Here we give simple theoretical arguments in favor of the predictive power of prediction markets, as seen in the article.
Assumptions. Suppose first that buyers enter an all-in-or-nothing contract (binary option) paying $1 if an event occurs, $0 otherwise. Suppose that trader $j$’s belief is given by $q_j$, and that beliefs are taken from a distribution $F(q)$. Suppose furthermore that beliefs are independent from wealth levels $y_j$ (taken from a distribution $y$), and that individuals are price-takers and trade so as to maximize their subjectively expected utility (no incentive to trade contracts).
Here, in deciding how many contracts to buy, individual buyers solve the following problem given that the price of the contract is $\pi$:
$$ \max_{x_j}EU_j=q_j\log(y_j+x_j(1-\pi))+(1-q_j)\log(y_j-x_j\pi) $$
which yields:
$$ \begin{equation}x_j^*=y_j\frac{q_j-\pi}{\pi(1-\pi)} \end{equation} $$
Observe that individual demand is:
The prediction market is in equilibrium when supply equals demand, i.e. when:
$$ \int_{-\infty}^\pi y\frac{q-\pi}{\pi(1-\pi)}f(q)dq=\int_\pi^{+\infty} y\frac{\pi-q}{\pi(1-\pi)}f(q)dq $$
By assumption, beliefs and wealth are independent, and therefore the previous equality yields:
$$ \pi=\int_{-\infty}^{+\infty}qf(q)dq=\overline{f} $$
In other words, the market price is the mean belief at equilibrium.
<aside> 💡 Interpretation. Individual subjective beliefs represent private but noisy signals of the likelihood of an event happening. If the noise term is normally distributed, then the mean of these private signals is an efficient estimate of the true likelihood of the event occurring, and hence these models yield conditions under which the prediction market price is a sufficient statistic for this private information.
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