Introduction to Philosophy
Lecture 6
Pascal’s wager
By David Kelsey
Pascal
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Blaise Pascal lived from 1623-1662.
He was a famous mathematician and a gambler.
He invented the theory of probability.
Probability and
decision theory
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Pascal thinks that we can’t know for sure whether God exists.
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Decision theory: used to study how to make decisions under uncertainty, I.e.
when you don’t know what will happen.
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Lakers or Knicks:
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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
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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: what is to be gained or lost if that outcome
occurs.
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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))
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Which game would you play?
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The Big 12: pay 1$ to roll two dice.
Lucky 7: pay 1$ to roll two dice.
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E.V. of Big 12:
E.V. of Lucky 7:
Payoff matrices
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Gamble: Part of the idea of decision theory is that you can think of any decision under
uncertainty as a kind of gamble.
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Payoff Matrix: used to represent a scenario in which you have to make a decision under
uncertainty.
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On the left: our alternative courses of action.
At the top: the outcomes.
Next to each outcome: add the probability that it will occur.
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Under each outcome: the payoff for that outcome
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Calling a coin flip:
– If you win it you get a quarter and if you lose it you lose a quarter.
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The coin comes up heads: ___
You call heads
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You call tails
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It comes up tails: ___
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The Expected Utility
of the coin flip
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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))
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For our coin tossing example:
– The EU of calling head:
• …
– The EU of calling tails:
– …
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Choose either action…
Another coin tossing game
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Different payoffs: what if the payoffs were greater when the coin comes up heads than if
the coin comes up tails.
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It comes up heads: ___ The coin comes up tails: ___
You call heads
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You call tails
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The EU if you call heads:
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And the EU if you call tails:
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So Call Heads!
Taking the umbrella
to work
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Do you take an umbrella to work? You live in Seattle. There is a 50% chance it will rain.
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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.
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It rains (___)
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Take umbrella
Don’t take umbrella
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EU (take umbrella) = …
EU (don’t take umbrella) = …
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Take the umbrella to work!
It doesn’t rain (___)
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Pascal’s wager
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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.
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It is a decision made under a time of uncertainty:
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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
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God exists (___)
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God doesn’t exist (___)
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Believe
Don’t believe
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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) = …
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Believe in God: …
Pascal’s argument
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Pascal’s argument:
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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
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The first move:
– Deny premise 1:
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The second move & Pascal’s reply:
– Believing for selfish reasons:
Denying premise 2
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Deny premise 2:
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Infinite payoff’s make no sense:
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Can we even assign a non-zero probability to God’s existence?
The Many Gods objection
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We could Deny premise 2 in another way:
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Many Gods & the Perverse Master…
The Perverse Master
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The new payoff matrix:
God exists (__) Perverse Master exists (__) Neither exists (___)
Believe
_____
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Don’t Believe _____
_____
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Disbelief seems no worse off than belief:
– EU (believe) = …
– EU (don’t believe) = …
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What if we thought it less likely that the perverse Master exists than does God: