Unbounded Utility

This map records logical connections between principles analysed in my Decision theory unbound (2024) and Symmetries of value (2026), as well as an unpublished work-in-progress called Unbounded Utility and Background Risk.

Setup

A gamble is a measurable outcome-valued random variable X on a fixed atomless standard probability space, which can be represented by [0,1] with Lebesgue measure or by countably many independent fair coin tosses. All such variables are available.

The primitive preference relation \succeq on gambles is always assumed reflexive and transitive. Outcomes are ordered by preference over certain gambles. Indifferent outcomes are identified. The optional Outcome Richness axiom says there are outcomes of every real utility value, and the optional Archimedean axiom says that there are no infinitesimal or infinite difference in utility levels.

The theory DU is the weakest background considered in my papers. In addition to Outcome Richness and Archimedean axioms, it has: (a) The Sure-Thing principle/Independence. (b) Stochastic Equivalence (equally distributed gambles are equally good). (c) Statewise Dominance.

Literature

Source literature

Colyvan, M. (2008). Relative expectation theory. Journal of Philosophy, 105(1), 37–44. Paper
Nover, H., & Hájek, A. (2004). Vexing expectations. Mind, 113(450), 237–249. Paper
Russell, J. S., & Isaacs, Y. (2021). Infinite prospects. Philosophy and Phenomenological Research, 103(1), 178–198. Paper
Seidenfeld, T., Schervish, M., & Kadane, J. (2009). Preference for equivalent random variables: A price for unbounded utilities. Journal of Mathematical Economics, 45, 329–340. Paper
Zachary Goodsell (2024). Decision theory unbound. Noûs, 58, 669–695. First published online in 2023. Paper
Zachary Goodsell (2026). Symmetries of value. Noûs, 60, 16–37. Paper
Zachary Goodsell (5 June 2026). Unbounded Utility and Background Risk. Unpublished working manuscript. Full text is not included. The record-specific references retain warnings about gaps and withdrawn arguments.
Hayden Wilkinson (2025). Flummoxing expectations. Noûs, 59(3), 700–728. First published online in 2024. Paper
Fred Espen Benth, Giulia Di Nunno and Dennis Schroers (2022). Copula measures and Sklar’s theorem in arbitrary dimensions. Scandinavian Journal of Statistics. Paper
AI-generated report, 30 September 2026, 4 pages; communicated by Branden Fitelson. The report attributes production to Claude agents without specifying model versions. Fitelson is the communicator, not the credited producer of the proofs. The source PDF is retained privately; only checked results are transcribed here. The report claims separate Codex refereeing, but those referee reports were not supplied. The public write-ups contain the arguments expanded and informally checked by GPT-6 (Codex), 2 October 2026; the stronger steered-conclosure claims in §4(a) are not imported as proved.

Other relevant literature

Arntzenius, F., Elga, A., & Hawthorne, J. (2004). Bayesianism, infinite decisions, and binding. Mind, 113(450), 251–283.
Bartha, P. (2016). Making do without expectations. Mind, 125(499), 799–827.
Dietrich, F., & List, C. (2005). The two-envelope paradox: An axiomatic approach. Mind, 114(454), 239–248.
Easwaran, K. (2008). Strong and weak expectations. Mind, 117(467), 633–641.
Easwaran, K. (2014a). Decision theory without representation theorems. Philosophers’ Imprint, 14.
Easwaran, K. (2014b). Principal values and weak expectations. Mind, 123(490), 517–531.
Fine, T. (2008). Evaluating the Pasadena, Altadena, and St Petersburg gambles. Mind, 117(467), 613–632.
Hájek, A. (2014). Unexpected expectations. Mind, 123(490), 533–567.
Lauwers, L. (2016). Why decision theory remains constructively incomplete. Mind, 125(500), 1033–1043.
Lauwers, L., & Vallentyne, P. (2016). Decision theory without finite standard expected value. Economics and Philosophy, 32(3), 383–407.
Lauwers, L., & Vallentyne, P. (2017). A tree can make a difference. Journal of Philosophy, 114(1), 33–42.
McGee, V. (1999). An airtight Dutch book. Analysis, 59(4), 257–265.
Meacham, C. J. G. (2019). Difference minimizing theory. Ergo, 6. Paper
Russell, J. S. (2020). Non-Archimedean preferences over countable lotteries. Journal of Mathematical Economics, 88, 180–186.
Russell, J. S. (forthcoming). Fixing stochastic dominance. British Journal for the Philosophy of Science.
Smith, N. J. J. (2014). Is evaluative compositionality a requirement of rationality? Mind, 123(490), 457–502.