# On the Statistical Differences between Binary Forecasts and Real World Payoffs

## TL;DR

This paper explores the fundamental differences between binary forecasts and real-world payoffs, revealing how conflating them leads to misconceptions in decision science, especially under complex distributions and tail risks.

## Contribution

It introduces a metric for pseudo-overestimation, clarifies the limitations of binary forecasting as performance indicators, and discusses implications for machine learning and tail risk analysis.

## Key findings

- Many psychological biases are due to mischaracterizing binary predictions as payoffs.
- Good binary forecasts do not necessarily imply good real-world performance.
- Differences between forecasts and payoffs are larger in fat-tailed distributions.

## Abstract

What do binary (or probabilistic) forecasting abilities have to do with overall performance? We map the difference between (univariate) binary predictions, bets and "beliefs" (expressed as a specific "event" will happen/will not happen) and real-world continuous payoffs (numerical benefits or harm from an event) and show the effect of their conflation and mischaracterization in the decision-science literature. We also examine the differences under thin and fat tails. The effects are: A- Spuriousness of many psychological results particularly those documenting that humans overestimate tail probabilities and rare events, or that they overreact to fears of market crashes, ecological calamities, etc. Many perceived "biases" are just mischaracterizations by psychologists. There is also a misuse of Hayekian arguments in promoting prediction markets. We quantify such conflations with a metric for "pseudo-overestimation". B- Being a "good forecaster" in binary space doesn't lead to having a good actual performance}, and vice versa, especially under nonlinearities. A binary forecasting record is likely to be a reverse indicator under some classes of distributions. Deeper uncertainty or more complicated and realistic probability distribution worsen the conflation . C- Machine Learning: Some nonlinear payoff functions, while not lending themselves to verbalistic expressions and "forecasts", are well captured by ML or expressed in option contracts. D- Fattailedness: The difference is exacerbated in the power law classes of probability distributions.