Rationality, Expected Value, and Pascal's Wager

There's been a lot of talk about probability in the context of Christian belief these days. From academic textbooks and journal articles, to message boards and blogs (including this one), material abounds on the probability of God's existence, the probability of miracles generally, and the probability of the resurrection of Jesus given certain facts of history in the light of Bayes' theorem. The guiding assumption behind all this, though not always explicitly stated, is that accepting the more probable hypothesis results in our holding the most rational belief. And most of us want to be rational.

 

In a recent post here it was argued that the probability of the resurrection of Jesus, given the specific evidence and general background knowledge bearing on the case, is high relative to competing hypotheses. At the least, I suggested, the resurrection should be considered not improbable (hence not irrational to believe). But some people objected that without specific calculations that suggestion is too vague. So now we will "plug in" some numbers. Recall again that per Bayes' Theorem, the probability of a hypothesis H, given evidence E and background knowledge K, equals the conjunction of its explanatory power and prior probability:

 

             P(E│H & K) x P(H│K)

P(H│E & K) = -----------------------------

                                   P(E│K)

If we estimate .25 to indicate a relatively low prior probability (low for me, high for most skeptics) of the truth of the resurrection as a hypothesis, as well as .25 for the probability of our having the particular sort of evidence for it that we do have (high for me, low for most skeptics), and .4 to mean somewhat modest "predictive power" of the hypothesis, we have: 

 

             .4 x .25

P(H│E & K) = ----------- = .4

                           .25 

So the resurrection in this scenario would be slightly improbable. Does this mean that we should be committed to believe with precisely 40% confidence that the resurrection actually occurred, or to withhold belief until probability exceeds .5? Not necessarily. 

 

A rational approach would surely consider not only the probability of hypotheses being true or false, but the consequences that would follow from those hypotheses being true or false. There would seem to be nothing especially rational about my taking a shortcut to work by crossing a bridge high over an icy lake where the probability of its collapsing sometime within the next year is "only" .3, for example; or moving into a neighborhood where a full 60 percent of the residents have never been physically assaulted because it's closer to my favorite shopping center. A good and prudent soldier does not lay down his arms whenever the probability of his survival dips below .5 in the heat of battle (indeed, he is far too busy to bother with such calculations, knowing that he maximizes his probability of survival by continuing to fight). A truly rational outlook, then, knows the difference not merely between true and false, or between probable and improbable, but between wisdom and folly.  

 

What this means it that a rational approach should take into account the expected value of a given decision, in addition to the probability of its success. Whereas expected value is a sophisticated statistical concept, it can be defined informally for non-mathematical sorts like me as "the weighted average of the values that X can take on, where each possible value is weighted by its respective probability"[1] – and where X is a random variable, meaning a variable with different probabilities corresponding to different possible outcomes. For example, a business owner might use an expected value approach to determine the more promising of two locations for a new plant, the one with the highest expected payoff given its probability of success and its profitability if successful. 

 

Now let us imagine an admittedly arbitrary but finite "payoff" scale of utils[2] to represent the level of happiness or satisfaction that believers and skeptics should expect to receive given the truth value of the resurrection – and given the traditional theological position that the resurrection of Jesus ensures eternal reward for believers, eternal judgment for unbelievers. Let's suppose that 1,000,000,000 here represents the ultimate prize of enjoying eternal life in the kingdom of heaven, in fellowship with Jesus Christ, along with a host of angels and redeemed believers. This signifies the reward of believers given that the resurrection hypothesis is true. 1, on the other hand, is the “loneliest number,” here meaning the despair of eternal judgment upon sin in the kingdom of darkness, in subjection to Satan, along with a host of demons and desolate unbelievers. In other words this is the complete, or virtually complete, absence of hope, joy or satisfaction – the reward of nonbelievers given that the resurrection hypothesis is true. In between are values representing various less extreme levels of expected satisfaction, depending on possible outcomes and their probabilities. In this way each decision outcome can be assigned an expected level of reward on the scale.

 

For the somewhat conservative (at least for a believer like me) probability estimate for the resurrection mentioned above, this leaves the following:

 

Posterior Probabilities:

 

P(R), probability that the resurrection hypothesis is true = .4

P(~R), probability that the resurrection hypothesis is false = 1 - .4 = .6 

 

Expected Value of Outcomes:

 

For believers, expected value of R = 1,000,000,000 x .4 = 400,000,000.  Expected value of ~R for believers = +/- 500,000 x .6 = 300,000. As mentioned above, the 1,000,000,000 represents the maximum level of satisfaction a soul can enjoy in principle. The admittedly arbitrary number of 500,000 for the believer means here something like: "life on earth is not too terrible, and still has its rewards – but it's not anything like what eternal life will be." (The important thing to keep in mind here is that this latter number is roughly equal for both believers and unbelievers.) So on our subjective-arbitrary scale the total expected value for believers = 400,000,000 + 300,000 = 400,300,000.

 

For nonbelievers, expected value for R = 1 x .6 = .6. (This is a level of satisfaction slightly lower than the lowest on our scale, only because the lowest whole number on the scale is multiplied by a probability of less than one, so we can round this up to one). Expected value of ~R for unbelievers = +/- 500,000 x .6 = 300,000 (the same as believers). Again, 1 represents the lowest possible level of happiness a soul can experience in principle. And for nonbelievers, the 500,000 means something like: "life on earth is not too terrible, and still has its rewards – it may not be paradise, but it could always be worse." So for the same arbitrary scale the total expected value for nonbelievers = 1 + 300,000 = 300,001.

 

The basic idea here is that in terms of the extreme consequences at stake, it would be more rational to accept even a slightly improbable position of faith in Christ because of the substantially higher potential reward. It's true that faith in Christ is costly, in that Jesus calls us to take up a cross, deny self and follow him (Luke 9:23). We are to invest our lives in heaven, not on earth (Matt. 6:19-21). But faith also pays compensating dividends in the way of fruit of the Spirit and a deep sense of purpose and calling. Circumstantially, there is no appreciable difference between the life experiences of believers and nonbelievers. Rain falls on the just (or justified) and the unjust (Matt. 5:45). Since life on earth has its ups and downs, pains and pleasures, joys and heartaches, etc., for everyone, the venture of faith, while costly, is relatively low-risk. 

 

Of course all this is essentially a restatement of Pascal’s Wager. Pascal developed his “wager” as something akin to the risk-reward principle that operates in all of life, from business and finance to romance. Given the relative scarcity of happiness on earth, the abundance of joy in the eternal kingdom of heaven, and the considerable prospect that the Christian gospel is true, it seems to me, as it did to Pascal, that to repent and believe in Christ would be the wisest investment one could ever make. I would say more about Pascal and his development of the wager, but unfortunately I am, like the rest of us, running out of time. 

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[1] "Expected Value," Statlect, https://www.statlect.com/fundamentals-of-probability/expected-value.

 

[2] “In microeconomics, happiness is measured by a concept called utility. The standard unit of measurement that microeconomics uses to measure utility is called the util…. The util has no concrete numerical value like an inch or a centimeter. It is merely an arbitrary, subjective and convenient way to assign value to consumer choices and to measure the consumer utility or utils of one choice against another choice.” – Marc Davis, “Microeconomics: Assumptions and Utility,” Investopedia, www.investopedia.com/university/microeconomics/microeconomics2.asp.