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Neuroscientist; Collège de France, Paris; Author, The Number Sense; Reading In the Brain

The Universal Algorithm For Human Decisions

The ultimate goal of science, as the French physicist Jean Perrin once stated, should be "to substitute visible complexity for an invisible simplicity". Can human psychology achieve this ambitious goal: the discovery of elegant rules behind the apparent variability of human thought? Many scientists still consider psychology as a "soft" science, whose methods and object of study are too fuzzy, too complex, and too suffused with layers of cultural complexity, to ever yield elegant mathematical generalizations.

And yet cognitive scientists know that this prejudice is simply wrong. Human behavior obeys rigorous laws of the utmost mathematical beauty and even necessity. I will nominate just one of them: the mathematical law by which we take our decisions.

All of our mental decisions appear to be captured by a simple rule that weaves together some of the most elegant mathematics of the past centuries: Brownian motion, Bayes' rule, and the Turing machine.

Let us start with the simplest of all decisions: how do we decide that 4 is smaller than 5? Psychological investigation reveals many surprises behind this simple feat. First, our performance is very slow: the decision takes us nearly half a second, from the moment the digit 4 appears on a screen to the point when we respond by clicking a button. Second, our response time is highly variable from trial to trial, anywhere from 300 milliseconds to 800 milliseconds, even though we are responding to the very same digital symbol "4". Third, we make errors—it sounds ridiculous, but even when comparing 4 with 5, we sometimes make the wrong decision. Fourth, our performance varies with the meaning of the objects: we are much faster, and make fewer errors, when the numbers are far from each other (such as 1 and 5) than when they are close (such as 4 and 5).

Well, all of the above facts, and many more, can be explained by a single law: our brain takes decisions by accumulating the available statistical evidence and committing to a decision whenever the total exceeds a threshold.

Let me unpack this statement. The problem that the brain faces when taking a decision is one of sifting the signal from the noise. The input to any of our decision is always noisy: photons hit our retina at random times, neurons transmit the information with partial reliability, and spontaneous neural discharges (spikes) are emitted throughout the brain, adding noise to any decision. Even when the input is a digit, neuronal recordings show that the corresponding quantity is coding by a noisy population of neurons that fires at semi-random times, with some neurons signaling "I think it's 4", others "it's close to 5", or "it's close to 3", etc. Because the brain's decision system only sees unlabelled spikes, not full-fledged symbols, it is a genuine problem for it to separate the wheat from the chaff.

In the presence of noise, how should one take a reliable decision? The mathematical solution to that the problem was first addressed by Alan Turing, when he was cracking the Enigma code at Bletchley Park. Turing found a small glitch in the Enigma machine, which meant that some of the German messages contained small amounts of information—but unfortunately, too little to be certain of the underlying code. Turing realized that Bayes' law could be exploited to combine all of the independent pieces of evidence. Skipping the math, Bayes' law provides a simple way to sum all of the successive hints, plus whatever prior knowledge we have, in order to obtain a combined statistic that tells us what the total evidence is.

With noisy inputs, this sum fluctuates up and down, as some incoming messages support the conclusion while others merely add noise. The outcome is what mathematicians call a "random walk" or "Brownian motion", a fluctuating march of numbers as a function of time. In our case, however, the numbers have a currency: they represent the likelihood that one hypothesis is true (e.g. the probability that the input digit is smaller than 5). Thus, the rational thing to do is to act as a statistician, and wait until the accumulated statistic exceeds a threshold probability value. Setting it to p=0.999 would mean that we have one chance in a thousand to be wrong.

Note that we can set this threshold to any arbitrary value. However, the higher we put it, the longer we have to wait for a decision. There is a speed-accuracy trade-off: we can wait a long time and take a very accurate but conservative decision, or we can hazard a response earlier, but at the cost of making more errors. Whatever our choice, we will always make a few errors.

Suffice it to say that the decision algorithm I sketched, and which simply describes what any rational creature should do in the face of noise, is now considered as a fully general mechanism for human decision making. It explains our response times, their variability, and the entire shape of their distribution. It describes why we make errors, how errors relate to response time, and how we set the speed-accuracy trade-off. It applies to all sorts of the decisions, from sensory choices (did I see movement or not?) to linguistics (did I hear "dog" or "bog"?) and to higher-level conundrums (should I do this task first or second?). And in more complex cases, such as performing a multi-digit calculation or a series of tasks, the model characterizes our behavior as a sequence of accumulate-and-threshold steps, which turns out to be an excellent description of our serial, effortful Turing-like computations.

Furthermore, this behavioral description of decision-making is now leading to major progress in neuroscience. In the monkey brain, neurons can be recorded whose firing rates index an accumulation of relevant sensory signals. The theoretical distinction between evidence, accumulation and threshold helps parse out the brain into specialized subsystems that "make sense" from a decision-theoretic viewpoint.

As with any elegant scientific law, many complexities are waiting to be discovered. There is probably not just one accumulator, but many, as the brain accumulates evidence at each of several successive levels of processing. Indeed, the human brain increasingly fits the bill for a superb Bayesian machine that makes massively parallel inferences and micro-decisions at every stage. Many of us think that our sense of confidence, stability and even conscious awareness may result from such higher-order cerebral "decisions" and will ultimately fall prey to the same mathematical model. Valuation is also a key ingredient that I skipped, although it demonstrably plays a crucial role in weighing our decisions. Finally, the system is ripe with a prioris, biases and time pressures and other top evaluations that draw it away from strict mathematical optimality.

Nevertheless, as a first approximation, this law stands as one of the most elegant and productive discoveries of twentieth-century psychology: humans act as near-optimal statisticians, and any of our decisions corresponds to an accumulation of the available evidence up to some threshold.