Layman’s Summary of Resolving Pascallian Decision Problems with Stochastic Dominance

Summary: Many people have the intuition that it is preferable to have a guaranteed small positive payoff than to gamble on a tiny chance of a huge outcome. This is sometimes used to justify avoiding speculative research in favor of more grounded approaches. But this intuition sometimes conflicts with standard decision theory tools like expected value. In a working paper, Christian Tarsney comes up with a clever resolution to this conflict: if you have extreme uncertainty about how much moral value there is in the universe (which we probably do) then paradoxically the “gamble” is actually better across the board than the supposed “guarantee” (for a certain technical definition of “across-the-board”). The result is a decision procedure which agrees with expected value calculations for non-Pascallian situations, but disagrees in Pascallian ones. His paper is somewhat technical, but I think the intuition can be understood by anyone with a grasp of basic probability theory so I’m attempting to write an explanation.

I’m trying to give an accurate summary of his paper while avoiding technical details, which is a challenging task. Any mistakes are mine, and concerned readers can find a rigorous version of these theorems in his paper. 

Intro

Pascal’s wager scenarios (where we are forced to choose between a guaranteed small outcome or an extremely small chance of a gigantic outcome) conflict with many people’s intuitions. Seemingly reasonable principles imply that we should prefer a small chance of doing something exceptionally great over a guarantee of doing something pretty good, but this seems counterintuitive. 

Tarsney argues that these scenarios seem counterintuitive because the wagers are phrased in a misleading way. In particular, they are asked in a vacuum: even though you care about maximizing the total amount of goodness in the universe, the scenario usually asks whether or not you personally would accept some wager, but doesn't state anything about what happens in the rest of the universe. Since your decision about accepting the wager doesn't affect those other parts of the universe, philosophers assume that information about them can be left out of the thought experiment. But Tarsney shows, quite counterintuitively, that this is not true in circumstances where we have extreme uncertainty about what’s going on outside of our actions. In these scenarios, not only are Pascal wagers not reliant on small probabilities, they are in some sense guaranteed to be better than the "sure thing".

Furthermore, he argues that we do, in fact, have extreme uncertainty about how good/bad the rest of the universe is (e.g. because there might be some unexplored branch of the galaxy that has huge levels of suffering or happiness). 

How do we reconcile the apparent usefulness of expected value (EV) with it giving counterintuitive conclusions in Pascallian cases? Tarsney argues that we should use an alternative decision criterion called stochastic dominance which agrees with EV in non-Pascallian situations, but, when combined with the above argument about uncertainty, disagrees with EV in Pascallian ones. 

Thus stochastic dominance gives us the best of both worlds: we can gain the benefits of EV in non-Pascallian situations without the counterintuitive implications in Pascallian ones.

Note: this paper assumes that we are trying to maximize the total amount of good in the universe, rather than the difference one individually makes to the total good in the universe. The argument may not apply to moral frameworks which aren’t doing this.

Intuition

(Lightly adapted from the original paper)

Suppose you can choose between:

• Safe: you will be guaranteed to make one person happier
• Risky: 10% chance of making 20 people happier, 90% chance of making zero people happier

You may prefer Safe over Risky, even if Risky is better in expectation.

But now someone comes in and says: I'm going to choose a possibly huge random number (which could be negative or positive) and add that to whatever you chose. So if my random number is 1 million and you chose Safe, 1,000,001 people will be made happier. If my random number is -1 million and you chose Safe, 999,999 people will be made sadder.

The original appeal of Safe was that you would be guaranteed to cause some positive amount of happiness, but now that guarantee is gone. In fact, if the random numbers are chosen in the right way, the probability of you causing a positive amount of happiness is actually higher if you choose Risky than if you choose Safe. (The next section gives a simple example of how you can choose random numbers to make this true.) Even more strongly: we can choose random numbers such that, for any value X, the probability that total happiness in the universe is greater than X is at least as high, and sometimes higher, if you choose Risky than if you choose Safe. (I.e. Risky stochastically dominates Safe.)

Tarsney claims that this is the moral situation we are all in: huge amounts of value and disvalue are possibly being created in distant branches of the universe (analogous to someone adding or subtracting huge numbers from our result), and there is no realistic scenario in which we can guarantee even a tiny positive outcome. Therefore, many (most? all?) supposedly Pascallian wagers are actually not wagers at all, and are strictly better than the supposedly "safe" options. 

Slightly more technical intuition

Again suppose you are choosing between Safe and Risky. Someone comes in and says that, whatever you choose, they are going to subtract one from the final value. So if you choose Safe you will make 1-1 = 0 people happier, and if you choose Risky there is a 10% chance that you will make 20-1 = 19 people happier, and a 90% chance that you will make 0-1 = -1 people happier (i.e. one person sadder).

What is the probability that you will end up helping some positive number of people? If you choose Safe, the probability is 0% (since you are guaranteed to get 0 units). But if you choose Risky there is a 10% chance of the final number being greater than zero.

So, somewhat counterintuitively, Risky is actually more likely to result in a positive outcome.

Now suppose we have the same scenario, but instead of definitely subtracting 1, the person will subtract a randomly chosen integer one through 19, chosen uniformly. Again, Risky has a greater chance of being greater than zero, for the same reason: Safe will always give a negative outcome, whereas in 10% of cases Risky will give a positive outcome.

But now it also has a greater chance of being greater than minus one: Safe will be greater than -1 only if the randomly chosen number is 1, which happens 1/19th of the time. But Risky will be greater than minus one 10% of the time.

Now Risky is more likely to result in an outcome greater than -1.

We can continue to add larger sets of possible random numbers, creating scenarios in which Risky will be more likely to result in an outcome greater than -2, -3, -4, … If we continue this process infinitely, we find that there is some probability distribution of random numbers such that Risky is more likely than Safe to result in a scenario greater than any number, i.e. it stochastically dominates.

One technical point is that this distribution of random numbers has to not just have infinite range but also be fat-tailed, because otherwise the really large numbers wouldn’t matter “enough”.In practical terms, this means that it’s not enough to be uncertain about what else is going on in the universe, but we have to give substantial credence to the possibility that something extremely bad or extremely good is happening.

Some Final Notes

1. Tarsney gives a concrete heuristic for deciding whether a small probability of a very large payoff actually stochastically dominates a smaller sure-thing payoff: If the probability of the large payoff is greater than the ratio of the sure-thing payoff to the interquartile range of the background distribution, then the 'long shot' option is probably stochastically dominant. He gives example numbers for certain longtermist interventions to see whether they stochastically dominate short-term interventions, but I don’t think anyone has done this rigorously.
2. Tarsney summarizes some implications: "An initially counterintuitive feature of the preceding arguments is their implication that what an agent rationally ought to do can depend on her uncertainties about seemingly irrelevant features of the world. To put the point as sharply as possible: Whether I am rationally required, for instance, to take a risky action in a life-or-death situation can depend on my uncertainties about the existence, number, and welfare of sentient beings in distant galaxies, perhaps outside the observable universe, with whom I will never and can never interact, on whom my choices have no effect, and whose existence, number, welfare, etc, make no difference to the local effects of my choices."
3. When given the choice between two options, it’s possible that neither will stochastically dominate the other. In contrast, EV will always tell us that one is better than another, or that they are equally valuable. So there may be some wager where stochastic dominance tells us that either accepting or rejecting the wager is permissible, but EV says one is strictly better than the other. This is either an advantage or disadvantage of stochastic dominance, depending on your perspective.

I would like to thank Christian Tarsney and Max Dalton for their comments on a draft.