2.1 Evolutionary origins of probability matching

Brennan and Lo [50] proposed an evolutionary framework for the origin of several behaviors that are considered “anomalous” in economic theories based on the assumption of rational behavior. In particular, probability matching—the tendency of the relative frequency of guesses of the outcomes of a sequence of independent random events to match the underlying probability distribution of events—can be explained when the uncertainty in environment is systematic across all individuals, an example demonstrating that natural selection is able to yield behaviors that may be individually sub-optimal but are optimal for the population. For expositional convenience, we present a brief review of this framework here, and then turn to our experimental design.

We begin with a population of individuals that live for one period, produce a random number of offspring asexually, and then die. During their lives, individuals make only one decision: they choose one of two possible courses of action, denoted a and b, and this choice results in one of two corresponding random numbers of offspring, x_a and x_b, given by:

where p is some probability between 0 and 1.

We further assume that x_a and x_b are independently and identically distributed over time, and identical for all individuals in a given generation. In other words, if two individuals choose the same action a, both will produce the same number of random offspring x_a. This implies that the variation in offspring due to behavior is wholly systematic, i.e., the link between action and reproductive success is the same throughout the population.

A “mindless” individual’s behavior in this world is fully specified by the probability of choosing action a. Following the notation in Brennan and Lo [50], we denote this probability as f. Each individual dies after one period, and we assume its behavior f is heritable: offspring will behave in a manner identical to their parents, i.e., they choose between the two actions according to the same probability f.

From the individual’s perspective, always choosing the action with a higher expected reproductive success (f = 0 or 1) will lead to more offspring on average. However, Brennan and Lo [50] showed that from the perspective of the population, this individually optimal behavior cannot survive. In fact, the evolutionarily dominant behavior will depend on the relationship between the probability p and the relative fecundity ratios r_j ≔ c_aj/c_bj for each of the two possible states of the world, j = 1, 2, where f can be anywhere between 0 and 1 in general, implying randomized behavior. See Proposition 3 of Brennan and Lo [50] for more detail.

Fig 1 illustrates the evolutionarily dominant behavior f* as a function of r₁ and r₂. If r₁ and r₂ are not too different in value—i.e., the ratio of fecundity between choices a and b is not very different between the two states of the world—then random behavior yields no evolutionary advantage over deterministic choice. In this case, the individually optimal behavior (f* = 0 or 1) will prevail in the population.

The asymptotes of the curved boundary line occur at r₁ = p and r₂ = q. Values of r₁ and r₂ for which exact probability matching is optimal are given by the solid black curve. Source: Brennan and Lo [50, Fig 1].

However, if one of the r variables is large while the other is small, then random behavior will be more advantageous for the population than a deterministic one. In such cases, there are times in which each choice performs substantially better than the other. Under those conditions, it is evolutionarily optimal for a population to diversify between the two choices, rather than always choosing the outcome with the highest probability of progeny in a single generation.

A simple numerical example from Brennan and Lo [50] will illustrate the basic mechanism of this model. Consider a population of individuals, each facing a binary choice between one of two possible actions, a and b. 70% of the time, environmental conditions are positive, and action a leads to reproductive success, generating 3 offspring for the individual. 30% of the time, environmental conditions are negative, and action a leads to 0 offspring. This corresponds to p = 70%, c_a1 = 3, c_b1 = 0 in the notation of (1). Suppose action b has exactly the opposite outcomes—whenever a yields 3 offspring, b yields 0, and whenever a yields 0, b yields 3. This corresponds to c_a2 = 0, c_b2 = 3 in the notation of (1). From the individual’s perspective, always choosing a, which has the higher probability of reproductive success, will lead to more offspring on average. However, if all individuals in the population behaved in this “rational” manner, the first time that a negative environmental condition occurs, the entire population will become extinct. Assuming that offspring behave identically to their parents, the behavior “always choose a” cannot survive over time. For the same reason, “always choose b” is also unsustainable. In fact, one can show that in this special case, the behavior with the highest reproductive success over time is for each individual to choose a 70% of the time and b 30% of the time, matching the probabilities of reproductive success and failure. Eventually, this particular randomizing behavior will dominate the entire population.

The key to understanding these behavioral predictions lies in the assumption of systematic reproductive risk. This dependence on risk has implications that go far beyond the current setting. For example, Zhang, Brennan, and Lo [51] show that environments with a mix of systematic and idiosyncratic reproductive risks cause different risk preferences to emerge. While our risk preferences may be determined by the nature of the risks to which we and our evolutionary ancestors have been exposed, we do not necessarily have the ability to distinguish between systematic and idiosyncratic risks in our day-to-day decision making.