On the Edge of Success: Tipping Points
Equilibrium values in network effects can be thought of when individuals’ reservation prices are equal to the market prices. Often three equilibrium values exist. One equilibrium value that always exists is at 0 (or when z=0). Another typical equilibrium value is z’. The third typical equilibrium value is z’’. The three equilibrium values 0, z’, and z’’ can be thought of as situations in which no one is using a particular product, when a small fraction (z’) is using the product, and when a large fraction (z’’) is using the product. In equilibrium, the fraction of the population you expect to be using the product should actually equal the fraction of the population using the product, so these are self-fulfilling expectations. For example, the reason why 0 is always a possible equilibrium value is because in this situation, everyone expects no one to buy the product. When everyone thinks no one will use the product, no one wants to buy it, so no one uses it. This is at equilibrium because people’s guesses as to what fraction of the population is buying the product were correct given the decisions people actually made.
Stable equilibria consist of values where the fraction of the individuals using a product converge toward the equilibrium value, even when other values close to, slightly above or below the equilibrium value rather than the exact equilibrium value, are chosen. One way to visualize this is with the graph and explanations below:
According to the graph above if people expect the fraction of the population using a product is anywhere between the two equilibrium values z’ and z’’, the curve of the graph is above p, meaning that a person is willing to pay more than the value p for the product. As a result, more and more people will buy the product than originally expected, and the fraction of people buying the product will keep growing until it reaches z’’. On the other hand, according to the graph above if people expect the fraction of the population using a product is anywhere between the two equilibrium values 0 and z’, the curve of the graph is below p, meaning that a person is willing to pay less than the value p for the product. As a result, this person will not buy the product since it is not worth the amount it costs, and less and less people will buy the product than originally expected until the fraction of people buying the product reaches 0. A similar idea applies if people expect the fraction of the population using a product to be anywhere between z’’ and 1. In this situation, the curve of the graph is below p, meaning that a person is willing to pay less than the value p for the product. As a result, this person will not buy the product since it is not worth the amount it costs, and less and less people will buy the product than originally expected until the fraction of people buying the product reaches back down to z’’. Since any predictions near 0 converge back down to 0 and any predictions close to z’’ converge back up to z’’, these two equilibria are stable. However, any prediction close to z’, that is not exactly z’, either converges down to 0 or up to z’’, making z’ unstable. For this reason z’ is called the tipping point.
In his article “Tipping Points in Social Networks”, Brad Hunter summarizes Malcolm Gladwell’s view of the concept of tipping points and what situations can lead to tipping points from a less mathematical and a more social view. Hunter starts by providing a analogy to the concept of a tipping point by comparing it to chemical phase changes. He explains how 32 degrees Fahrenheit is the tipping point, or in other words z’. This is because at any temperature below 32 degrees Fahrenheit, water molecules will be in the solid phase as ice. However, at any temperature above 32 degrees Fahrenheit, water molecules will be in the liquid phase as water. The idea of the molecules being in the solid phase at temperatures lower than 32 degrees Fahrenheit is comparable to any value below z’ converging at 0. The idea of the molecules being in the liquid phase at temperatures higher than 32 degrees Fahrenheit is comparable to any value above z’ converging at z’’.
In terms of reaching and getting over tippings points, so as to reach the phase of exponential growth towards the higher equilibrium, Hunter provides three of Gladwell’s suggestions from a more social view. Instead of thinking about surpassing a tipping point in terms of more and more people buying a product than expected, thus making this product more successful, Gladwell thought of it terms of spreading a message among people. He described how if you want to spread a message to a larger audience and if you want this message to surpass its tipping point, then you need to consider the “context”, the “stickiness”, and the “few” of the message. In other words, in order for the message to be spread in terms of “context”, it has to be delivered to people at the right time, meaning when they are prepared and willing to hear the message. In order for the message to be spread in terms of “stickiness”, it has to be an idea that will grab people’s attention and will be remembered. In order for the message to be spread in terms of “the few”, it has to be delivered to a select group of people (the “right” people) rather than a large group that will not think the message is worth spreading and will not pass it on.
Equilibria concepts are useful when examining tipping points in a social network. By examining social networks based on the fundamental ideas of equilibria concepts and knowledge of how stable and unstable equilibria interact, it is easier to determine how a network may be swayed in favor of one stable equilibria or another.
Sources and Links:
Easley, David, and Jon Kleinberg. Networks, Crowds, and Markets Reasoning about a Highly
Connected World. Cambridge University Press, 2016.
Hunter, Brad. Tipping Points in Social Networks, web.stanford.edu/class/symbsys205/tipping_point.html.