We consider the estimation of binary election outcomes as martingales and
propose an arbitrage pricing when one continuously updates estimates. We argue
that the estimator needs to be priced as a binary option as the arbitrage
valuation minimizes the conventionally used Brier score for tracking the
accuracy of probability assessors.

We create a dual martingale process $Y$, in $[L,H]$ from the standard
arithmetic Brownian motion, $X$ in $(-\infty, \infty)$ and price elections
accordingly. The dual process $Y$ can represent the numerical votes needed for
success.

We show the relationship between the volatility of the estimator in relation
to that of the underlying variable. When there is a high uncertainty about the
final outcome, 1) the arbitrage value of the binary gets closer to 50\%, 2) the
estimate should not undergo large changes even if polls or other bases show
significant variations.

There are arbitrage relationships between 1) the binary value, 2) the
estimation of $Y$, 3) the volatility of the estimation of $Y$ over the
remaining time to expiration. We note that these arbitrage relationships were
often violated by the various forecasting groups in the U.S. presidential
elections of 2016, as well as the notion that all intermediate assessments of
the success of a candidate need to be considered, not just the final one.

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## How to Forecast an Election

Nassim Nicholas Taleb

posted on 21 March 2017

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