Utilitarian Risk (Non)aversion

One of the reasons for which people often focus on direct charity instead of longer-term advocacy is that the results of charity are certain. A person can immediately see pets being taken care of when she volunteers at an animal shelter; it's less clear that she'll actually prevent animal suffering by handing out a few leaflets on factory farming. Yet, I think the latter option is the better of the two. Indeed, in this essay, I'll argue that high-risk, high-payoff projects can often be ethically optimal and that small probabilities deserve greater consideration than they are usually afforded.

To begin, it will be helpful to introduce a few basic concepts about probability. A random variable is a quantity that takes one of several possible values according to chance. For instance, one might roll a die three times and define the random variable X to represent the number of times the die came up six. X might take one of several different values: 0 if a six was never rolled, 1 if a six was rolled once, etc.

The expected value of a random variable is the average of the values that the variable might take, weighted in proportion to their likelihood. It's calculated by multiplying each value of the variable by its corresponding probability and taking the sum.

Example. Alice is taking the SAT test. She comes upon a question for which she has no clue about the answer, so she decides to guess randomly. By doing so, will her score increase, decrease, or stay the same on average? If Alice guesses correctly, she'll gain one point. With five choices, there's a 1/5 probability that this will happen, so her expected gain is (1)(1/5) = 1/5. However, if Alice guesses a wrong answer, she'll lose 1/4 of a point. There's a 4/5 chance of that outcome, so her expected loss is (-1/4)(4/5) = -1/5. Alice's overall expected value is her expected gain plus her expected loss: 1/5 - 1/5 = 0.

Economists typically assume that, as people earn more and more income, the additional utility (satisfaction) that each dollar gives them decreases. So for instance, if you make $100 a year and get a raise to $101 a year, your increase in utility is probably bigger than if you earned $1,000,000 a year and got a raise to $1,000,001 a year. This is called the law of diminishing marginal utility of income.

Economists also typically assume that people act so as to maximize the expected value of their utility (or "expected utility"). Combined with diminishing marginal utility of income, this implies that people are financially risk averse--that is, when they're presented with two options that have the same expected value of income, they choose the one with lower risk.

Example. Bob currently doesn't have health insurance. If he doesn't get sick this year, he'll earn $30,000. But if he does get sick, he'll have to pay for hospital expenses of $10,000. There's a 1/2 chance that he'll get sick. Bob's expected income is thus (1/2)($30,000)+(1/2)($20,000) = $25,000.

Bob stumbles upon a health-insurance plan that costs $5,000. If insured, Bob won't have to pay any expenses for getting sick. So whether he buys the insurance or not, Bob will have an expected income of $25,000. Will he buy the insurance?

Imagine that Bob has $25,000. If Bob has diminishing marginal utility of income, a $5,000 drop in income is more of a loss than a $5,000 increase in income is a gain. Since the loss and gain are equally likely, Bob's expected utility is lowered by risk. Thus, if Bob acts to maximize expected utility, he will buy the insurance.

The law of diminishing marginal utility usually doesn't apply for society as a whole, however (where I'm defining "society" to comprise people and sentient animals). For instance, suppose donating $1 to Vegan Outreach prevents an expected 5 years of suffering in factory farms. Well, presumably $100 donated will prevent an expected 500 years of suffering. In this case, marginal social utility doesn't diminish with respect to dollars donated. This fact can lead to some unexpected results.

Example. Utilitarian Charles is investing in the stock market to accumulate wealth that he will donate to Vegan Outreach. Should he choose a low-risk or high-risk stock?

Most investors in the stock market are risk averse. Therefore, riskier stocks have risk premiums--i.e., they have higher expected payoffs to compensate for greater variability. But if the marginal social utility of donating to Vegan Outreach doesn't decrease, then expected suffering prevented is directly proportional to the stock's expected payoff. The high-risk stock is thus the option of greater expected social utility.

Question. But what if the high-risk stock will only pay off, say, 1/5 of the time? Then, four times out of five, investing in the stock market didn't accomplish anything. Four times out of five, there's more suffering as a result of choosing the riskier stock!

Response. I think there is diminishing marginal utility on the part of individuals who want to effect change with respect to the magnitude of change effected. That is, bringing about ten times as much change probably doesn't feel ten times as good to those who accomplish it. So there is diminishing marginal individual utility for donating income. But utilitarianism is not about maximizing your own utility; it's about maximizing social utility. And the option of higher expected social utility is the one that's truly most compassionate.

I should note that one's individual utility, while negligible by comparison to the changes in social utility that one can effect, may be important for instrumental reasons. For example, utilitarian Dorothy may be more productive if she feels good about her accomplishments; she may give up entirely if she tries to work toward a project with a tiny probability of making a difference. This effect needs to be incorporated into actual consideration of the expected values of risky and nonrisky endeavors.

After all of this, the reader might wonder, Why do we care about maximizing expected values? The expected value of a random variable is just some mathematical function, so who cares about it?

Well, consider the following scenario.

Scenario. You are presented with two buttons, E and F, and are allowed to press only one. E will prevent two million units of suffering with probability one in a million (expected value = 2). F, on the other hand, will prevent one unit of suffering with certainty (expected value = 1). Which do you press?

First imagine you were allowed to continue pressing the button that you chose for 10^100 times. According to the law of large numbers, as the number of repetitions of button pressing approaches infinity, the actual average value of suffering prevented per button press approaches the expected value. So if you could repeat yourself 10^100 times, it's practically guaranteed that you'd prevent twice as much suffering by choosing E over F.

So how does the situation change when you can only press the button once? I don't think it does. Before your single press of the button, you don't know that you won't get lucky by pressing E. Who knows, you might end up doing something two million times as good as pressing F. Since suffering is suffering, and one unit is (by definition) just as bad as the next, choosing the highest expected value is truly the most merciful option.