Axiomatic treatments of utility from von Neumann & Morgenstern onward

by Warren D. Smith March 2007, with some stimulation by Jobst Heitzig. See also some utility-history.

Notation: If a and b are events, let "a⊇b" mean "in the view of any fixed individual: a is preferable to b, or that individual does not care which of a or b happen." We write a∼b ("the individual is indifferent about whether a or b happens") to mean "a⊇b and b⊇a." We write a⊃b to mean "a⊇b and not a∼b." Let "p?A:B" mean "a lottery causing A to happen with probability p and B with probability 1-p." Then here are some interesting possible axioms which ⊇ could satisfy.

Totality Axiom:
For all events a,b either a⊇b, or b⊇a, or both, i.e. a∼b.
Continuity Axiom:
If A⊃B and B⊃C then B ∼ p?A:C for at least one p with 0<p<1.
Dominance (via convexity) Axiom:
For all p with 0<p<1:
Transitivity Axiom:
If a∼b and b∼c then a∼c.
Herstein-Milnor Lottery Axiom:
If A∼B then ½?A:C ∼½?B:C.

Here are some theorems.

Existence Theorem for monotonic utility function (P.C.Fishburn 1983): Totality, Continuity, Dominance, and Transitivity imply that there exists a real-valued function u() so A⊇B ⇔ u(A)≥u(B), where u( p?A:B ) is a continuous increasing function of p when 0≤p≤1 if A⊃B (and is constant in p if A∼B).

Existence Theorem for linear utilities (J.von Neumann & O.Morgenstern 1947): Totality, Continuity, Dominance, and Herstein-Milnor imply that there exists a "utility function" u which assigns to each lottery L a real number u(L) with the following three properties:

Strict monotonicity (MonT):
u(a) ≥ u(b) if and only if a⊇b.
Lottery utilities are expected utilities (Lin):
u(p?a:b) = p·u(a) + (1-p)·u(b).
Uniqueness up to transformations of scale (AffUniq):
For any two functions u, v that both satisfy the preceding two properties, there are real numbers r,s such that v = r + s·u.

These are essentially the properties economists usually take for granted when speaking about individual "utility."

This second theorem is a standard starting point. What do we think of it? Some (not I, but apparently some) would dispute the Continuity axiom. E.g. to paraphrase a dispute on the subject:

Anti-Continuist: Any probability p>0 that my child dies (no matter how small p is), is worse than me losing 1 penny of money. Therefore the Continuity axiom is false.
Continuist: I know you are lying, because I know you bent down to pick up a penny and therefore stopped watching your child for 30 milliseconds. Also, you are not presently tossing pennies over your child's head to protect her against incoming meteorites.

If (essentially) we discard the continuity axiom, then it still is possible to prove analogues of the Theorem but which involve utility values in Nelson's "non-standard" reals (the field extension that arises by including an "infinitesimal" quantity ε).

Some (not I, but perhaps some) also might dispute the Totality axiom.

The axiom that people seem to be the most willing to discard is the Lottery Axiom since, e.g, they regard uncertainty as somehow bad. Fishburn in 1983 brilliantly considered the consequences of that and proved the following therem:

Theorem of nonlinear utilities (Fishburn 1983): Totality, Continuity, Dominance, and the "Symmetry Axiom" below ⇔ there exists a skew-symmetric bilinear functional SSBF on 2-tuples of lotteries such that A⊇B ⇔ SSBF(A,B)≥0 and that SSBF() is unique up to "similarity transformation." [Skew-symmetry means SSBF(A,B) = -SSBF(B,A) and bilinearity means SSBF(p?A:B, C) = p·SSBF(A,C) + (1-p)·SSBF(B,C) and SSBF(C, p?A:B) = p·SSBF(C,A) + (1-p)·SSBF(C,B).]

Fishburn's Symmetry Axiom:
If A⊇B, B⊇C, A⊇C, and B ∼ ½?A:C, then p?A:C ∼ ½?A:B ⇔ p?C:A ∼ ½?C:B.

You are warned that these skew-symmetric bilinear nonlinear utility functionals can be rather strange and hard to use. (They allow intransitive preference "cycles" and generally disobey the Herstein-Milnor axiom. Quick sanity check: have you, personally, ever felt that A was better than B was better than C which in turn was better than A for three options A,B,C?) Almost nobody besides Fishburn ever has used it. Also, anybody disputing the Herstein-Milnor Lottery axiom would quite plausibly dispute Fishburn's symmetry axiom too. Finally, it is important to keep in mind that Fishburn's utilities are the same thing as ordinary utilities if we are talking about deterministic events.


  1. Peter C. Fishburn: Nontransitive measurable utility, J. Math'l Psychology 26 (1982) 31-67.
  2. Peter C. Fishburn: Transitive measurable utility, J. Economic Theory 31 (1983) 293-317.
  3. I.N.Herstein & J.Milnor: An axiomatic approach to measurable utility, Econometrica 21,2 (1953) 291-297.
  4. Niels Erik Jensen: An Introduction to Bernoullian Utility Theory: I. Utility Functions, Swedish Journal of Economics 69,3 (1967) 163-183.
  5. J. von Neumann & O.Morgenstern: Theory of Games and Economic Behavior (Princeton Univ. Press); Axiomatic treatement of utility in pages 13-31 and appendix 617-632 or so (page numbers may depend on your edition, and only present in editions after 1947).

Harsanyi's argument "Social Utility" is the average of individual utilities

Let uk(e) denote the utility (according to individual k) of an event e. We want to investigate how to aggregate the individual utilities into a "social utility" saying "how good e is for all of society." For what reason should we claim that social utility is just the average of individual utilities?

J.C.Harsanyi, in a 2-page article involving no mathematics whatever [J.Political Economy 61,5 (1953) 434-435], came up with the following nice idea: "Optimizing social welfare" means "picking the state of the world all individuals would prefer if they were in a state of uncertainty about their identity." I.e. if you are equally likely to be anybody, then your expected utility is the summed utility in the world divided by the number of people in it – i.e. average utility. Then by the linear-lottery property (Lin) of von Neumann utility, it follows that social utility is averaging.

Actually, Harsanyi nowhere connected to von Neumann's axiomatic development, but he should have because it connects beautifully. Indeed, Harsanyi's paper can justifiably be criticized as the work of a mathematical idiot, because he also neglected to note the key point that von Neumann utility is only unique up to, and only defined up to, scaling. The whole idea of averaging individual von Neumann utilities, then, can only even be talked about at all if all individuals somehow have uniquified their utility functions by adopting a common scale.

One way to do that would be to assume the existence of two magical special events, which everybody can be assumed to view as having the same utility-difference, e.g.

  1. The world is destroyed by a sudden giant explosion,
  2. Everybody is magically transplanted into identical bodies in identical pleasant situations on a new planet that magically appears.

And for a yet another approach to proving utility is additive starting from a set of axioms, see [Uzi Segal: Let's agree all dictators are equally bad, Journal of the Political Economy 108,3 (June 2000) 569-589]. Segal basically uses the approach we just outlined, except for a sleight of hand where he uses his "dictator indifference" axiom to play the role of our "two magical events" to set up a common scale. Anyhow, I for one do not agree that "all dictators are equally bad"; it appears to me that some have been good.

Social utility = averaging via new approaches not depending on lotteries

Some would dispute the axioms underlying all that. For example, some might dispute the Lottery axiom:

Objector: A lottery awarding a million dollars to A with probability 1/2 and to B with probability 1/2, is better than just giving A the money or just giving B the money because it is more "fair" and I also value fairness.
Or in the other direction, many investors argue the lottery is worse than the deterministic award because we are "risk averse" and regard uncertainty itself as a bad thing.

If Herstein-Milnor Lottery axiom is discarded, linearity (Lin) fails whereupon many social utility aggregation methods become permitted besides just averaging.

Fortunately, there are also other axiom sets (by Warren D. Smith) which also yield the "averaging=social aggregation" (or one can say "summing" in place of "averaging" since the two yield equivalent preferences) correspondence but via reasoning that never involves probabilities and lotteries: See puzzles 36 and 37 here.

But these still do not really settle the matter because there also is room for people to dispute some of those axioms, as is discussed in puzzle 38. The question of which axioms about "social goodness" to accept, is probably not something that can be settled by mathematics. Still, I think every reasonable person would agree that all these axioms are usually at least approximately true.

Social utility = averaging – yet another approach via "unexploitability" and "Darwinian evolution"

See this by Warren D. Smith (with help from Steve Omohundro, Marcus Pivato, and building on works of others including Harsanyi).

About the history of utility concepts

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