When deciding between two products, you likely gather information about the options by, for example, comparing specifications. But your choice might also be influenced by the choices of those around you. Such group decision making is the focus of a new statistical model developed by Krešimir Josić of the University of Houston and his colleagues. In particular, they analyzed how decision makers learn from one another in homogeneous and heterogeneous populations. They found that the best decision was reached when some population members are impulsive and some are cautious. The results apply to a wide range of behaviors, from social hierarchy in animals to how pedestrians decide when it’s safe to cross a street.
In the researchers’ model, each rational agent in the population gathers private evidence and, unlike in earlier models from other groups, the data accumulate over time. The agents are given two options, which are quantified by a positive value (correct decision) or a negative value (wrong decision). The agents’ inclination is calculated as a belief y, which evolves as a random walk with a general trend toward the correct decision. (For more on random walks, see Physics Today, September 2019, page 18.) When y is beyond some positive or negative threshold θ (the red and green lines in the graphs), the agent makes the correct or wrong decision, respectively. When the threshold value is the same for each agent, the agents are considered to be in a homogeneous population, and when the value is different for each agent or for groups of agents, they are part of a heterogeneous population.
After a first agent (agent 1, red in the top part of the image) makes a choice—that is, the agent’s belief y has exceeded its threshold θ—the rest of the group adjusts their beliefs by θ. In homogeneous populations, when every agent adds θ to their belief value, those agents whose y is the same sign as agent 1’s exceed their identical threshold value. By the time agent 1 has made a selection, typically more than half the population has y greater than 0 as a result of y‘s trend toward the correct choice. So if agent 1 selects the correct choice, usually just over half follow suit. But if agent 1 selects the wrong choice, typically just under half go along (red agents in the center part of the image). In either scenario, the remaining agents (blue) who don’t follow agent 1 in that first wave decide whether agent 1 made the right decision based on the fraction of the population still undecided, and they then select the best option (green agents in the bottom part of the image).
In diverse populations, agents have two or more threshold values. Thus if agent 1 picks the wrong option, not every agent with y less than 0 will reach their threshold value and take the plunge. They instead learn from the rapid but foolhardy decisions of low-threshold agents. The high-threshold cautious agents know that agent 1’s choice is unreliable, so only when enough low-threshold agents follow along do they decide. In the end, the fraction who end up making the wrong decision is smaller. Low-threshold agents speed up the process by choosing quickly and thus offering rapid information. And the wary agents keep the bulk of the group from going down an incorrect path.
Josić and his colleagues’ model can shed light on, for example, social hierarchy in animals. It suggests that even though agents who decide first may emerge as leaders, they potentially are just group members who make quick decisions with little information. To apply their model more directly to real-world scenarios, Josić’s group will need to modify some of the model’s features. For example, they could have agents accumulate information at different rates. (B. Karamched et al., Phys. Rev. Lett. 125, 218302, 2020.)
