I did a Philosophy Independent Study class, and I wrote a term paper. It’s my first term paper. I’m posting it here because I figure I did write something in the month of March, and it’s not code or coursework that will be scooped up by an LLM and or a fellow student hoping to play hooky with their homework.

As an aside, I’d forgotten how rewarding the process of solidifying ideas through writing with the intent of it being publicly viewable is in comparison to writing inane garbage for my own journal or school notes. Maybe I should do that more often.

Introduction

Suppose you are going about your day when suddenly Satan comes by to offer you a deal: he has an apple of infinite slices, and you can choose with each slice – including the first – whether to take that slice or to end the bonanza. If you manage to take a finite number of slices, then you get to go to heaven upon death. However, if you end up taking infinite slices you will go to hell. Your goals are twofold: get as many apple slices as you can, but not infinitely many since you prefer heaven to hell. This infinite decision problem is known as Satan’s Apple. What should you do?

It seems that only way to avoid going to hell when faced with Satan’s Apple is to stop at an arbitrary point. I will explore the use of risk aversion to avoid the diachronic tragedy of this problem.

Dominance and Diachronic Tragedy

Satan’s Apple seems paralysing since, from the perspective of utility maximisation, there is a simple dominance argument in favour of eating the next apple slice at each synchronous (instantaneous) decision point. Suppose you’ve eaten $n$ slices so far, where $0 \leq n \ll \infty$. You then face the following choices:

1. Eat the $(n + 1)^{\text{th}}$ apple slice

2. Reject the $(n + 1)^{\text{th}}$ apple slice

An action is deemed to dominate another if it always provides at least as much utility as the other in every possible state of the world. If you always prefer going to heaven or going to hell, i.e the utility of heaven is always greater than hell, then going to heaven dominates going to hell, and we notate $\left( \text{go to heaven} \right) \succ \left( \text{go to hell} \right)$.

However, if we consider that for some finite natural number $n$, $(n) \neq \infty \Rightarrow (n + 1) \neq \infty$, and that you always prefer to have more apple slices than you already have, it seems very compelling to say that $\left( \text{eat }n + 1\text{ slices} \right) \succ \left( \text{eat }n\text{ slices} \right)$. So, for every possible $n$, it is always preferrable to take $(n + 1)$ apple slices over merely $(n)$ since just one more apple slice does not push you over the brink to infinity. However, if you were to follow this line of dominance reasoning for every apple slice offered, it will undoubtedly lead to hell since you end up choosing every apple slice offerred!

This pattern of behaviour has been dubbed by [@hedden_options_2015] as diachronic tragedy. This is because a series of decisions that synchronically seem rational (such as taking $n + 1$ slices) ends up leading to a course of action that diachronically (through time) seems undesirable, yet entirely predictable, hence the tragedy. The problem might lie in accounting for the contribution of individual actions towards the diachronic tragedy. If we notice that each choice to take an apple slice entails the possibility of taking all (i.e hell), whereas ending the deal will guarantee avoiding hell, there may be a way to counter the dominance argument.

Acts and Utilities

At the start of the problem, we can divide the possible sequences of actions as being either $$\begin{array}{rlr} ALL & \equiv \text{ Take all the slices, }\text{ or }S_{n} & \equiv \text{ Take }n\text{ slices }\text{ where }0 \leq n \leq \infty \end{array}$$

We may then consider two possible ways of assigning utilities to these actions.

Unbounded Utility

Based on [@hajek_waging_2003]’s invalidating Pascal’s wager, assigning infinite positive utility to heaven, infinite negative utility to hell, and any other infinite utility assignment, is untenable. It would lead to an inability to decide between actions. In the case of Satan’s Apple, consider the following:

If we assign $U\left( \text{heaven} \right) = + \infty$, $U\left( \text{hell} \right) = - \infty$, and $U\left( S_{n} \right) = 1$, then the utility of $U\left( S_{0} \right) = U\left( \text{heaven} \right) = + \infty$ while $U(ALL) = U\left( \text{hell} \right) + \sum_{n = 0}^{\infty}U\left( S_{n} \right) \times n = + (\infty) + - (\infty) = \text{ undefined}$. It would be arbitrary to assign infinite utility to only one of heaven or hell, so that does not seem like a desirable route to arriving at a decision (say if only $U\left( \text{hell} \right) = - \infty$ and $U\left( \text{heaven} \right) = 0$ utility instead). Thus, it seems necessary that to arrive at any sort of decision that is rationally justified by expected utility maximisation, we must assume bounded utility.

Diminishing Marginal Utility

In [@bartha_satan_2014] the authors lay the utilities as follows: Taking $0$ apple slices and guaranteeing heaven is the status quo so $U\left( S_{0} \right) = 0$, and $U\left( \text{hell} \right) = - 1000$. The actual number doesn’t matter as long as $U\left( \text{hell} \right) < U\left( \text{heaven} \right)$. $U\left( S_{n} \right) = 10 - \frac{5}{n}$ so that $\lim\limits_{n \rightarrow \infty}\left( U\left( S_{n} \right) \right) = \lim\limits_{n \rightarrow \infty}\left( 10 - \frac{5}{n} \right) = 10 - \lim\limits_{n \rightarrow \infty}\left( \frac{5}{n} \right) = 10 - 0 = 10$. With these we get the utility of taking all to be $U(ALL) = \text{ }10 - 1000 = - 990$.

This implements bounded utility through diminishing marginal utility, so each additional slice gives less additional utility than the previous slice. For example, $U\left( S_{2} \right) - U\left( S_{1} \right) = 10 - \frac{5}{2} - 10 - \frac{5}{1} = 2.5$ whereas $U\left( S_{3} \right) - U\left( S_{2} \right) = 0.8\overline{3} < 2.5$ The reason diminishing marginal utility alone doesn’t give a rationale for stopping at any $n$ is that the marginal utility never goes below zero, so there is still always the dominance argument for taking an additional apple slice, thus succumbing to the problem of diachronic tragedy.

Deliberative dynamics

The most successful approach to Satan’s Apple presented by [@bartha_satan_2014] is deliberative dynamics. As the authors describe it, deliberative dynamics involves making a decision using calculations of future expected utilities as evidence about future choices, and revising credences / calculations until reaching an equilibrium where one can finally carry out their decision:

1. First assign credences (degrees of belief) $q_{n}$ to the occurrence of all possible sequences $S_{n}$.
   One possible way of assigning credences is based on cardinalities as follows:

   $$q_{n} = P\left( \frac{S_{n}}{S_{1} \cdot S_{2} \cdot \ldots \cdot S_{n - 1}} \right)\text{ if }n > 1\text{ otherwise }q_{1} = P\left( S_{1} \right)$$

   Since $S_{n}$ occurring depends on choosing to continue at each previous state $S_{1},\ldots,S_{n - 1}$

2. For each $n$, compute expected utilities for the two choices:

   $$\begin{aligned} EU\left( \neg S_{n} \right) & = 10 - \frac{5}{n - 1}\text{ if }n > 1\text{ otherwise }EU\left( \neg S_{n} \right) = 0\text{ if }n = 1 \end{aligned}$$

   These expected utility calculations come from the traditional expected utility formula multiplying probabilities of outcomes with their utilities. We also say $T_{k} \equiv \text{ take exactly }k\text{ slices}$, more precise than $S_{n} \equiv \text{ Take }n\text{ slices}$. For $EU\left( S_{n} \right)$ the relevant probabilities are:

   $$P\left( \frac{T_{n}}{S_{n}} \right) = \left( 1 - q_{n + 1} \right),P\left( \frac{T_{n + 1}}{S_{n}} \right) = q_{n + 1}\left( 1 - q_{n + 2} \right),\ldots,P\left( \frac{ALL}{S_{n}} \right) = \prod_{i = n + 1}^{\infty}q_{i}$$

   $$\begin{aligned} EU\left( S_{n} \right) & = \sum_{k \geq n}P\left( \frac{T_{k}}{S_{n}} \right) \cdot U\left( T_{k} \right) + P\left( \frac{ALL}{S_{n}} \right) \cdot U(ALL) \end{aligned}$$

   The status quo will have the following expected utility at the outset:

   $$EU\left( SQ_{n} \right) = q_{N} \times EU\left( S_{n} \right) + \left( 1 - q_{n} \right) \times EU\left( \neg S_{n} \right)$$

1. Revision of each credence $q_{n}$ is done as per the following revision rule $\phi$//, applied $\forall n$ ?sequentially over and over? as part of the deliberation process $$\phi\left( q_{n} \right) = q’{n} = q{n} \times \left( \frac{EU\left( S_{n} \right)}{EU\left( SQ_{n} \right)} \right)$$

   Which is repeated by applying these updated credences to update expected utility calculations.

2. The agent stops deliberating after reaching an equilibrium (the status quo stops changing) so

   $$\phi(q_{n}) = q’{n} = q{n}\forall n$$

Rational credences

The way deliberative dynamics works, the only initial credences that reach any equilibria are $0$ and $1$ [@bartha_satan_2014], since those are the only credences that remain unaltered by the updating rule, just like with Bayesian updating of credences. [@bartha_satan_2014] also points out that these equilibria are unstable since relatively small changes to degrees of belief that led to landing at an equilibrium credence $1$, could lead to an equilibrium credence of $0$, and vice versa. So, reaching equilibrium requires extreme credences, and raises the question of whether the problem of Satan’s Apple necessitates extreme degrees of belief.

One intuitively appealing constraint on rational credences is what [@bartha_satan_2014] calls practically consistent. I would prefer not have a set of credences (like $q_{n} = 0\forall n \geq 1$) that would lead me to perform directly contradictory actions i.e, taking every apple slice while believing I certainly won’t.

It is also pragmatically undesirable since $q_{n} = 0\forall n \geq 1$ will guarantee the lowest utility outcome of hell. On this basis, holding credence $q_{n} = 0$ for any $n$ seems undesirable as it would lead to contradictory action if one were to maximise expected utility, as is widely assumed to be required for rationality.

Thus, if deliberative dynamics provides a local expected utility maximising justification for a stopping point based on one’s credences, then it can be used to reduce Satan’s Apple down to a problem of holding rationally permissible credences. If an agent doesn’t hold initial credences $\text{ }q_{n} = 1\forall n$, are they unable to reach any equilibria, and thus unable to act out on any decision?

Risk aversion

[@buchak_risk_2009] defines risk aversion to be a global sensitivity and aversion to a higher degree of variance between outcomes, and introduces risk-weighted-expected-utility as an improvement over traditional expected utility theory. Briefly, it allows for rational agents to have their own risk function $r$ which applies to their probabilities and maps them to a value still between $0$ and $1$. This risk function reflects how an agent might perceive the expected utility of an outcome differently if it is offered next to a riskier or surer bet, instead of in isolation.

For example, risk aversion allows for a rational agent to prefer a bet that surely pays out $$ 50$ over one that has a $\frac{1}{2}$ chance of paying $$ 0$ and a $\frac{1}{2}$ chance of paying $$ 100$, even though the traditional expected utility of both is the same. Traditional expected utility theory dictates any preference between these two would be impermissible.

Any sufficiently risk averse agent may have their risk function such that even extremely low probabilities are mapped to $1$ in their expected utility calculation. It is also assumed that their risk function would never be decreasing, so that an increase in the probability of an outcome never decreases their risk weighted credence in an outcome. This would be especially useful for Satan’s Apple, since each decision point involves choosing between a guarantee of heaven and a greater utility with a risk of hell.

Suppose an agent doesn’t hold $q_{n} = 1\forall n$ at the start of Satan’s Apple, so ordinary deliberative dynamics would not help them reach a decision. However, they are globally sensitive to the property of Satan’s Apple that choosing $n + 1$ slices over ending at $n$ slices carries a chance of ending up in hell whereas foregoing the increase in utility will result in heaven for sure. This could be represented by a risk function that maps their credences such that they reach a deliberative dynamical equilibrium and can justify foregoing the $(n + 1)^{\text{th}}$ slice.

An extreme risk function that assigns every non-zero credence $r(q) = 1$ so only $r(0) = 0$ would be convenient for this purpose. However, [@buchak_risk_2009] suggests that $r$ must be a continuous function. If that is the case, it makes the task more difficult. A continuous function that assigns every credence $0 \leq q \leq 1$ a risk-weighted credence of $1$ seems undesirable, since a credence of $0$ gives certainty that something will not occur so mapping it to assume occurrence would be contradictory. However, if a risk function is continuous, and does not map every possible credence onto $1$, then we still face the same issue of inability to reach equilibria.

What we are looking for through the risk function is a means of quantifying that despite one’s credences and expected utility, their risk averse preference is towards a guaranteed outcome. It seems like a discontinous risk function is the most convenient means of arriving at this.

Timidity

[@beckstead_paradox_2024] argues that what I propose is a timid approach to this infinite decision problem. They point out many problems with timidity, such as that of extreme sensitivity to minor increases in risk irrespective of possible utility gained, and a “[s]trange dependence on distant space and time” [@beckstead_paradox_2024]. There does not seem to be a reasonable means of avoiding diachronic tragedy while also avoiding the above issues.

In the case of Satan’s Apple, the structure of the infinite decision problem exploits traditional utility maximisation to induce diachronic tragedy. Thus, if one wishes to avoid diachronic tragedy, it seems that they must accept the features of timidity as the only means of justifying stopping.

Time

As an aside, [@bartha_satan_2014] makes an assumption of computation time being a negligible factor in the deliberative dynamics involved to arrive at a decision. However, it also assumes that the infinite decision problem of Satan’s Apple is posed as a supertask (a way of executing infinite tasks in a finite amount of time). This distinction is important, since it relates to the expected utilities of the status quo. If Satan’s Apple is presented as a truly infinite problem, then one could just continue to eat apple slices forever since there is always utility to be gained.

The only way to realise the consequences is through the supertask, hence a desire to avoid hell. Similarly, deliberative dynamics also might end up taking infinite time, except that the deliberation process itself doesn’t confer any concrete utility. However, if we are performing it, then it too could be assumed to done in a supertask; it does not seem that far-fetched to perform a supertask within a supertask. Otherwise, being stuck in the deliberation process itself seems to be an outcome to be avoided.

Conclusion

The problem of Satan’s Apple is inherently difficult for a rational agent. Traditional expected utility maximisation condemns them to the lowest utility outcome if they synchronically choose to maximise utility. Accounting for risk aversion in deliberative dynamics enables agents with a wider range of initial credences to avoid diachronic tragedy with a rational justification. However, this requires adherence to extreme risk aversion. These costs seem to be an unavoidable characteristic of the infinite decision problem, so if one wishes to beat Satan’s Apple they face the potential losses of extreme risk aversion when it is applied elsewhere.