Reinforcement learning in professional basketball players

Autonomous behaviour and the limits of human volition

Humans and some other animals can autonomously generate action choices that contribute to solving complex problems. However, experimental investigations of the cognitive bases of human autonomy are challenging, because experimental paradigms typically constrain behaviour using controlled contexts, and elicit behaviour by external triggers. In contrast, autonomy and freedom imply unconstrained behaviour initiated by endogenous triggers. Here we propose a new theoretical construct of adaptive autonomy, meaning the capacity to make behavioural choices that are free from constraints of both immediate external triggers and of routine response patterns, but nevertheless show appropriate coordination with the environment. Participants (N = 152) played a competitive game in which they had to choose the right time to act, in the face of an opponent who punished (in separate blocks) either choice biases (such as always responding early), sequential patterns of action timing across trials (such as early, late, early, late…), or predictable action-outcome dependence (such as win-stay, lose-shift). Adaptive autonomy was quantified as the ability to maintain performance when each of these influences on action selection was punished. We found that participants could become free from habitual choices regarding when to act and could also become free from sequential action patterns. However, they were not able to free themselves from influences of action-outcome dependence, even when these resulted in poor performance. These results point to a new concept of autonomous behaviour as flexible adaptation of voluntary action choices in a way that avoids stereotypy. In a sequential analysis, we also demonstrated that participants increased their reliance on belief learning in which they attempt to understand the competitor's beliefs and intentions, when transition bias and reinforcement bias were punished. Taken together, our study points to a cognitive mechanism of adaptive autonomy in which competitive interactions with other agents could promote both social cognition and volition in the form of non-stereotyped action choices.

Subpopulations of neurons in lOFC encode previous and current rewards at time of choice

Studies of neural dynamics in lateral orbitofrontal cortex (lOFC) have shown that subsets of neurons that encode distinct aspects of behavior, such as value, may project to common downstream targets. However, it is unclear whether reward history, which may subserve lOFC's well-documented role in learning, is represented by functional subpopulations in lOFC. Previously, we analyzed neural recordings from rats performing a value-based decision-making task, and we documented trial-by-trial learning that required lOFC (Constantinople et al., 2019). Here, we characterize functional subpopulations of lOFC neurons during behavior, including their encoding of task variables. We found five distinct clusters of lOFC neurons, either based on clustering of their trial-averaged peristimulus time histograms (PSTHs), or a feature space defined by their average conditional firing rates aligned to different task variables. We observed weak encoding of reward attributes, but stronger encoding of reward history, the animal's left or right choice, and reward receipt across all clusters. Only one cluster, however, encoded the animal's reward history at the time shortly preceding the choice, suggesting a possible role in integrating previous and current trial outcomes at the time of choice. This cluster also exhibits qualitatively similar responses to identified corticostriatal projection neurons in a recent study (Hirokawa et al., 2019), and suggests a possible role for subpopulations of lOFC neurons in mediating trial-by-trial learning.

Stochastic model for football's collective dynamics

In this paper, we study collective interaction dynamics emerging in the game of football (soccer). To do so, we surveyed a database containing body-sensor traces measured during three professional football matches, where we observed statistical patterns that we used to propose a stochastic model for the players' motion in the field. The model, which is based on linear interactions, captures to a good approximation the spatiotemporal dynamics of a football team. Our theoretical framework, therefore, can be an effective analytical tool to uncover the underlying cooperative mechanisms behind the complexity of football plays. Moreover, we showed that it can provide handy theoretical support for coaches to evaluate teams' and players' performances in both training sessions and competitive scenarios.

Dichotomous dopaminergic and noradrenergic neural states mediate distinct aspects of exploitative behavioral states

The balance between exploiting known actions and exploring alternatives is critical for survival and hypothesized to rely on shifts in neuromodulation. We developed a behavioral paradigm to capture exploitative and exploratory states and imaged calcium dynamics in genetically identified dopaminergic and noradrenergic neurons. During exploitative states, characterized by motivated repetition of the same action choice, dopamine neurons in SNc encoding movement vigor showed sustained elevation of basal activity that lasted many seconds. This sustained activity emerged from longer positive responses, which accumulated during exploitative action-reward bouts, and hysteretic dynamics. Conversely, noradrenergic neurons in LC showed sustained inhibition of basal activity due to the accumulation of longer negative responses in LC. Chemogenetic manipulation of these sustained dynamics revealed that dopaminergic activity mediates action drive, whereas noradrenergic activity modulates choice diversity. These data uncover the emergence of sustained neural states in dopaminergic and noradrenergic networks that mediate dissociable aspects of exploitative bouts.

Competition Rather Than Observation and Cooperation Facilitates Optimal Motor Planning

Humans tend to select motor planning with a high reward and low success compared with motor planning, which has a small reward and high success rate. Previous studies have shown such a risk-seeking property in motor decision tasks. However, it is unclear how to facilitate a shift from risk-seeking to optimal motor planning that maximizes the expected reward. Here, we investigate the effect of interacting with virtual partners/opponents on motor plans since interpersonal interaction has a powerful influence on human perception, action, and cognition. This study compared three types of interactions (competition, cooperation, and observation) and two types of virtual partners/opponents (those engaged in optimal motor planning and those engaged in risk-averse motor planning). As reported in previous studies, the participants took a risky aim point when they performed a motor decision task alone. However, we found that the participant's aim point was significantly modulated when they performed the same task while competing with a risk-averse opponent (p = 0.018) and that there was no significant difference from the optimal aim point (p = 0.63). No significant modulation in the aim points was observed during the cooperation and observation tasks. These results highlight the importance of competition for modulating suboptimal decision-making and optimizing motor performance.

Modeling ball possession dynamics in the game of football

In this paper, we study interaction dynamics in the game of football-soccer in the context of ball possession intervals. To do so, we analyze a database comprising one season of the five major football leagues of Europe. Using this input, we developed a stochastic model based on three agents: two teammates and one defender. Despite its simplicity, the model is able to capture, in good approximation, the statistical behavior of possession times, pass lengths, and number of passes performed. In the last section, we show that the model's dynamics can be mapped into a Wiener process with drift and an absorbing barrier.

Searching for momentum in NBA triplets of free throws

The literature on momentum is still undecided, with mixed results whether momentum exists or is only perceived to exist ("hot hand fallacy"). We explore whether momentum exists by looking at cases in which a basketball player has three consecutive free throws. A free throw is a well-defined task executed in a stable environment, allegedly giving momentum optimal chances to occur. Taking 14 NBA seasons we collected over 4500 three-free-throw sets (triplets). We obtained the outcomes of the shots as well as some additional variables about the player and the game: the player's average free-throw percentage, home or away, and the game score and the quarter when the free throws were attempted. We first analyse the hit rates in the three shots and then proceed to regression analysis that also controls for the abovementioned variables. We address several concerns raised in the literature. All comparisons and analyses yield the same conclusion that there is no evidence for momentum in the data.

Lateral orbitofrontal cortex promotes trial-by-trial learning of risky, but not spatial, biases

Individual choices are not made in isolation but are embedded in a series of past experiences, decisions, and outcomes. The effects of past experiences on choices, often called sequential biases, are ubiquitous in perceptual and value-based decision-making, but their neural substrates are unclear. We trained rats to choose between cued guaranteed and probabilistic rewards in a task in which outcomes on each trial were independent. Behavioral variability often reflected sequential effects, including increased willingness to take risks following risky wins, and spatial 'win-stay/lose-shift' biases. Recordings from lateral orbitofrontal cortex (lOFC) revealed encoding of reward history and receipt, and optogenetic inhibition of lOFC eliminated rats' increased preference for risk following risky wins, but spared other sequential effects. Our data show that different sequential biases are neurally dissociable, and the lOFC's role in adaptive behavior promotes learning of more abstract biases (here, biases for the risky option), but not spatial ones.

Risk aversion in the adjustment of speed-accuracy tradeoff depending on time constraints

Humans are often required to make decisions under time constraints and to adjust speed-accuracy tradeoff (SAT) based on time constraints. Previous studies have investigated how humans adjust SAT depending on the time discount rate of expected gain. Although the expected gain of actions can be determined by both gain and probability, only situations where gain decreases over time have been tested. Considering the effect of risk on decision-making, the difference in time discount factors may modulate the response strategies for SAT, since temporal changes in variance of possible outcomes differ when gain or probability decreases over time. Here, we investigated the response strategies for SAT under different time discount factors. Participants were required to select one of the two options with different initial values in situations where the expected gain of options declined over time by a linear decrease in gain or probability. Comparison of response strategies between conditions revealed that response times in the gain condition were longer than those in the probability condition, possibly due to risk-aversion. These findings indicate the existence of common rules underpinning sensorimotor and economic decision-making.

Deviation from the matching law reflects an optimal strategy involving learning over multiple timescales

Behavior deviating from our normative expectations often appears irrational. For example, even though behavior following the so-called matching law can maximize reward in a stationary foraging task, actual behavior commonly deviates from matching. Such behavioral deviations are interpreted as a failure of the subject; however, here we instead suggest that they reflect an adaptive strategy, suitable for uncertain, non-stationary environments. To prove it, we analyzed the behavior of primates that perform a dynamic foraging task. In such nonstationary environment, learning on both fast and slow timescales is beneficial: fast learning allows the animal to react to sudden changes, at the price of large fluctuations (variance) in the estimates of task relevant variables. Slow learning reduces the fluctuations but costs a bias that causes systematic behavioral deviations. Our behavioral analysis shows that the animals solved this bias-variance tradeoff by combining learning on both fast and slow timescales, suggesting that learning on multiple timescales can be a biologically plausible mechanism for optimizing decisions under uncertainty.

Recent experience can only provide limited information to guide decisions in a volatile environment. Here, the authors report that the choices made by a monkey in a dynamic foraging task can be better explained by a model that combines learning on both fast and slow timescales.

Evidence for Sequential Performance Effects in Professional Darts

Objectives: The study of sequential effects in aiming tasks might shed light on the organization of repetitive motor performances over time. To date, investigations of such effects in sports have been limited and yielded mixed results. Given the relatively short time intervals between successive attempts, and the absence of defensive interventions, dart throwing provides a potentially fruitful testing ground for examining the presence of sequential performance effects in the motor domain.

Methods and Results: A total of 80 competitive darts matches of 10 of the world’s best players were scored from publicly available video footage in terms of sequences of hits and misses of triple 20. In darts, throws are organized in legs, i.e., a rapid succession of three throws by the same player, allowing us to investigate various transitions in performance (throw 1 → 2, 2 → 3, and 3 → 1). The resulting binary sequences were analyzed statistically in terms of independence and stationarity. Across players significant statistical evidence was found for sequential dependence from the first throw in a leg to the second throw, but not for the other transitions. As regards to stationarity, a significant decline in performance was observed in the course of the match.

Conclusions: In professional darts, evidence can be found for both sequential dependence as well as for non-stationarity, implying that performance does not, or at least not always, constitute a stationary random independent process. More research is needed on the motor control mechanisms underlying the observed carry-over effects within triplets as well as the possible causes of non-stationarity.

A Switching Observer for Human Perceptual Estimation

Human perceptual inference has been fruitfully characterized as a normative Bayesian process in which sensory evidence and priors are multiplicatively combined to form posteriors from which sensory estimates can be optimally read out. We tested whether this basic Bayesian framework could explain human subjects' behavior in two estimation tasks in which we varied the strength of sensory evidence (motion coherence or contrast) and priors (set of directions or orientations). We found that despite excellent agreement of estimates mean and variability with a Basic Bayesian observer model, the estimate distributions were bimodal with unpredicted modes near the prior and the likelihood. We developed a model that switched between prior and sensory evidence rather than integrating the two, which better explained the data than the Basic and several other Bayesian observers. Our data suggest that humans can approximate Bayesian optimality with a switching heuristic that forgoes multiplicative combination of priors and likelihoods.

Association of Catastrophic Neonatal Outcomes With Increased Rate of Subsequent Cesarean Deliveries

OBJECTIVE

To evaluate whether full-term deliveries resulting in neonates diagnosed with hypoxic-ischemic encephalopathy are associated with a significant increase in the rate of subsequent unscheduled cesarean deliveries.

METHODS

We conducted a retrospective chart review study and examined all deliveries in the Department of Obstetrics and Gynecology at Hadassah University Hospital, Mt. Scopus campus, Jerusalem, Israel, during 2009-2014. We reviewed all cases of hypoxic-ischemic encephalopathy in singleton, term, liveborn neonates and identified seven such cases, three of which were attributed to obstetric mismanagement and four that were not. We measured the rate of unscheduled cesarean deliveries before and after the events and their respective hazard ratio.

RESULTS

Before a mismanaged delivery resulting in hypoxic-ischemic encephalopathy, the baseline rate of unscheduled cesarean deliveries was approximately 80 unscheduled cesarean deliveries for every 1,000 deliveries. In the first 4 weeks immediately after each of the three identified cases, there was a significant increase in the rate of unscheduled cesarean deliveries by an additional 48 unscheduled cesarean deliveries per 1,000 deliveries (95% confidence interval [CI] 27-70/1,000). This increase was transient and lasted approximately 4 weeks. We estimated that each case was associated with approximately 17 additional unscheduled cesarean deliveries (95% CI 8-27). There was no increase in the rate of unscheduled cesarean deliveries in cases of hypoxic-ischemic encephalopathy that were not associated with mismanagement.

CONCLUSION

The increase in the rate of unscheduled cesarean deliveries after a catastrophic neonatal outcome may result in short-term changes in obstetricians' risk evaluation.

A stable brain from unstable components: Emerging concepts and implications for neural computation

Neuroscientists have often described the adult brain in similar terms to an electronic circuit board- dependent on fixed, precise connectivity. However, with the advent of technologies allowing chronic measurements of neural structure and function, the emerging picture is that neural networks undergo significant remodeling over multiple timescales, even in the absence of experimenter-induced learning or sensory perturbation. Here, we attempt to reconcile the parallel observations that critical brain functions are stably maintained, while synapse- and single-cell properties appear to be reformatted regularly throughout adult life. In this review, we discuss experimental evidence at multiple levels ranging from synapses to neuronal ensembles, suggesting that many parameters are maintained in a dynamic equilibrium. We highlight emerging hypotheses that could explain how stable brain functions may be generated from dynamic elements. Furthermore, we discuss the impact of dynamic circuit elements on neural computations, and how they could provide living neural circuits with computational abilities a fixed structure cannot offer. Taken together, recent evidence indicates that continuous dynamics are a fundamental property of neural circuits compatible with macroscopically stable behaviors. In addition, they may be a unique advantage imparting robustness and flexibility throughout life.

Sub-optimality in motor planning is retained throughout 9 days practice of 2250 trials

Optimality in motor planning, as well as accuracy in motor execution, is required to maximize expected gain under risk. In this study, we tested whether humans are able to update their motor planning. Participants performed a coincident timing task with an asymmetric gain function, in which optimal response timing to gain the highest total score depends on response variability. Their behaviours were then compared using a Bayesian optimal decision model. After 9 days of practicing 2250 trials, the total score increased, and temporal variance decreased. On the other hand, the participants showed consistent risk-seeking or risk-averse behaviour, preserving suboptimal motor planning. These results suggest that a human's computational ability to calculate an optimal motor plan is limited, and it is difficult to improve it through repeated practice with a score feedback.

Towards a Decision Quality Model for Shot Selection in Basketball: an Exploratory Study

We take the first steps towards a shot selection quality model in basketball that incorporates decisional cues that might be predictive, not only of proximal results (e.g., scoring), but also of distal results (e.g., winning/losing the match). 2976 jump-shots from 50 Euroleague matches were sampled, following systematic observation guidelines. The decisional cues under scrutiny were shooting opposition, distance and lateral angle, disposition to offensive rebound and disposition to defensive balance at the moment of shooting. A first set of regressions between decisional cues and proximal results showed higher opposition and distance to decrease the probability of scoring (OR = .81; p < .001 and OR = .89; p = .013); a better disposition towards rebound to increase the chances of catching rebound (OR = 1.57; p < .001); and better defensive balance disposition to decrease the probability of a fast break (OR = 1.27; p < .036). A second set of regressions between proximal and distal results showed shooting and offensive rebound effectiveness to predict total points scored (β = .62; p < .001 and β = .32; p < .001) and game result (winning/losing the game; OR = 1.12; p < .001 and OR = 1.05; p = .021). Finally, an analysis of the impact of decisional cues on distal results showed a positive relationship between likelihood of winning and average team's disposition to offensive rebound (OR = 1.18; p = .018). These results cast light on the actual weights (validities) of the different cues involved in predicting outcomes of shooting decisions. This evidence could help coaches provide objective feedback about players' shooting performance beyond hit percentages.

Heterogeneous Suppression of Sequential Effects in Random Sequence Generation, but Not in Operant Learning

There is a long history of experiments in which participants are instructed to generate a long sequence of binary random numbers. The scope of this line of research has shifted over the years from identifying the basic psychological principles and/or the heuristics that lead to deviations from randomness, to one of predicting future choices. In this paper, we used generalized linear regression and the framework of Reinforcement Learning in order to address both points. In particular, we used logistic regression analysis in order to characterize the temporal sequence of participants' choices. Surprisingly, a population analysis indicated that the contribution of the most recent trial has only a weak effect on behavior, compared to more preceding trials, a result that seems irreconcilable with standard sequential effects that decay monotonously with the delay. However, when considering each participant separately, we found that the magnitudes of the sequential effect are a monotonous decreasing function of the delay, yet these individual sequential effects are largely averaged out in a population analysis because of heterogeneity. The substantial behavioral heterogeneity in this task is further demonstrated quantitatively by considering the predictive power of the model. We show that a heterogeneous model of sequential dependencies captures the structure available in random sequence generation. Finally, we show that the results of the logistic regression analysis can be interpreted in the framework of reinforcement learning, allowing us to compare the sequential effects in the random sequence generation task to those in an operant learning task. We show that in contrast to the random sequence generation task, sequential effects in operant learning are far more homogenous across the population. These results suggest that in the random sequence generation task, different participants adopt different cognitive strategies to suppress sequential dependencies when generating the "random" sequences.

Motor planning under temporal uncertainty is suboptimal when the gain function is asymmetric

For optimal action planning, the gain/loss associated with actions and the variability in motor output should both be considered. A number of studies make conflicting claims about the optimality of human action planning but cannot be reconciled due to their use of different movements and gain/loss functions. The disagreement is possibly because of differences in the experimental design and differences in the energetic cost of participant motor effort. We used a coincident timing task, which requires decision making with constant energetic cost, to test the optimality of participant's timing strategies under four configurations of the gain function. We compared participant strategies to an optimal timing strategy calculated from a Bayesian model that maximizes the expected gain. We found suboptimal timing strategies under two configurations of the gain function characterized by asymmetry, in which higher gain is associated with higher risk of zero gain. Participants showed a risk-seeking strategy by responding closer than optimal to the time of onset/offset of zero gain. Meanwhile, there was good agreement of the model with actual performance under two configurations of the gain function characterized by symmetry. Our findings show that human ability to make decisions that must reflect uncertainty in one's own motor output has limits that depend on the configuration of the gain function.

"Hand down, man down." Analysis of defensive adjustments in response to the hot hand in basketball using novel defense metrics

The hot-hand phenomenon, according to which a player’s performance is significantly elevated during certain phases relative to the expected performance based on the player’s base rate, has left many researchers and fans in basketball puzzled: The vast majority of players, coaches and fans believe in its existence but statistical evidence supporting this belief has been scarce. It has frequently been argued that the hot hand in basketball is unobservable because of strategic adjustments and defensive interference of the opposing team. We use a dataset with novel metrics, such as the number of defenders and the defensive intensity for each shot attempt, which enable us to directly measure defensive pressure. First, we examine how the shooting percentage of NBA players changes relative to the attributes of each metric. We find that it is of lesser importance by how many defenders a player is guarded but that defensive intensity, e.g., whether a defender raises his hand when his opponent shoots, has a larger impact on shot difficulty. Second, we explore how the underlying metrics and shooting accuracy change as a function of streak length. Our results indicate that defensive pressure and shot difficulty increase (decrease) during hot (cold) streaks, so that defenders seem to behave according to the hot-hand belief and try to force hot players into more difficult shots. However, we find that shooting percentages of presumably hot players do not increase and that shooting performance is not related to streakiness, so that the defenders’ hot-hand behavior cannot be considered ecologically rational. Therefore, we are unable to find evidence in favor of the hot-hand effect even when accounting for defensive pressure.

The effect of perceived streakiness on the shot-taking behaviour of basketball players

We examine behavioural changes of basketball players arising from the hot-hand belief and use data of 1216 National Basketball Association games to measure the effect of cold and hot streaks on three proxies of shot difficulty. We find that the more consecutive shots players make (miss), the more difficult (easier) shots become along the three dimensions. Furthermore, most players' performance seems to improve during hot streaks because they attempt more difficult shots while no significant decrease in shooting accuracy takes place. This might explain why most previous studies could not find empirical evidence for the hot-hand belief in basketball when considering in-game field goal shooting.

Spatial generalization in operant learning: lessons from professional basketball

In operant learning, behaviors are reinforced or inhibited in response to the consequences of similar actions taken in the past. However, because in natural environments the “same” situation never recurs, it is essential for the learner to decide what “similar” is so that he can generalize from experience in one state of the world to future actions in different states of the world. The computational principles underlying this generalization are poorly understood, in particular because natural environments are typically too complex to study quantitatively. In this paper we study the principles underlying generalization in operant learning of professional basketball players. In particular, we utilize detailed information about the spatial organization of shot locations to study how players adapt their attacking strategy in real time according to recent events in the game. To quantify this learning, we study how a make \ miss from one location in the court affects the probabilities of shooting from different locations. We show that generalization is not a spatially-local process, nor is governed by the difficulty of the shot. Rather, to a first approximation, players use a simplified binary representation of the court into 2 pt and 3 pt zones. This result indicates that rather than using low-level features, generalization is determined by high-level cognitive processes that incorporate the abstract rules of the game.

According to the law of effect, formulated a century ago by Edward Thorndike, actions which are rewarded in a particular situation are more likely to be executed when that same situation recurs. However, in natural settings the same situation never recurs and therefore, generalization from one state of the world to other states is an essential part of the process of learning. In this paper we utilize basketball statistics to study the computational principles underlying generalization in operant learning of professional basketball players. We show that players are more likely to attempt a field goal from the vicinity of a previously made shot than they are from the vicinity of a missed shot, as expected from the law of effect. However, the outcome of a shot can also affect the likelihood of attempting another shot at a different location. Using hierarchical clustering we characterize the spatial pattern of generalization and show that generalization is primarily determined by the type of shot, 3 pt vs. 2 pt. This result indicates that rather than using low-level features, generalization is determined by high-level cognitive processes that incorporate the abstract rules of the game.

Psychological Momentum: Why Success Breeds Success

Whether trying to win Presidential primaries, trading stocks, or playing sports, performance-enhancing effects of psychological momentum (PM) are widely accepted. But, does initial success (S1) lead to subsequent success (S2) in and of itself due to increased know-how on one's and opponents’ performance or because it creates psychological force (momentum) that mediates this relationship? We review research on the phenomenon and show its strong empirical foundations in various domains of human performance. To advance research, we present an organizing theoretical framework that proposes both mediating and moderating effects of PM as mechanisms to explain why success breeds success in general. Initial success is critical for PM and has 3 types of effects: intensity, frequency, and duration. Whether performing alone (trader) or against an opponent (tennis player), perceptions of self as a performer (Sp) and of opponent as a performer (Op) are at the center of PM. The theory posits that the more the initial success separates the two (Op/Sp), the greater the PM. These and associated perceptions, however, have to turn into an increased subjective probability of winning or succeeding before PM becomes a psychological force. Evidence supports the mediating mechanism since initial success increases PM, which in turn enhances subsequent success. When initial success with PM leads to a greater likelihood of subsequent success than without PM, PM then modifies (“moderates”) the S1–S2 relationship without PM's independent effect on S2. There is also tentative evidence for a moderated mediation effect as the influence of PM seems to be greater for male than female performers. Areas of future research are highlighted.

Complex population response of dorsal putamen neurons predicts the ability to learn

Day-to-day variability in performance is a common experience. We investigated its neural correlate by studying learning behavior of monkeys in a two-alternative forced choice task, the two-armed bandit task. We found substantial session-to-session variability in the monkeys’ learning behavior. Recording the activity of single dorsal putamen neurons we uncovered a dual function of this structure. It has been previously shown that a population of neurons in the DLP exhibits firing activity sensitive to the reward value of chosen actions. Here, we identify putative medium spiny neurons in the dorsal putamen that are cue-selective and whose activity builds up with learning. Remarkably we show that session-to-session changes in the size of this population and in the intensity with which this population encodes cue-selectivity is correlated with session-to-session changes in the ability to learn the task. Moreover, at the population level, dorsal putamen activity in the very beginning of the session is correlated with the performance at the end of the session, thus predicting whether the monkey will have a "good" or "bad" learning day. These results provide important insights on the neural basis of inter-temporal performance variability.

How recent history affects perception: the normative approach and its heuristic approximation

There is accumulating evidence that prior knowledge about expectations plays an important role in perception. The Bayesian framework is the standard computational approach to explain how prior knowledge about the distribution of expected stimuli is incorporated with noisy observations in order to improve performance. However, it is unclear what information about the prior distribution is acquired by the perceptual system over short periods of time and how this information is utilized in the process of perceptual decision making. Here we address this question using a simple two-tone discrimination task. We find that the “contraction bias”, in which small magnitudes are overestimated and large magnitudes are underestimated, dominates the pattern of responses of human participants. This contraction bias is consistent with the Bayesian hypothesis in which the true prior information is available to the decision-maker. However, a trial-by-trial analysis of the pattern of responses reveals that the contribution of most recent trials to performance is overweighted compared with the predictions of a standard Bayesian model. Moreover, we study participants' performance in a-typical distributions of stimuli and demonstrate substantial deviations from the ideal Bayesian detector, suggesting that the brain utilizes a heuristic approximation of the Bayesian inference. We propose a biologically plausible model, in which decision in the two-tone discrimination task is based on a comparison between the second tone and an exponentially-decaying average of the first tone and past tones. We show that this model accounts for both the contraction bias and the deviations from the ideal Bayesian detector hypothesis. These findings demonstrate the power of Bayesian-like heuristics in the brain, as well as their limitations in their failure to fully adapt to novel environments.

In this paper we study how history affects perception using an auditory delayed comparison task, in which human participants repeatedly compare the frequencies of two, temporally-separated pure tones. We demonstrate that the history of the experiment has a substantial effect on participants' performance: when both tones are high relative to past stimuli, people tend to report that the 2^nd tone was higher, and when they are relatively low, they tend to report that the 1^st tone was higher. Interestingly, only the most recent trials bias performance, which can be interpreted as if the participants assume that the statistics of stimuli in the experiment is highly volatile. Moreover, this bias persists even in settings, in which it is detrimental to performance. These results demonstrate the abilities, as well as limitations, of the cognitive system when incorporating expectations in perception.

Is coaching experience associated with effective use of timeouts in basketball?

Experience is an important asset in almost any professional activity. In basketball, there is believed to be a positive association between coaching experience and effective use of team timeouts. Here, we analyze both the extent to which a team's change in scoring margin per possession after timeouts deviate from the team's average scoring margin per possession-what we called timeout factor, and the extent to which this performance measure is associated with coaching experience across all teams in the National Basketball Association over the 2009-2012 seasons. We find that timeout factor plays a minor role in the scoring dynamics of basketball. Surprisingly, we find that timeout factor is negatively associated with coaching experience. Our findings support empirical studies showing that, under certain conditions, mentors early in their careers can have a stronger positive impact on their teams than later in their careers.

The problem of shot selection in basketball

In basketball, every time the offense produces a shot opportunity the player with the ball must decide whether the shot is worth taking. In this article, I explore the question of when a team should shoot and when they should pass up the shot by considering a simple theoretical model of the shot selection process, in which the quality of shot opportunities generated by the offense is assumed to fall randomly within a uniform distribution. Within this model I derive an answer to the question “how likely must the shot be to go in before the player should take it?” and I show that this lower cutoff for shot quality depends crucially on the number of shot opportunities remaining (say, before the shot clock expires), with larger demanding that only higher-quality shots should be taken. The function is also derived in the presence of a finite turnover rate and used to predict the shooting rate of an optimal-shooting team as a function of time. The theoretical prediction for the optimal shooting rate is compared to data from the National Basketball Association (NBA). The comparison highlights some limitations of the theoretical model, while also suggesting that NBA teams may be overly reluctant to shoot the ball early in the shot clock.