20th WCP: The Rationality of Probabilities for Actions in Decision Theory

Spohn's (1977, 1978) decision model, an advancement of Fishburn's theory (1964), is valuable for making explicit a principle which is used by Savage (1954/1972) and Fishburn (1964). The principle is the following: "Any adequate quantitative decision model must not explicitly or implicitly contain any subjective probabilites for acts." (1) (Spohn 1977, p. 114) This principle is not used in the decision theories of Jeffrey (1965, 1983) and of Luce and Krantz (1971). According to Spohn (1977) this principle is important, because it has implications for the concept of action, Newcomb's problem, theory of causality and freedom of will. On the one hand I will try to argue against Spohn (1977, 1978) with Jeffrey (1965, 1968, 1977, 1983) that the principle has to be given up. On the other hand I will try to argue against Jeffrey (1965, 1968, 1977, 1983) that the decision-maker ascribes subjective probabilities to actions on the condition of the given decision situation.

The importance of this principle will be illustrated by Newcomb's problem. In 1969 Robert Nozick introduced Newcomb's problem to the philosophic community as a conflict between the principle of expected utility maximization and the principle of dominance. Nozick's introduction led to a "Newcombmania" (Levi 1982), because philosophers have decisively different opinions about the correct solution to this problem. The "Newcombmania" showed itself in the development of causal and evidential decision theories and other proposals. Because the evidential theories (for example Jeffrey 1965, 1983) do not use the principle, they cannot give a solution to Newcomb's problem in case you accept the principle. The causal theories which use subjunctive conditionals (for example Lewis 1981) are problematical, because they still have to provide a logic of subjunctive conditionals, a probability theory for subjunctive conditionals and a corresponding decision theory. Because Skyrms' (1980) causal theory and Kyburg's (1980) proposal of epistemic vs. stochastic independence also don't use the principle, only Spohn's solution (1978) to Newcomb's problem is left. This solution which recommends taking both boxes is valuable for its simplicity in contrast to the theories with subjunctive conditionals. According to Spohn it is a mistake to use probabilities conditionalized on actions for the results of the prediction, if the prediction is earlier than the choice. According to Spohn it is right that the subjective probabilities of the decision-maker have a fixed value independently of his decision. Therefore the maximizer of the expected subjective utility should take both boxes, for taking both boxes dominates taking the second box.

According to Spohn (1978) the principle doesn't refer to past actions and actions of other persons, but only to actions which are open to the decision-maker in his decision model that is to future actions of the decision-maker. Spohn concedes however that decision-makers frequently have and utter beliefs about their future actions like the following:

(I) "I believe it is unlikely that I will wear shorts during the next winter."

Spohn (1977) wants this utterance to be understood in such a way that it doesn't express a probability for an action, but a probability for a decision situation: "I believe it is unlikely that I will get into a decision situation during the next winter in which it would be best to wear shorts."

Conceding this opinion to Spohn for a while, consider what happens with the following utterance:

(II) "I believe it is unlikely that I will run 100 meters in 9 seconds during the next year".

Spohn's reformulation would look like the following: "I believe it is unlikely that I will get into a decision situation during the next year in which it would be best to run 100 meters in 9 seconds." This reformulation seems nonsensical, for it wouldn't matter how much I tried I never would be able to run 100 meters in 9 seconds. If the Olympic Games were to happen next year and I were in this decision situation, it would be best for me to run 100 meters in 9 seconds, but nevertheless I wouldn't be able to do it because of my bodily constitution. I could at most try to run 100 meters in 9 seconds. With this we have reached what Jeffrey (1965, 1968, 1983) terms probabilistic acts and tries. Jeffrey postulates actions with probabilistic results in analogy to observations with probabilistic results and comments this in the following way (1983, p. 177-178):

In the simplest cases, where n = 2, where B1 is some good proposition B, and where B2 is the bad proposition , we speak of the agent as trying to make B true; and where PROB B ... is very close to 1, we may speak of the agent as believing it to be in his power to make B happen if he chooses ... Rather, to speak of the agent's trying to make B true is to speak of his performing an act of which he takes the net effect to be an increase in the probability of B ... the agent may speak in this way without thereby assuming the existence of a proposition E in his preference ranking for which we have (11-9) PROB E = 1 PROB B = prob(B| E) where PROB is the agent's belief function after he decides to perform the act. Thus, trying to hit the bullseye may be an act without there being any proposition that plays E to hitting the bullseye's B, above." (2)

In the light of Jeffrey's remarks, it seems plausible to suppose that Jeffrey and Spohn are talking about two different things, when they speak of probabilities for actions. While Jeffrey considers utterances like (II), Spohn looks at utterances like (I). A difference can be pointed out between both utterances: While "wearing shorts" is not quantified, "running 100 meters" is quantified by 9 seconds. Cases which are quantified seem to be exactly the actions with probabilistic results à la Jeffrey. This can also be shown by pointing out that the third utterance in comparison to the fourth utterance sounds strange:

(III) "I will try to wear shorts during the next winter."

(IV) "I will try to run 100 meters in 9 seconds during the next year."

One can object to that by showing that there are also actions with probabilistic results which are not quantified such as in the following utterance:

(V) "It is unlikely that I will climb Mount Everest during the next winter."

The following reformulation à la Jeffrey sounds appropriate, too:

(VI) "I will try to climb Mount Everest during the next winter."

The last example suggests that the utterances (I), (II), and (V) nevertheless do not differ from each other. Jeffrey (1965, 5.8; 1968; 1983, 5.8) takes exactly this position. Jeffrey believes that all actions are probabilistic, so that my trying to wear shorts during the next winter is probably successful, (3) while my trying to run 100 meters in 9 seconds during the next year is unlikely to lead to success. Spohn however doesn't explain how utterances of type (II) should be understood.

From Spohn's and Jeffrey's remarks you can analyse the utterances (I) and (II) in the following way: In the first example it is on the one hand unlikely that the decision-maker will get into a decision situation in which it would be best to wear shorts during the next winter, on the other hand trying to perform this action leads probably to success. In the second example, however, it is on the one hand likely that the decision-maker will get into a decision situation in which it would be best to run 100 meters in 9 seconds during the next year, on the other hand trying to perform this action is unlikely to lead to success. Utterances like "It is unlikely that ..." can therefore refer to different objects like decision situations, actions, events, etc. I propose to speak of probabilities for actions just in case the probability statement refers to the action, for this corresponds to our natural usage of language.

Against this you could object that the decision-maker can only ascribe probabilities to actions with reference to the given decision situation. For utterances like

(VII) "It is unlikely that I - regardless of the given decision situation - will fly to the next galaxy"

are false, because you can always imagine decision situations in which it would be possible and best to fly to the next galaxy. For example, somewhere in the future it could be possible that we would be able to build very fast space shuttles, so that we could reach the next galaxy within a few years and that we would need a substance which we could only get from a planet of the next galaxy, so that it would best for us to fly to the next galaxy. Thus I have to propose that the decision-maker ascribes probabilities to actions on the condition of the given decision situation. Because every action is embedded in a decision situation, this conclusion seems to be correct.

Spohn's argumentation (1977, 1978) for the principle continues. According to Spohn it is commonly accepted that subjective probabilites for particular events manifest itself in the willingness to bet on these events with appropiate betting odds and that this doesn't apply to probabilities for actions. For the willingness of the agent to bet on his own action just depends on his gain that is to say on the stake of his betting partner, according to Spohn, so that his own stake is quite irrelevant. He will accept the bet only if his gain is so high that his action together with his gain is to be prefered to all alternatives. Spohn (1978) illustrates this by the following example:

"If someone ... offered me DM 30 to watch a particular movie, I would presumably accept, while I wouldn't accept it, if someone offered me only DM 5 - both being independent of what the other person would get from me, if I didn't watch the movie." (p. 73)

Considering Spohn's example I have to ask whether it is a bet; Spohn concedes in a footnote that it is only a bet-like agreement. If we suppose that it is a bet, one has to ask whether it is really the case that I would join a bet at DM 30 and wouldn't join a bet at DM 5, if my own stake was > DM 30, ³ DM 5 and £ DM 30, or < DM 5. Let's consider the case that my stake is DM 10,000 (negative change from reference point (4) ). If money were important to me and if I owned just a little bit more than DM 10,000 (reference point), then I would neither at DM 30 nor at DM 5 agree (positive change from reference point), for there could always happen something, which could hinder me from watching the movie, and the risk to loose DM 10,000 would be too high for me even if I were highly riskprone. By considering this case we see that the willingness of the agent to bet on his own action doesn't depend solely on the stake of his betting partner, but also on his own stake. Furthermore such factors as riskaversion, reference point, and the amount of change from the reference point seem to influence the willingness of the agent to bet.

In my opinion the willingness of the agent to bet on his own action given a real bet depends in the following way on his own stake and on the stake of his betting partner: Suppose that I bet with X that I will run 100 meters in 9 seconds during the next year. If the stake of X were DM 1,000, we would nevertheless argue without considering the reference point and the amount of change from the reference point that my stake given a moderate riskaversion would depend in the following way on my subjective probability for reaching the goal: The first maxim is to keep my own stake as low as possible, for I want to keep my own possible loss as low as possible. But the more improbable I think it is that I will reach my goal, the lesser my stake should be in comparison with your stake; and the more probable it is that I will reach my goal, the higher my stake could be in comparison to your stake; if however the probability of my goal reaching is P = 0.5, my stake could equal the stake of my betting partner.

But even if we concede that Spohn's principle is valid, we would have difficulties on another level, for according to Spohn (1977) the principle requires that actions are things which are under the decision-maker's full control relatively to the decision model describing him. Spohn illustrates this by the following example: When the decision-maker wants to go from here to there, this action isn't under his full control, because there could be obstacles. But for reasons of simplicity a decision model can assume that the actions of the decision-maker are under the decision-maker's full control. Thus Spohn assumes ideally that actions are under the decision-maker's full control. From this starting-point it is absurd to have subjective probabilities for things which are under the decision-maker's full control and which he can actualize as he pleases, Spohn claims. But Spohn doesn't give reasons for justifying the idealization. The idealization leads to a simplification of the modeling of decision theory; this however cannot be the justification for the idealization, for an idealization is only permitted, if there is no relevant loss in information.

You can also question in another respect whether the principle requires the decision-maker's full control over his actions. For if Spohn wants utterances like (I) to be understood in such a way that they express probabilities for decision situations, this doesn't entail anything about control over actions yet. But for the validity of the principle he could also presuppose the decision-maker's full control over his actions. For if he reformulates utterances like (I) in such a way that they say the following "I believe it is unlikely that I will get into a decision situation during the next winter in which it would best to wear shorts", he presupposes that you can perform such actions as "wearing shorts" without having any problems. But isn't it rather unrealistic to assume that we have full control over our actions? Couldn't it be a much more realistic point of view to presuppose just a partial control over our actions? If you, for example, just consider how many men and women try to loose weight day by day, doubts about full control over our actions come up. In connection with this I want to ask whether there aren't any actions which remain only probable after the removal of all outer and inner obstacles (= full control). Exactly this seems to be the case in my view. As the atom decay of radioactive substances happens only probably within a particular time, my trying to hit the bullseye leads only probably to success. Although the outside conditions (for example the light conditions) and the interior conditions (for example the concentration capability) are at an optimum, it is nevertheless only probable that I will hit the bullseye, for the target is 100 meters distant and moves along a carrier with a breadth of 5 meters.

Notes

(1) Trivial conditional probabilities like P(A| A) = 1 for an action A or P(A'| A) = 0 for two disjunctive actions A and A' are not considered.

(2) E is the following proposition: The agent tries to make B true. prob B is the probability of B before the agent has decided to perform B.

(3) That is also the reason for thinking that utterances like (III) sound strange; for if PROB B is near to 1, the agent believes that he can make B true as he pleases.

(4) Kahnemann and Tversky (1979) say that you should consider the value as a function with two arguments: The asset position that serves as reference point, and the magnitude of the change (positive or negative) from that reference point.

Bibliogaphy

Fishburn, P. C. (1964), "Decision and Value Theory", Wiley, New York.

Jeffrey, R. C. (1965), "The Logic of Decision", McGraw-Hill, New York.

Jeffrey, R. C. (1968), "Probable Knowledge", in I. Lakatos (ed.), The Problem of Inductive Logic, North-Holland Publishing Company, Amsterdam: 166-180.

Jeffrey, R. C. (1977), "A Note on the Kinematics of Preference", Erkenntnis 11: 135-141.

Jeffrey, R. C. (1983), "The Logic of Decision", The University of Chicago Press, Chicago, London, second edition.

Kahnemann, D. and Tversky, A. (1979), "Prospect theory: An analysis of decision under risk", Econometrica 47: 263-291.

Kyburg, H. E. (1980), "Acts and Conditional Probabilities", Theory and Decision 12: 149-171.

Levi, I. (1982), "A Note on Newcombmania", The Journal of Philosophy 79: 337-342.

Lewis, D. (1981), "Causal Decision Theory", Australasian Journal of Philosophy 59: 5-30.

Luce, R. D. and Krantz, D. H. (1971), "Conditional Expected Utility", Econometrica 39: 253-271.

Nozick, R. (1969), "Newcomb's Problem and Two Principles of Choice", in N. Rescher et al. (eds.), Essays in Honor of Carl G. Hempel, Reidel, Dordrecht: 114-146.

Savage, L. J. (1954/1972), "The Foundations of Statistics", Wiley, New York, Dover.

Skyrms, B. (1980), "Causal Necessity", Yale University Press, New Haven, London.

Spohn, W. (1977), "Where Luce and Krantz Do Really Generalize Savage's Decision Model", Erkenntnis 11: 113-134.

Spohn, W. (1978), "Grundlagen der Entscheidungstheorie", Monographien: Wissenschaftstheorie und Grundlagenforschung vol. 8, Scriptor Verlag, Kronberg/Ts.