Information cascade

Information cascades occur when external information obtained from previous participants in an event overrides one's own private signal, irrespective of the correctness of the former over the latter. The experiment conducted by Anderson[10] is a useful example of this process. The experiment consisted of two urns labeled A and B. Urn A contains two balls labeled "a" and one labeled "b". Urn B contains one ball labeled "a" and two labeled "b". The urn from which a ball must be drawn during each run is determined randomly and with equal probabilities (from the throw of a dice). The contents of the chosen urn are emptied into a neutral container. The participants are then asked in random order to draw a marble from this container. This entire process may be termed a "run", and a number of such runs are performed.

Each time a participant picks up a marble, he is to decide which urn it belongs to. His decision is then announced for the benefit of the remaining participants in the room. Thus, the (n+1)th participant has information about the decisions made by all the n participants preceding him, and also his private signal which is the label on the ball that he draws during his turn. The experimenters observed that an information cascade was observed in 41 of 56 such runs. This means, in the runs where the cascade occurred, at least one participant gave precedence to earlier decisions over his own private signal. It is possible for such an occurrence to produce the wrong result. This phenomenon is known as "Reverse Cascade".

A person's signal telling them to accept is denoted as H (a high signal, where high signifies he should accept), and a signal telling them not to accept is L (a low signal). The model assumes that when the correct decision is to accept, individuals will be more likely to see an H, and conversely, when the correct decision is to reject, individuals are more likely to see an L signal. This is essentially a conditional probability – the probability of H when the correct action is to accept, or ${\displaystyle P[H|A]}$. Similarly ${\displaystyle P[L|R]}$ is the probability that an agent gets an L signal when the correct action is reject. If these likelihoods are represented by q, then q > 0.5. This is summarized in the table below.[11]

The first agent determines whether or not to accept solely based on his own signal. As the model assumes that all agents act rationally, the action (accept or reject) the agent feels is more likely is the action he will choose to take. This decision can be explained using Bayes' rule:

${\displaystyle {\begin{aligned}P\left(A|H\right)&={\frac {P\left(A\right)P\left(H|A\right)}{P\left(H\right)}}\\&={\frac {P\left(A\right)P\left(H|A\right)}{P\left(A\right)P\left(H|A\right)+P\left(R\right)P\left(H|R\right)}}\\&={\frac {pq}{pq+\left(1-p\right)\left(1-q\right)}}\\&>p\end{aligned}}}$

If the agent receives an H signal, then the likelihood of accepting is obtained by calculating ${\displaystyle P[A|H]}$. The equation says that, by virtue of the fact that q > 0.5, the first agent, acting only on his private signal, will always increase his estimate of p with an H signal. Similarly, it can be shown that an agent will always decrease his expectation of p when he receives a low signal. Recalling that, if the value, V, of accepting is equal to the value of rejecting, then an agent will accept if he believes p > 0.5, and reject otherwise. Because this agent started out with the assumption that both accepting and rejecting are equally viable options (p = 0.5), the observation of an H signal will allow him to conclude that accepting is the rational choice.

The second agent then considers both the first agent's decision and his own signal, again in a rational fashion. In general, the nth agent considers the decisions of the previous n-1 agents, and his own signal. He makes a decision based on Bayesian reasoning to determine the most rational choice.

${\displaystyle P(A|{\text{Previous}},{\text{Personal signal}})={\frac {pq^{a}(1-q)^{b}}{pq^{a}(1-q)^{b}+(1-p)(1-q)^{a}q^{b}}}}$

Where a is the number of accepts in the previous set plus the agent's own signal, and b is the number of rejects. Thus, ${\displaystyle a+b=n}$. The decision is based on how the value on the right hand side of the equation compares with p.[11]

The original model makes several assumptions about human behavior and the world in which humans act,[7] some of which are relaxed in later versions[11] or in alternate definitions of similar problems, such as the diffusion of innovations.

1. Boundedly Rational Agents: The original Independent Cascade model assumes humans are boundedly rational[12] – that is, they will always make rational decisions based on the information they can observe, but the information they observe may not be complete or correct. In other words, agents do not have complete knowledge of the world around them (which would allow them to make the correct decision in any and all situations). In this way, there is a point at which, even if a person has correct knowledge of the idea or action cascading, they can be convinced via social pressures to adopt some alternate, incorrect view of the world.
2. Incomplete Knowledge of Others: The original information cascade model assumes that agents have incomplete knowledge of the agents which precede them in the specified order. As opposed to definitions where agents have some knowledge of the "private information" held by previous agents, the current agent makes a decision based only on the observable action (whether or not to imitate) of those preceding him. It is important to note that the original creators argue this is a reason why information cascades can be caused by small shocks.
3. Behavior of all previous agents is known

1. Cascades will always occur – as discussed, in the simple mode, the likelihood of a cascade occurring increases towards 1 as the number of people making decisions increases towards infinity.
2. Cascades can be incorrect – because agents make decisions with both bounded rationality and probabilistic knowledge of the initial truth (e.g. whether accepting or rejecting is the correct decision), the incorrect behavior may cascade through the system.
3. Cascades can be based on little information – mathematically, a cascade of an infinite length can occur based only on the decision of two people. More generally, a small set of people who strongly promote an idea as being rational can rapidly influence a much larger subset of the general population
4. Cascades are fragile – because agents receive no extra information after the difference between a and b increases beyond 2, and because such differences can occur at small numbers of agents, agents considering opinions from those agents who are making decisions based on actual information can be dissuaded from a choice rather easily.[7] This suggests that cascades are susceptible to the release of public information.[7] also discusses this result in the context of the underlying value p changing over time, in which case a cascade can rapidly change course.

Information cascades have become one of the topics of behavioral economics, as they are often seen in financial markets where they can feed speculation and create cumulative and excessive price moves, either for the whole market (market bubble) or a specific asset, like a stock that becomes overly popular among investors.

Marketers also use the idea of cascades to attempt to get a buying cascade started for a new product. If they can induce an initial set of people to adopt the new product, then those who make purchasing decisions later on may also adopt the product even if it is no better than, or perhaps even worse than, competing products. This is most effective if these later consumers are able to observe the adoption decisions, but not how satisfied the early customers actually were with the choice. This is consistent with the idea that cascades arise naturally when people can see what others do but not what they know.[18]

An example is Hollywood movies. If test screenings suggest a big-budget movie might be a flop, studios often decide to spend more on initial marketing rather than less, with the aim of making as much money as possible on the opening weekend, before word gets around that it's a turkey.

Information cascades are usually considered by economists:

• as products of rational expectations at their start,
• as irrational herd behavior if they persist for too long, which signals that collective emotions come also into play to feed the cascade.

Dotey et al.[19] state that information flows in the form of cascades on the social network. According to the authors, analysis of virality of information cascades on a social network may lead to many useful applications like determining the most influential individuals within a network. This information can be used for maximizing market effectiveness or influencing public opinion. Various structural and temporal features of a network affect cascade virality. Additionally, these models are widely exploited in the problem of Rumor spread in social network to investigate it and reduce its influence in online social networks.

In contrast to work on information cascades in social networks, the social influence model of belief spread argues that people have some notion of the private beliefs of those in their network.[20] The social influence model, then, relaxes the assumption of information cascades that people are acting only on observable actions taken by others. In addition, the social influence model focuses on embedding people within a social network, as opposed to a queue. Finally, the social influence model relaxes the assumption of the information cascade model that people will either complete an action or not by allowing for a continuous scale of the "strength" of an agents belief that an action should be completed.

• Small protests began in Leipzig, Germany in 1989 with just a handful of activists challenging the German Democratic Republic.[21] For almost a year, protesters met every Monday growing by a few people each time.[21] By the time the government attempted to address it in September 1989, it was too big to quash.[21] In October, the number of protesters reached 100,000 and by the first Monday in November, over 400,000 people marched the streets of Leipzig. Two days later the Berlin Wall was dismantled.[21]

• The adoption rate of drought-resistant hybrid seed corn during the Great Depression and Dust Bowl was slow despite its significant improvement over the previously available seed corn. Researchers at Iowa State University were interested in understanding the public's hesitation to the adoption of this significantly improved technology. After conducting 259 interviews with farmers[22] it was observed that the slow rate of adoption was due to how the farmers valued the opinion of their friends and neighbors instead of the word of a salesman. See[23] for the original report.

In addition to the examples above, Information Cascades have been shown to exist in several empirical studies. Perhaps the best example, given above, is.[10] Participants stood in a line behind an urn which had balls of different colors. Sequentially, participants would pick a ball out of the urn, looks at it, and then places it back into the urn. The agent then voices their opinion of which color of balls (red or blue) there is a majority of in the urn for the rest of the participants to hear. Participants get a monetary reward if they guess correctly, forcing the concept of rationality.

Other examples include

• De Vany and Walls[24] create a statistical model of information cascades where an action is required. They apply this model to the actions people take to go see a movie that has come out at the theatre. De Vany and Walls validate their model on this data, finding a similar Pareto distribution of revenue for different movies.
• Walden and Browne also adopt the original Information Cascade model, here into an operational model more practical for real world studies, which allows for analysis based on observed variables. Walden and Browne test their model on data about adoption of new technologies by businesses, finding support for their hypothesis that information cascades play a role in this adoption[25]