Background
Probability Theory
Pascal's Triangle & Probability
Application of Probability Theory
Pascal's wager
Objections
Homework
Joyce Lam Nga Ching
2001714828
Phil1007
12-4-2002
27-4-2002
Decision
Theory
With the thought on gambling and the wager, Pascal brought about the advent of decision theory. The core of Pascal's Wager argument,at least from a logical point of view,lies in decision theory (i.e. in the theory of decision-making in situations of uncertainty about outcomes) The wager in its various guises introduces some of the decision-theoretic principles that are still useful as guides to action when faced with a decision of what to do in an uncertain situation. The principle of maximizing expected utility is found in at least one of Pascal’s version of the wager. The idea of expected utility is that of a weighted average of probability and gain.
Decision theory is the theory of deciding what to do when it is uncertain what will happen. Given an exhaustive list of possible hypotheses about the way of world is, the observations or experimental data relevant to these hypotheses, together with an inventory of possible decisions, and the various utilities of making these decisions in various possible states of the world, determine the best decision.
In any decision problem, the way the world is, and what an agent does, together determine an outcome for the agent. We may assign utilities to such outcomes, numbers that represent the degree to which the agent values them. It is typical to present these numbers in a decision matrix, with the columns corresponding to the various relevant states of the world, and the rows corresponding to the various possible actions that the agent can perform.
In decisions under uncertainty, nothing more is given---in particular, the agent does not assign subjective probabilities to the states of the world. Still, sometimes rationality dictates a unique decision nonetheless. Consider, for example, a case that will be particularly relevant here. Suppose that you have two possible actions, A1 and A2, and the worst outcome associated with A1 is at least as good as the best outcome associated with A2; suppose also that in at least one state of the world, A1’s outcome is strictly better than A2’s. Let us say in that case that A1 superdominates A2. Then rationality surely requires you to perform A1.
In decisions under risk, the agent assigns subjective probabilities to the various states of the world. Assume that the states of the world are independent of what the agent does. A figure of merit called the expected utility, or the expectation of a given action can be calculated by a simple formula: for each state, multiply the utility that the action produces in that state by the state’s probability; then, add these numbers. According to decision theory, rationality requires you to perform the action of maximum expected utility (if there is one).
Pascal's wager & Decision Theory
A special case of this problem occurs when no experiments made. In the though abut concerns us, Pascal deliberately "tie his hands" and refuses to look at any observations or experimental data bearing on the existence of a Christian God. He is writing for those who will not countenance miracles, or the doctors of the church, or the witness of the faithful. So we may restrict our attention to the logic of decision when there is no experimental data.
Among the valid argument forms investigated by decision theory, there are the three argument forms that concern us follow.
Dominance
The simplest special case occurs when one course of action is better no matter what the world is like. Schematically, suppose that we have some exhaustive set of possible states of affairs: we label the states S1,S2....Suppose that in some state Si, the utility Ui1 of performing act A1 is greater than the utility of Ui2 of performing act A2. In no other state of affairs is the utility of performing A1 less than A2. Then under no circumstance could A2 have better consequences than A1 , and under some circumstance A1 could be better than A2. A1 is said to dominance A2. If one act dominates all others, the solution to our decision problem is " perform the dominant act."
Expectation
The argument from dominance does not consider how likely are various states of affairs. Even if dominating A1 is better only in a very unlikely state of affairs, then, because of A1 can never fare worse than any other act, it is worthwhile performing A1.But suppose no action dominates, although we think we know which states of affairs are more likely than others. Suppose we can assess the probability of each states of affairs. Then (no matter what one means by "probability")one argues as follows. We have assigned a probability p1 to each possible state of affair Si in some exhaustive set. Let Uij stand for the utility of doing Aj if Si actually obtains. The expected value, or expectation, of Aj is the average5 value of doing Aj: namely,Σ piUij. An argument from expectation concludes with the advice "Perform an act with highest expectation."
Dominating expectation
It may happen that we do not know, or cannot agree on, the probabilities of various states of affairs. We can at best agree on a set of probability assignments to the state Si. For example, suppose we agree that the coin is biased towards heads but disagree how great is the bias; at least we agree that the probability of " next toss gives haed" exceeds one in two. If in some admissible probability assignment, the expectation of A1 exceeds that of any other act, whereas in no admissible assignment is the expectation of A1 less than that of any other act, then A1 has dominating expectation. The argument form dominating expectation concludes: "Perform an act of dominating expectation."
The three argument schemes are mutually consistent. If one act does dominate the rest, then it will be recommended by all three arguments. If there is no dominating act but an act of highest expectation, then that act will also be the act of dominating expectation. The argument from dominance is the rarest, most specialc case. The argument from dominating expectation is more widely applicable.
Pascal's procedure in the thought "Infini-rien" is to offer an argument from dominance. Then if its premises be rejected, to offer an argument from expectation. Then if the second lot of premises be rejected, to offer an argument from dominating expectation.
Go to Rationality
Reference
1. Hacking,Ian,"The Logic of Pascal's Wager"Jordan, Jeff(Ed),Gambling on God: essays on Pascal's Wager, (London, Rowman & Littlefield Publishers,Inc.,1994)
2. http://plato.stanford.edu/entries/pascal-wager/#1