Surreal Decisions
This work addresses foundational issues in decision theory for infinite scenarios, offering a novel mathematical framework that could impact philosophical and economic analyses of transfinite decisions.
The paper tackles the problem of extending expected utility theory to infinite values, which has led to paradoxes, by proposing a surreal decision theory based on Conway's surreal numbers, proving a representation theorem and showing it respects dominance reasoning, and applies it to Pascal's Wager to analyze objections like Mixed Strategies and Many Gods, concluding that while the wager is coherent, it does not compel religious life regardless of credences.
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.