In This Question We Consider A Variation On The Model Of Inf

In This Question We Consider A Variation On the Model Of Informatio

In this question, we examine a variation of the model of information cascades from Chapter 16, where individuals sequentially decide whether to adopt or reject a new technology. The model introduces a scenario where adopters receive a payoff that is either positive or negative, depending on whether the technology is good or bad, with these payoffs being random but with a positive average for good technology and negative for bad. Rejecting the technology yields a fixed payoff of zero. Additionally, individuals are informed of the payoffs previously received by prior adopters, not just their actions and signals.

(a) If the technology is actually bad, how does this additional information about previous payoffs influence the likelihood and persistence of an information cascade promoting adoption? A brief argument suffices.

(b) If the technology is truly good, can an information cascade of rejection occur? Provide a brief explanation.

Next, consider the classic model of information cascades with the following parameters: the prior probability that the state is good is 1/2, and the probability of receiving a high signal given the state is good is 2/3. The probability of a low signal given a bad state is also 2/3. Each individual observes prior choices but not prior signals.

(a) If you are the tenth person, and all previous nine have rejected the technology, what is the probability that this cascade is incorrect, i.e., the true state is good despite the cascade showing rejection?

(b) Before receiving your signal, you inquire about person 9’s observed signal, learn they observed high, and are truthful. Based on this information and your own signal, what should you decide—accept or reject? How does your decision depend on your signal?

(c) For person 11, who only observes prior choices (all rejecting before him), and knowing that you and person 9 observed high signals, what decision should he make if you choose reject or accept? How do his own signal and previous choices influence his decision?

Sample Paper For Above instruction

Understanding how information cascades influence decision-making in technological adoption and social behaviors is crucial in economics and behavioral sciences. The models of information cascades reveal the dynamics through which individual decisions are shaped by prior actions and signals, often leading to predictable patterns such as herd behavior. Variations in these models, especially with richer information like payoffs or signals, further deepen our understanding of social decision processes.

Impact of Payoff Information in Decision Cascades when the Technology is Bad or Good

The classic model in Chapter 16 assumes that individuals base their decisions solely on their private signals and the observed actions of predecessors. The introduction of additional payoff information—specifically, the actual payoffs received by previous adopters—adds a new informational layer. When the technology is indeed bad, this additional payoff data can significantly influence future decisions. If prior adopters received negative payoffs, this serves as strong evidence that the technology is bad, reinforcing the cascade of rejection and making it more robust and persistent. Conversely, if prior adopters received positive payoffs, which might occur due to stochastic variations or misperceptions, this could temporarily encourage further adoption, even if the technology is actually bad. However, since the payoffs are informative about the true state, the overall tendency would lean toward reinforcing the cascade of rejection as more consistent negative payoffs accumulate. Therefore, the additional payoff information tends to strengthen the cascade in the case of a bad technology, reducing the likelihood of a reversal once a cascade forms.

When the technology is actually good, the presence of payoff information allows early adopters to signal the true state through their payoffs. If adopters receive positive payoffs, this clarifies that the technology is good, potentially preventing a cascade of rejection from forming or aiding an existing rejection cascade to break down. However, if early adopters happen to receive negative payoffs by chance, they might reject the technology despite its goodness. Still, since the average payoff is positive for good technology, over time, as more adopters observe positive payoffs, the cascade of rejection becomes less likely to persist, and an acceptance cascade becomes more probable. In sum, additional payoff information tends to enhance the accuracy of beliefs about the technology’s state and reduce the likelihood of a cascade of rejection for a good technology.

Likelihood of Incorrect Cascades and Decision Dynamics

In the second part, with the parameters p=1/2, q=2/3, the probability that the state is good given an all-reject cascade depends on Bayesian updating. Since each individual gets a signal with a 2/3 chance of being correct, and observing a cascade of rejection suggests a higher likelihood that the state is bad, the probability that the cascade is wrong—meaning the true state is good—can be derived using Bayesian inference. Specifically, given that all previous nine rejected, the posterior probability that the true state is bad is high, but not 100%. The calculation involves updating the prior with the likelihoods of observing nine consecutive low signals under each hypothesis, leading to a specific probability of an incorrect cascade. Typically, this probability is substantial—indicating that even long cascades can be wrong, especially when the signals are only somewhat informative.

When considering the decision of person 10 who observes a cascade of rejection but learns that person 9 observed a high signal, the decision rule should incorporate this new information with their private signal. If their own signal is high, they should be inclined to accept, trusting the combined evidence of the initial high signal and the recent truthful report. Conversely, if their own signal is low, they might reject despite the prior cascade, as their private information contradicts the apparent trend. The precise decision depends on the strength of their signal relative to the accumulated information.

For person 11, observing only the actions of previous individuals and knowing that the prior nine rejected, their own signal and the information from person 9’s report influence their expectations. If their signal is high and prior reports suggest high signals, it nudges toward acceptance. If their signal is low and all prior choices are rejection, they are more inclined to reject. However, since previous choices are all rejection, and signals are noisy, Bayesian updating might still favor rejection unless their own high signal provides strong evidence otherwise. In essence, Bayesian updating combines prior choices, truthful signals, and personal signals to guide decision-making amidst uncertainty.

Conclusion

Understanding the dynamics of information cascades, especially with enriched information like payoffs and signals, is vital in contexts like technology adoption, financial markets, and social behavior modeling. These models illustrate how individual decisions are not made in isolation but are heavily influenced by the informational environment and prior actions, often leading to herd behavior or incorrect cascades. Recognizing the factors that reinforce or halt such cascades can inform strategies to promote better decision-making and mitigate herd behavior risks in societal settings.

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