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Pascal’s Wager
Blaise Pascal was one of several founder’s of the Theory of
Probability, which lead to modern statistics, and founded the
Theory of Rational Choice.
The Theory of Rational Choice includes two sub-theories:
Decision Theory and Game Theory.
Suppose one must decide among a range of alternative options
about what to do in a given situation, where each alternative can
lead to one of several different outcomes, how should one decide
which alternative to pick? There are three kinds of cases:
Decision Under Certainty:
One knows for sure what outcomes will occur, depending on what
one does.
Decision Under Risk:
One doesn’t know which outcomes will occur, but you know,
given your actions, what the chances or probability is that each
outcome will occur.
Decision Under Uncertainty:
One knows what the various possible outcomes are, but one does
not know what their probabilities are.
Decision Under Certainty.
Simplest Case: Solvable Zero-Sum Games against a Rational
Opponent.
Game1: Each player has two strategies: Rock and Scissors. Both
players choose a strategy ‘in the blind’, i.e. without information
about how the other has chosen. There are four combinations of
strategies, RR, RS, SR, RR. Players tie if they choose the same
strategy; if RS or SR occurs, the player choosing R wins. A win is
worth $2, a tie is worth $1. How should you play?
1
2
1
0
One Strategy Dominates another if and only if
1. The first never does worse than the second.
2. The first sometimes does better than the second.
Here R dominates S for both players. Both players will therefore
always choose R, and each will receive $1 each time the game is
played.
Rational Rule 1: Never choose a dominated strategy.
Decision Under Risk.
Game 2: You are choosing where to spend spring break. You can
go skiing in Colorado or to the beach in Florida. Skiing is much
the best option if there is good snow, being, in your judgment,
twice as much fun as sunning on the beach in good weather.
Sunning on the beach is twice as much fun as merely hitting the
clubs at Daytona Beach, which is all there is to do if the weather
turns bad in Florida. But the clubs at Daytona are three times as
good as those in Colorado, so the worst possible option is to go to
Colorado and get bad weather. You know (from reliable
forecasts) that the chance of rain in Florida is 50%, while the
chance of good snow in Colorado is 30%. What should you do?
12
1
6
3
Calculate the expected utility of each option, and choose the option
with the highest expected utility.
EU(S)=Pr(O1)V(O1)+Pr(O2)V(O2)
EU(Co)=.3(12) + .7 (1) = 3.6+.7=4.3
EU(Fl)=.5(6) + .5 (3) = 3 + 1.5 = 4.5
So, go to Florida.
Rational Rule 2: If you can calculate expected utilities (this
means you must know two things: how much you value each
possible outcome on all available strategies, and what the
probability of those outcomes is), always choose the strategy with
the highest expected utility.
Decision Under Uncertainty.
Game 3: You are offered the following wager: for $100 you make
buy a contract which pays out various amounts depending on
whether the US invades Iraq before June 1 2003.
If you decline the contract, you get 0 whatever happens. If you
accept the contract (so you pay $100) and the US invades before
June 1 you will be paid $110 (for a net gain of $10), while if the
US does not invade before June 1 you will be paid nothing (for a
net loss of $100). What should you do?
+10
-100
0
0
If you new that it was very very likely that the US will invade
before June 1, you might take the gamble, on the grounds that
while you might loose money, the overwhelming probability is that
you would win money. But perhaps you aren’t so sure about the
probabilities. In this case, you might reason as follows—if you
take the gamble, you risk loosing 100 bucks, which would be very
bad. On the other hand, if you decline, you risk nothing, so it is
the preferred option.
The reasoning here employs a principle called Maximin: choose
the strategy whose worst possible result is the least bad. Maximin
decision making is Risk Averse—it says avoid the worst possible
result at all costs, even if this means giving up on a chance at a
very large gain.
A strategy S is a Maximin strategy iff there is no alternative
strategy R such that the worst possible outcome on R is at least as
good as the worst possible outcome on S.
Rational Rule 3: If there is a Maximin strategy, choose it.
Pascal’s Wager.
Pascal reasons as follows. We do not know whether God exists or
not, and there is no way to discover this with certainty. But we
might assume that it is even odds whether or not He does, since
there are two, perhaps equally likely, alternatives: He exists, or He
doesn’t. We have two strategies open to us: choose to believe
God exists (on faith), or not to do this. A choice is forced, since to
remain agnostic is in effect not to believe in God’s existence. We
then consider the value of each possible outcome on each strategy.
We can believe correctly, believe wrongly, not believe when God
does exist, or not believe when he doesn’t exist. Suppose we
believe, and God does exist. We are rewarded by an eternity in
heaven, which we might regard as an infinite gain. Suppose we
believe and God does not exist. Then we are out some time (spent
in Church), some money (in tithes), and some fun (in the good
stuff we have to give up). So there is some finite cost in this case.
What is the expected utility of believing?
EU(B)=.5(+)+.5(-F)=+
What, conversely, is the expected utility of not believing? Well, if
we don’t believe and God exists, we spend eternity in hell, which
we might reasonably regard as a disaster of infinite magnitude.
But if we don’t believe and God doesn’t exist, then we are out
nothing.
EU(~B)=.5(-) + .5(0)=-
The choice is clear—believe.
One need not even use the assigned probabilities. Game theory
yields the same result for decision under uncertainty:
+
-F
-
0
We ought choose a maximin strategy if there is one, since this is
decision under uncertainty. Clearly, the worst possible outcome is
to not believe when God exists. No outcome that is possible if one
believes is nearly as bad. So believing is a maximin strategy, and
we ought to adopt it.
Problem. It is crucial when using decision theoretic machinery to
make sure you have correctly: assigned values to each outcome,
noted all the possible outcomes, and considered all the possible
strategies. In this case, Pascal has failed to consider all the
possible outcomes. In particular, he assumes that the only possible
God is the Christian God. But there are others (in fact an infinity
of others). Explicit consideration of 1 omitted alternative will do
the trick here. It is possible that the Christian God doesn’t exist,
but the philosopher’s God does. The philosopher’s God is
peculiar—She rewards with Hell those and only those who believe
on good evidence, and withhold belief when such evidence is
lacking. All others she punishes to eternal damnation. So now our
payoff matrix looks like this:
+
-F
-
-
0
+
Now there is no longer any game theoretic reason to prefer
believing over not believing. And if we assign equal probabilities
to each alternative outcome, each strategy will have the same
expected utility.
So Pascal was wrong to think that we have rational grounds for
believing in God’s existence even if we lack evidence that He
exists. But the formulation of his argument lead to important new
theories about rational decision, to which we will recur latter in the
course.
The Problem of Evil.
There are two versions of the problem of evil. Both challenge
belief in God by noting the fact that the world contains evil; in
particular, people suffer without having done anything to deserve
that suffering. The traditional problem of evil can be given in the
form of an argument, and it goes like this.
1. If God exists, he is omni-benevolent, omniscient and
omnipotent.
2. Benevolent beings do not allow innocents to suffer if they can
prevent this.
3. Innocents suffer.
4. Assume God exists.
5. God knows that innocents suffer (since he is omnicscient).
6. God could if He chose prevent this suffering (since he is
omnipotent).
7. Hence, God is not omni-benevolent.
This contradicts the definition of God in 1, so
8. God does not exist.
The problem of evil, in this form, is a traditional puzzle in both
Theology and Philosophy of Religion. Attempts to rebut the
argument are called ‘Theodicies’. Two important Theodicies are
reprinted in part in the text. In general, a Theodicy attempts to
show that it is possible for the suffering of innocents to serve some
greater good which an omni-benevolent being is bound to pursue
(i.e. it is possible that there is some good, such that a being who
permits evil in order to make it possible to achieve this good is in
fact acting benevolently).
A second form of the problem of evil is different. It allows that a
Theodicy might be correct, in that the suffering of innocents might
serve some larger and overriding good. But it goes on to deny that
this good could in any way make up for the evil necessary to
secure it, so that it is wrong to endorse the project, no matter how
great the good, which requires the suffering of innocents. This is
Dostoevsky’s complaint-- the children who suffer as part of God’s
plan are seriously wronged. We may forgive those who have
harmed them for whatever wrong they have done us, but we cannot
forgive them for the wrongs they have done the children—this is
not in our power to do, for these evils are not inflicted on us. But
God could have prevented these evils and didn’t. To endorse
God’s project, his plan, is to endorse the doing of this evil to the
children, and to forgive the wrongdoers and to encourage others so
to do. But to do this would be wrong.