salvaging pascal’s wager liz jackson and andy rogers

Salvaging Pascal’s Wager

Liz Jackson and Andy Rogers

Pascal’s Wager

God Exists (G) God doesn’t exist (~G)

Believe in God (BG) Infinite Pleasure Nothing

Don’t Believe in God (~BG)

Infinite Pain Nothing

BG weakly dominates ~BG if there is at least one possible state where the outcome given BG is better and no state where the outcome given ~BG is better. This is true if you either only include the afterlife or if you think that living as an atheist provides equal happiness to living as a theist.

Pascal’s Wager


Believe in God (BG) Infinite Pleasure Finite Pleasure or Pain


Infinite Pain Finite Pleasure or Pain

BG does not dominate ~BG if you think that living as an atheist provides more happiness than living as a theist.

BG strictly dominates if you think that living as a theist provides more happiness than living as an atheist (William James).

Pascal’s Wager


Believe in God (BG) Infinite Positive Utility

Finite Utility


Infinite Negative Utility

Finite Utility

For most of the argument we will assume that positive utility is pleasure and negative utility is pain. There are many people who object to thinking of utility this way and we will address that issue at the end.

Wager with Probabilities

God Exists (G)

God doesn’t exist (~G)

Expected Value (EV)

Believe in God (BG)

Infinite Pleasure *

Pr (G) = ∞

Pr (~G) * 0 =

0∞


Infinite Pain *

Pr (G) = -∞Pr (~G) * 0 =

0-∞

Wager with Probabilities

God Exists (G)

God doesn’t exist (~G)

Expected Value (EV)

Believe in God (BG)

Infinite Pleasure *

Pr (G) = ∞ Pr (~G) * f = f ∞


Infinite Pain *

Pr (G) = -∞Pr (~G) * f = f -∞

f = some finite number

As long as your credence in G is more than zero you end up with the EVs seen above.

Objection 1: Many Gods Objection (Sober and Mougin)

One common objection to Pascal’s Wager is to point out that Pascal’s version of the Christian God isn’t the only God possible; the Gods of other religions need to be included in the decision matrix. Many of these religions are mutually exclusive, and believing the truth of one religion will often not give you the payoff of another. For example, if one adds a Muslim God who sends Christians like Pascal to hell, then the results become inconclusive. Any set of non-zero values for Pr (Christianity) and Pr (Islam) will give us the following, somewhat confusing, results.

Objection 1: Many Gods Objection (Sober and Mougin)

(C) (I) (A) EV

B(C) ∞ -∞ f ∞ + -∞ B(I) -∞ ∞ f ∞ + -∞ B(A) -∞ -∞ f -∞ + -∞

(Christianity)

(Islam) (Atheism)

B(C) Pr (C) * ∞ Pr (I) * -∞ Pr (A) * f

B(I) Pr (C) * -∞ Pr (I) * ∞ Pr (A) * f

B(A) Pr (C) * -∞ Pr (I) * -∞ Pr (A) * f

Atheism +(C) (I) (A+) EV

B(C) ∞ -∞ -∞ ∞ + -∞ + -∞

B(I) -∞ ∞ -∞ ∞ + -∞ + -∞

B(A) -∞ -∞ ∞ ∞ + -∞ +-∞

(A+): all theists go to hell and all atheists go to heaven

Objection 2: Mixed Strategies Objection (Hájek)

Any decision you make that includes the positive probability that you will eventually come to believe in God has an infinite expected value.

Coin Flip Example

Probability Money EV($)

Play the game 0.5 ∞ ∞

Flip 1x to play 0.25 ∞ ∞




Someone offers you a betting game where you can flip a coin: if you get heads you receive an infinite amount of money and if you get tails nothing happens.

Deciding to play the game has the same expected value as deciding to first flip a coin and then only play the game if you get heads or deciding to flip any finite number of coins and then only play the game if they are all heads.

Mixed Strategies

.

“Wager for God if and only if a die lands 6 (a sixth times infinity equals infinity); if and only if your lottery ticket wins next week; if and only if you see a meteor quantum-tunnel its way through the side of a mountain and come out the other side ... Pascal has ignored all these mixed strategies - probabilistic mixtures of the "pure actions" of wagering for and wagering against God - and infinitely many more besides. And all of them have maximal expectation. Nothing in his argument favors wagering for God over all of these alternative strategies.”

No Deciding Between Infinities

.

What we’re left with after Sober&Mougin and Hajek’s objections is a situation where it appears that we can’t rationally prefer one option over another when infinite utlities are involved.

Thought Experiments

ONE(A) receiving $1 per day for an infinite amount of time (B) receiving $100 per day for an infinite amount of time

TWO(A) Something similar to a relatively happy moment lived over and over (B) Something similar to the best moment of your life lived over and over

THREE(A) Heaven: 2 pleasure 1 pain(B) Heaven’: 2 pleasure 0 pain

If we can’t rank infinite utilities then we cannot capture the intuition that in each case mentioned above B seems preferable to A.

Our Solution

Pleasure Per Period: First, we will distinguish the amount of pleasure experienced in the moment from the duration of time for which one gets to experience pleasure

Ratio in the Limit: The second way in which we want to deal with infinity differently is that we want to focus on finding the ratio in the limit between two (or more) rewards, instead of simply multiplying everything by infinity

Our Solution

THESIS: The infinite utilities in Pascal’s Wager can be expressed as finite utilities per unit of time experienced over an infinite amount of time. Then, they can be ranked by looking at the ratio of the utility per unit of time between the religions as the length of time approaches infinity. Other factors can be included in the calculation as well, such as the probability that the religion is true, ease of conversion, moral and epistemological objections, and whether the religion allows you to hedge your bets by continuing to research other religions.

‘Religion’ – a worldview that provides some information about future gains and/or losses based on one’s actions and/or beliefs

Our Solution

Is this still “Pascal’s Wager” in any important sense? When we say we are salvaging the wager, we take the important part of the wager to be that it is a decision theoretic apparatus that favors religions which promise/threaten an infinite afterlife over those which do not. Using our approach, an infinite religion with a non-zero credence will be favored over a finite religion.

Our Solution: Example 1

Religion A: If you meet the standards of the religion, then God rewards you with 10 units of pleasure per hour for an infinite length of time. Peter assigns a credence of .3 to this religion.

Religion B: If you meet the standards of the religion, then God rewards you with 20 units of pleasure per hour for an infinite length of time. Peter assigns a credence of .7 to this religion.

x: Amount of time (substitute for ∞)

(10*(.3)*x) equals 3 (20*(.7)*x) 14

We are left with a 14:3 (B:A) ratio in favor of religion B. Peter ought to choose to believe religion B in order to maximize EV.

Example 2

Peter must choose between 8 religions

Religion Credence

Pleasure

Suffering

Time Period

Atheism (A) (0.5) 10p 10s 100 yrs

Universalism (U)

(0.2) 10p 0

Plato (Pl) (0.1) 10p 10s 10,000 yrs

Buddhism (B) (0.1) 10p 10s 100 Trillion yrs

Anglican (An) (0.03) 10p 10s

Catholicism (Ct)

(0.03) 10p 10s

Islam (Is) (0.03) 10p 20s

Calvinism (Cv) (0.01) 10p 20s

For 1 year

For 100 years

For 10,000 years

(k=1,000)

For 100 Trillion years(T=Trillion)

For 100 googol years

(g=googol)

For 100 googolplex years

(gp=googolplex)

1 year 100 years 10,000 years 100 trillion years 100 googol years 100 googol-plex years

-700

-600

-500

-400

-300

-200

-100

0

100

200

300

AtheismUniversalismPlatoBuddhismAnglicanismCatholicismIslamCalvinism

The ratios in the limit

As we multiply by larger and larger numbers the ratio of the EVs will even out to 6:6:6:6:12:12:15:9 or 2:2:2:2:4:4:5:3.

The ratio as the length of time approaches infinity is 2:2:2:2:4:4:5:3 for A:U:Pl:B:An:Ct:Is:Cv

In order to maximize EV, Peter ought to choose Islam.

Responding to Sober and Mougin

One could add Atheism+ and, if the credence were high enough, it could cause atheism to have an expected value that competes with the traditional religions. However, the objection only applies when the credence for Atheism+ is sufficiently high.

Responding to Hajek

Directly believing the top religion (the one with the highest EV) is always going to be preferable to the mixed strategy of flipping a coin and believing the top religion only if heads. This is because one now needs to multiply (the value per year * credence) by 0.5 [the probability of getting heads]. This mixed strategy will change the final ratios.

Objection 1:Psychological objection Objection: It is difficult or impossible to believe or convert to some religions.

Response: This can be incorporated into the framework by multiplying the (pleasure per year * credence) by the probability that attempting to convert will be successful.

Objection 2:Moral objection

Objection: It is morally wrong to believe on the basis on expected value (instead of on the basis on evidence).

Response: This can be incorporated into the framework by assigning a cost to breaking this moral rule and subtracting that cost from (pleasure per year * credence * probability of successful conversion).

Objection 3: Bet-hedging (Kenny Boyce)

Objection: This framework does not account for the cost associated with a religion that does not allow you to ‘hedge your bets.’ In other words, many religions do not allow you to think critically about the religion or research other worldviews.

Response: This can be incorporated into our framework by assigning a value to the degree to which a religion will allow you to hedge your bets by thinking critically and researching other religions. For example, you could do this by adding ((the probability that thinking critically and researching would lead to you finding sufficient evidence such that the religion with the highest EV changes) * (the probability that you are wrong) * (the suffering you would receive for being wrong)) to ((pleasure per year) * (credence) * (probability of successful conversion)) – (the cost of breaking a moral rule by believing based on expected value).

Objection 4:Super-religion (Salvatore Florio)

Objection: Someone could create a religion that promises a heaven where you receive the highest amount of pleasure promised by any religion so far times a million. A “super-religion” could stack the deck such that they always come out with the highest EV.

Response: There is a finite, physical limit to how much pleasure a finite human can experience in one moment. Many religions (Islam, Christianity) probably already promise an amount of pleasure close to this limit, if not meeting it. In other words, the traditional religions are super-religions.

Objection 5:Lukewarm religions (Amy Seymour)

Objection: This framework favors religions that only require a very low credence to get the rewards of a religion. For example, if 10 religions all only required a 0.1 credence on order to receive their rewards, then the expected value of believing each of those religions at 0.1 could be very high.

Response: This is true if you have a high enough credence for these lukewarm religions, but this could be avoided by assigning a low credence to those religions.

Objection 6:Diminishing Marginal Utility (Elliott

Wagner)

Objection: You could reject the pleasure/pain notion of utility and assign utility in a different manner. One way in which this could be an issue is if you assign a lower utility to future pleasure than to present pleasure. For example, you could assign X units of utility to X units of pleasure today, 1/2X units of utility to X units of pleasure tomorrow, 1/4X units of utility to X units of pleasure the day after that and keep decreasing by half per day.

Response: Long-term we would like to address this because we believe the pleasure/pain notion of utility we have been using is correct but this would require substantial further argument.

Conclusion

In this presentation, we have argued for a method of decision making in regards to competing worldviews which takes seriously two powerful objections to Pascal’s Wager: that there are many possible Gods offering infinite rewards (Sober and Mougin) and that one can use mixed strategies when choosing between infinite religions (Hajek). As in the standard version of Pascal’s Wager, our proposed method will advantage religions with infinite afterlives, but unlike the standard version of Pascal’s Wager, our proposed method will also take into account other factors such as credence and ease of conversion.

salvaging pascal’s wager liz jackson and andy rogers

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