This post summarises a paper by Eddy Chen and Daniel Rubio on using Surreal numbers to resolve problems of Infinite Ethics. Future posts will argue that surreals are the correct approach to this problem before extending upon this work. However, this post merely aims to summarise this paper.

Background

The problem of Infinite Paralysis is best described as follows. Suppose that there are infinite people and that they are happy so that there is infinite utility. I then come along and punch 100 people destroying 100 utility. Since there was infinite utility at the start and infinity minus 100 is still infinity, so arguably I’ve done nothing wrong. However, this seems to be a reductio ad absurdum if I’ve ever seen one.

One approach mentioned by Bostrom is to use hyperreals to represent infinite sequences of utility. In particular, he sets the ith index of the hyperreal representing utility to the sum of the first i numbers.

Unfortunately, there is not a unique definition of the hyperreals - they require what’s called a non-principle ultrafilter to be defined in order to determine the ordering. Our choice of this seems essentially arbitrary and therefore hard to justify principally. Additionally, summation requires a preferred location around which to sum, which can be hard to justify philosophically.

Surreals Decisions

Chen and Rubio outline a surreal decision theory by adapting the the Von Neumann-Morgenstern axioms. They then use it to analyse Pascal’s Wager to demonstrate that the validity of the argument depends on particular infinite values assigned in the problem and the various deities that exist.

They note that Expected Utility Theory with standard infinities (cardinal numbers) seems to produce absurd results. In particular, it is indifferent between the following, when most people would prefer them in order

Infinity or something: Infinite utility if heads, 10000 utility if tails
Infinity or nothing: Infinite utility if heads, 0 if tails
Infinity or bust: Infinite utility if heads, -10000 if tails

They provide another example, where they argue that the ordering is obvious, but that this is undefined for Expected Utility Theory:

Biased positive infinity: 9/10 times plus infinity, 1/10 minis infinity
Fair infinity: 50/50 chance of plus infinity or negative infinity
Biased negative infinity: 9/10 times minus infinity, 1/10 plus infinity

They then argue that surreal numbers can correctly solve these problems. At this stage I’ll note that the “obvious” solution requires an additional assumption that the infinities in the above problem all have the same magnitude. Without this assumption, the answer really is undefined.

Surreal Numbers

They then outline what surreal numbers are and how they are constructed. For our purposes, all that matters is that they have the following two properties:

x+1 ≠ x
Standard arithmetic works the same, ie. addition and multiplication are commutative, associativity, distributivity, additive inverses, multiplicative inverses for non-zero values, ect

Surreal Von Neuman-Morganstern Axioms:

Just like the finite version, this theorem states that a set of preferences can be represented by a utility function under certain assumptions such that the ordering always prefers the option with higher utility.

The paper lists the following four assumptions (link to image if the text below is broken):

Completeness: ∀x, y ∈ X, either x ≼ y or y ≼ x
Transitivity: ∀x, y, z ∈ X, if x ≼ y and y ≼ z, then x ≼ z
Continuity*: ∀x, y, z ∈ X, if x ≼ y ≼ z, then there exists a surreal p ∈ *[0, 1] such that px + (1 − p)z ∼ y.
4. Independence*: ∀x, y, z ∈ X, ∀p ∈ *(0, 1], x ≼ y if and only if px + (1 − p)z ≼ py + (1 − p)z.

(Here *[0,1] means the range 0 to 1 in the surreal numbers)

They then consider the potential for probability theory to be extending to the surreals by considering the compatibility of the Kolmogorov axioms. Most are compatible, but the countable additivity of events can't be maintained if we insist that probabilities remain normalised. They suggest that an alternative formulation of countable additivity might be able to work around this limitation, but they aren't too concerned about this as finite additivity is sufficient for this paper.

Pascal’s Wager:

One response to Pascal’s Wager is the Mixed Strategy approach. Using typical infinities, any finite chance of an infinity with no chance of negative infinity causes the expected value to be infinite. Since regardless of our decision at this point in time, we will still have a non-zero chance of us eventually becoming Christian, we should therefore be indifferent between all actions. This argument seems absurd and this can be see to be the case once we use Surreal Numbers as a smaller chance of becoming Christian leads to smaller infinite value.

Another response is the Many Gods response. This response argues that the original Pascal’s Wager assumes without reason that there is only one possible God, when there might be multiple possible Gods offering different levels of plus or minus infinite utility. This paper is able to make this argument more precise than it is normally made thanks to surreal numbers. They then conclude that Pascal’s Wager doesn’t deliver what it promises: it says that you should believe in God regardless of the evidence, when in fact it depends on the likelihood of each deity existing and your expectation of their punishments.

Relevance to Infinite Ethics:

This paper doesn't discuss infinite ethics, but the application is rather trivial. In the Surreals, X+1 is different from X, so we don’t run into infinite paralysis. This will be discussed in more detail in future posts.