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Decision Theory, Philosophical Perspectives
Introduction
Decision theory is concerned with how agents should act when the consequences of their actions are uncertain. The central principle of contemporary decision theory is that the rational choice is the choice that maximizes subjective expected utility. This entry explains what this means, and discusses the philosophical motivations and consequences of the theory. The entry will consider some of the main problems and paradoxes that decision theory faces, and some of responses that can be given. Finally the entry will briefly consider how decision theory applies to choices involving more than one agent.
Decision Theory
The history of decision theory is traced to a gambling puzzle that Chevalier de Mr posed to Blaise Pascal. Here is a simpler version of the same kind of puzzle:
Suppose someone offers you a bet that pays $10 if a fair coin lands Heads but costs you $5 if the fair coin lands Tails. Should you take the bet?
To motivate Pascals answer, consider two flawed strategies. First, we might assume the worst is going to happen. Then we would make sure the worst result is as good as possible. If the bet is taken, the worst result is that the coin lands Tails, as we would lose $5. If the bet is not taken there is no change in wealth. So if we follow this maximin strategy i.e. maximize the minimum payoff, we shouldnt take the bet, as no change is better than losing $5.
Alternatively, we might assume the best result will happen, and seek to maximize our gain. If we take the bet, the best result is that the coin lands Heads. This gives us a profit of $10. If the bet is not taken there is no change in wealth. So if we follow this maximax strategy i.e. maximizing the maximum payoff, we should buy the ticket.
Both of these strategies are flawed because they dont take into account the probability of each outcome. Pascals key insight was that the decision should be based on the size of each payoff weighted by the probability that it would occur. This gives us the expected monetary value of each action. As the coin has a probability of of landing either Heads or Tails, taking the bet has an expected monetary value of $10 * - $5 * = $2.5. This is greater than the expected monetary value of not taking the bet, which is 0. So the rational choice in this case is to buy the ticket. In general, Pascal argued that the rational choice is that which maximizes expected monetary value.
The first important modification to Pascals decision theory came from Gabriel Cramer and Daniel Bernoulli. They introduced the concept of utility, which can be thought of as ones overall level of well-being. Cramer and Bernoulli discovered that money has diminishing marginal utility. This means that each extra dollar added to your wealth adds a little less to your utility. For example, if a millionaire finds a dollar on the street, it wont make him as happy as if a homeless person had found it. Thus, decisions should not be based on expected monetary value, but on expected utility. So modified, decision theory says that the rational choice is that which maximizes expected utility.
To use the above example, suppose the utilities of winning $10, winning $0 and losing $5 are 100, 0 and -80 respectively. Then the expected utility of taking the bet is 100 * - 80 * = 10. As this is greater than the expected utility of not taking the bet (0), the bet should be taken.
Insert Figure 1 here. Please polish
The diminishing marginal utility of money
Representation theorems
We need one more modification to get to contemporary decision theory. We saw that Pascal recommended weighting the outcomes by the probability that they occur. But where does this probability come from? The probability is a representation of the agents degree of belief that the outcome will occur; it is subjective probability. Thus, we arrive at contemporary decision theory: the rational choice is that which maximizes subjective expected utility.
Subjective probability connects decision theory with the beliefs of the agent. Similarly, utilities connect decision theory with the desires of the agent. An agents strong desire for an outcome can be understood in terms of the agent assigning a high utility to that outcome. So we see that decision theory systematizes the way that rational choices are based on beliefs (probabilities) and desires (utilities).
But does decision theory presuppose that rational agents have precise numerical degrees of belief and strength of desires? No. Representation theorems show that if an agents preferences (choices) satisfy certain constraints, then the agent can be represented as having certain numerical beliefs and desires. These constraints include transitivity, continuity and independence. Transitivity says that if a is preferred to b, and b is preferred to c, then a is preferred to c. Continuity says that if a is preferred to b, and b is preferred to c then there is some gamble between a and c such that the agent is indifferent between the gamble and b. Independence says that an agents choice between two options should not be affected by an outcome that is independent of the choice (Allais paradox below offers an example). The most influential representation theorems are those of Frank Ramsey, John von Neumann & Oskar Morgenstern, Leonard Savage and Richard Jeffrey. These differ in their constraints and ontologies, but they all show that agents whose preferences have a sufficiently rich and coherent structure can be represented numerically.
The main import of representation theorems is that they provide the most compelling arguments for the central principle that rational choices maximize subjective expected utility. Representation theorems derive this principle from minimal and intuitively plausible constraints on preferences.
Lets now move on to challenges and applications of decision theory, starting with a direct challenge to the assumption of independence.
Allais Paradox
Maurice Allais offered the following case as a counter-example to decision theory. Suppose you are offered the following two choices. In Gamble 1 you can choose either Risky1 or Safe1. In Gamble 2 you can choose either Risky2 or Safe2. What would you do in each case?
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Allais Paradox
Decision theory tells us that we should either choose the top (risky) option for both or the bottom (safe) option for both. This is because the only difference between the tables is the third (89%) possibility, and the payoff in this possibility is independent of the choice. The axiom of independence says that your choice should not be affected by payoffs that are independent of that choice. The problem is that many people violate this axiom. Many people choose Safe1 and Risky2, and continue to do so after careful consideration. The reason is that humans are risk averse, leading people to prefer Safe1, where risk can be avoided, and Risky2, where risk cannot be avoided.
One response to this paradox is to maintain that decision theory remains the correct normative theory, and risk-aversion is confused or irrational. An alternative response is to develop a theory, such as the prospect theory of Daniel Kahneman and Amos Tversky, that matches actual behaviour more closely.
Pascals Wager
A famous early application of decision theory is Pascals Wager.
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Pascals Wager
If you believe in God, you get either eternal salvation if God exists, or a false belief if he doesnt. If you dont believe in God, you get either eternal damnation if God exists, or a true belief if he doesnt. As the stakes are so much higher if God exists, Pascal argued that the rational choice is to believe in God as long as there is some positive probability, however small, that God exists.
One objection is that there are many Gods one could believe in, whereas Pascals argument assumes there is only one. Alternatively, Richard Dawkins points out that God might reward evidence-based belief, rather than blind faith in his existence. Thirdly, Alan Hjek argues that if there remains a non-zero chance of eternal salvation even if you dont believe in God, then there is no benefit to believing in God, as any non-zero number multiplied by infinity is still infinity i.e. (100 * infinity) = (1 * infinity) = (0.1 * infinity). Hjeks response assumes that the utility of eternal salvation is infinite. Infinite utilities also lead to other problems, as the next case shows.
St. Petersburg Paradox
Suppose a fair coin will be flipped until it lands Tails. You will win $2n,where n is the number of flips. So if the coin lands Tails on the first toss, you win $2. If it lands Tails on the second toss, you win $4, and so on. The game is played just once, and ends when Tails lands. The expected value of this game is infinite i.e. 1/2*$2 + 1/4*$4 + 1/8*$8 So according to decision theory, anyone should be willing to pay any amount of money to play it. But it is clearly not rational to give up everything you own to play this game.
One response to this paradox is to claim that it overlooks the diminishing marginal utility of money. But although this response was historically important for motivating the concept of utility, it is ultimately unsatisfying. The reason is that we can simply replace the dollars won with units of utility. The expected utility of playing the game is now infinite, but it is still not rational to give up everything you own to play.
Another response is to argue that utilities are bounded. In other words, there is a level of utility so high that no additional utility is possible. But given the insatiable appetites of humans, this assumption seems problematic. Moreover, the claim would need to be that a bounded utility function is a constraint on rationality, not just a feature of human psychology.
Newcombs Problem and Causal Decision Theory
Perhaps the most important paradox of recent years is Newcombs problem. Suppose you are faced with a transparent box that contains $1000 and an opaque box that contains either $1,000,000 or $0. You may take either both boxes or just the opaque one. The twist is that there is a Predictor that you know to have an excellent track record. If this Predictor has predicted you will take both boxes, the opaque box is empty. If it has predicted you will take only the opaque box, then that opaque box contains the million dollars. What would you do?
Insert Figure 4 here
Newcombs Problem
Traditional decision theory (called evidential decision theory in this context) says that you should take only one box, as this maximizes expected utility. But this is considered counter-intuitive by many, as the million dollars is already either there or not there at the moment of decision. This worry has led to the development of causal decision theory, which says that you should perform the action that causes the outcomes with the greatest expected utility. Thus causal decision theory endorses taking both boxes.
Newcombs problem is a case where the outcome partially depends on the actions of someone (or something) else. The final two problems are more familiar cases where more than one person is involved.
Game Theory
The outcomes of many of our decisions depend on the decisions of others. This interaction is dramatized by the Prisoners Dilemma. Suppose you and a partner are charged with committing a crime together. Interrogated separately, each of you can either stay silent or confess.
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Prisoners Dilemma
No matter what the other does, each agent is better off confessing; both confessing is the dominant strategy. It is also a Nash equilibrium, meaning that neither agent has any incentive to change his action, given the action of the other.
The philosophical significance of this result is that rational agents combine to generate a sub-optimal outcome. Each would be better if they both stayed silent, yet each should confess according to game theory. Some philosophers, such as David Gauthier and Douglas Hofstadter, argue that in fact the rational decision is to stay silent, but this remains a minority view.
If the agents face this kind of situation more than once, there is more scope for co-operation. This has led to research among game-theorists into the evolution of co-operation and morality.
Social Choice Theory
Whereas in game theory each agent has its own decision and payoff, in many important cases, such as elections, a group has to make a decision and live with the consequences together. The central question of social choice theory is how individual preferences should be aggregated to result in a group decision.
An important result in social choice theory is Arrows Impossibility Result. This states that no voting procedure other than dictatorship can satisfy certain reasonable requirements, for example, that if every voter prefers X over Y, then the group prefers X over Y. One avenue of research this has led to is finding a voting procedure that best satisfies the constraints we think voting procedures should satisfy.
See also
Belief and judgment
Decision
Group decision making
Prisoners Dilemma
Rationality
References
Arnauld, Antoine and Nicole, Pierre (1662/1964) Logic or the Art of Thinking, Bobs-Merrill Company.
Jeffrey, Richard (1983) The Logic of Decision, 2nd edition, University of Chicago Press.
Ramsey, F. P. (1926) Truth and Probability. In Philosophical Papers, ed. D. H. Mellor. Cambridge: Cambridge University Press.
Resnick, Michael (1987) Choices: An Introduction to Decision Theory, Minneapolis, University of Minnesota Press.
Savage, Leonard (1954) The Foundations of Statistics. New York: Wiley.
Joyce, James (1999) The Foundations of Causal Decision Theory. Cambridge: Cambridge University Press.
Darren Bradley
City College of New York
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