Expected Value (and beyond)

To make the best choices, we face the impossible task of evaluating the future. Until the invention of "expected value," people lacked a simple way to quantify the value of an uncertain future event. Expected value was famously hit upon in a 1654 correspondence between polymaths Blaise Pascal and Pierre de Fermat. Pascal had enlisted Fermat to help find a mathematical solution to the "problem of points:" namely, how can a jackpot be divided between two gamblers when their game is interrupted before they learn of its final outcome?

A gamble's value obviously depends upon how much one can win. But Pascal and Fermat further concluded that a gamble's value also should be weighted by the likelihood of a win. Thus, expected value is computed as a potential event's magnitude multiplied by its probability (thus, in the case of a single gamble "x," E(x) = x*p). This formula is now so common that it is taken for granted. But I remember a fundamental shift in my worldview after my first encounter with expected value—as if an impending fork in the road transformed into a broad landscape of potentials, whose hills and valleys were defined by goodness and likelihood. This open view of all possible outcomes implies optimal choice—to maximize expected value, simply head for the highest hill. Thus, expected value is both elegant in its computation and deep in its implications for choice.

Even today, expected value forms the backbone of dominant theories of choice in fields including economics and psychology. More recent replacements have mainly tweaked the key ingredients of expected value—adding a curve to the magnitude component (in the case of Expected Utility), or flattening the probability component (in the case of Prospect Theory). But beyond its longevity, what amazes me most about this seventeenth century innovation is that the brain may faithfully represent something like it. Specifically, not only does activity in mesolimbic circuits appear to correlate with expected value before the outcome of a gamble is revealed, but this activity can be used to predict diverse choices—ranging from what to buy, to which investment to make, to whom to trust.

Thus, expected value is beautiful in its simplicity and utility—and almost true. Like any good scientific theory, expected value is not only quantifiable, but also falsifiable. As it turns out, people don't always maximize expected value. Sometimes they let potential losses overshadow gains or disregard probability (as highlighted by Prospect Theory). These quirks of choice suggest that while expected value may prescribe how people should choose, it does not always describe what people do choose. On the neuroimaging front, emerging evidence suggests that while subcortical regions of the mesolimbic circuit are more sensitive to magnitude, cortical regions (i.e., the medial prefrontal cortex) more heavily weight probability. By implication, people who have suffered prefrontal damage (e.g., due to injury, illness, or age) may be more seduced by attractive but unlikely offers (e.g., lottery jackpots).

Indeed, thinking about probability seems more complex and effortful than thinking about magnitude—requiring one not only to consider the next best thing, but also the one after that, and after that, and so on. Neuroimaging findings suggest that more recently evolved parts of the prefrontal cortex allow us not to "be here now"—but instead to transport ourselves into the uncertain future. Mental and neural evidence for differentiating magnitude and probability suggest a limit on the explanatory power of expected value. To some, this limit paradoxically makes expected value all the more intriguing. Scientists often love explanations more for the questions they raise than the questions they answer.