Sunday, 21 October 2012

The Unpredictability of Humans

There is a common belief among the general public that humans are unpredictable. This seems to stem from the intuitive understanding that, at any point, we could simply choose to behave in a completely different way - so how could such a thing possibly be predicted? In contrast to this, I like to think of humans as meaty, irregular-shaped billiard balls.

This extends the billiard ball construct that is often used to characterise and demonstrate principles of physics but adds the component of irregularity (the "meaty" part is just for artistic effect). The importance of this distinction is that it describes the illusion of unpredictability, in that a regular billiard ball is said to be predictable as it travels in a way that is consistent with the direction of the initial force acting upon it, but an irregular shaped billiard ball will appear to almost have "a mind of its own" as multiple forces and impacts drive it in various directions. To illustrate this, I will appeal to my dog's favourite toy:
(For any hardcore behaviorists, I recommend you turn away now as I am about to engage in some mild anthropomorphism).

My dog loves this ball (currently sans a number of nodules that have been gnawed off) and I believe it is because, unlike a tennis ball, it will bounce in an unpredictable way when thrown. Sometimes it just bounces forwards like a tennis ball, but often it will swerve wildly to the left or right, and then bounce off in some other direction after making contact with the floor again. I can't say for sure but it seems to me that this captivates my dog as it almost mimics the unpredictability of living creatures (which, incidentally, seem to be the only other thing that can hold his attention for any extended period of time). 


My above analogy was a rather roundabout way of trying to distinguish between true unpredictability and pseudo-unpredictability. The former refers to aspects or agents in the world which cannot be predicted due to some inherent stochastic component, whereas the latter refers to things which only appear to be unpredictable due to our ignorance of the details of the situation. In the case of the irregular-shaped billiard balls, the "unpredictability" comes about as a result of us not having direct accessibility to the variables affecting its behavior at any point in time; that is, we don't know which nodule is being acted upon, or at what angle, or with what force, and this prevents us from making simple predictions like "if I hit the ball on this angle, it will travel in this direction". 

So what are these 'nodules' on humans? As you might have expected, humans have slightly more nodules than the ball my dog loves, and these nodules are composed of far more intricate substances than the rubber of the ball. They are the complex interactions between our genetics and environment, our histories and current situations, our perceptions and reality, and so on. When we are pushed by a force, we are not only behaviorally thrust in one simple direction, but instead we are essentially pushed into another force (e.g. our genetics) which pushes us into another direction that crashes into another force (e.g. our reinforcement history) which pushes into another direction again, ad infinitum


One of my favourite examples of how these imperceptible nodules only give the appearance of unpredictability instead of producing real predictability is an equation in the field of choice theory called the matching law. Originally proposed by Herrnstein in 19611, the matching law is a quantitative description of Thorndike's law of effect (the non-quantitative relationship between an action and its future probability) and was formulated as:
where B refers to behavior, R refers to reinforcement, and the numbers refer to the possible options in an experimental setting (the above only gives us two options; for example, a left and right button to press). What this very simple equation says is that the distribution of choice between two options is proportional to the reinforcement that is received after choosing each option. So if a subject is faced with a left and right button, with the right option providing twice as much reinforcement as the left option, then what we find is that the subject will allocate twice as many responses to the right option. 

This formulation of the matching law is known as "strict matching", where there is a direct correlation between behavior and reinforcement. The reality of the situation is that choice behavior is not perfectly described by this equation and to accurately describe behavioral results we need to refine this in the form of the generalised matching law2 or the contingency-discriminability model3, that include variables to account for things like sensitivity to reinforcement, inherent bias, and confusion over allocating reinforcement to a behavior. But, for the purposes of this article, we'll just focus on strict matching as it is the easier concept to wrap our heads around.


Most of the groundbreaking research using the matching law is done with animals because laboratory animals give us cleaner results - we can control their environments, histories, calorie intake, and to a degree we can also control their genetics. Humans, on the other hand, are these meaty, irregular-shaped billiard balls where some nodules will forever remain invisible to us due to the lack of data we have on them. However, what we can do is gather as much data as we can on the known variables and make some predictions. 

"An Application of the Matching Law to Social Dynamics"4: This study extended an earlier study5 by applying the matching law to the behavior of people having a discussion with other people; specifically, it examined the behavior of a subject when discussing the topic of juvenile delinquency with two confederates. These confederates had been given instructions by the experimenter to make statements of agreement during the discussion, according to a variable schedule, and the responses measured were things like the eye contact made with the confederates or the orientation of their body towards them, and the numbers were plugged into the equation presented above. Here is an example of the results from one subject:

What this graph tells us is that the behavior of the subject very closely matched what was predicted by strict matching - with strict matching being described by the dotted line and the actual behavior described by the solid line. The prediction is imperfect, due to the invisible variables mentioned above and the fact that strict matching is not as accurate as more current theories of choice, but the accuracy of the equation in such a messy and complicated situation speaks volumes to the predictability of humans.

The matching law and sports: Some of the more interesting studies looking at the application of the matching law to humans has been done by looking at the behavior of amateur and professional sports players. For instance, a group of researchers looked at the shot selection of basketballers6 to see if the success of their shots would predict whether they would be more or less likely to take two-point or three-point shots. The results for each team showed that their shot selection was almost entirely accounted for by the consequences of their shots; that is, the matching law was very successful at predicting the behavior of the players. 

Another study looked at the 2004 National Football League data of the types of plays called by the coaches to see how their behavior was affected by the consequences of their calls7. They found that the types of calls and plays dictated by the coaches was again well-described by the matching law, and the relative risk of events like turnovers affected the future probability of their calls in ways that is accurately predicted by the matching law.


It means that humans are in fact just meaty, irregular-shaped billiard balls. They are highly predictable even when we observe their behavior in noisy situations where information on some important variables is unknown. The kind of behavior that we would normally assume to be "random" or "unpredictable", like the apparent autonomous choice to direct your conversation towards person A instead or person B, or the choice to try a three-point shot instead of dribbling into the key for the two-point shot, are simply illusions of unpredictability. 

When we take into account basic scraps of information and plug them into our behavioral laws, we gain some insight into these fascinating human nodules. Instead of being perplexed at these meaty billiard balls bouncing in baffling ways, we are instead amazed at how accurately we can judge the effect that these nodules will have on the direction of their behavior.


1. Herrnstein, R.J. (1961). Relative and absolute strength of responses as a function of frequency of reinforcement. Journal of the Experimental Analysis of Behavior, 4, 267–72.

2. Baum, W.M. (1974). On two types of deviation from the matching law: Bias and undermatching. Journal of the Experimental Analysis of Behavior, 22, 231–42.

3. Davison, M., & Jenkins, P. E. (1985). Stimulus discriminability, contingency discriminability, and schedule performance. Animal Learning & Behavior, 13, 77- 84.

4. Borrero J.C, Crisolo S, Tu Q, Rileand W.A, Ross N.A, Francisco M.T, et al. (2007). An application of the matching law to social dynamics. Journal of Applied Behavior Analysis. 40:589–601.

5. Conger R, Killeen P.R. (1974). Use of concurrent operants in small group research: A demonstration. Pacific Sociological Review. 17:399–416.

6. Alferink L.A, Critchfeld T.S, Hitt J.L, Higgins W.J. (2009). Generality of the matching law as a descriptor of shot selection in basketball. Journal of Applied Behavior Analysis. 42:595–608.

7. Reed D.D, Critchfield T.S, Martens B.K. (2006). The generalized matching law in elite sport competition: Football play calling as operant choice. Journal of Applied Behavior Analysis. 39:281–297.


  1. "meaty, irregular-shaped billiard balls"

    Awesome! It is a good analogy.

    Though I'm still iffy on Bayesian statistics (because backwards probability is incoherent).... I do like the idea that, when we talk about things as probabilistic, much of the time we are just expressing our ignorance. Bayes himself understood probability to be an expression of confidence, not a measure of universe-mysteriousness.

    1. Thanks! I'm glad the analogy makes sense to people who don't live inside my head.

      That's an interesting way of describing Bayes' understanding of probability and I think it's quite a good way to think about it.

  2. The matching law is very interesting. Do you know what kind of strategies are used to establish possible sources of under/over matching and bias in empirical studies?

    1. There are a number of studies looking at over- and undermatching, and the most common finding is that organisms undermatch (I think generally they find a sensitivity of about .8, where strict matching is obviously 1.0). As for testing ideas of these findings, originally it was argued that undermatching is simply imperfect matching. So strict matching is the optimal behavior but the fallibility of cognition just gets in the way.

      This is (in my opinion) best explained by a model proposed by Davison and Jenkins known as the contingency-discriminability model, and this includes a "discriminability" or "confusion" parameter which suggests that undermatching occurs due to mistakes in attributing a reinforcer to a behavior. So if a rat is tapping away at a lever and occasionally it swaps over to another lever and it gets the reward for its responding on the original lever, then it can easily mistakenly attribute the reinforcer to its current behavior. To test this you can set up experiments that see what happens when you make the stimuli harder and harder to distinguish (closer colours, no change-over delay, etc) and we do find that behavior approaches matching when stimuli are highly discriminable, but we find undermatching when they are harder to discriminate.

      Overmatching is much rarer but it seems to only really occur when there is a cost to switching between alternatives. So if switching requires some physical obstacle or extra responding on a changeover key, then the organisms will be less willing to switch which means that they will tend to overmatch. There's a good article here which discusses some of the research on the topic in the intro: Overmatching in rats: The barrier choice paradigm.

      The "bias" parameter in the generalised matching law is argued to represent inherent biases in response to particular situations - for example, it might just be the case that subject102 prefers the colour yellow instead of red due to some biological component of their eye, which would be unaffected by environmental contingencies. I'm not really sure if this has been tested in any rigorous way, to be honest, but there have been a couple of good books on the topic that might explain it better that I can. For example:

      The Matching Law: Papers in Psychology and Economics


      The matching law: A research review

      (I think the latter is out of print but if you were interested, email me at and I can send you a pdf copy).

  3. Thanks!! Excellent info as usual.

  4. Beyond enlightening...thank you.

    Sort of invites us all to laugh at our games while ultimately battling our demons.