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Pascal’s Wager

The Economics of Evangelism: The Insights of Decision Theory Mark Pingle Department of Economics University of Nevada, Reno, USA pingle@unr.edu. Pascal, 1670, Pensees 194 : …I know …I must soon die, but what I know least is the very death I cannot escape.

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Pascal’s Wager

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  1. The Economics of Evangelism: The Insights of Decision TheoryMark PingleDepartment of EconomicsUniversity of Nevada, Reno, USApingle@unr.edu

  2. Pascal, 1670, Pensees 194:…I know …I must soon die, but what I know least is the very death I cannot escape.

  3. Adam Smith, Theory of Moral Sentiments, 1759, Part I, paragraph I.I.13: “The foresight of our own dissolution is so terrible to us …. [that it] makes us miserable while we are alive… . The dread of death [is] the great poison to happiness…. it afflicts and mortifies the individual.” • Adam Smith, Theory of Moral Sentiments, 1759, Part I, paragraph I.II.11: Fear … is a passion derived altogether from the imagination, which represents, with an uncertainty and fluctuation that increases our anxiety, not what we really feel, but what we may hereafter possibly suffer.

  4. Pascal’s Wager Pascal, 1670, Pensees 233: “God is or He is not. But to which side will we incline? … A game is being played at the extremity of this infinite distance where heads or tails will turn up. What will you wager? … You must wager. It is not optional… Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you win, you win everything, if you lose you lose nothing. Do not hesitate then; wager that he does exist.”

  5. Pascal’s wager: Presents faith as a choice made under uncertainty • Jeff Jorden, Chapter 1, Gambling on God: Essays on Pascal’s Wager: “Pascal’s wager brought about the advent of decision theory. • Iannaccone, L.R., Journal of Economic Literature, 1998: “For the most part, however, the problem of religious uncertainty has received little attention and scarcely any formal analysis.” • Theory Question: What theoretical conclusions about the faith decision can we derive from decision theory? • Policy Question: What should evangelists do to win converts?

  6. Is Faith a Choice? • Calvanism: No, God selects those who will be saved. • Arminianism: Yes, God wants all to be saved, and those who choose God will be saved.

  7. THE CONSENTOF THE PASTORS OF THE CHURCH OF CHRIST AT GENEVA, CONCERNING THE ETERNAL PREDESTINATION OF GOD, BY WHICH HE HAS CHOSEN SOME MEN UNTO SALVATION, WHILE HE HAS LEFT OTHERS TO THEIR OWN DESTRUCTION, AND ALSO THEIR CONSENT CONCERNING THE PROVIDENCE OF GOD, BY WHICH HE GOVERNS HUMAN AFFAIRS, SET FORTH ---BY JOHN CALVIN January 1st, 1552. “because the Gospel is called ‘the power of God unto salvation to everyone that believeth,’ [there are some who] would blot out under this pretext the election of God; whereas it ought to have entered into the minds of such to think from whence faith comes!”“whosoever shall hold faith to be the earnest and pledge of adoption, will assuredly confess that it flows from Divine election as its eternal source.”

  8. Where did Pascal Stand on Predestination? • Pascal (Pensees, 242): “God is a hidden God. … This is what the Scripture points out when it says in so many places that those who seek God find Him.” • Pascal (Pensees, 248): “Faith is different from proof; the one his human, the other is a gift of God.” • Pascal (Pensees, 253): “Two extremes: To exclude reason, to admit reason only.”

  9. Pascal (Pensees, 257): “There are only three kinds of persons; those who serve God having found Him; others who are occupied in seeking Him; while the remainder live without seeking Him and without knowing Him. The first are reasonable and happy, the last are foolish and unhappy; those between are unhappy and reasonable.” • Pascal (Pensees, 195): [To] live in indifference to the search for truth in a matter which is so important…, according to the principles of reason, … is wholly unreasonable. … Eternity exists, and death, which must open to it, … threatens …every hour … .

  10. Azzi and Ehrenberg (1975): First effort to explain religious participation in life as a deterministic function (i.e., a choice) of the perceived afterlife reward.

  11. Dilemma Version of Pascal’s Wager • The Dilemma: Better to believe in God if God exists, and better to not believe in God if God does not exist. God either exists or does not exist. What to choose?

  12. Decision-Making Under Knightian Uncertainty (Ignorance): • The decision maker cannot subjectively assign probabilities to the two states, so “probabilistically unsophisticated” decision rules must be applied • Optimist (or extremely wishful thinker) • Assumes the best of all worlds will arise • Acts to obtain the maximum payoff, valuing improvement • Applies the max-max decision rule • Seeking to obtain heaven, the choice is to believe in God

  13. Pessimist (or extremely prudent) • Assumes the worst of all worlds will arise • Acts to avoid the minimum payoff, valuing security • Applies the max-min decision rule • Seeking to avoid hell, the choice is to believe in God

  14. St Petersberg Paradox: (Nicholas Bernoulli, 1713) Flip a coin, and get one Euro if head. Then,Flip a coin twice, and get 2 Euros if both heads. Then …Flip a coin n times, and get Euros if all heads, …Expected value of the Original St Petersberg Game is • Paradox: People are not willing to pay an infinite amount of money to play this game. • Similarly, people might only ascribe finite value to infinite afterlife rewards.

  15. Modified St Petersberg Game: Let k>2 be the finite base reward for success, rather than 2. As before, the heavenly reward is infinite

  16. Discounting Heavenly Rewards

  17. As long as k is finite, then is finite when the discount rate r is large enough • No matter how large k gets, as long as k is finite, then a very high discount rate r can make an infinitely large heavenly reward seem small

  18. Non-Infinite Payoff Version of Pascal’s Wager • Hurwicz (Parameterizes optimistic to pessimistic decision makers) • Suppose payoff for heaven is not infinite, but rather is +M, and suppose payoff for hell is -M rather than negative infinity. Belief parameter measures degree of optimism (versus pessimism) , • Expected payoff for belief: • Expected payoff for non-belief: • Belief chosen over non-belief if and only if: • All will choose belief, from optimists to pessimists, if the perceived afterlife payoff (reward and punishment) is large enough.

  19. Let be the reward for heaven, and be the punishment for hell, to examine the more general case. • 1990 Gallop Poll (From Iannaccone, 1998): Life after Death: 71% Heaven: 75% Hell: 60% • Belief chosen over non-belief if and only if • A person tending toward optimism (α near one) more likely choose non belief when they also believe there is little reward for belief (MR near zero). • A person tending toward pessimism (α near zero) will more likely choose non belief when they also believe there is little punishment for non-belief (MP near zero).

  20. Implications for Evangelism: • Need to stress afterlife rewards for belief to attract the optimistic, or wishful. • Need to stress afterlife punishments for non-belief to attract the pessimistic, or prudent.

  21. Other Decision Rules to use Under Uncertainty (Ignorance) Regret Avoider • Considers regret, rather than utility level attained • Prefers to avoid regret • Applies min-max regret rule • Regret for mistaken belief: • Regret for mistaken unbelief: • Belief chosen over non-belief if and only if • To avoid the regret of getting hell rather than heaven, • the choice is to believe in God

  22. The Principle of Insufficient Reason Resolve the uncertainty by allocating equal probability to each mutually exclusive alternative • Expected value of Belief: • Expected value of Non-Belief: • Belief chosen over non-belief if and only if • Because the expected net reward for belief is greater than the expected net reward for non-belief, the choice is to believe in God.

  23. God Does Not Exist God 1 Exists God 2 Exists Not Believe in God Believe in God 1 Believe in God 2 Many Gods Version of Pascal’s Wager

  24. Pessimist chooses belief in God 1 because of magnitude of • Optimist chooses belief in God 1 because of magnitude of • Hurwicz decision maker chooses belief in God 1 because • Regret Avoider chooses to belief in God 1, facing • regret , in order to avoid the greater potential regret • that would be faced if either non-belief or belief in • God 2 were chosen.

  25. Applier of Principle of Insufficient Reason: • Expected value of Non-Belief: • Expected value of Belief in God 1: • Expected value of Belief in God 2: • Belief in God 1 chosen because and Alternative Assumption: and • Religion 1 will attract the pessimists, while religion 2 will attract the optimists. • A change in the degree of optimism can therefore lead to a change in religions

  26. Which religion attracts the regret avoiders and those applying the Principle of Insufficient Reason depends upon the size of relative to the size of Implications for Evangelism: • One religion will dominate if most people perceive that the religion offers the highest potential reward for belief and the highest potential punishment for non-belief. • Heaven, or a high reward concept like it, is needed to attract optimists • Hell, or a high punishment concept like it, is needed to attract pessimists

  27. Atheism will be adopted by: • Optimists who conceive the reward of a life of non-belief greater than any potential afterlife reward • Regret avoiders, and those applying the Principle of Insufficient Reason, who conceive the regret for not rightly being an Atheist to be greater than the regret for not rightly being a believer. • Pessimists who conceive the cost of a life of belief to be greater than any potential afterlife punishment

  28. Expected Utility “Probablistically Sophisticated” decision makers can construct a probability distribution over the different states of nature.

  29. Expected utility of belief: Expected utility of non-belief: Choose to believe if and only if Choose to not believe if and only if

  30. Even if the conceived net afterlife benefit associated with belief is extremely high relative to the perceived net benefit associated with non-belief, it is rational to choose non-belief if the perceived probability that God exists is small enough. • Conversely, if the perceived probability that God exists is high enough, then the benefit structure is also inconsequential. • When is extremely high relative to (B+C), a small change in the probability can change the choice from non-belief to belief or vice versa.

  31. Implications for Evangelism: • One set of non-believers will be those who believe it is not likely that God exists. • Convincingly stressing that the net afterlife benefit is extremely large, relative to any foregone earthly net benefit, will make it easier to convert a decision maker from non-belief to belief by increasing the probability that God exists. • A fundamental evangelistic objective should be to increase the subjective probability that people assign to the existence of God.

  32. Investment, Gambling, and Insurance An example payoff structure: (Assume values, not utilities) Expected value of Atheism: [0.10][-8]+[0.90][1] = 0.1 > 0

  33. Expected value of Theism: [0.10][8]+[0.90][-1] = -0.1 <0 • Cannot buy insurance against the possibility Atheism is not true • Risk averse decision-makers may choose to gamble on God rather than accept the downside risk associated with Atheism

  34. Theism as an investment: [0.10][10]+[0.90][-1]=0.1<0 • If expected value is positive, then Theism is not a gamble, but rather is an investment. • Subjective Risk of associated “investing in God” decreases when the subjective probability that God exists increases.

  35. Implications for Evangelism: • The concept of Hell, or some similar penalty for non-belief, has evangelistic value for attracting people who are risk averse • Non-belief may have perceived positive expected value, but it can be criticized as a risky investment • Even if belief is perceived to have negative expected value, Gambling on God is the only way to effectively insure against a large loss that may be associated with non-belief.

  36. Expected Value of Perfect Information • If you obtained perfect information that God does not exist, then you would choose to not believe, and you would receive the benefit B • If you obtained perfect information that God exists, then you would choose to believe, and you would receive the benefit • You believe God exists with probability , and that God does not exist with probability . Therefore, the “expected payoff of perfect information is:

  37. The expected payoff of belief is: • Therefore, the “expected value of perfect information,” when belief is chosen, (the difference between expected payoff of perfect information and the expected payoff of belief) is : • The expected value of perfect information will be small for believers who (1) are very confident that God exists, or (2) do not perceive much net benefit to the life of a non-believer. • The expected value of perfect information will be small for non-believers who (1) are very confident that God does not exist, or (2) do not perceive much net benefit to the afterlife of a believer. • The expected payoff of non-belief is: • Therefore, the “expected value of perfect information,” when non-belief is chosen, is:

  38. Implications for Evangelism: • Neither confident believers nor confident non-believers will be very motivated to listen to any message about the truth, because they expect to gain very little • Conversely, less confident believers and less confident non-believers will be more motivated to listen to evangelistic messages.

  39. Bayesian Updating When the conceived net afterlife benefit associated with belief is extremely high relative to the perceived net benefit B+C associated with non-belief, a small change in the probability can change the choice from non-belief to belief or vice versa. Bayes Rule has become synonymous with rational belief formation Let denote “posterior” probability that God exists, given some new information I. Let denote the “prior” probability that God exists, and let be the subjective probability the decision maker assigns to receiving information I, under the presumption that God exists. Let be the subjective probability the decision maker assigns to receiving information I, under the presumption that God does not exist.

  40. The Bayesian Updating Rule is:

  41. When will new information generate strong confidence that God exists? • When is large relative to , or when the likelihood of receiving the information if God exists is large relative to receiving the information when God if God does not exist. • When is large relative to , or when the person already is very confident that God exists

  42. Implications for Evangelism: • The information that will particularly increase the belief that God exists will be information that people would not expect to receive if God does not exist but would expect to receive if God exists (i.e., a supernatural experience). • For a person who is very confident that God does not exist, new information to the contrary will not dramatically increase the belief that God exists.

  43. Religion 1 True Religion 2 True Adopt Religion 1 Adopt Religion 2 A Model of Religious Choice Definition of Religion: A set of beliefs about life and the afterlife, such that adopting the religion yields a different set of perceived payoffs than non-adoption • Does not presume belief in the supernatural • Atheism: A null religion • Fundamental Assumptions (To which varying decision criteria • may be applied): • Religions are mutually exclusive: If one is true, • then the others are not true. • For each religion, finite expected payoffs can be assigned to adoption and non-adoption.

  44. Dominance Definition and Ignorance Theorem Policies that can facilitate dominance: • Provision of significant earthly benefits (carrot) • Earthly persecution for non-adoption (stick)

  45. A Model of Agnosticism • Probabilistically sophisticated decision maker • Probability that Religion 1 is true: P • Probability that Religion 2 is true (same as probability that Religion 1 is false): 1-P • Model agnostics as people who: • recognize they are not certain about the values of the probabilities they ascribe to the truth of the different religions • recognize the potential significance of just a small change in these probabilities First Alternative: Give up Agnosticism, and Choose

  46. Second Alternative: Remain Agnostic Assume this extracts an (anxiety) cost C The probability q is the likelihood of getting information that will change your subjective assessment of the probability that Religion 1 is true In this model, a decrease in the probability that Religion 1 is true implies an increase in the probability that Religion 2 is true

  47. Intuitive Results • Regret occurs if no new information is received because the waiting cost must be paid without their being an expected benefit • Regret occurs if new information is received, but the value added from the new information does not cover the waiting cost • Agnosticism is preferable to Theism if the expected change in probability can cover the cost of waiting in one of two ways: • Case 1 (Greater confidence): Increase in P is large enough to increase expected payoff of Religion 1 by more than enough to compensate for the waiting cost • Case 2 (Choice Reversal): Decrease in P is large enough to reverse the choice, making Religion 2 preferable to Religion 1, and compensate for the waiting cost

  48. Case 1: Technical Details Agnosticism preferred to Theism if • Since C>0, must be positive • Left side of inequality is expected increase in perceived well-being obtained from waiting • is the net benefit of choosing Religion 1 over Religion 2, under the assumption that Religion 1 is true, and Agnosticism more likely to be preferred if this net benefit is large • No matter how large the net benefit , the probability q of getting new information must be “large enough”, as must the increase in the subjective probability estimate that Religion 1 is true. • A high expected cost C of waiting discourages Agnosticism

  49. Case 2: Technical Details Agnosticism preferred to Theism if • Since C>0 and , must be negative • Left side of inequality is expected increase in perceived well-being obtained from waiting • Same issues as with case 1, but must also overcome the previously perceived net benefit

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