Nash Equilibrium: Can Belief-Weighted Social Preferences Lead to Fairer Outcomes?

The core problem that belief-weighted Nash social welfare functions address is the aggregation of individual preferences in situations involving uncertainty, where individuals hold differing beliefs about the likelihood of various outcomes. Traditional methods like the Pareto criterion often fail to provide fair or robust solutions in such scenarios. These functions aim to provide a structured way to combine these diverse preferences to arrive at a collective decision. They go beyond simple majority voting and account for the relative importance of each individual's beliefs, offering a more nuanced and potentially fairer framework for social decision-making.

Belief-weighted Nash social welfare functions differ significantly from the Pareto criterion, especially in the context of uncertain environments. The Pareto criterion states that if everyone prefers one option over another, society should also prefer that option. This approach can lead to uncompromising outcomes. In contrast, belief-weighted Nash social welfare functions don't necessarily adhere to the Pareto efficiency. They incorporate a more sophisticated process that considers each individual's subjective expected utility and the relative importance of their beliefs. The functions use a 'belief profile' that captures the range of viewpoints. They assign individual weights to each belief profile, determining the contribution of each individual's preferences to the final social preference. This approach is designed to provide a more balanced and nuanced perspective.

The calculation of social preferences using belief-weighted Nash social welfare functions involves a few key steps. First, each individual's preferences are represented using a 0-normalized subjective expected utility function, assigning a numerical value to each possible outcome. Second, all possible combinations of individual beliefs are considered, forming 'belief profiles.' Third, a vector of individual weights is assigned to each belief profile, influencing how much each individual's preferences matter. Finally, the social preference is determined by calculating the weighted product of individual utilities for each option. The option with the highest weighted product is deemed the most socially desirable. This approach considers both individual valuations and the relative importance of each individual's beliefs, providing a balanced aggregation.

Using Savage's theory of choice under uncertainty is critical within belief-weighted Nash aggregation. Savage's theory provides the framework for understanding how individuals make decisions when outcomes are uncertain. It models uncertain prospects as 'acts', which are mappings from states of nature to outcomes. An individual's preferences are summarized by their subjective assessment of the likelihood of events and the utility they assign to outcomes. Belief-weighted Nash aggregation builds upon this framework by incorporating these individual subjective expected utilities into a social welfare function. This allows the function to account for differing beliefs and preferences under uncertainty, which is essential for creating fair outcomes. By starting with Savage preferences, the aggregation process is grounded in a coherent and established model of individual decision-making.

Belief-weighted Nash social welfare functions are considered more robust because they are designed to handle the complexities of social decision-making in the face of uncertainty and diverse beliefs. They offer a more nuanced aggregation process than traditional methods like the Pareto criterion, which can lead to uncompromising results. These functions consider both individual utilities and the relative importance of each individual’s beliefs. This robustness is reflected in their ability to incorporate a wide range of viewpoints and to offer a more balanced approach to social welfare, even if they might not always align with traditional notions of Pareto efficiency or produce Savage-rational social preferences. This makes them valuable tools for economists and policymakers who need to navigate collective choice in uncertain environments.