Covariance of Wealth and Cost-Effectiveness

Summary

In considering strategies for making and investing money to reduce suffering, it's often easiest to think about accumulating wealth to donate as a separate issue from figuring out where to donate that wealth. However, in some cases, the correlation between the two should not be ignored and may actually allow donors to achieve extra expected benefit from their donations. In particular, I consider as examples the correlation of one's career success with technological progress and the use of prediction markets to achieve an expected "free lunch" from covariance between financial returns and cost-effectiveness of donations.

Note that this discussion may have more theoretical than practical interest, though there could be real-life examples in which these ideas are important.

Why covariance could matter

Suppose you want to donate some of your wealth to a charitable cause t years from now. Let X_t denote the dollar amount you'll donate, and let Y_t denote the cost-effectiveness of your donation at that time in utils / dollar or (suffering reduced) / dollar or (some unit you care about) / dollar. Cost-effectiveness is assumed constant regardless of how much is donated.^a The total good you do is X_t * Y_t.

Of course, you don't know for sure what X_t and Y_t will be; instead, you have subjective probability distributions for their values. So X_t and Y_t are really random variables, and your goal is to maximize E[X_t * Y_t], where E[ ] denotes expected value. It's not hard to show that

E[X_t * Y_t] = E[X_t] * E[Y_t] + Cov[X_t, Y_t],

where Cov[ ] stands for covariance.

In most cases, Cov[X_t, Y_t] is likely to be close to 0. For instance, suppose you plan to donate to Animal Ethics after investing your money in the stock market for a few years. The return of the stock market probably has little correlation with how effectively Animal Ethics reaches people through lectures and videos.

However, there are some cases in which Cov[X_t, Y_t] may be big enough to be worth considering.

Example 1: Job markets

Suppose you want to donate toward a cause that will address some uncertain future technological development (such as in the areas of AI, nanotechnology, or biotechnology) or speculative societal trend (such as the advent of a "technological singularity"). Your donations in this area will be highly cost-effective if the technology or societal trend materializes but nearly useless if it doesn't. The expected payoff of the career path you pursue in order to accumulate wealth to donate may be dependent on how future developments in the above technologies or societal trends turn out, because those factors will affect the job market (and, potentially, the success of the company you work for, if it makes money from the same areas of technology about which you hope to donate).

Naively one might say, "If societal trend T happens, my donations will be far more cost-effective than if it doesn't happen. I'm going to plan on putting my money toward work related to T. However, just because donating regarding T has high expected value doesn't mean T is highly likely. In assessing the job market, I shouldn't be biased toward thinking that T is any more likely than it objectively is." However, this reasoning ignores the potential benefits of covariance between your career success and T. Other things being equal, a job that is likely to pay better if and only if T occurs is a better choice, because Cov[X_t, Y_t] > 0 in that case.

Example 2: Prediction markets

Suppose you want to donate toward humane insecticides, to relieve the expected suffering of insects on cropland. Imagine that a major study is being conducted by an international, independent body of scientists and animal welfarists to provide more conclusive evidence than is currently available as to whether insects can feel pain. You plan to invest your money in the stock market and donate it after the results of the study come out.

Suppose a liquid prediction market is offering bets as to whether the upcoming study will say "yes" or "no" on insect pain.^b Instead of investing in the stock market, you should buy "yes" shares in the prediction market. That way, if the study concludes that insects do feel pain (which increases your own subjective probability for insects being able to feel pain and, hence, the subjective cost-effectiveness of humane insecticides), you'll have lots of money to spend; if the study concludes that insects can't feel pain, your loss of money is not that severe.

In practice, use of prediction markets may be limited, at least in the US, by legal restrictions on gambling. Still, some markets like Intrade do use real money. Others, like bet2give and Long Bets, are legal because winnings are donated to charity; if the cause you wanted to donate toward was a formal nonprofit anyway, then restrictions on gambling wouldn't matter, except that the donations might not be tax-deductible.

A major problem with prediction markets—especially those that donate profits to charity—is that they're unlikely to be liquid enough to handle the quantity of share purchases that it would otherwise be optimal to make. If you have $500,000 to donate toward relieving expected insect suffering, maybe you could make an arrangement with an insurance company, such that you'll only receive a payoff from your "insect-suffering insurance" if the study comes out saying insects do feel pain. This type of "hole-in-one insurance" was used by the X Prize Foundation to pay for the Ansari X Prize.

It would be interesting to explore the possibility of creating a new prediction market or insurance company specifically designed for donors who want to exploit covariance. Presumably this class of people is not limited to utilitarians; most donors to projects whose cost-effectiveness is substantially correlated with external events should be interested. How big is the set of such donors?

Notes

A donation cause that has positive covariance with donatable wealth isn't automatically a good choice; it's just a better choice than a cause with similar expected cost-effectiveness that has no such covariance. The criterion for comparison is still E[X_t * Y_t] for all projects.

As is indicated by the subscript _t, exploiting covariance requires postponing donations until the future, which brings in considerations of the time-value of donating now instead of later, wisdom gained by waiting to donate until later, probabilities of disaster between now and t years from now, expected investment returns, expected returns on prediction-market securities, expected inflation, and tax effects.

A relatively simple comparison to make is between (a) investing money until t years from now in a capital market that's uncorrelated with cost-effectiveness and (b) putting that money into a prediction market or bet that will expire in t years. Let X_t be the random variable for your donatable wealth under (a) and X'_t be the same under (b). Y_t is the same in either case. Ignoring differential tax treatment, the expected benefit from (b) divided by the expected benefit from (a) is

E[X'_t * Y_t] / E[X_t * Y_t] = ( E[X'_t]* E[Y_t] + Cov[X'_t, Y_t] ) / (E[X_t] * E[Y_t] )
= E[X'_t] / E[X_t] + Cov[X'_t, Y_t] / ( E[X_t] * E[Y_t] ).

Covariance and "acting as if"

This covariance idea is another way of framing a general point about expected-value calculations: When expected values are dominated by one branch of possible outcomes, you should generally "act as if" that branch is true, because in that case, the impact to the overall expected value is highest.

For example, suppose you're on a game show in which there are two doors, of which you're allowed to open only one. Exactly one door has $1 behind it. If the $1 is in door A, you just keep the $1. If the $1 is in door B, then the game-show host multiplies your winnings by 999. The $1 is equally likely to be behind door A and door B.

1. Suppose you open door A. If it contains the money (50% chance), you get $1; if not, you get $0. The expected value is (0.5)($1) + (0.5)($0 * 999) = $0.5.
2. Now suppose you open door B. The expected value is (0.5)($0) + (0.5)($1 * 999) = $499.5.

The calculations are dominated by the case where door B contains the $1, so you should generally "act as if" door B does contain the $1. That's the branch of outcomes where your actions would matter most.

To see the same in covariance terms, let X be an indicator variable for whether the door you picked has money behind it, while Y represents the multiplication factor that the game-show host uses (1 or 999 depending on whether the money is behind door A or B, respectively). Your total expected payout is E[X * Y]. Regardless of which door you pick:

E[X] = (0.5)($1) + (0.5)($0) = $0.5, and
E[Y] = (0.5)(1) + (0.5)(999) = 500.

Now, suppose you pick door A. Then:

Cov[X,Y] = Prob(door A has $)(X - E[X])(Y - E[Y]) + Prob(door B has $)(X - E[X])(Y - E[Y])
= (0.5)($1 - $0.5)(1 - 500) + (0.5)($0 - $0.5)(999 - 500)
= -$249.5.

This means that if you pick door A,

E[X * Y] = E[X] * E[Y] + Cov[X, Y] = ($0.5) * 500 + -$249.5 = $0.5,

just as we computed earlier. Next, suppose you choose door B. Now:

Cov[X,Y] = Prob(door A has $)(X - E[X])(Y - E[Y]) + Prob(door B has $)(X - E[X])(Y - E[Y])
= (0.5)($0 - $0.5)(1 - 500) + (0.5)($1 - $0.5)(999 - 500)
= $249.5

So if you pick door B,

E[X * Y] = E[X] * E[Y] + Cov[X, Y] = ($0.5) * 500 + $249.5 = $499.5,

which is again exactly what we saw before.

Footnotes

1. This assumption isn't necessary, though the analysis becomes more complicated if it's not maintained. Suppose C(X_t, Y_t) represents the average cost-effectiveness of donation to a cause when X_t dollars are donated; Y_t represents some other variable that affects cost-effectiveness. The total benefit of donating X_t dollars is X_t * C(X_t, Y_t). If X_t and Y_t are taken as random variables, the expected benefit is E[X_t * C(X_t, Y_t)].  (back)
2. Perhaps it's fanciful to imagine a market for such a question. Still, prediction markets do sometimes trade bets on the outcomes of studies. A recent market on Intrade allows users to bet on whether "The next independent publicized study of adult talkativeness will find Women talk 10% more then Men."  (back)