Quadratic Payments with constrained probabilities

TL;DR

This paper explores quadratic payment models in voting scenarios, explicitly modeling probability functions and analyzing how realistic probability shapes influence the quadratic nature and introduce new trade-offs.

Contribution

It generalizes quadratic payments to incorporate realistic probability functions, revealing trade-offs absent in idealized models.

Findings

Quadratic payments can be extended to non-constant probability functions.

Realistic probability shapes affect the quadratic nature of payments.

Trade-offs emerge when moving from ideal to realistic probability models.

Abstract

Dealing with quadratic payments, marginal probability is usually considered ideally constant, maybe for the sake of initial simplicity. Considering the voting scenario depicted in "Quadratic Payments: A Primer" by Vitalik Buterin, firstly its math foundations are made explicit. Developing a simple referendum model, more realistic outcome probability and marginal probability qualitative shapes are introduced. Enforcing seemingly reasonable assumptions, quadratic payments are then generalized to take into account these new functions shapes, and the way they are still quadratic is discussed. Closing remarks underline the emerging of trade-off constraints not existing in ideal case.