I think most atheists are dumbfounded by evangelical types. The odds are stacked fairly strongly against them. It’s essentially guaranteed that they’ll fail. Thinking about this absurdity, I decided to do some very basic math. How much utility does a believer get by evangelizing? But first I need to construct a payout matrix.

Above, I show the two possible actions a believer can take, the two possible states of the nonbeliever, and three possible consequences.

The believer’s preferences are C₁PC₂PC₃

That is, the believer prefers the atheist changes her mind and joins the church to the status quo. And he prefers the status quo to the status quo and frustration. This is the set of ordinal preferences over outcomes.

Next, we must determine the utility the believer receives for each of the three consequences. Let’s assume that he receives a full unit of utility for convincing the atheist. That is, u(C₁) = 1. Because I do not want to use negative numbers here, the rest of the consequences will contain a percent of a full unit of utility. Therefore, u(C₂) = 0.5. That is, the status quo will serve as a bass line. Anything above it is a plus, and anything beneath it is a minus. Finally, u(C₃) = 0.1. This is the worst utility one can receive.

Next, we need to make some assumptions about the probability that the believer will be successful. I was unable to find statistics online about evangelical groups’ success rates, so we will have to take our best guess. I’d imagine their success rate can’t be better than 1%. Therefore, p(S₁) = .01, and p(S₂) = .99* (because these are the only two possibilities, they must add up to 1).

Now we just have to put this into mathematical terms:

EU(A₁) = p(S₁)u(C₁) + p(S₂)u(C₃) =

In English, the expected utility of evangelizing is equal to the probability of the atheist being convertible, multiplied by the utility of the atheist changing her mind and joining your church, plus the probability that the atheist is not convertible, multiplied by the utility of being frustrated and maintaining the status quo.

To solve:
(.01)(1) + (.99)(.1) = ~0.11

Next,

EA(A₂) = p(S₁)u(C₂) + p(S₂)u(C₂) =

That is, the expected utility of not evangelizing is equal to the probability that the atheist is convertible, multiplied by the utility of maintaining the status quo, plus the probability that the atheist is not convertible, multiplied by the utility of maintaining the status quo.

To solve,
(.01)(.5) + (.99)(.5) = 0.5

The expected utility of evangelizing is a mere ~0.11, whereas the expected utility of leaving atheists alone is 0.5. In other words, they have more to gain from not evangelizing.

Discussion

To be fair this is not taking into consideration the believer’s future utility. In many religions it is a duty to spread the word. Therefore, they have hope for a future reward in a life after this one. Unfortunately, that’s just not the way utility works. Generally we speak of utility in terms of how much we can get while still alive. There’s no way to measure utility in any hypothetical afterlife, so it’s pointless to try.

I’m sure some evangelical types will maintain that, yes, they receive their reward after this life, but with no way to demonstrate the reward, we have no way to measure the reward.

*After playing around with the numbers, trying to find a point where the believer gets more utility from evangelizing, I found that if we change p(S1) = .1, and p(S2) = .9, it’s more rational for the believer to try to convert atheists, but those probabilities are way off of reality.