Prospects for a Theory of Non-archimedean Expected Utility: Impossibilities and Possibilities

In this talk I examine the prospects for a theory of probability aspiring to support a decisiontheoretic interpretation by which principles of probability derive their normative status in virtue of their relationship with principles of rational decision making. More specifically, I investigate the possibility of a theory of expected utility abiding by distinguished dominance principles which have played a pivotal role in the development of subjective probability and critical discussion thereof. I focus on dominance principles that a theory of real-valued expected utility cannot support, motivating proponents of these principles to develop theories admitting a non-Archimedean range to meet the demands which these principles exact. Thus, specifically, in this talk I examine the prospects for a theory of non-Archimedean expected value and more generally, expected utility. I am certainly not the first to entertain the non-Archimedean possibility. Non-Archimedean representations of uncertainty have captivated the interest of those who wish for rational credal probabilities and expectations either to respect laws supplementing the familiar set of putatively binding laws or to accommodate credal states complementing the familiar set of putatively rational credal states. In other words, at least two distinct considerations have either motivated authors to appeal to a nonArchimedean representation or stirred their excitement about the possibilities such a representation creates. One motivation seeks conformity to a longer list of laws governing credal expressions of uncertainty within a framework facilitating compliance with these laws. A second motivation seeks conformity to the familiar laws possibly along with others within a framework capable of registering putatively rational credal expressions of uncertainty. For example, some authors, like Shimony [1955], Kemeny [1955], Stalnaker [1970], Carnap [1971], and Skyrms [1980, 1995], have motivated a need to admit a non-Archimedean range to comply with a principle flowing from a decision-theoretic consideration: If credal probabilities and expectations are to track an agent’s preferences over gambles, then a gamble the agent evaluates across states as sometimes worse and never better than another gamble ought to receive strictly lower expected value than the other gamble, a quantitative manifestation of an injunction called the principle of weak dominance. Consequently, where random quantities are understood as gambles with which the agent may be faced in a decision problem, these authors have in particular demanded that each event an agent judges possible receive positive probability, a constraint called regularity. A well-known problem, however, is that an account of probability confined to real numbers cannot meet such a requirement without imposing substantive constraints on admissible credal judgments. To make room for regular probabilities, these authors have suggested that credal probabilities and expectations be permitted to take values in some non-Archimedean extension of the system of real numbers.