Analyzing Prediction Markets and Their Limitations

Prediction markets have proven to be valuable tools for estimating probabilities of claims that can be resolved at a specific point in time. These markets excel in predicting uncertainties related to real-world events and even values of primitive recursive functions. However, their direct application to questions without a fixed resolution criterion is challenging, leading to predictions about whether a sentence will be proven rather than its truth.

When it comes to questions that lack a fixed resolution criterion, a different approach is necessary. Such questions often involve countable unions or intersections of more basic events or are represented as First-Order-Logic sentences on the Arithmetical Hierarchy. In more complex cases, they may even transcend First-Order Logic and fall into the realm of hyperarithmetical sentences.

In this paper, the authors propose an alternative approach to betting on events without a fixed resolution criterion using options. These options can be viewed as bets on the outcome of a “verification-falsification game,” offering a new framework for addressing logical uncertainty. This work stands in contrast to the existing framework of Garrabrant induction and aligns with the constructivist stance in the philosophy of mathematics.

By introducing the concept of options in prediction markets, this research has far-reaching implications for both philosophy and mathematical logic. It provides a fresh perspective on addressing uncertainties in a broader range of questions and challenges the traditional methods by offering an alternative framework that accommodates events lacking fixed resolution criteria. These findings encourage further exploration and could lead to significant advancements in our understanding and utilization of prediction markets.

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