Introduction to Philosophy Lecture 6 Pascal’s wager By David Kelsey

Pascal

• • • Blaise Pascal lived from 1623-1662.

He was a famous mathematician and a gambler.

He invented the theory of probability.

Probability and decision theory

• Pascal thinks that we can’t know for sure whether God exists.

•

Decision theory:

used to study how to make decisions under uncertainty, I.e. when you don’t know what will happen.

–

Lakers or Knicks:

–

Rain coat:

•

Rule for action:

when making a decision under a time of uncertainty always perform that action that has the highest expected utility!

Expected Utility

•

The expected utility for any action

: the payoff you can expect to gain on each trial if you continued to perform trials...

– It is the average gain or loss per trial.

–

A trial:

is a an attempt at achieving success. • Example… –

The payoff or value of an outcome:

occurs.

what is to be gained or lost if that outcome • •

To compute the expected value of an action:

– ((The prob. of a success) x (The payoff of success)) + ((the prob. of a loss) x (the payoff of a loss))

Which game would you play?

–

The Big 12:

pay 1$ to roll two dice. –

Lucky 7

: pay 1$ to roll two dice. – –

E.V. of Big 12: E.V. of Lucky 7

:

Payoff matrices

•

Gamble

: Part of the idea of decision theory is that you can think of any decision under uncertainty as a kind of gamble.

•

Payoff Matrix

: used to represent a scenario in which you have to make a decision under uncertainty.

– – –

On the left:

our alternative courses of action.

At the top:

the outcomes.

Next to each outcome:

add the probability that it will occur.

–

Under each outcome

: the payoff for that outcome •

Calling a coin flip:

– If you win it you get a quarter and if you lose it you lose a quarter.

• • • The coin comes up heads: ___ It comes up tails: ___ You call heads ___ ___ You call tails ___ ___

The Expected Utility of the coin flip

• •

So when making a decision under a time of uncertainty

: construct a payoff matrix

Which action:

– Perform the action with the highest expected utility!

–

To compute the expected value of an action:

• ((The prob. of a success) x (The payoff of success)) + ((the prob. of a loss) x (the payoff of a loss)) • •

For our coin tossing example:

–

The EU of calling head:

• … –

The EU of calling tails:

– …

Choose either action…

Another coin tossing game

•

Different payoffs:

what if the payoffs were greater when the coin comes up heads than if the coin comes up tails.

• • • It comes up heads: ___ The coin comes up tails: ___ You call heads ___ ___ You call tails ___ ___ •

The EU if you call heads:

– … •

And the EU if you call tails:

– … •

So Call Heads!

Taking the umbrella to work

•

Do you take an umbrella to work?

You live in Seattle. There is a 50% chance it will rain. – – –

Taking the Umbrella

: a bit of a pain. You will have to carry it around.

• Payoff = -5.

If it does rain & you don’t have the umbrella

: you will get soaked • payoff of -50.

If it doesn’t rain then you don’t have to lug it around:

• payoff of 10. • • • • • It rains (___) It doesn’t rain (___) Take umbrella ___ ___ Don’t take umbrella ___ ___

EU (take umbrella)

= …

EU (don’t take umbrella)

• Take the umbrella to work!

= …

Pascal’s wager

•

Choosing to believe in God

: Pascal thinks that choosing whether to believe in God is like choosing whether to take an umbrella to work in Seattle.

–

It is a decision made under a time of uncertainty:

–

But We can estimate the payoffs

: •

Believing in God is a bit of pain

whether or not he exists: •

An infinite Reward: …

•

Infinite Punishment

: …

Pascal’s payoff matrix • • • Believe Don’t believe God exists (___) God doesn’t exist (___) ____ ____ ____ ____ •

Assigning a probability to God’s existence:

– A bit tricky since we don’t know.

– For Pascal: • since we don’t know if God exists we know the probability of his existence is greater than 0.

– EU (believe) = … – EU (don’t believe) = …

•

Believe in God: …

Pascal’s argument

•

Pascal’s argument:

– 1. You can either believe in God or not believe in God.

– 2. Believing in God has greater EU than disbelieving in God.

– 3. You should perform whatever action has the greatest EU.

– 4. Thus, you should believe in God.

•

Not existence but Belief:

…

Denying premise 1

•

The first move:

–

Deny premise 1:

•

The second move & Pascal’s reply:

–

Believing for selfish reasons

:

Denying premise 2

• Deny premise 2: –

Infinite payoff’s make no sense:

–

Can we even assign a non zero probability to God’s existence?

The Many Gods objection

•

We could Deny premise 2 in another way:

–

Many Gods & the Perverse Master…

The Perverse Master

• • • •

The new payoff matrix:

God exists (__) Perverse Master exists (__) Neither exists (___) Believe _____ _____ ___ Don’t Believe _____ _____ ___ •

Disbelief seems no worse off than belief:

–

EU (believe)

= … –

EU (don’t believe)

= … • What if we thought it

less likely that the perverse Master exists than does God: