http://www.jstor.org/view/00346527/di990706/99p03304/0?frame=noframe&userID=80fdba30@cornell.edu/01cce4406400501bc33

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The Economics of Rumors

In this work, the authors explore a family of models of information transmission processes in regard to an investment opportunity in which expected returns are known to few people. Each individual has a cost of undertaking the project and only receives information about which other agents invested. From this information, he would like to infer the expected returns of the project and decide whether or not to invest. Specifically, the basic model has two assumptions: (i) each person chooses to invest based only on what the previous investor decided and (ii) only the first investor knows the opportunity exists, and has a private “signal” of what the expected returns are. Through various relaxations on these assumptions, the authors obtain models in which the optimal investment decisions exhibit interesting cascade effects.

Assume that the first investor, learns of the opportunity and its expected returns. If he decides to invest, then each other investor gets an opportunity to invest sequentially and only observes the decision of the previous investor (he does not have his own private “signal” of returns). Now, each investor knows the prior probability of another investor’s having high costs and the prior of an investment yielding high returns. He knows the action that each investor will take conditioned on his cost and knowlege of returns. So, if he observes the previous investor’s investing, he makes use of a bayesian posterior calculation of returns and uses this to make an investment decision based on his costs. Certain settings of the priors yield different gaurantees on the optimal investment decisions for each investor. For this model, we assume that they are set such that the second investor invests if the first investor invested (it can be shown that such parameter values exist). Now, the third investor knows how this decision process works. Therefore, his observing that the investor2 invested is equivalent to knowing that investor1 invested and therefore knowing the state of the world. It follows that everyone invests if and only if the first person invests. This model is the extreme form of our information cascade.

Next, we consider the effects of relaxing the assumption that not all the agents know of the opportunity, while still assuming that only the first investor knows of the returns. Now, since everyone knows of the opportunity, all low cost investors will always invest (low returns > low cost). However, all high cost investors invest only when high returns are sufficiently likely (determined by parameter values). Under the assumption that a high-cost investor would not invest solely on his private information, the authors show that there exists a high-cost investor that will not invest even if the previous investor invests. The reason is that if it the investor knew that the previous investor had low costs it would not give any information about the returns. It was shown that the infered probability of high returns decreases with time due to the increasing expected number of low-cost investors and therefore decreasing informativeness of the previous investment. Under this model, the market is robust to a runaway information cascade in the sense that a few early investment decisions are not enough to sway the entire population.

The previous two models fail to capture the fact that speed of transmission should depend on the importance of the information - it should spread slowly if it affects few people. Here the authors relax the assumption that the rumor is heard sequentially and consider a model where the probability of an investor learning of an oportunity in a given time interval is proportional to the number of people who have invested in the past. The analysis of this model is much more complex than the previous two, so only the main results are discussed here. Under this assumption, the rumor propagates faster if the investment yields high returns. Furthermore, the ratio of the probability that a rumor is heard at time s given low returns state to that in high returns state increases with time and is unbounded. So, as time goes on, if an investor had just heard the rumor for the first time, it is more likely to be a low returns state. In this sense the rumor gives us very precise information. This is result is in sharp contrast to both the results of the first model, in which the informativeness of the rumor remained constant throughout time, and the second model, in which the informativeness of the rumor decreased with time.

The three models considered here provide different interperetations of a rumor for each individual as time progresses. The first model exhibits dynamics similar to the model we discussed in class in which observing two people’s descisions could affect everyone else. While for the second model, the optimal strategy of certain individuals (high-cost investors) was to rely on the previous investment less and less as time went on. Finally, in our third model, the optimal investment choice as time went on was to rely entirely on having heard the rumor at that time (and realizing that it was a low returns state), and was not infulenced by private “signals” (high/low cost investor).