# Election Predictions as Martingales: An Arbitrage Approach

## TL;DR

This paper models election outcome predictions as martingales and introduces an arbitrage pricing framework that accounts for continuous updates and uncertainty, providing a novel approach to election forecasting.

## Contribution

It proposes a new arbitrage-based valuation method for binary election predictions using martingale processes, linking volatility and uncertainty to estimate accuracy.

## Key findings

- Arbitrage relationships between binary value, estimator, and volatility are established.
- High uncertainty leads to binary estimates approaching 50%.
- Forecasting groups often violated arbitrage relationships in 2016 elections.

## Abstract

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.