Budget Constraints in Prediction Markets

TL;DR

This paper characterizes how individual budget constraints influence trading impact in prediction markets with cost-function-based market makers, introducing the concept of budget additivity and analyzing specific scoring rules.

Contribution

It provides a detailed characterization of optimal trades under budget constraints and introduces the property of budget additivity in prediction markets.

Findings

Budget additivity holds under certain conditions.

Logarithmic market scoring rule is budget additive with affine independence.

Quadratic market scoring rule is not budget additive.

Abstract

We give a detailed characterization of optimal trades under budget constraints in a prediction market with a cost-function-based automated market maker. We study how the budget constraints of individual traders affect their ability to impact the market price. As a concrete application of our characterization, we give sufficient conditions for a property we call budget additivity: two traders with budgets B and B' and the same beliefs would have a combined impact equal to a single trader with budget B+B'. That way, even if a single trader cannot move the market much, a crowd of like-minded traders can have the same desired effect. When the set of payoff vectors associated with outcomes, with coordinates corresponding to securities, is affinely independent, we obtain that a generalization of the heavily-used logarithmic market scoring rule is budget additive, but the quadratic market scoring rule is not. Our results may be used both descriptively, to understand if a particular market maker is affected by budget constraints or not, and prescriptively, as a recipe to construct markets.