Emergent Properties of Choice
I start from a brief presentation of Allais' paradox; yet, I am not primarily concerned with the question how to solve it. The paradox provides a convenient way to demonstrate that distribution of alternatives we face in a situation of choice may give rise to new factors. These emergent properties may need to influence a one time choice of rational decision-makers, although they should not be taken into account in long reiterative games.
Let me introduce to you decisiotheoretic emergentism.
According to the independence axiom an outcome of the choice shall be neutral if a constant value is added to each alternative. But if we consider the table of preferences presented by Allais this presumption seems intuitively questionable.
In the choice between g1 and g2 (where M stands for one million crowns), most people choose g1 over g2, although g2 gives higher expected value. Yet, if we choose between g3 and g4, almost everybody prefers g4 over g3. But the problem may be seen as two identical alternatives g1=g3 and g2=g4 just in the choice between g1 and g2 in column R an outcome of one million crowns has been added to each alternative whereas in the second case the constant added equals zero.
These results contradict with the independence axiom. The first solution is to go Savage's way and, after reconsideration, to change one's mind in the g1/g2 choice. But strong intuitiveness of the Allais paradox makes this solution less than attractive.
It might seem better to search for some troublesome decisio-theoretic axioms easy to replace. This is the way decision theorists usually go. But they have a problem in finding axioms to be eliminated.
My 'modest' proposal is to look how to solve Allais paradox not strictly in theory of decision, but into pragmatics of theory of decision. We must consider how the system (here the theory of utility) satisfies the requirements and needs of users.
I shall call the idea proposed decisio-theoretic emergentism. This idea has affinities to the well known bundle-effect which consists in the fact that objects may change in value if combined (or separated). Let us look at this issue with a more ontological eye than decision-theorists usually do. Wine and fish, for example, may be seen in two different ways: as two different products or as one product, namely a dinner. This is not just a question of an epistemic choice whether to "see" the new bundle as a dinner or as fish plus wine. These products actually create a new object, namely a dinner, whose qualities are not reducible to the qualities of its parts. But this "bundling effect" happens only in some situations; in other situations (for instance in a shopping basket) the two products remain separate. This is an important difference since the bundling effect results in partial independence of the value of the bundle from the values of its components. Whether objects create a bundle or not depends on pragmatic aspects of the case. The value of dinners comes from relations between products and their users. For instance dinners made of fish and wine emerge since there are people who: First, eat fish and wine dinners. Second, would rather use dinners than fish alone and wine alone. Bundling effects are important for theory of decision but they can not be predicted by this theory alone with no recourse to pragmatics.
This last observation can help us deal with the Allais paradox. We may look at the table of monetary values above and try to find some of its possible pragmatic peculiarities. Our final evaluation over g1 and g2 occurs on the pragmatic level after all. What is the difference between the choices g1/g2 and g3/g4? I wish to propose that in the alternative g1 a new property emerges. The emerging property of the alternative g1 is of course certainty. Consequently, the choice between g1 and g2 involves a new object, hard to grasp but fully ontologically separable from other objects of this choice. We choose not only an expected utility of a certain alternative, but alternative g1 brings about also certainty, unlike any other alternative. Certainty is a new product which is worth a certain price. Hence, the emergent property of certainty produces a deviation of the choice most people make from the expected utility theory.
We should observe that the value of certainty is different that an agent's attitude towards risk. One's attitude towards risk, in the standard sense introduced in theory of decision by Hurwitz, is a constant value of one's choices it can be included as a factor which influences the value of expected utility. On the other hand, certainty is a feature of the collective set of alternatives.
The only affinity between decisio-theoretic attitude towards risk, so called risk aversion, and the emergent property of certainty is this: A risk-averse agent is very likely to care more about certainty than a risk-seeking one; hence, he would attribute a higher value to certainty. (1)
Certainty is not regret.
Frederick Schick has a somewhat similar solution to the Allais' paradox; yet, there are important formal differences between the two. Schick proposes to add to the choice g2 a negative value of regret factor (if Y happens we do not receive just zero, we acquire zero+regret). This negative factor bends utility distribution from the consequentialist 'norm'.
To the contrary, I add a positive value of certainty to the column g1. It is added to the value of the whole column, not to any single outcome (Schick adds regret to the outcome Y only).
Our philosophical explanations differ even further then the formal accounts. Schick explains the regret factor as "a fuller way of reporting the outcome of getting no money". The outcome can be stated as: "zero-when-I-could-have-had-1,000,000-for-sure". (2) He treats zero as not equally bad in every situation. The actual value of getting zero depends on what economists call the opportunity cost (what could have happened instead) as well as of a psychological factor (how much would I regret not getting at least the minimum prize).
On my account, certainty is a formal feature of a distribution of options. It is an objective property of the set of available options. As I shall explain in the final section of this paper, it is rational for agents to pay a certain bonus for acquiring certainty. In short games a choice to buy certainty is not a psychological deficiency but a rational choice of strategy.
Although both regret and certainty owe their value to pragmatic considerations, the value of certainty supervenes on strictly formal features of the collective set of available alternatives whereas the value of regret depends on psychological features. (Schick could talk of rational regret, but it is also based primarily on psychology of human beings not primarily on formal features of the choice).
To conclude the argument, it is not the utility theory that needs to be revised (or ignored) because of the Allais' Paradox; it is rather our interpretation of situations of choice that should be made more complete. An apparently paradoxical outcome suggested by Allais is due to the emergence of a new object: certainty.
The cost of certainty.
The value of certainty, if taken into account, results in deviation from the value of expected utility. Hence, in terms of expected utility theory buying certainty is a sub-optimal choice. Certainty may have to be seen as a deceitful Siren, promissing peace of mind yet leading to losses. Indeed, agents who buy certainty are being exploited in a way since the sum of all such agents is bound to have sub-optimal results; insurance companies do just this. But isn't it rational to buy an insurance despite a reasonable expectation that we would lose some money on it? The answer depends in part on the pragmatic features of our predicament, and in part on the size of our business compared to the size of a possible loss. Concerning the first factor, in long games one has a good chance to achieve roughly the results predicted by the theory of expected utility. But if the game is short, we are likely to end up much below, or much above the value of expected utility. Now the pragmatic factor comes into play. Nagel is right when he argues in a different context that it makes a difference that some losses are losses for me. If a given game is very important for me, I may have good pragmatic reasons to pay a substantial bonus for avoiding the worst result and this is the price of certainty. Referring to the second factor, if one time damages account for a small fraction of our financial worth we can self-insure ourselves (by doing nothing). But if the cost of damage is high for the business because the size of business is small or possible damage high buying insurance is a rational thing to do. Although we still have good reasons to prefer rather more money than less which gives us some reasons to follow closely the expected utility (by choosing g2 in our example), we may have dominant reasons to buy certainty.
The point whether it is rational to buy certainty needs to be stated clearly. In repetitive games the results are very close to the probability distribution. In these cases, as the expected utility theory teaches, emergent properties should be ignored since any concern for them leads to sub-optimal outcomes. But in one time games (or in short sequences) the situation is different. One may have good reasons to worry rather about his or her particular game than about general likelihoods. If an agent has the one upon her lifetime chance of making a million dollars for sure, he or she should not risk it even for a very good, though uncertain, prospect of a still higher profit. (3) This is because, due to uniqueness of one time opportunities, probability laws should not be applied to such opportunities if certain pragmatic conditions are satisfied.
A critic may wonder whether an emergent property would be eliminated, if zero columns were replaced by fairly small amounts of money, like one British pound. Obviously, it could not. The amount of one pound is a negligibly small compensation for the loss of one million dollars. The boundary between negligibly small amounts of money and the amounts big enough to allow an emergent property of certainty to play a role (if this amount replaces zero in each column), is often vague. Its value should to be based on pragmatic considerations: one person's fortune is another's misery.
By the way, we may also face a problem of 'negligibly small' uncertainties. We may find certain likelyhoods negligible (say, a likelyhood of getting killed by a car accident in normal conditions). Just like too small compensations do not produce certainty, these small likelyhoods of disasters do not impede the value of certainty which, in usual circumstances, we attribute to so many things we value in our life.
(1) Kahneman and Tversky emphasize the effect of certainty in changing empirical attitudes towards risk in their: "Rational Choice and the Framing of Decisions", in Hogarth R. M. and Reder M. W. (eds.) Rational Choice: The Contrast between Economics and Psychology, Chicago Univ. Press 1987 p. 456
(2) Schick, Frederic: Understanding Action: Cambridge University Press 1991 p.126
(3) A part of the explanation is declining value that most people attitute to money increases above a certain level. But this is by no means a sufficient explanation since Allais' paradox can be constructed even in the situation in which the attitude to money increases (tested in other situations) is too weak to explain the paradox.