How Not to Calculate Utilities in an Infinite Universe

Everything you do causes almost everything -- or so I have argued (blog post version here, more detailed and careful version collaborative with Jacob Barandes in my forthcoming book).  On some plausible cosmological assumptions, each of your actions ripples unendingly through the cosmos (including post-heat-death), causing infinitely many good and bad effects.

Assume that our actions do have infinitely many good and bad effects.  My thought today is that this would appear to ruin some standard approaches to action evaluation.  According to some vanilla versions of consequentialist ethics and ordinary decision theory, the goodness or badness of your actions depends on their total long-term consequences.  But since almost all of your actions have infinitely many good consequences and infinitely many bad consequences, the sum total value of almost all of your actions will be ∞ + -∞, a sum which is normally considered to be mathematically undefined.

Suppose you are considering two possible actions with short-term expected values m and n.  Suppose, further, that m is intuitively much larger than n.  Maybe Action 1, with short-term expected value m, is donating a large some of money to a worthwhile charity, while Action 2, with short-term expected value n, is setting fire to that money to burn down the house of a neighbor with an annoying dog.  Infinitude breaks the mathematical apparatus for comparing the long-term total value of those actions: The total expected value of Action 1 will be m + ∞ + -∞, while the total expected value of Action 2 will be n + ∞ + -∞.  Both values are undefined.

Can we wiggle out of this?  An Optimist might try to escape thus: Suppose that overall in the universe, at large enough spatiotemporal scales, the good outweighs the bad.  We can now consider the relative values of Action 1 and Action 2 by dividing them into three components: the short-term effects (m and n, respectively), the medium-term effects k -- the effects through, say, the heat death of our region of the universe -- and the infinitary effects (∞, by stipulation).  Stipulate that k is unknown but expected to be finite and similar for Actions 1 and 2.  The expected value of Action 1 is thus m + k + ∞.  The expected value of Action 2 is n + k ∞.  These values are not undefined; so that particular problem is avoided.  The values are, however, equal: simple positive infinitude in both cases.  As the saying goes, infinity plus one just equals infinity.  A parallel Pessimistic solution -- assuming that at large enough time scales the bad outweighs the good -- runs into the same problem, only with negative infinitude.

Perhaps a solution is available for someone who holds that at large enough time scales the good will exactly balance the bad, so that we can compare m + k + 0 to n + k + 0?  We might call this the Knife's Edge solution.  The problem with the Knife's Edge solution is delivering that zero.  Even if we assume that the expected value of any spatiotemporal region is exactly zero, the Law of Large Numbers only establishes that as the size of the region under consideration goes to infinity, the average value is very likely to be near zero.  The sum, however, will presumably be divergent – that is, will not converge upon a single value.  If good and bad effects are randomly distributed and do not systematically decrease in absolute value over time, then the relevant series would be a + b + c + d + ... where each variable can take a different positive or negative value and where this is no finite limit to the value of positive or negative runs within the series -- seemingly the very archetype of a poorly behaved divergent series whose sum cannot be calculated (even by clever tools like Cesaro summation).  Thus, mathematically definable sums still elude us.  (Dominance reasoning also probably fails, since Actions 1 and 2 will have different rather than identical infinite effects.)

This generates a dilemma for believers in infinite causation, if they hope to evaluate actions by their total expected value.  Either accept the conclusion that there is no difference in total expected value between donating to charity and burning down your neighbor's house (the Optimist's or Pessimist's solution), or accept that there is no mathematically definable total expected value for any action, rendering proper evaluation impossible.

The solution, I suggest, is to reject certain standard approaches to action evaluation.  We should not to evaluate actions based on their total expected value over the lifetime of the cosmos!  We must have some sort of discounting with spatiotemporal distance, or some limitation of the range of consequences we are willing to consider, or some other policy to expunge the infinitudes from our equations.  Unfortunately, as Bostrom (2011) persuasively argues, no such solution is likely to be entirely elegant and intuitive from a formal point of view.  (So much the worse, perhaps, for elegance and intuition?)

The infinite expectation problem is robust in two ways.

First, it affects not only simple consequentialists.  After all, you needn't be a simple consequentialist to think that long-term expected outcomes matter.  Virtually everyone think that long-term expected outcomes matter somewhat.  As long as they matter enough that an infinitely positive long-term outcome, over the course of the entire history of the universe, would be relevant to your evaluation of an action, you risk being caught by this problem.

Second, the problem affects even people who think that infinite causation is unlikely.  Even if you are 99.99% certain that infinite causation doesn't occur, your remaining 0.01% credence in infinite causation will destroy your expected value calculations if you don't do something to sequester the infinitudes.  Suppose you're 99.99% sure that your action will have the value k, while allowing 1 0.01% chance that it's value will be ∞ + -∞.  If you now apply the expected value formula in the standard way, you will crash straightaway into the problem.  After all, .9999 * k + .0001 * (∞ + -∞) is just as undefined as ∞ + -∞ itself.  Similarly, .9999 * k + ∞ is simply ∞.  As soon as you let those infinitudes influence your decision, you fall back into the dilemma.