Philosophy 1600 - Honors Seminar

 

Topic 1: Faith, Rationality, and the Ethics of Belief

Pascal's Wager

        Necessary Background Material: The Expected Value of an Action

 

Possible Outcomes:                         Denver Wins           Green Bay Wins                  Some team other than
                                                                                                                                                Denver or Green Bay wins

            Your subjective probability:                              0.2                               0.3                                               0.5

            Possible bets (below), and
            Payoffs (to the right)

            Bet that Denver wins:                                         $6                                   -$1                                             -$1

            Bet that Green Bay wins                                    -$1                                   $2                                             -$1

  Make no bet at all                                                $0                                    $0                                              $0

 

(The payoffs are determined by the odds offered by the bookmaker, which, in the table above, are assumed to be 6 to 1 in the case of a bet on Denver, and 2 to 1 in the case of a bet on Green Bay.  In all cases, one is betting only $1.)

How does one calculate which of the three possible actions listed above is best?  The thing to do is to calculate what is called the "expected value" of each action, and see which has the highest expected value.  To calculate the expected value of an action, one looks at each possible outcome, multiplies the gain or loss given that outcome by the probability of the outcome, and then adds up the results to get the expected value of the action in question.

Illustrations

The expected value of betting that Denver will win  =

[(Probability that Denver will win) x (Gain or loss if Denver wins and one has bet that Denver will win) + (Probability that Green Bay will win) x (Gain or loss if Green Bay wins and one has bet that Denver will win) + (Probability that some other team will win) x (Gain or loss if some team other than Denver or Green Bay wins and one has bet that Denver will win)]  =

[(0.2 x $6) + (0.3 x -$1) + (0.5 x -$1)  =  $1.2 - $0.3 - $0.5  =  $0.4.

Similarly, the expected value of betting that Green Bay will win  =

[(0.2 x -$1) + (0.3 x $2) + (0.5 x -$1)  =  -$0.2 + $0.6 - $0.5  =  -$0.1.

 

 One Version of Pascal's Wager Argument

 

Possible Ways the World Might Be:        Pascal's God Exists              Pascal's God Does Not Exist                                                                                                             

Probability:                                                          Perhaps very small,                    Perhaps very large,
                                                                                but at least not zero.                   but less than one.

Possible bets (below)                                                            Payoffs in Each Case  (below)

Believe that Pascal's                                             + Infinity                                   Finite   
God exists                                                                                                                  (May be either positive or negative.)

Refrain from believing                                       – Infinity                                    Finite   
that Pascal's God exists                                                                                          (May be either positive or negative.)

 

Comment

Pascal's God" means a deity that conforms to Pascal's conception of God, and, in particular, a deity that (1) provides anyone who believes in his existence, for whatever reason, and however good or bad the person may be, with eternal happiness, and, similarly, (2) causes anyone who does not believe in his existence, for whatever reason, to suffer forever.

One way in which Pascal's God may fail to exist is if there is an anti-Pascalian God, defined as a deity that (1) causes anyone who believes in Pascal's God, for whatever reason, to suffer forever, and (2) provides anyone who does not believe in Pascal's God, for whatever reason, with eternal happiness.  So consider the following table, where this possibility is made explicit:

Possible Ways the          Pascal's God          The Anti-Pascalian         Neither Pascal's God
World Might Be:             Exists                      God Exists                        nor the Anti-Pascalian
                                                                                                                        Exists                                                                                                                                                                                                                    

Probability:                          Perhaps very ,           Perhaps very,                          Perhaps very                                                
                                                small, but at              small, but at  large,               large, but
                                                least not zero.           least not zero.                         less than one.

                                                Let this                       Let this                                   Then this
                                                probability be p       probability be q                   equals 1 – (p+q)

Possible bets (below), and
Payoffs (to the right)

Believe that Pascal's                       + Infinity                   - Infinity                      Finite
God exists                                                                                                                   (May be either
                                                                                                                                      positive or negative.)

Refrain from believing                   - Infinity                    + Infinity                   Finite               
that Pascal's God exists                                                                                           (May be either
                                                                                                                                     positive or negative.)

When we notice that one of the possibilities that may obtain if Pascal's God does not exist is that of the anti-Pascalian God, we see that there may be infinite rewards and punishments that run in exactly the opposite direction from those that Pascal considered.  The two sets of infinite rewards and infinite punishments will then exactly cancel one another out, unless there is some reason for thinking that p is greater than q, or that q is greater than p.