COMPARATIVE EXPECTATIONS ARTHUR PAUL PEDERSEN ABSTRACT. I introduce a mathematical account of expectation based on a qualitative crite- rion of coherence for qualitative comparisons between gambles (or random quantities). The qualitative comparisons may be interpreted as an agent’s comparative preference judgments over options or more directly as an agent’s comparative expectation judgments over random quantities. The criterion of coherence is reminiscent of de Finetti’s quantitative criterion of coherence for betting, yet it does not impose an Archimedean condition on an agent’s comparative judgments, it does not require the binary relation reflecting an agent’s compar- ative judgments to be reflexive, complete or even transitive, and it applies to an absolutely arbitrary collection of gambles, free of structural conditions (e.g., boundedness, measura- bility, closure under algebraic or topological operations, etc.). Moreover, unlike de Finetti’s criterion of coherence, the qualitative criterion respects the principle of weak dominance,a standard of rational decision making that obliges an agent to reject a gamble that is possi- bly worse and certainly no better than another gamble available for choice. Despite these weak assumptions, I establish a qualitative analogue of de Finetti’s Fundamental Theorem of Prevision, from which it follows that any coherent system of comparative expectations can be extended to a weakly ordered coherent system of comparative expectations over any collection of gambles containing the initial set of gambles of interest. The extended weakly ordered coherent system of comparative expectations satisfies familiar additivity and scale invariance postulates (i.e., independence) when the extended collection forms a linear space. In the course of these developments, I recast de Finetti’s quantitative account of coherent prevision in the qualitative framework adopted in this article. I show that comparative pre- visions satisfy qualitative analogues of de Finetti’s famous bookmaking theorem and his Fundamental Theorem of Prevision. The results of this article complement those of another article [Pedersen, 2013]. I explain how those results entail that any coherent weakly ordered system of comparative expecta- tions over a unital linear space can be represented by an expectation function taking values in a (possibly non-Archimedean) totally ordered field extension of the system of real num- bers. The ordered field extension consists of formal power series in a single infinitesimal, a natural and economical representation that provides a relief map tracing numerical non- Archimedean features to qualitative non-Archimedean features. I NTRODUCTION An influential view in the foundations of probability portrays an intimate connection be- tween, on the one hand, rational credences and expectations and, on the other hand, rational decisions made under conditions of uncertainty. On this view, criteria of rational decision I would like to extend my gratitude to Kenny Easwaran, Frederik Herzberg, Konstantinos Katsikopoulos, Marcus Pivato, Alex Pruss, Dana Scott, Stanislav Speranski, Peter Wakker, and Greg Wheeler for valuable comments and discussion. I would also like to thank the anonymous reviewers for their helpful remarks. This paper is dedicated to the memory of Horacio Arl ´ o-Costa. 1