Critical Thinking Lecture 14 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 attempt if you continued to make attempts... It is the average gain or loss per attempt. 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 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: Which action has the higher expected utility?

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 God exists (___) God doesn’t exist (___) Believe ____ ____ Don’t believe ____ ____ 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) = … Which action has greater expected utility?

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.

Denying premise 1 The first move: Can you choose to believe? The second move: Would God reward selfish believers?

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) = … Is it less likely that the perverse Master exists than does God?