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A couple of years ago, I blogged about an information cascade problem I’d read about in David Easley and Jon Kleinberg‘s textbook on Networks, Crowds, and Markets. To recall the problem (which they themselves borrowed from Lisa Anderson and Charles Holt:

The experimenter puts an urn at the front of the room with three marbles in it; she announces that there is a 50% chance that the urn contains two red marbles and one blue marble, and a 50% chance that the urn contains two blue marbles and one red marble…one by one, each student comes to the front of the room and draws a marble from the urn; he looks at the color and then places it back in the urn without showing it to the rest of the class. The student then guesses whether the urn is majority-red or majority-blue and publicly announces this guess to the class.

The fascinating result is that the sequence of guesses locks in on a single color as soon as two consecutive students agree. For example, if the first two marbles drawn are blue, then all subsequent students will guess blue. If the urn is majority-red, then it turns out there is a 4/5 probability that the sequence will converge to red and a 1/5 probability that it will converge to blue.

Let me explain why I find this problem so fascinating.

Consider a scenario where you are among a group of people faced with the single binary decision — let’s say, choosing red or blue — and that each of you is independently tasked with recommending the best decision given your own judgement and all available information. Assume further that each of you is perfectly rational and that each of your prior decisions (i.e., without knowing what anyone else thinks) is based on independent and identically distributed random variables. Let’s follow the example above, in which each participant in the decision process has a prior corresponding to a Bernoulli random variable with probability p = 2/3.

If each of you makes a decision independently, then the expected fraction of participants who makes the right decision is 2/3.

But you could do better if you have a chance to observe others’ independent decision making first. For example, if you get to witness 100 independent decisions, then you have a very low probability of going wrong by voting the majority. If you’d like the gory details, review the cumulative distribution function of binomial random variables.

On the other hand, if the decisions happen sequentially and every person has access to all of the previous decisions, then we see an information cascade. Rationally, it makes sense to let previous decisions influence your own — and indeed 4/5 > 2/3. But there’s still a one in five chance of making the wrong decision, even after you witness 100 previous decisions. We are wasting a lot of independent input because of how participants are incented.

I can’t help wondering how changing the incentives could affect the outcome of this process. What would happen if participants were rewarded based, in whole or in part, on the accuracy of the participants who guess after them?

Consider as an extreme case rewarding all participants based solely on the accuracy of the final participant’s guess. In that case, the optimal strategy for all but the last participant is to ignore previous participants’ guesses and vote based solely on their own independent judgements. Then the final participant combines these judgements with his or her own and votes based on the majority. The result makes optimal use of all participants’ independent judgments, despite the sequential decision process.

But what if individuals are reward based on a combination of individual and collective success? Consider the 3rd participant in our example who draws a red marble after the previous participants guess blue. Let’s say that there are 5 participants in total. If the reward is entirely based on individual success, the 3rd participant will vote blue, yielding an expected reward of 2/3. If the reward is entirely based on group success, the 3rd participant will vote red, yielding an expected reward of 20/27 (details left as an exercise for the reader). If we make the reward evenly split between individual success and group success, the 3rd participant will still vote blue — the benefit from helping the group will not be enough to overcome the cost to the individual reward.

There’s a lot more math in the details of this problem, e.g. “The Mathematics of Bayesian Learning Traps“, by Simon Loertscher and Andrew McLennan. But there’s a simple take-away: incentives are crucial in determining how we best exploit our collective wisdom. Something to think about the next time you’re on a committee.

## By Daniel Tunkelang

High-Class Consultant.

## 9 replies on “Information Cascades, Revisited”

On a related note, I’ve always been interested in how people choose to represent a choice given partial and full information. For instance, before the England-France game yesterday, we could ask 500 people which they think is more likely: England win, draw, France win. And after the game (which we now know was a draw), we could ask another 500 people in the same population “before the game, which would you have thought was more likely?” In effect: measuring how a population thinks given partial information, then given full information but still asking about a choice given partial information. The difference between the two measurements would place numerical value on “being right” versus accuracy of representing one’s prior opinions.

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Daniel,
you are right that incentives are very important here, but there are also some basic questions about information processing. Note that in the original setup, people get to see the ‘signal’ provided by others (the announced guess), but not the ‘data’ (the color of the ball). If they had access to *all* the data, the signal would be ignored -or, I guess technically what would happen if *everyone* had access to the data is that their signal would be “faithful” to it. There would never be a cascade.

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Antonio, I agree that access to all data makes things easier. But what I’m after in this example is that the data lives in the heads of the people making the decision. So giving everyone access to all of the data access requires everyone to disclose what is in their heads. That requires particular incentives. With different incentives, people provide signals that leads to worse collective utility.

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Tanjasays:

Context and risk matter a lot-hence, not only positive, but also negative incentives (sanctioning) may come into play. Furthermore, the pool of available choices also influences the degree to which diversity of opinions can be achieved, and diversity is the key-factor when it comes to exploiting the wisdom of crowds.

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Antonio:

We may have a bit of a problem here because ‘data’ and ‘information’ are pretty slippery concepts. To me, the data in this example is the color of the ball. So it’s not in the heads of people -but it is *private*, as it is not shown. This is especially relevant in a social environment in that the data may be out there, but there are acquisition costs so there are ‘intermediaries’ from which most of us get our data -including news agencies, etc. But I agree with you that in the end incentives play a central role, since to share their private data, in most situations people need incentives.

Daniel:

I agree that the original example I cite is about data. But I deliberately transformed this example into
a scenario where you are among a group of people faced with a common binary decision where all participants are perfectly rational and hold priors (i.e., without knowing what anyone else thinks) is based on independent and identically distributed random variables. Sorry if that segue was confusing.

Antonio:

I see what you are saying. You fit a nice mathematical model and yes, there everything would depend on incentives to encourage people to disclose ‘what is in their heads’ -since that’s the only way to update the priors on this model. The question left, of course, is whether modeling people as ‘perfectly rational’ is a good idea 🙂

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