Experienced Utility and normative decision making

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XU
Loren K. Fryxell
City St George’s, University of London
loren.fryxell@citystgeorges.ac.uk
July 30, 2025
2025 World Cong...

Introduction
Introduction XU XU for Multiple States LELO

Overview
Part I: XU
• Is a hedonometer a real thing?
• Yes, and I’m going to tell you how to construct it
• This is a theo...

Overview
Part II: LELO
• The LELO principle is an ethical principle I propose in this
paper.
• It says that the morally be...

Overview
Part III: XU + LELO
• An ethical observer who respects LELO acts as if they seek to
maximize the total XU across ...

XU for a Single State
Introduction XU XU for Multiple States LELO

XU (Experienced Utility)
There is a long history of thinking about utility as a measure of
pleasure over time. For example...

Main Idea
The main idea is simple. A hedonometer should represent an
individual’s preferences over living these experience...

Experiences
Definition. An experience is a triple (ω, i, I) where
• ω is a complete description of everything that happens...

Fix the State of the World and the Individual
Fix a state of the world ω and an individual i
We seek to construct a hedono...

Half-Day Example
time
pleasure
0
To fix ideas, let’s consider the half-day time slice of some universe
in which you exerci...

Preferences over Experiences
Normalize your half-day to lie on the unit interval
Experiences are then intervals lying in t...

Interpretation of ⪰
1 How you define what it means to say I ≻ J completely
changes the interpretation of the axioms (and i...

Desired Representation Theorem
We seek axioms which are necessary and sufficient for the existence
of an XU representation...

Axioms
There are three axioms:
1 Rationality
2 Monotonicity
3 Continuity
Introduction XU XU for Multiple States LELO

Axiom 1. Rationality
Axiom 1. An individual has complete and transitive preferences ⪰
over I.
Introduction XU XU for Multi...

Axiom 2. Monotonicity
Suppose I and K are adjacent, and J and L are adjacent
If I ⪰ J and K ⪰ L
0 1
I K J L
then I ⊕ K ⪰ J...

Axiom 3. Continuity
Small changes in an interval result in small changes in preference
If I ≻ J
0 1
I J
then I′ ≻ J′
0 1
I...

Theorem 1 Preliminaries
Before stating the results, we must first define the concept of a
positive, zero, and negative exp...

Positive, Zero, and Negative Incomplete Experiences
0 1
I J
Definition. An incomplete experience I ∈ I [0, 1) is
1 positi...

Positive, Zero, and Negative Complete Experiences
0 1
I J
Definition. The complete experience [0, 1) is
1 positive if ther...

These Definitions Make Sense with the Axioms
As defined, an experience could, in principle, be positive, zero, and
negativ...

Uniformly Non-Negative and Non-Positive Experiences
Definition. An experience I ∈ I is
1 uniformly non-negative if all sub...

Regularity
Definition. [0, 1) is regular if there exists a finite interval partition
P of I such that each cell P ∈ P is u...

Additive and Continuous Representation
Definition. U : I → R is additive if for any I, J ∈ I,
U(I ⊕ J) = U(I) + U(J).
Defi...

Theorem 1
Theorem 1. Suppose [0, 1) is regular. Axioms 1, 2, and 3 hold if
and only if there exists an additive and contin...

Axiom 3′
.
What about an integral?
If we strengthen the continuity axiom, then we can get an integral
representation (whic...

Theorem 2
Theorem 2. Suppose [0, 1) is regular. Axioms 1, 2, and 3′ hold if
and only if there exists an integral represent...

Constructing the Hedonometer
The proof is constructive
I will now show you how to construct the XU representation, i.e.,
t...

Constructing the Hedonometer
1
1
2
t
v
Cut it in half (n = 1)
Find the dividing point such that the first segment is indif...

Constructing the Hedonometer
1
1
2
t
v
Cut each in half again (n = 2)
Do this again for each of the previous segments. Ass...

Constructing the Hedonometer
1
1
2
t
v
And again (n = 3)
Do this again for each of the previous segments. Assign each
util...

Constructing the Hedonometer
1
1
2
t
v
And again (n = 4)
Do this again for each of the previous segments. Assign each
util...

Constructing the Hedonometer
1
1
2
t
v
And again and again.. (n = ∞)
Introduction XU XU for Multiple States LELO

Why an Additive Representation?
Just as it is intuitive to have a measure of weight that is additive
in the concatenation ...

Why an Additive Representation?
We could just as easily measure weight with a measurement that is
not additive.
Nothing wo...

XU for Multiple States of the
World
Introduction XU XU for Multiple States LELO

Many States of the World
We can do the same exercise for each state of the world ω ∈ Ω,
making comparisons both within and...

Preference Intensity
This gives us an XU for each state of the world, and for each time
slice within each state of the wor...

Preference Intensity
An apple is twice as good as a banana means that “half” the
experience of an apple, by preference, is...

Preference Intensity
In general, we quantify how much better A is than B by asking:
How many equal pieces must we cut the ...

LELO
Introduction XU XU for Multiple States LELO

Fundamental Theories
In the previous sections, I discussed a fundamental theory of
well-being
By fundamental theory, I mea...

Extended Sympathy
To measure the well-being of an experience, we looked at your
preferences over living each time slice wi...

Extended Sympathy
I’m not the first to propose the idea of living in other people’s
shoes. This is known as extended sympa...

XU Across Individuals
Hence, in principle, we can do the same exercise for each individual
in each state of the world ω ∈ ...

Quote
“Ethics is a theory of rational behavior in the service of the
common interests of society as a whole” (Harsanyi, 19...

The LELO Principle
The LELO Principle (Live Every Life Once). A state of the
world ω1 is morally preferred to ω2 if and on...

XU and LELO
Proposition. Suppose social preferences ⪰∗ satisfy the LELO
principle. Then
⪰∗
is utilitarian in U ⇐⇒ U is an ...

Thank you for listening!
Questions, comments, or concerns?
Introduction XU XU for Multiple States LELO

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