of learning asymmetries, such as between positive and negative^{24,25} or confirmatory and disconfirmatory outcomes^{28}. The onesided learning policy highlighted here in the context of transitive inference is orthogonal to these other asymmetries but may play a similar role in leveraging a biased but advantageous learning strategy (see also refs. ^{27,43,44}). Unlike with ‘optimal’ cognitive biases reported previously^{45,46,47,48,49}, we did not find the benefit of the present learning asymmetries to emerge from general limitations (noise) in decisionmaking (Extended Data Fig. 10). We speculate that human learners may adopt the present biases more strategically, in settings where the availability of only sparse feedback presages the requirement of future inferential judgements.
Previous theories have proposed richer and more complex cognitive mechanisms for transitive inference, often with an emphasis on the key role of the hippocampus in representing relational knowledge^{15,50}. Early research appealed to the idea that individuals used spatial representations to learn ordered value sequences^{1,8,51}. More recently, various models have been proposed that use associative learning mechanisms to describe how interactions between episodic memories in the hippocampus can generalize relational knowledge from local to distant comparisons^{17,52}. In our present experiments, we found no evidence for transitive inference through such ‘associative linking’ and failed to observe its key empirical prediction (an inverse symbolic distance effect; Fig. 3b and Extended Data Fig. 1b, right). We show instead that simpler mechanisms of value learning^{21,53,54} combined with clever biases (that is, asymmetric learning rates) can be sufficient for performing transitive inference and for accurately describing human learners.
While our model explains transitive inference via learning of individual item values, our findings do not preclude the emergence of a more explicit (for example, maplike) mental model of the items’ relational structure^{55} or the possibility that participants develop direct action policies for each pairing^{56} after learning progress. A related question for future work is to what extent learning from relational feedback may transfer to comparisons with new items that were not contained in the learning set.
To summarize, we report evidence for pronounced asymmetries in transitive relational learning, where observers selectively update their beliefs only about the winner (or the loser) in a pair. Although asymmetric learning yields distorted value representations, it proves beneficial for generalization to new, more distant relationships. This biased learning regime thus seems well adapted for navigating environments with relational structure on the basis of only sparse and local feedback.
Methods
Participants
The participants in Exps. 1 and 2 were recruited from a participant pool at the Max Planck Institute for Human Development. Of these, n = 20 participated in Exp. 1 (13 female, mean age 27.15 ± 3.91 years), and n = 35 participated in Exp. 2 (14 female, 27 ± 3.80 years). The participants in Exps. 3 and 4 were recruited online via Prolific Academic (www.prolific.co), with n = 76 completing Exp. 3 (23 female, 24.73 ± 5.40 years) and n = 60 completing Exp. 4 (23 female, 25.92 ± 4.54 years). The participants in Exps. 1 and 2 received compensation of €10 per hour and a bonus of €5 depending on performance. Payment in Exps. 3 and 4 was £4.87 (£1.46 bonus) and £3.75 (£1.12 bonus), respectively. We obtained written informed consent from all participants, and all experiments were approved by the ethics committee of the Max Planck Institute for Human Development.
Participants who did not reach abovechance learning levels were excluded from analysis. The threshold for inclusion was set to 60% correct judgements in the last two blocks of the experiment, which corresponds to a binomial test probability of P < 0.01 compared with chance level (50%). After exclusion, n = 17 (Exp. 1), n = 31 (Exp. 2), n = 48 (Exp. 3) and n = 49 (Exp. 4) participants remained for analysis.
Stimuli, task and procedure
In Exps. 1 and 2, eight pictures of everyday objects and common animals were used as stimuli (Fig. 1a). In Exps. 3 and 4, we included 12 additional pictures of objects and animals and selected for each participant a new subset of 8 images as stimuli. An additional set of 8 pictures was used for instructions and practice purposes in each experiment. All images were from the BOSS database^{57}, with the original white background removed.
All experiments involved learning the latent relations between the eight stimuli (A > B > C > D > E > F > G > H) through pairwise choice feedback, where the latent value structure was pseudorandomly assigned to the pictures for each participant. On each trial, a pair of pictures was presented, and the observers were asked to choose the highervalued stimulus (twoalternative choice with timeout). All possible stimulus pairings (7 neighbouring and 21 nonneighbouring) were randomly intermixed across trials, with randomized ordering of the elements in a pair (for example, A–B or B–A). Prior to all experiments, the participants were given written instructions and were asked to complete two brief practice blocks to become familiar with the task.
Experiment 1 (full feedback, n = 17)
On each trial in Exp. 1, two items were presented one after the other at fixation (0.5 s per item) with an interstimulus interval of 2–3 s (randomized). After the second item, Arabic digits ‘1’ and ‘2’ were displayed to the left and right of fixation (the positions were randomized across trials), and the participants were asked to choose the highervalued item by pressing the corresponding arrow key (left or right) within 2 s. A written feedback message (‘great’ for correct responses, ‘incorrect’ for errors) was shown after each choice (neighbouring and nonneighbouring pairs). The items’ latent values in Exp. 1 were probabilistic (with a Gaussian distribution) and designed such that feedback was truthful on approximately 80% of neighbour trials (probabilistic feedback). Each participant performed 448 learning trials with all possible stimulus pairings (n = 56) presented in each of eight consecutive blocks. Experiments 1 and 2 were conducted in lab, using Psychophysics Toolbox Version 3 (ref. ^{58}) running in MATLAB 2017a (MathWorks).
Experiment 2 (partial feedback, n = 31)
The design of Exp. 2 was nearly identical to that of Exp. 1, but choice feedback was given only after neighbouring pairs. After nonneighbouring pairs, a neutral ‘thank you’ message was displayed instead. Neighbouring pairs were presented more often (2.5 times as often as nonneighbouring pairs), resulting in 616 trials (presented in 8 blocks of 77). In Exp. 2, we additionally recorded EEG, and the participants performed a brief pictureviewing task before the experiment. These data were collected for the purpose of a different research question and are not reported here.
Experiment 3 (partial feedback, n = 48)
The basic design of Exp. 3 was identical to that of Exp. 2, except for the following changes. Both pair items were displayed simultaneously on the screen for 2.5 s, one to the left and the other to the right of a centred fixation cross. The participants were instructed to quickly select the highervalued item using the left or right arrow key. After neighbouring pairs, a feedback message (‘win’ or ‘loss’) was presented. After nonneighbouring pairs, no feedback message was shown. Experiments 3 and 4 were programmed in PsychoPy v.2020.1.3 (ref. ^{59}) and conducted online (pavlovia.org), with intermittent attention checks.
Experiment 4 (partial feedback, deterministic, n = 49)
The design of Exp. 4 was identical to that of Exp. 3, but feedback was always truthful (deterministic feedback). As learning expectedly proceeds faster with deterministic feedback, neighbouring pairs were presented only two times as often as nonneighbours, and we reduced the number of trials to 420 (presented in 6 blocks of 70 trials).
Models
Itemlevel learning
To model how observers update their value estimates about the winning item i and the losing item j after relational feedback, we assume a simple delta rule^{60} (Model Q1):
where Q_{t} is the estimated item value at time t, and α is the learning rate.
Transitive inference is enabled by a modified updating rule (similar to refs. ^{14,22}) based on the relative difference d_{t}(i, j) between the value estimates for the winner i and the loser j in a pair:
where η is a scaling factor. Value updating is then moderated by the extent to which feedback is consistent (or inconsistent) with d_{t}(i, j) (Model Q1*):
for the winning item i and the losing item j, respectively. Note that equation (1) is a special case of equation (3) when η = 0.
We can allow asymmetric updating of winners and losers by introducing separate learning rates, α^{+} and α^{−} (Models Q2 and Q2*):
where the winning item i is updated via α^{+}, and the losing item j is updated via α^{−}.
To convert the value estimates from itemlevel learning into pairwise choice probabilities for any two items i and j, we use a logistic choice function to define the probability of choosing i > j on the basis of the difference between the estimated item values:
where τ_{item} is the (inverse) temperature parameter controlling the level of decision noise in choices based on itemlevel learning.
Pairlevel learning
For the partialfeedback scenario, we also define an alternative learning model (Model P) that learns pairwise preferences between neighbouring items (rather than the individual items’ values). For each neighbouring pair n (1…7), we can describe the preference between its members (for example, p_{A>B}) probabilistically in terms of a beta distribution:
Following truthful feedback (for example, ‘correct’ when A > B was chosen), the upper value of the beta distribution is updated, increasing the preference in favour of the higherranking pair member:
whereas following untruthful feedback (only in experiments with probabilistic feedback; see Exps. 2 and 3), the lower value is updated, reducing the preference for the higherranking member:
with γ acting as a learning rate. We can thus define the learned neighbour preference at time t on the basis of the expectation of the beta distribution (Model P):
where p_{n,t} = 0.5 reflects indifference, and values of p_{n,t} larger (or smaller) than 0.5 reflect a preference for the higher (or lower) ranking pair member. While this mechanism can learn the relations between neighbouring items under partial feedback, it fails to learn the relations between nonneighbouring items, for which there is no direct feedback signal. However, transitive inference of preferences between nonneighbouring items is possible through associative recall of those neighbour preferences that ‘link’ the two nonneighbour items in question. To allow for this possibility, we define the inferred preference between any two items i and j via the set M of intermediate neighbour preferences \(p_{n,t}\) separating i and j (Model Pi):
where i − j is the rank distance between the items’ true values, and λ is a free parameter reflecting failure to retrieve linking pair preferences in the range [0, ∞]. If λ = 0, preferences between nonneighbours will be a lossless average of all intermediate neighbour preferences (that is, perfect memory). As λ grows, the preference between nonneighbours will shrink to indifference with increasing distance between j and i. In other words, this model performs perfect transitive inference if λ = 0 and no transitive inference as λ → ∞. Note that for neighbour pairs (where i − j = 1), equation (8) is equivalent to equation (7).
We again use a logistic choice rule to define the probability of choosing item i over j on the basis of pair preference p_{i>j,t} subject to decision noise τ_{pair}:
From equations (6)–(9), we constructed alternative models incorporating basic pairlevel learning (Model P, where λ is fixed at a large value) and pairlevel transitive inference (Model Pi, where λ is a free parameter).
To combine itemlevel (equations (1)–(5)) and pairlevel (equations (6)–(9)) learning, we assume that choices are triggered by whichever of the two models provides a stronger preference on a given trial. Choices are thus based on itemlevel learning (CP_{item}) if:
and are based on pairlevel learning (CP_{pair}) if:
This effectively implements a mixture of item and pairlevel learning.
Model space
From equations (1)–(10), we constructed a nested model space (Extended Data Fig. 1c) with either one or two learning rates (1, symmetric; 2, asymmetric updating; equation (4)). One set of models allows for simple itemlevel RL only (Models Q1 and Q2) or additionally for itemlevel transitive inference (Models Q1* and Q2*, equations (1)–(5)). Alternative models (equations (6)–(9)) incorporated pairlevel learning (Model P) and pairlevel inference (Model Pi). Mixture models (equation (10)) combined itemlevel and pairlevel learning (Q1* + P, Q2* + P, Q1* + Pi and Q2* + Pi). Technically, all models under study were derived from the most flexible model, Q2* + Pi, with individual parameter restrictions (for example, γ = 0 yields Model Q2*, or α^{+} = α^{−} yields symmetric updating).
Performance simulations
We simulated the performance of our itemlevel learning models (Q1, Q2, Q1* and Q2*) in tasks akin to those used in the human experiments, with full and partial feedback (Fig. 2 and Extended Data Figs. 1 and 2). The performance simulations were run in MATLAB R2020a (MathWorks). The models were initialized with flat priors about the item values (all Q_{1}(i) = 0—that is, the first choice was always a random guess with CP_{1} = 0.5). As in the human experiments, choice feedback was provided either for all pairs (full feedback) or only for neighbour pairs (partial feedback). We simulated model performance over a range of learning rates (α^{+} and α^{−}, 0 to 0.1 in increments of 0.001). Relational differenceweighting (η) was set to either 0 (Models Q1 and Q2) or 8 (Models Q1* and Q2*), and decision noise (τ_{item}) was set to 0.2 and 0.04 (full and partial feedback), which resembles the noise levels estimated in our human observers in the respective experiments. Mean choice probabilities (for example, Fig. 2a, bottom) and performance levels (for example, Fig. 2b) were simulated using the same number of trials and replications (with a new trial sequence) as in the respective human experiments. Simulation results under partial feedback (Fig. 2e and Extended Data Figs. 2 and 3) were qualitatively identical when inspecting performance on nonneighbouring pairs only.
Parameter estimation and model comparison
Model parameters were estimated by minimizing the negative loglikelihood of the model given each observer’s singletrial responses (from all trials in the experiment) across values of the model’s free parameters (within bounds (lower;upper): α/α^{+}/α^{−}(0;0.2), η(0;10), τ_{item}(0;1), γ(0;1), λ(0;100), τ_{pair}(0;1), with a uniform prior). The bestfitting parameter estimates are shown in Extended Data Fig. 4. Model fitting was performed in R (ref. ^{61}). Minimization was performed using a differential evolution algorithm^{62} with 200 iterations. We then computed the BIC of each model for each participant and evaluated the models’ probability of describing the majority of participants best (pxp)^{63}. In Fig. 3e,f, we also provide a pseudoR^{2} computed as R^{2} = 1 − (BIC_{model}/BIC_{null}), which quantifies goodness of fit relative to a null model of the data, with larger values indicating better fit (similar to ref. ^{64}). Model comparisons for Exp. 1 (full feedback) were restricted to itemlevel learning models, as the availability of direct feedback for every pairing would equate pairlevel learning models (P and Pi) to homogenous learning of all pairs, obviating contributions from transitive inferences.
To quantify modelestimated asymmetry (Fig. 5), we computed an index of the normalized difference in learning rates, A = (α^{+} − α^{−})/(α^{+} + α^{−}), which ranges from −1 (updating of losers only) to 1 (updating of winners only), with A = 0 indicating symmetric updating. For comparison between full and partialfeedback experiments, we contrasted the absolute A estimated from the winning model in Exps. 2–4 (Q2* + P, see Fig. 3f) with that estimated from Model Q2 in Exp. 1.
Model and parameter recovery
To establish whether the individual models can be distinguished in model comparison, we simulated, for each participant and model, 100 experiment runs using the individuals’ empirical parameter estimates under the respective model. We then fitted the generated datasets (binomial choice data) with each model and evaluated how often it provided the best fit (in terms of BIC). This way, we estimated the conditional probability that a model fits best given the true generative model (p(fitgen)). However, a metric more critical for evaluating our empirical results is p(genfit), which is the probability that the data was generated by a specific model, given that the model was observed as providing the best fit to the generated data^{65}. We compute this probability using Bayes’s theorem, with a uniform prior over models (p(gen)):
To mimic the level of inference in our human data fitting, we examined mean p(fitgen) and p(genfit) on the experiment level, on the basis of full simulations of all participants in Exp. 1 (full feedback) and Exp. 2 (partial feedback). Critically, under partial feedback (Exps. 2–4), all our models were robustly recovered with this approach (Extended Data Fig. 5).
Under full feedback (Exp. 1), human participant behaviour was best characterized by symmetric learning rates (α^{+} ≈ α^{−}), even when both learning rates were free parameters (Figs. 3e and 4). To test whether we could have detected asymmetric learning had it occurred in Exp. 1, we enforced asymmetry in the simulation by setting α^{−} to values near zero (by drawing from a rectified Gaussian with µ = 0 and σ = 0.01). We likewise enforced differenceweighted updating (η > 0) when simulating Model Q2*, by setting η to similar levels as empirically observed in the partialfeedback experiments (µ = 3 and σ = 0.5). With this, the model recovery for Exp. 1 successfully distinguished between symmetric (Q1 and Q1*) and asymmetric learning models (Q2 and Q2*; Extended Data Fig. 6). However, models with differenceweighted updating (Q1* and Q2*; equations (2) and (3)) were partly confused with Models Q1 and Q2. In other words, our empirical finding of Q1 as the winning model in Exp. 1 (Fig. 3e) does not rule out the possibility of Q1* as the generative process under full feedback.
To establish whether our inferences about model parameters (for example, Fig. 5) are valid, we simulated choices under partial feedback (Exp. 2) using our winning model (Q2* + P). Choice datasets were simulated using each participant’s empirical parameter estimates and iteratively varying each parameter over 20 evenly spaced values within the boundaries used in ‘Parameter estimation and model comparison’ (see above). We then fit the model to the simulated datasets and examined the correlations between generative and recovered parameters (Extended Data Figs. 7 and 8). All fitted parameters correlated most strongly with their generative counterparts (min 0.59, max 0.93), while correlations with other generative parameters were generally weaker (min −0.44, max 0.43).
Statistical analyses
The behavioural and modelling results were analysed using nonparametric tests (twosided), as detailed in the Results. In the case of multiple tests, the maximum P value (uncorrected) is reported in the main text, while the individual test results are detailed in Supplementary Tables 1–4.
Reporting Summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
Data availability
The data that support the findings of this study are available at https://doi.org/10.5281/zenodo.5561411.
Code availability
The experiment and analysis code is available at https://doi.org/10.5281/zenodo.5561411.
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Acknowledgements
We thank K. Tsetsos, S. Nelli, C. Summerfield and N. Schuck for helpful feedback and discussion, and S. Appelhoff and J. Wäscher for technical support. This work was supported by a Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) research grant to B.S. (DFG SP1510/61). C.M.W. is supported by the German Federal Ministry of Education and Research (BMBF): Tübingen AI Center, FKZ: 01IS18039A, and funded by the DFG under Germany’s Excellence Strategy—EXC 2064/1–390727645. The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.
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J.L.D. and B.S. designed the experiments. J.L.D., C.W. and I.P. performed the experiments. B.S. designed the modelling approach. S.C. and B.S. performed the simulations and analyses with contributions from C.M.W. and J.L.D. S.C. and B.S. visualized the simulations and results. B.S., C.M.W., S.C. and J.L.D. wrote the paper.
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Extended data
Extended Data Fig. 1 Additional model simulations and illustration of the model space.
a, Simulation of model Q1 under partial feedback. Same conventions as in Fig. 2. The simple Qlearning models (Q1/Q2) can only learn about the extreme items (here, A and H) under partial feedback. b, Choice matrices predicted by pairlevel learning without (left, model P) or with associative recall of ‘linking’ pair relationships (right, model Pi). Choice behaviour was simulated with a pairlevel learning rate γ=1. Associative recall in model Pi (right) was enabled by additionally lowering parameter λ to 1 (see Methods for details). c, Schematic overview of the model space. See Methods, Models.
Extended Data Fig. 2 Performance simulations with probabilistic choice outcomes.
Same conventions as in Fig. 2. Left, full feedback (for all pairs; cf. Figure 2b). Right, partial feedback (only for nonneighbouring pairs; cf. Figure 2e). Optimal learning under partial feedback is characterized by asymmetric updating (α^{+} ≠ α^{−}), just as was observed with deterministic outcomes (cf. Figure 2e).
Extended Data Fig. 3 Pilot experiment with partial feedback (n=11).
The design was identical to Exp. 2, except that all item pairs (neighbours and nonneighbours) were presented equally frequently (like in Exp. 1). Left, Performance simulation shows a similar benefit of asymmetric updating as we observed in simulation of Exp. 24 (where neighbouring pairs were presented more frequently, cf. Figure 2e and S2, right). Right, Mean proportions of correct choices in the pilot experiment. The overall learning level was relatively low, with n=9 (of 20) pilot participants not meeting our inclusion threshold for abovechance performance (cf. Methods: Participants). The descriptive choice data of the remaining 11 pilot participants (shown in right) indicate a similar learning asymmetry as we observed in our main experiments.
Extended Data Fig. 4 Individual parameter estimates for the winning model in each experiment (cf. Figure 3).
a, Experiment 1 (model Q1). b, Experiments 24 (model Q2*+P). Coloured dots show individual participant estimates, where blue, orange and pink colours refer to Experiment 2, 3 and 4, respectively. The diamond shape denotes the mean across experiments. While the estimates of η showed large variability with many values close to the upper bound, we observed no improvement in model fit when increasing the upper bound further.
Extended Data Fig. 5 Model recovery results, partial feedback (cf. Figure 3f).
The models were generally well distinguished both in \(p(fitgen)\) and in \(p(genfit)\). Of particular importance, our best asymmetric models (Q2* and Q2*+P, see Results) were well distinguished from their symmetric counterparts (Q1* and Q1*+P), with confusion rates no higher than 5%. Columns (left) and rows (right) may not always sum to 1 due to rounding of cell entries.
Extended Data Fig. 6 Model recovery results for Exp. 1 with full feedback (cf. Figure 3e).
Simple Qlearning (models Q1/Q2) could not be confidently distinguished from models Q1*/Q2*. However, symmetric (Q1/Q1*) and asymmetric learning (Q2/Q2*) were distinguished relatively well. See Methods: Model and parameter recovery for details.
Extended Data Fig. 7 Parameter recovery results under partial feedback for our bestfitting model (Q2*+P).
All fitted parameters correlate most strongly with their generative counterparts (diagonal) while correlations with other generative parameters (offdiagonal) are generally weaker.
Extended Data Fig. 8 Detailed parameter recovery results for the individual parameters.
The parameter values used to simulate choice data are plotted on the xaxes and the parameter estimates obtained from fitting the model to the simulated data are plotted on the yaxes. Light blue: mean recovered parameter values with bootstrapped 95% confidence intervals. Dark blue line shows linear fit. Results from individual recovery runs are shown as halftransparent black dots.
Extended Data Fig. 9 Model comparison (Exp. 24) analogous to Fig. 3f, but additionally allowing for differential learning from confirmatory vs. disconfirmatory choice feedback.
Here, all models included an additional parameter ω ∈ (0;10), by which beliefconfirming learning rates were modelled as: α_{conf} = α * ω. Same conventions as in Fig. 3f. Markers show model fits using a pseudoRsquared (Rsq, left yaxis; diamonds and error bars show mean ± s.e.m., dots show individual participants). Overlaid red bar graphs indicate each model’s probability of describing the majority of subjects best (right yaxis, pxp: protected exceedance probability). While the extra parameter ω led to general improvements in fit (note overall higher Rsq compared to Fig. 3f), the model comparison result with respect to winner/loser asymmetries was identical, both in terms of Rsq/BIC and pxp. The estimates of parameter ω in the winning models were larger than 1 (mean=4.72, SE=0.27, p < 0.001, r=0.83, Wilcoxon signedrank test against 1, collapsed across Exp. 24), indicating an overall bias towards confirmatory feedback.
Extended Data Fig. 10 Performance simulations under partial feedback (analogous to Fig. 2e) for different levels of decision noise (τ_{item}).
Asymmetric learning is beneficial regardless of decision noise level and accordingly, across a wide range of overall performance levels. Note that different colour scales for each panel are used to increase interpretability (see colour bars). Simulations with probabilistic outcomes yielded a qualitatively very similar pattern.
Supplementary information
Supplementary Information
Supplementary Tables 1–4 and Methods.
Supplementary Video
Simulations of Qlearning under partial feedback for models Q1 (left) and Q1* (centre) with the same learning rate for winning and losing items, and for model Q2* (right) with asymmetric learning about winners only (α^{−} set to 0). The coloured dots indicate the momentary Qvalues for items A–H. Simulations are shown for a trial sequence with deterministic feedback for illustration purposes. The video first plays 200 learning trials (top) and then repeats the same trials more slowly (bottom). Nonneighbour trials (on which no feedback is given) are fast forwarded. The green and red discs indicate the winning and losing items on every trial. Model Q1 (left) effectively learns only about the extreme items (A and H), while intermediate item values fluctuate unsystematically around the preexperiment baseline. In models with differenceweighted updating (Q1* and Q2*, centre and right), value differences propagate through the item series, which leads to a more monotonic value structure that enables transitive inferences about intermediate items (B–G). In model Q1* (centre), with symmetric learning, propagation can occur in both directions, which results in partly conflicting updates for midrange items (for example, C–F). This induces residual nonmonotonicity in the evolving value structure, which can compromise transitive inference. In model Q2* (right), with asymmetric learning, conflicting updates are reduced, leading to a more strictly monotonic value structure that enables superior transitive inference.
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Ciranka, S., LindeDomingo, J., Padezhki, I. et al. Asymmetric reinforcement learning facilitates human inference of transitive relations. Nat Hum Behav 6, 555–564 (2022). https://doi.org/10.1038/s4156202101263w
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DOI: https://doi.org/10.1038/s4156202101263w
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