<p>I don't know if this will be useful to anyone else but I sorted it out.</p>
<p>To get equation 12, take the ratio of the denominator of equation 8 to (denominator of equation 8 + denominator of equation 9). These two denominators are equal, which gives us ½.</p>
<p>To get equation 13, take the ratio of the denominator of equation 10 to (denominator of equation 10 + denominator of equation 11).</p>
<p>The notation is a bit strange as <span class="math-container">$\hat{\theta}$</span> is a posterior probability. The denominators in equations 8-11 are all possible cases in which we can see that permutation of signals so that is the probability of seeing the signal and the state of the world.</p>
<p>In equations 12 and 13, we need the same joint probability as is computed in the denominators of the previous equations but now we use the ratio of the joint probabilities to compute the conditional probability of a state given signals. E.g.,</p>
<p><span class="math-container">$Prob (x_H|s_B, s_G) = \frac{Prob (s_B, s_G, x_H)}{Prob (s_B, s_G, x_H) + Prob (s_B, s_G, x_L)}$</span>.</p>
<p>The above is a shortcut to the computations.</p>

CC BY-SA 4.0
2023-04-16T01:00:30.767Z
2023-04-16T01:00:30.767Z
Anirban Mukherjee
<p>I am working my way through Scharfstein and Stein's seminal paper on herding.</p>
<p>Herd Behavior and Investment
Author(s): David S. Scharfstein and Jeremy C. Stein
Source: The American Economic Review , Jun., 1990, Vol. 80, No. 3 (Jun., 1990), pp. 465- 479
Published by: American Economic Association Stable URL: <a href="https://www.jstor.org/stable/2006678" rel="nofollow noreferrer">https://www.jstor.org/stable/2006678</a></p>
<p>I am unable to derive equations 12 and 13. It should be almost trivially simple but I guess I am misunderstanding something. Would anyone have notes or slides on this paper? Any suggestions? I would write the equations down but they don't make sense without the model and then this post would get way too long.</p>

<game-theory><bayesian-game>
Equations 12 and 13 in Scharfstein and Stein (AER, 1990)
Anirban Mukherjee
# Equations 12 and 13 in Scharfstein and Stein (AER, 1990)

(Asked by: username_0 on 2023-04-15)

I am working my way through Scharfstein and Stein's seminal paper on herding.

Herd Behavior and Investment
Author(s): David S. Scharfstein and Jeremy C. Stein
Source: The American Economic Review , Jun., 1990, Vol. 80, No. 3 (Jun., 1990), pp. 465- 479
Published by: American Economic Association Stable URL: <https://www.jstor.org/stable/2006678>

I am unable to derive equations 12 and 13. It should be almost trivially simple but I guess I am misunderstanding something. Would anyone have notes or slides on this paper? Any suggestions? I would write the equations down but they don't make sense without the model and then this post would get way too long.

**Tags:** game-theory, bayesian-game

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## 1 answers

### Answer by username_0 on 2023-04-16. Score: 0 | Accepted answer

I don't know if this will be useful to anyone else but I sorted it out.

To get equation 12, take the ratio of the denominator of equation 8 to (denominator of equation 8 + denominator of equation 9). These two denominators are equal, which gives us ½.

To get equation 13, take the ratio of the denominator of equation 10 to (denominator of equation 10 + denominator of equation 11).

The notation is a bit strange as $\hat{\theta}$ is a posterior probability. The denominators in equations 8-11 are all possible cases in which we can see that permutation of signals so that is the probability of seeing the signal and the state of the world.

In equations 12 and 13, we need the same joint probability as is computed in the denominators of the previous equations but now we use the ratio of the joint probabilities to compute the conditional probability of a state given signals. E.g.,

$Prob (x\_H|s\_B, s\_G) = \frac{Prob (s\_B, s\_G, x\_H)}{Prob (s\_B, s\_G, x\_H) + Prob (s\_B, s\_G, x\_L)}$.

The above is a shortcut to the computations.


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