From One to Many: Heterogeneity
One Ring to rule them all, One Ring to find them,
One Ring to bring them all and in the darkness bind them.
—J. R. R. Tolkien, The Fellowship of the Ring
Economists are fond of the “representative agent,” a theoretical convention that makes the math much, much easier. The idea behind the representative agent is that instead of having to worry about, say, every consumer in the economy, we can substitute a single consumer to represent everyone—one agent to rule them all, as it were. Obviously, such an assumption greatly simplifies the resulting model, as the representative agent can stand in for a vast horde of individually quirky consumers who might be difficult to track one by one. Indeed, theoretical economists and policy makers often use such a trick in models that influence the lives of hundreds of millions. As long as individual behaviors average out appropriately, using such an approach seems like an obvious choice.
Whether we can use representative agents to model complex systems is really a question about whether heterogeneity matters. If it doesn’t matter, then assuming average behavior embodied in the form of a representative agent suffices: the same behavior will emerge from a system modeled by a population of actors as from one consisting of a single representative agent. If it does matter, then we need a new approach to understand, predict, and control our world.
Jane Jacobs, in her remarkable book Cities and the Wealth of Nations, admonishes economists to get the answer right here:
We think of the experiments of particle physicists and space explorers as being extraordinarily expensive, and so they are. But the costs are as nothing compared with the incomprehensibly huge resources that banks, industries, governments and international institutions like the World Bank, the International Monetary Fund and the United Nations have poured into tests of macro-economic theory. Never has a science, or supposed science, been so generously indulged. And never have experiments left in their wake more wreckage, unpleasant surprises, blasted hopes and confusion, to the point that the question seriously arises whether the wreckage is reparable; if it is, certainly not with more of the same. Failures can help set us straight if we attend to what they tell us about realities. But observation of realities has never, to put it mildly, been one of the strengths of economic development theory.
Consider a honeybee hive. Every egg laid by the queen goes through a delicate sequence of development from egg to larva to pupa to finally emerging from its honeycomb cell as a fully formed bee. For this sequence to be successful, it requires a narrow range of temperatures to be maintained inside the hive (close to 94 degrees Fahrenheit). Of course, the temperature outside the hive varies wildly, so how can bees keep the inside temperature confined to such a small range?
It turns out that worker bees have two temperature-related behaviors. When a worker gets too cold, it seeks out other bees and rapidly buzzes its wings to generate heat. When it gets too warm, it moves away from others and fans its wings to form air currents that will cool things down (see Figure 4.1).
The temperature in the hive depends on the actions of its workers. There is no central command center in the hive, and it is only through the decisions and actions of each individual bee that things get done. It turns out that an individual honeybee’s temperature-related behavior is given by a genetically determined set point. Temperatures much above or below this set point cause the bee to undertake cooling or warming behavior, respectively.
Temperature control seems like a situation in which a population would benefit from homogeneity—in which nature would evolve a representative agent. Researchers at the University of Sydney (see Jones et al., “Honey Bee Nest Thermoregulation: Diversity Promotes Stability,” Science, 2004) investigated this question and found a surprising result.
As a thought experiment, suppose we observe a hive of bees in which every bee’s genetic thermostat is set to the same ideal temperature. You might think that since all of the bees are so precisely calibrated, the hive will maintain a constant temperature. This is not what happens. When the temperature creeps below the set point, large numbers of bees instantly huddle together and buzz their wings, causing a large increase in the temperature. As the temperature rises, it quickly goes past the ideal point, and all of the bees switch to their cooling behavior and scatter and fan, inducing a rapid drop in temperature. As the temperature plummets below the ideal point, the mass of bees switches behavior yet again. What emerges is not a hive with a tightly controlled temperature but one that experiences wild swings in temperature.
As an alternative, suppose that we have a hive of heterogeneous bees, each with a slightly different set point around the ideal temperature. In this hive, as the temperature starts to creep below the ideal point, only a few bees start to huddle together and provide a little additional warmth, slowly raising the temperature. Indeed, any time the temperature overshoots or undershoots the ideal point, there is a graduated response by the bees, with only a few joining in at first, and more joining in only if things start to stray further from the ideal. Ultimately, this heterogeneous strategy allows the hive to maintain a precise temperature with only minimal oscillations.
Thus, having a heterogeneous population of honeybees is adaptive to the hive, leading to a much more tightly controlled temperature and greater success in brood rearing. In real hives the virgin queen spends her first few days going out on flights where she mates with around eight to twenty drone (male) honeybees from different hives, rather than just one. Once the queen is back in her hive, she produces worker bees that are either sisters or half-sisters to one another, guaranteeing some heterogeneity among them.
The average temperature set point of the honeybees was the same in both our homogeneous and heterogeneous hives. The difference was that in the heterogeneous hive there was some variance of the set points around this average, whereas in the homogeneous hive every worker had the same set point. So, at least in terms of a honeybee hive, the representative agent model would be very misleading, implying hive temperatures that oscillate wildly when in fact they are actually quite stable.
Now consider a model of a market. Let’s assume that the market is populated by homogeneous representative traders who decide to buy or sell based on incoming information. Just as we saw with the honeybees, this type of model is going to result in some unusual market behavior. As the information in the market begins to change, at some point the representative trader is going to want to buy. Since all of the traders use the same rule, this is going to cause a drastic increase in demand, and prices will experience a rapid rise. As prices go up, the information eventually changes to a point where, in perfect synchronicity, all of the traders want to sell, inducing a price crash. As in the case of the hive, a market with homogeneous traders leads to wild price oscillations.
Stable markets emerge only with heterogeneous agents. With many types of traders, responses to changing information are graduated, with slight changes in information influencing only the most sensitive traders, and more extreme changes provoking responses from the less sensitive traders. Such a market will be much better behaved than a homogeneous one, experiencing milder price swings and more reasonable “price discovery.”
In hives and markets heterogeneity provides needed stability, but this is not always the case in other systems. Suppose we want to model the dynamics of a social movement, ranging from a neighborhood-level riot to the overthrow of a national government. Let’s assume that each of, say, one hundred people in our society has a sensitivity level, S, such that if she observes S or more people participating in the movement, then she will join. Finally, let’s assume that there is a group of outside rabble-rousers that tries to start the movement.
Assume that our one hundred people all have the same sensitivity level set at, say, 50. How many rabble-rousers will it take to trigger an all-out social movement? If the number of rabble-rousers is less than fifty, then no one else joins in the fray. If the number of rabble-rousers is fifty or above, then everyone joins. Thus, in a homogeneous world, it takes at least as many rabble-rousers as the fixed sensitivity level to catalyze a movement. In this example, we need a fairly large number of rabble-rousers—equal to half of the population—before we see a full-blown social movement.
Alternatively, assume that we have a very heterogeneous population, with each of our one hundred people having a unique sensitivity. To make this an extreme example, line up the population and give the first person a sensitivity of 1, the second a sensitivity of 2, and so on down the line, until the last person is assigned a sensitivity of 100. In this world, how many rabble-rousers are needed to catalyze a society-wide social movement? The answer, of course, is one. One single rabble-rouser is enough to get the person with a sensitivity of 1 to join in, and once we have two people in the movement, that is enough to get the person with a sensitivity of 2 to join, and this triggers the third (which, according to Arlo Guthrie’s song “Alice’s Restaurant,” constitutes an organization), and so on down the line, until all one hundred members of our society have joined the movement.
Both of the social worlds above are characterized by a critical tipping point, whereby below this point no one joins the movement and above it everyone does. Of course, this tipping point is dramatically different in the two worlds, being equal to fifty (half the population) in the first and only one in the second. Note that in both worlds, the average threshold for the population is about fifty, so the different tipping points are due to the variations in the thresholds of the two worlds. In the first world, the presence of homogeneous agents implies no variance, while in the second agent heterogeneity induces a lot of variance.
Thus, in the social movement model, we find a case where heterogeneity leads to instability rather than stability. However, both the bees and the protest share important characteristics that underlie the dramatic difference in outcomes. In both cases, heterogeneity leads to a graduated response, where slight changes in the environment cause slight changes in the system’s behavior. The difference between the models is in the type of feedback they engender. In the case of hive temperature regulation, the system is governed by negative feedback, and having a graduated response tends to stabilize the system. In the case of the social movement, there is positive feedback, and a graduated response is like a rolling snowball, where the accumulation of snow makes it bigger and heavier and more likely to pick up additional snow.
Despite the type of feedback in play, both models make the same essential point about representative agents: they can be quite misleading, as the mean is not the message. If we consider systems with all agents acting at the mean, we will often make bad predictions, expecting too little stability in the case of the honeybees and too much in the case of social movements.
Policy can often influence the level of heterogeneity in the system and thus determine the system’s overall behavior. Heterogeneity is likely to be a stabilizing force in markets, and therefore we might want to encourage diversity by ensuring that we have many moderately sized trading houses competing with one another using proprietary trading algorithms. However, if you want to quash a social rebellion, having a homogeneous population with a high threshold will prevent small events from growing into revolutions. While policy can’t dictate a homogeneous population, it can influence the feedback loops by, say, altering the information individuals receive about reasonable threshold levels or the number of activists. Alternatively, if you want to initiate a social movement from a small spark, then you want to encourage a diversity of views and a sense that everyone is participating, so that a single spark can lead to a cascade that ignites a full-blown movement—it takes just one ring to bring them all.