Surreal Decisions

TL;DR

This paper introduces a surreal number-based framework for transfinite decision theory, resolving paradoxes in infinite utility models and applying it to analyze Pascal's Wager with new insights.

Contribution

It develops a surreal number-based decision theory, proves a representation theorem, and applies it to Pascal's Wager, demonstrating its advantages over previous approaches.

Findings

Surreal decision theory respects dominance reasoning with infinite values.

Pure Pascalian strategy outperforms mixed strategies in the surreal framework.

The theory shows that rational decision depends on one's credence, challenging the persuasive power of Pascal's Wager.

Abstract

Although expected utility theory has proven a fruitful and elegant theory in the finite realm, attempts to generalize it to infinite values have resulted in many paradoxes. In this paper, we argue that the use of John Conway's surreal numbers shall provide a firm mathematical foundation for transfinite decision theory. To that end, we prove a surreal representation theorem and show that our surreal decision theory respects dominance reasoning even in the case of infinite values. We then bring our theory to bear on one of the more venerable decision problems in the literature: Pascal's Wager. Analyzing the wager showcases our theory's virtues and advantages. To that end, we analyze two objections against the wager: Mixed Strategies and Many Gods. After formulating the two objections in the framework of surreal utilities and probabilities, our theory correctly predicts that (1) the pure Pascalian strategy beats all mixed strategies, and (2) what one should do in a Pascalian decision problem depends on what one's credence function is like. Our analysis therefore suggests that although Pascal's Wager is mathematically coherent, it does not deliver what it purports to, a rationally compelling argument that people should lead a religious life regardless of how confident they are in theism and its alternatives.