Unbounded Utility and Axiomatic Foundations

Abstract

In utility theory, certain infinite gambles pose problems, of which the St. Petersberg paradox is the most well-known. Various methods have been used to eliminate the paradoxes, including the supposition that utility is bounded, or that probabilities must decline faster than utility so as to always result in finite expectation values, or simply excluding the divergent gambles by definition.

We present a unified resolution of the matter that allows for unbounded utility and accommodates convergent infinite gambles.

We also argue that merely excluding the divergent gambles, by whatever method, does not entirely avoid the associated paradoxes, which in practical terms already show up for finite approximations to them.

Full text

Unbounded Utility and Axiomatic Foundations
Richard Kennaway
r.kennaway@uea.ac.uk
School of Computing Sciences, University of East Anglia, Norwich, U.K.
March 18, 2021
Abstract
In utility theory, certain infinite gambles pose problems, of which the St. Petersberg paradox
is the most well-known. Various methods have been used to eliminate the paradoxes, including the
supposition that utility is bounded, or that probabilities must decline faster than utility so as to always
result in finite expectation values, or simply excluding the divergent gambles by definition.
We present a unified resolution of the matter that allows for unbounded utility and accommodates
convergent infinite gambles.
We also argue that merely excluding the divergent gambles, by whatever method, does not en-
tirely avoid the associated paradoxes, which in practical terms already show up for finite approxima-
tions to them.
1 Introduction
Many axiomatisations have been given of preference among actions, which imply that the preference is
equivalent to numerical comparison of a real-valued function of actions, called a utility function. There
are too many to survey here, and we base our exposition on Savage’s system [10,11].
A consequence of Savage’s axioms, and several other systems, is that utility must be bounded. Savage
raised this as a possibility in the first edition of his book, but Fishburn proved it [4,3], and the result was
included in the second edition. The proof supposes utility to be unbounded, constructs two versions of
the St. Petersburg gamble, and derives from the axioms a contradiction.
This has often been felt unsatisfactory, and the response has been to ask instead, “what is wrong with
Savage’s axioms?” [16]. Attention has typically focused on modifying or weakening particular axioms,
or simply excluding all infinite gambles. We take a more principled approach here, motivated by Jaynes’s
dictum [8] that infinities must be understood as limits of finite processes. If the limit is problematic, the
limiting process must be re-examined.
2 A modified version of Savage’s axioms
In Savage’s setting, there is a state space S, a set Xof consequences, and a set Aof acts, functions from
Sto X. We refer to [11] for Savage’s axioms, which he names P1–P7 (P for postulate).
We first make some minor modifications of these axioms, for reasons independent of the present subject.
Savage assumed finitely additive probability, but we prefer to work in the countably additive framework.
Savage himself wrote that the development could likely be carried out in that context, and cites Ville-
gas [15] as having established the necessary groundwork. So while Savage considers Ato be the set
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of all functions from Sto X, we take Sand Xto be measurable spaces, equipped with σ-fields Sand
X, and define Aas the set of measurable functions from Sto X. To connect measurability with the
preference relation we need two further axioms. In axiom PC an “event” is a measurable subset of S.
PM (measure): Xis the σ-algebra generated by intervals of the preference ordering on X.
PC (continuity): Given events A1⊆A2⊆. . . with union A,Ais the least upper bound of
A1, A2, . . . for the ≤Prelation (“not more probable than”) defined by Savage.
We write Pfor the axioms P1–P7 together with PM and PC. Pimplies the existence of a unique proba-
bility measure Pon Sand a utility function U:X→R, from which a utility function on acts is defined
by U(f) = RS(U◦f)dP , such that preference is equivalent to comparison of expected utility.
The one change that we shall make to Savage’s axioms in order to accommodate unbounded utility is to
impose on all of them a limitation to finite acts, i.e. acts with only finitely many different consequences.
In particular, the preference relation is (at first) only taken to be defined on finite acts. We call the revised
set PF.
It is routine to go through Savage’s argument and confirm that the axioms PF yield the same construction
of probability and utility on finite acts. Furthermore, every model of Pis a model of PF.
We can then consider RS(U◦f)dP for arbitrary acts f. Some of these expectation values may be finite,
some may be assigned symbolic values of ±∞, and for some, the integral may not converge even to
an infinite value. When the integral is finite we call fwell-behaved. When it is finite or infinite, fis
well-defined. For other acts, fis ill-defined.
Let PWB be the axioms modified to range over well-behaved acts. The proof of the following theorem
is again routine.
THE OR EM 2.1. PF implies PWB.
This justifies identifying the expectation value of well-defined acts with their utility and defining pref-
erence on those acts from the utility function. Alternatively, we could take PWB as an additional set of
axioms and demonstrate that they imply that the preference is modelled by the extended utility function
just defined. That is just a difference of framing. What is important is to proceed in these two stages: first
construct utility for finite acts, then extend preference and utility as far as possible by taking the obvious
limits.
The ill-defined acts are excluded, problems beginning with Savage’s axiom P1, that the preference re-
lation is total. An ordering of all measurable functions from Sto X, compatible with the utility where
defined, is not to be hoped for. Some have considered utility functions ranging over lexicographically
ordered vector spaces or non-standard reals [7,12,13], or using a different concept of expectation called
“weak expectation” [2,5,6]. However, these make only limited progress towards a total preference re-
lation. There are severe limitations on what can be achieved while maintaining some basic desiderata of
such an ordering [1].
The present approach may seem simple, almost trivial, but it extends utility to exactly all of the acts
for which it is intuitively meaningful, without having to give up real-valuedness or exclude any acts
that intuitively seem to have finite values. Other approaches either fail to extend it as far (for example,
by allowing only finite acts), or extend it a little way into the space of ill-defined acts at the cost of
surrendering some of the motivating ideas. We consider the fact that the ill-defined acts cannot be
approached by a limiting process to be exactly the reason that they should be excluded.
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3 Finite approximations to paradoxical games
As a general rule, whenever a theoretical problem appears at a limit, a corresponding practical problem
will already arise at a finite stage of approaching that limit. One such problem arises for a truncated
version of the St. Petersburg game [14]: with a fair entry fee, the player is exposed to a large downside
risk before first seeing a profit.
We have simulated several St. Petersburg-like games, in which a coin is tossed until it comes up heads,
and the payout depends on the number of tosses. (The code for all of these simulations is available on
Github at https://github.com/rkennaway/StPetersburg.) Such games can be truncated
after the nth toss: if it has not terminated by then, the game ends and the payout is zero. Truncated games
are finite. All versions of utility theory assign them finite utilities.
We choose a truncation length of 16 coin-flips. With payoffs typically doubling or less with the number
of flips, penny stakes at the start amplify to a maximum payoff in a single game of less than $1000. These
games are practical to play.
For each game we assume an entry fee of 97% of the expected value. This gives the player a 3% expected
profit, a figure we have chosen as similar to the house edge at roulette, which is 2.7% (1/37). The game
of roulette turns a steady profit for the house. These games, however, do not turn a steady profit for the
player, except over impractically large numbers of repetitions.
For St. Petersburg, the payout for heads on the nth coin-flip is 2n. The expectation value of the truncated
game is 16, giving an entry fee of 15.52. In a run of 1000 games, the player has about a 50% chance of at
some point owing the banker at least 4000 units. The expected profit for such a run of 1000 is 480. The
average profit for 10000 runs of 1000 games is close to this, but its standard deviation is over 11000, or
23 times the mean.
The Pasadena game [9] pays out (−1)n+12n/n when it ends at the nth coin-flip, which is a little slower
than exponentially growing. The series of expected values is 1−1
2+1
3−1
4+. . . This is only conditionally
convergent (to log(2) = 0.693). Rearranging the order of the terms can make it converge to anything
or nothing. In an indefinitely long sequence of truncated Pasadena games, the player will experience
swings of fortune almost proportional to the number of games played, up to the largest payout available.
We have simulated several other paradoxical games, with similar results.
Convergent games do not show these alarming phenomena. In the Convergent St. Petersburg game, the
payout for heads on the nth coin-flip is n. Truncated to ksteps the expectation value is 2−(k+ 2)/2n.
With the same parameters as before, in a run of 1000 games the player ends with a profit more than 90%
of the time. A plot of the player’s winnings over time is close to a Wiener process with a drift velocity
equal to the expected profit per game.
All of these games, viewed on a long enough time scale, behave like Wiener processes with drift. But
for these non-well-behaved games, the required time scale grows exponentially with truncation length
(roughly as the square of the game’s maximum payout). The game can stay unprofitable longer than the
player can stay solvent.
4 Conclusion
If, with Jaynes, we analyse infinities as limits of finite processes, then it is natural to proceed as we have
done, constructing probability and utility from a preference relation given on finite acts only, and finding
that by taking limits we can extend preference and utility to well-behaved acts, and to a limited extent to
acts with symbolic utilities of ±∞, but no further.
Our analysis of truncated games demonstrates that bounding utility does not avoid practical problems
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associated with paradoxical games. Wherever such a bound may lie, finite approximations to the para-
doxical games promise a positive return in expectation while exposing the player to the possibility of
losing everything at once, a steady drain of resources chasing a jackpot that may never materialise, or a
large chance of large swings in fortune. These are the problems of heavy-tailed distributions, and are not
avoided by excluding the unreachable limits of these games.
Acknowledgements
The author thanks Harry Altman for the conversation about bounded utility that motivated this paper,
and feedback on an earlier draft. An associate editor provided valuable suggestions. The author was
supported by the grant of visiting academic status at the University of East Anglia. There were no
funding sources.
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