Guess High

I came across an example for teaching statistics from Gelman and Nolan (2002), Teaching Statistics: A Bag of Tricks that was interesting (here is a link to a journal article published on it in The American Statistician). Suppose that there is a jar of quarters filled to a certain line. We don't know how many are in the jar, but after pooling information from a lot of guesses we have an average "guess" of 160, and the standard deviation of the guesses is 60, so let's take that as the distribution, supposing it is "normal." Now, if I want to guess the number of quarters in the jar (and the payoff is fixed at, say $50), it is a simple matter of maximizing the likelihood function, L() (which is equivalent to minimizing your squared losses):
max{(1/sqrt(2²))*exp((-1/2²)*(x-)²)}.
Maximizing, and plugging in 160 and 60 for  and , respectively, you get back the intuitive guess – 160, the average you started with.

But that's not how these carnival games typically work. The guesser gets what's in the jar, usually. So, to find the guess that maximizes x times the likelihood function:
max{(x/sqrt(2²))*exp((-1/2²)*(x-)²)},
which is equivalent to maximizing the natural log of xL(q), i.e.
max{ln(x)- (1/2²)*(x-)²)},
since the natural log is a monotonic function. The first-order condition is:
1/x - (1/²)*(x-) = 0,
and plugging in 160 and 60 for  and , we get:
x = 180.

This is related to the moral hazard involved with stock and asset pricing (or, the recommendations and ratings put out to customers by banks, brokerages, and ratings institutions). Sure, we could estimate an accurate value for an asset, but when your payoff is positively correlated with the value of your guess, you'll have a systematic incentive to guess high. If enough of these "high" guesses accumulate over time, eventually it becomes obvious that the "guesses" being collected in the market are way off from the "true" expected value.