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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.