On the status of Savage’s Postulates

On the status of Savage’s Postulates Author: Matthias Würtenberger Abstract This essay will focus on assessing the axioms, or rather postulates as they are called, of Savage’s (1954) representation theorem for the Subjective Expected Utility theory. What his representation theorem achieves is to prove the existence of an expected utility representation of the preferences of a rational agent. Getting clear about the status of Savage’s axioms is of high relevance when one considers the overall project of the theory of Subjective Expected Utility. It is essentially a normative theory, spelling out how choices in the face of uncertainty are ought to be made. Choices among actions should maximise expected utility. Which action to choose from in an uncertain environment or action space, is determined not solely by the consequence of an action but also by the likelihood of the state of the world in which the consequence might be realised. Building upon the works of von Neumann & Morgenstern [1944] who showed that probabilities can be used to construct a utility measure on consequences such that preferences amongst gambles cohere with their expected utilities, Savage’s main achievement is determining conditions which are jointly sufficient to establish the existence of a unique probability representation of a rational agent’s beliefs. These conditions are Savage’s postulates and they represent certain constraints imposed upon preferences among acts. In case all these constraints are met by an agent’s preferences, he is said to be rational. The structure of our essay is as follows: first of all Savage’s strategy in setting up his representation theorem will be discussed. Here the postulates figuring in his representation theorem will be critically assessed. In assessing them we will be discussing which of his axioms represent genuine requirements of rationality and which are mere ‘structure’ axioms motivated by mathematical or theoretical concerns [Suppes (2002)]. The aim is to give an overview of the different reconstructions and discussions, most notably the accounts of Broome [1991], Fishburn [1970], and Joyce [1999]. Next, once their status is approximately settled, will be a critical discussion of all the candidates for rationality axioms. Here the aim is to contrast the arguments for and against each particular rationality-axiom. This is not to repeat the discussion about whether these can be taken to be requirements of rationality or not, this issue is taken for granted at this stage, but rather whether each rationality-axiom under discussion should be taken as a requirement of a theory of practical rationality. CONTENTS Contents 1 Savage’s strategy 0 2 The Axioms - their roles and status 4 3 Rationality Axioms? 3.1 Transitivity . . . . . . . . . . 3.2 Completeness . . . . . . . . . 3.3 Allais’ Paradox & Separability 3.4 State-Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 12 15 19 21 4 Conclusion 23 References 24 1 Savage’s strategy The main point of decision theory is to make clear the nature of meansends constraints. More specifically, it specifies the connection between an individual’s preferences about ends and her preferences about means that rationality requires. The aim of Savage, or any decision theory, is to construct an intelligible axiomatisation of a theory of preferences. If successful, such an axiomatisation would achieve to represent the preferences of rational agents. In doing so one is confronted with, what Jeffrey [1965:144] calls, the existence problem. Namely, formulating qualitative conditions on preferences which jointly imply the existence of a quantitative probability- and utility-function representing these preferences. These two kinds of functions merge together, yielding an expected utility function. Savage’s [1954] proof of the existence of such an expected utility representation of preferences involves two major stages. The first one consists in postulating a set of axioms1 that are jointly suffcient to establish the existence of a unique probability representation of the agent’s beliefs. Secondly, it is shown that such probability representations can be used to construct utility measures on outcomes such that preferences amongst lotteries cohere with their expected utilities. Savage’s main achievement in establishing the proof lies in the derivation of a qualitative probability relation over events which is equivalent to the first stage. The construction of an expected utility representation where the probabilities of states are intertwined with the utilities of the consequences has originally been carried out by von Neumann and Morgenstern [1944]. The advancement of Savage’s theory therefore lies in the disentanglement of states’ probabilities from the utilities of their outcomes. Savage constructs a representation theorem for rational action. The strategy here is to go from accepted, local principles of rationality to a global one.2 1 2 In the course of this essay we will use the notions of axiom, postulate and principle interchangeably. In more mathematical terms, a representation theorem for a specific class of mathematical 0 1 Savage’s strategy By local principles are meant certain rational patterns of preference, which (arguably) are Savage’s seven (or eight) postulates. His theorem shows that these postulates are individually necessary and jointly sufficient for an expected utility representation of an individual’s beliefs and desires. The global principle is exactly this, the maximisation of expected utility (hereafter MEU). Therefore this representation theorem is primarily of interest for those who already accept MEU in the first place, commitment to it stands prior to the theory. We will therefore not enter the principal debate about the pros and cons of the doctrine. Nevertheless, there is, vitally, a lot of room for interpreting the MEU. This is because the principle of MEU on its own, as a mere global principle, does not reveal much as to what actions, what preferences among them, are rational. The quintessential question of practical rationality ‘How should I act?’ cannot be satisfactorily answered by that global principle alone. In fact attempting to answer the previous question by saying ‘go and maximise your expected utility’ is close to giving an utterly meaningless answer, since it is not at all clear how one should act in specific cases. And this is where the local principles come into the picture. Not only do these give substance or meaning to the doctrine of MEU - what it demands at the level of individual preferences -, but also give some argument for MEU’s plausibility. The principle of MEU rises and falls with the strength and plausibility of its local principles. It is therefore of upmost importance that the local principles lend their plausibility to independent reasons. Assessing their individual justification will be the main theme of this essay. Before getting into the interplay of Savage’s postulates, we briefly note some peculiar facts about the ontology of his framework. There is a distinction between acts, states of the world, consequences and events.3 Intuitively one might consider the second and third to effectively be the same - an outcome is just a particular way the world turns out. But in the Savage framework they are, strictly speaking, entities of a separate kind, in so far states of the world do not correspond to outcomes or consequences.4 The conceptual clarifications and distinction made in the remainder of this section, will be relevant throughout the upcoming discussion. As we will see Savage was not always precise and definite about what acts, consequences, and states actually stand for. Sure, they definitely relate to our pre-theoretical understanding of them as used in every day language, but besides that they have not much in common. • Acts: or actions are defined as functions from the set of all possible worlds 3 4 structures, establishes “that every structure in that class is isomorphic to some structure in a distinguished proper subclass. Typically, the subclass of structures is in some sense more ‘concrete’ than the class as a whole.” [Makinson 2007:4] A widely used synonym for consequences, which we will here use interchangeably is outcomes. On the contrary Jeffrey’s [1965] account of decision theory is representation-independent. Meaning his ontology consists only of propositions. For instance, actions are simply propositions that can be made true. 1 1 Savage’s strategy to the set of consequences. Acts are assumed to be the relata of an agent’s preferences. When choosing an act one “decides ‘now’ once and for all; there is nothing for [one] to wait for, because [one’s] decision provides for all contingencies” [Savage 1954:17]. Accordingly, one has to interpret acts as choices between complete courses of future actions. Courses of actions determine everything one will eventually care about. This definition might strike one as being very unintuitive and far-removed from the every-day understanding and usage of the concept ‘act’. The definition gives rise to a particular problem. According to Jeffrey [1965:224] the Savage-acts, or possible acts as he called them, lead to the situation in which one “never knows just what act one is performing; for to know that, one would have to know how the act would turn out in every possible world.” It is easy to see that such acts are virtually impossible to conceive. If taken literally, one could never form a preference on any act, since acts would have to be evaluated with respect to every possible world. One would have to consider every eventuality which might become relevant, be it ever so small. This fact led Jeffrey to conclude that Savage-acts cannot be objects of choice [1965:224]. Joyce [1999:61] on other hand, does not so readily dismiss the account and gives a more benevolent reading of Savage-acts. He tries to secure plausibility by interpreting them “as objects of instrumental desire whose realization may or may not lie within the agent’s control.” Acts are instrumental because they are means to the end, the consequences. In addition, there is another problem coming in through the existence of so called constant acts [Savage 1954:25]. As we will see in due course they are made possible by a controversial postulate, namely P0. The merit of this construction is that through it, preferences not only range over acts but also events. These are special cases of acts with the striking feature of yielding the same consequence in each state of the world - thereby extending preferences to outcomes. It is very unclear whether we ever come to face such acts in real life. One plausible candidate for a constant act, would be choosing to commit suicide via a bullet proof method of say, taking in cyanide poison. In such a limiting case an individual could be sure that his act ‘oral intake of Xmg cyanide poison’ will always lead to the desired consequence, namely his death. Apart from such a case there seem to be none which we could bring about.5 This is one reason why we should be careful to interpret aspects of Savage’s framework too literally without good qualification. • States: are descriptions “of the world leaving no relevant aspect undescribed” [Savage 1954:9]. Crucially they have to contain some object about which one is uncertain about, i.e. what the temperature will be in Munich 5 Moreover, if constant acts were all we could choose from, the point of decision theory would be lost. This is because we could choose outcomes directly, the act guarantees a particular outcome. Maximising ones expected utility would not be possible anymore since actions do not come with risk anymore. Instead one should act so as to maximise non-expected utility by choosing the act which yields the best consequences [Joyce 1999:67]. 2 1 Savage’s strategy tomorrow at noon.6 In addition, they are descriptions of situations in the world over which an individual does not have direct control. The crucial part in this characterisation is what exercising ‘no direct control’ precisely means. In a first approximation this could mean that the probabilities of the states obtaining are independent of the relevant individual - more specifically, independent of her desires and choices. A typical example of a ‘state’, in the context of wanting to make an omelette, could be the circumstance of whether a given egg is good or rotten [Savage 1954:14]. The fact of the matter whether a given egg is rotten or not, should neither depend on ones desire that it is good, nor on ones action to break it into the bowl to the other good eggs. • Outcomes: are the consequences of actions, they describe “anything that might happen to the person” performing the act [Savage 1954:13]. A description of an outcome should encompass all the information relevant to the decision maker.7 In other words (descriptions of) outcomes should be specific to everything an individual cares about with respect to the given choice situation consequences are the set of events we care about. • Events: are sets of states [Savage 1954:10], where each event forms a subset of the set of states. How these sets are partitioned, i.e. what events are relevant depends on the interest of the decision maker. Returning to the omelette example above, one might be interested in the event that one egg out of a dozen is rotten - meaning the event would comprise 12 states where in each state a different egg is rotten. Consequently, an event forms a vector of possible outcomes [Broome 1991:90], i.e. all the outcomes which contain one rotten egg out of the twelve under consideration. In such a case, where the same consequence prevails in every state of the world, acts are functions from the set of events to consequences. As we have observed, Savage’s framework comprises elements which - often in their limiting cases but also in principal - do not have any obvious correlate in ordinary language. We arguably never face constant acts or are able to choose among completely specified courses of actions. We face these restrictions in principal, it is important to realise that this fact is not a result of ones bounded rationality. In the following one should be aware of these special devices - at times it will be advisable to abstract from these peculiarities and refrain to interpret certain aspects literally. We hold that, apart from technical considerations, the main challenge for 6 7 This seemingly proves Arrow’s [1971:45] characterisation of Savage-states as an “encryption of the world so complete that, if true and known, the consequences of every action would be known” wrong. States have to contain some uncertainty. An outcome O is said to be underspecified if there exists “a possible circumstance C such that the agent would prefer having O in the presence of C to having O in C’s absence. Whenever there is such a circumstance O must be replaced by two more specific outcomes O1 = (O&C) and O2 = (O&¬C)” [Joyce 1999:52]. 3 2 The Axioms - their roles and status the plausibility of Savage’s theory consists in giving good reasons for holding that each one of his postulates presents a genuine requirement of rationality. For those that do not stand the test of rationality, the question of how essential they are for the Savage theory to hold, remains. The next section will go through each of the postulates, look at their respective roles in the representation theorem with the aim of finding out which ones can be considered genuine principles of rationality. Once an approximate (interim) division among them has been made, the subsequent discussion will turn to a more in-depth discussion of the (more controversial) rationality postulates. 2 The Axioms - their roles and status The exact presentation and formulation of Savage’s axioms varies considerably in the contemporary literature discussing them. Originally Savage presented seven postulates–P1 up to P7– to which one additional postulate has been added; it was found to be implicit in Savage’s theory. Here we denote it by P0. We will stick to Savage’s ordering of his postulates with the exception of adding P0. Additionally, we will make use of a slightly more up to date notation, which has the advantage of being more concise and handy for the task in hand. This section will go through all postulates and discuss their function. Savage’s original Postulates:8 For all states s, s′ ∈ S and subsets A, B, C; consequences f, g, h ∈ F ; acts which are functions from states to consequences - f, g, h from S to F . The relation is not preferred to between acts . P1 Ordering Principle: The relation  is a simple ordering. P2 Separability/Sure-Thing Principle: For every f, g and B, f  g given B or g  f given B. P3 State Independence: If f (s) = g, f ′ (s) = g ′ for every s ∈ B, and B is not null; then f  f ′ given B, if and only if g  g ′ . P4 Probability Principle: For every A, B, A  B or B  A. P5 Non-Triviality: It is false that for every f, f ′ , f  f ′ —— (alternatively) There is at least one pair of consequences f, f ′ s.t. f ≻ f ′ . P6 Non-Atomicity: Suppose it is false that g  h, then for every f there is a (finite) partition of S s.t, if g ′ agrees with g and h′ agrees with h except on an arbitrary element of the partition, g ′ and h′ being equal to f there, then it will be false that g ′  h or g  h′ . P7 Averaging: If f  g(s) given B for every s ∈ B, then f  g given B. 8 Savage [1954:End papers]. 4 2 The Axioms - their roles and status Suppes [2002:250f.] uses a distinction to categorise axioms of a theory. Which axioms are a requirement of rationality and which ones (solely) play a structural role in the proof of Savage’s representation theorem? Structural axioms make existential demands on the environment of a theory, while not being implied by the existence of an expected utility representation of preference. According to Suppes only P5 and P6 are structural ones, Joyce [1999] adds P0 and part of P19 to the list of structural postulates, these are furthermore not essential, that is necessary, for the existence of a numerical expected utility representation. The former because it makes claims about the richness of the action space. The latter, the completeness requirement, because it makes claims about the judgemental state of the agent. One way to argue for the rationality requirement of completeness is to take Suppes’ condition, of it being implied by the expected utility representation, into account. In view of this condition, completeness seems to be a requirement of rationality. But we would not want to stop here and just accept the validity of Suppes’ argument insofar he takes the theory of expected utility for granted. In fact it is not clear whether he can. As we argued above, since the individual postulates give substantive meaning to MEU, each individual postulate should have independent reasons speaking in its favour. In order to really get behind the question of whether completeness is a requirement of rationality we need some other argument. We will come back to this point below (section 3.2). Aside from a rough, preliminary characterisation in this chapter, we will try to avoid entering the discussion about the alleged rationality axioms. At this point we will rather classify some of the axioms as potential rationality candidates whereas we discard the more obvious uncontroversial cases. Here we will point to some of their general implications. In criticising some of the postulates we will draw on arguments regarding their general plausibility. P0 - The Rectangular Field Assumption Broome [1991:115f.] found the the Rectangular Field Assumption (hereafter RFA) to be implicitly contained in Savage’s formulation of P1. According to the RFA each state of the world is arbitrarily assigned one outcome out of the set of all possible outcomes. Any arbitrary prospect constructed that way figures in an individual’s preference ordering! In other words the RFA defines a preference relation over all functions ( = acts) from states of the world to consequences. The reason Broome coined the term in question is because it defines a product set, and “a product set occupies a rectangle or a series of rectangles in a vector space” [1991:80]. The set of functions coming out of this procedure is a set of very rich acts. Consequently Joyce [1999:83] calls the RFA Act Richness. What it comes down to is that any possible combination of states of the world and consequences is contained in the set of acts. The intuitively 9 Namely the completeness condition. Joyce’s ordering of Savage’s postulates is structured differently to the way we present them here. In the terminology of Joyce P0 is called Act-Richness denoted by SAV0 , the completeness part of P1 is denoted by SAV3 , P5 Nontriviality by SAV1 , and P6 Event Richness is SAV7 [1999:83–95]. 5 2 The Axioms - their roles and status ‘artificial’ ontological distinction between states and consequences gives rise to such very implausible acts.10 An example would be ‘making the world hot (the act) when it is cold (the state of the world)’. Or alternatively, ‘arriving dry at work (act) while being soaked with rainwater (state of the world)’. Could this assumption be a candidate requirement of rationality? On a literal reading it seems that it couldn’t. A principle which gives rise to nonsensical prospects cannot be a plausible requirement of rationality. Moreover, one might object that the the examples presented are blatant contradictions. Making the world hot when it is cold is certainly a physical impossibility, nevertheless in Savage’s framework this presents no logical contradiction. The reason for this is that the set of states is ontologically distinct from the set of consequences. More specifically, our example comprises two distinct worlds - one being hot the other one being cold. In ordinary language there exists no such distinction among domains - there is just one domain or world. Acts like ‘making the world hot when it is cold’ present indeed a logical, conceptual contradiction in ordinary language, though in Savage’s formal language it does not. Nevertheless, only the logician would say that P0 cannot yield logical contradictions. Clearly then the motivation for this assumption is theoretical, and cannot be a genuine requirement of rationality. In fact P0 makes the proof of Savage’s representation theorem easier. This is because the proof for separability (P2) depends on this axiom, via the construction of constant acts. Broome [1991:115] raises attention to another problem implied by the RFA. Consider an outcome-description of loosing in a lottery on behalf of a risk-avers individual. An intuitive description of the outcome ‘End up loosing the lottery’ is: ‘Receive no prize and feel disappointed for not having won.’ Broome’s argument for why such a description is in conflict with Savage’s theory is the following: “The rectangular field assumption says your preference ordering includes all arbitrary prospects. Amongst them is the prospect that leads to this particular outcome for sure. This prospect determines, whatever lottery ticket you draw, that you get no money and also feel disappointment. But this feeling of disappointment is supposed to be one you get as a result of bad luck in the draw. It is hard to see how you could feel it if every ticket in the lottery would lead to the same boring result. So this prospect seems causally impossible, and that may make if doubtful that it will have a place in your preferences.” [Broome 1991:116] What Broome’s argument comes down to is that the (random) attribution of consequences involving risk sentiments to states of the world is not permitted by the RFA, because such risk properties intimately depend on the precise combination of states and consequences involved in an act. Steele [2007:145] argues that Broome11 over-interprets the RFA, thus unfairly targets Savage’s 10 11 The RFA also gives rise to constant acts, as mentioned above. Next to Weirich [1986] 6 2 The Axioms - their roles and status theory with respect to risk- and regret-sentiments. Although the RFA “strongly suggests that the description of outcomes should preclude risk sentiments”, she holds that the principle ought not to be interpreted literally. She gives three reasons to support her claim. First, because the RFA is an idealisation which constitutes an ideal agent’s preference space in such a way that it can be represented by “a continuous utility function (unique up to positive linear transformation) and a corresponding unique probabilistic belief function for an agent.” Consequently one could not assume the existence of such an individual “with such extraordinary discerning powers [. . . ] It is impossible for us ordinary mortals to entertain a complete preference ordering over the infinitely rich option space that Savage’s theory requires.” Although her argument does not directly target the completeness assumption of P1, this attack nevertheless feeds from it. We will postpone a critical discussion of completeness until later on. Does the RFA rise and fall with the completeness part of P1? Is the completeness assumption central to the RFA? Not necessarily so, since one of its implausible features comes in through the random, arbitrary assignment of outcomes to states of nature. To us, that seems to be the main factor which makes P0 such a strange thing. Nevertheless it might be argued that completeness is needed, because if it fails what determines which arbitrary combinations of outcomes to states go into one’s incomplete preference relation? Second, she points the finger at the above mentioned physical (and conceptual) impossibility of some Savage-acts. “The actual world constrains the set of actions that any agent, ideal or otherwise, is able to carry out. Just because we can conceive of an abstract map from states to some combination of outcomes doesn’t mean that the act in question is, will be, or ever was, a viable possibility in the actual world.” Third, she adds that many state-consequence combinations will also be “outright contradictory”. This point was already discussed above. For her it is not clear why Broome invokes causal impossibility as an insurmountable obstacle for incorporating risk sentiments into outcomes. Given the ideal nature of the RFA, she concludes that “any attempt to draw from it concrete conclusions about the contents of act outcomes is questionable.” [Steele 2007:145f.]. We will keep her advice in mind. P1 - Ordering This postulate defines the basic properties of the preference relation of a rational decision maker should display [Savage 1954:19]. Savage’s characterisation of the preference relation can be described to involve two parts. Defining a partial order, which is a binary relation  over the set of acts A, yields transitivity, i.e. such that for three acts a, b, c if a ≻ b and b ≻ c then it should be the case that a ≻ c.12 To give an example if I prefer an apple to a banana, and a pineapple 12 A partial order also generates reflexivity - a  a - and antisymmetry - if a  b and b  a then a = b. 7 2 The Axioms - their roles and status to an apple I should therefore prefer a pineapple to a banana. As we will see further below, intransitive preferences expose one to be taken advantage of, hence intransitivity is seen as a requirement of rationality. This constitutes the first part of P1. The second part consists in adding completeness to the properties of the preference relation which generates a complete ordering. It effectively implies that a decision maker has to have a definite preference over every act a in the set of acts A. Joyce [1999:84] argues that this constraint is not strictly required for the existence of an expected utility representation. Furthermore, he argues that it would even be unreasonable to place this requirement on rational agents. While Joyce may see the completeness assumption as not even controversial, we take note of his arguments but will discuss both it and transitivity more extensively in the next chapter (section 3.1). Our motivation for doing so is twofold, first and trivially the lengthy discussion is better placed in a separate section, second because completeness seems to be assumed to be a requirement of rationality by many authors.13 This fact renders completeness, contrary to Joyce, controversial and therefore demands careful examination. P2 - Separability, Independence or the Sure-Thing principle The preference relation is further built up such that it is separable across events. In other words, one’s desirability14 of the consequence of one act in one state of the world should be independent of the desirability of its consequences in any other state of world. It entails that “on no account, should preferences among consequences be modified by the discovery of which event obtains” [Savage 1954:32]. In other words, rational preference among acts should not be sensitive towards situations where these yield identical outcomes. Joyce [1999:86] puts it “that a rational agent should make judgments about the relative desirabilities of acts by treating their common outcomes as “dummy variables” whose effects “cancel out” in her deliberations.” In discussing Allais’ Paradox below (section 3.3) we will see a particular application of how this cancelling out is supposed to work. As P2 is thought to form the core of Savage’s theory, it is fiercely defended as a requirement of rationality. But more about that later on. The upshot of the separability axiom is that, jointly with P0, P1 and a theorem about conjoint additive structures, it implies that there exists an additive utility representation of preferences over acts that is unique up to positive affine transformation. Put differently, the value of each act is the sum of the state-dependent utilities of its consequences. 13 14 Such as Binmore [2009] and any subscriber to revealed preference theory. Meaning the utility one assigns to an outcome. 8 2 The Axioms - their roles and status P3 - State-Independence This postulate ensures the ordinal comparability of state-dependent utilities [Savage 1954:26f.]. In order to make cardinal comparability possible, P3 would have to be stronger. The source of this lies at the fact that utility is interpreted ordinal. Fishburn [1970:193] describes state independence as a companion to P2, which states if an act f yields consequence x, and g yields y given an event E where the probability P r(E) 6= 0, then act f is preferred to g given E if and only if act f ′ is preferred to g ′ when f ′ yields x and g ′ yields y. P3 achieves to establish a correspondence between preferences on consequences (through constant acts) and conditional preferences on events whose obtaining is regarded as possible, ie. non-null. Joyce [1999:87] sees P3 as some form of noncontextuality constraint on an individual’s beliefs and desirabilities. It is constraining in so far ones judgements about the probabilities of states or desirabilities of outcomes are not permitted be dependent on what act happens to be true. This makes it possible that a preference relation is effectively extended to consequences in terms of the relation among acts. P1 up to P3 determine such a fundamental preference relation between acts. Although most authors in the literature see P3 as an obvious requirement of rationality, its status was prominently challenged by Aumann [1971]. We will look at his critique in the next chapter (section 3.4). P4 - Probability Principle P4 introduces a new relation, namely a probability relation with states of the world as its relata [Savage 1954:33]. The binary relation, which we denote by ☎, should be read as ‘more probable than’. Since it is defined on the set of states (or events), it serves to make a probability representation of an agent’s attitude to states possible, e.g. how probable an agent judges the event E ‘It is raining in Munich at 12 am tomorrow’ in contrast to ¬E ‘It is not raining in Munich at 12 am tomorrow’. In order to fully achieve this, Savage constructs certain circumstances which are meant to provide a test for when one event is judged to be more (or less, or equally) probable than another. It is important to note that the notion of probability being used here is a purely subjective one. Savage is not concerned about determining the objective probability of it raining in Munich tomorrow at 12 am, but individuals’ probability. The prescriptive part of P4 is that individuals should prefer the alternatives that are more likely to yield the more desirable outcomes. In other words, given two alternatives with equal desirabilities any rational individual should prefer the more probable one. Once this method of determining ones credence in a state’s obtaining is established, P4 ensures that any two events can be compared to each other with respect to their relative chance of obtaining. One property of the probability relation, which is implicitly contained in this procedure is the completeness 9 2 The Axioms - their roles and status condition. Only with it, all events are comparable.15 The probability measure also needs to be transitive and quasi-additive. The latter property is one of ratios and absolute scales. The latter is needed to establish the link to (mathematical) probability. All these properties, which define a weak order, are sufficient in order to apply a representation theorem for probability [Fishburn 1970:193]. P4 is generally assumed to be a requirement of rationality, and quite uncontroversially so. If satisfied it guarantees coherence among ones beliefs. Since it acts as a constraint on the relation between beliefs and preferences, and not so much on rational preference, we will not discuss it in further detail here. P5 - Non-Triviality P5 demands that agents are not indifferent between at least one pair of (constant) acts [Savage 1954:17]. This postulate is commonly accepted to be purely structural since it, next to P6, is solely needed to ensure that the probability measure ☎ can be represented numerically. It is also required for it to be unique. If P5 were not satisfied then the probability relation would be reflexive which is needed for mathematical reasons [Fishburn 1970:cf.193]. Further evidence that P5 is a structural axiom comes from the fact it can easily be conceived to fail without there being some irrational act implied by doing so. Joyce [1999:84] argues that there are plenty of acts between which one can justifiably be indifferent. Imagine that a selection of outcomes are equally desirable to an agent, i.e. whether having a beer or a wine for breakfast. In this case there is just no decision to make, because both drinks, being alcoholic, seem equally detestable in the morning. P4 and P5 taken together, make it possible that an agent’s belief in the chance of a state’s obtaining can be extrapolated by offering him prizes in a specific way. Together they lead to the introduction of a notion of qualitative personal probability. It is qualitative in so far it signifies what it means for an individual to consider one event more probable than another in terms of the preference relation among acts already introduced. P6 - Non-Atomicity P6 allows the set of states to be partitioned as finely as one wants [Savage 1954:38f.]. Accordingly, there do not exists atomistic, fundamental descriptions of states. It is always permitted to add further details to descriptions of states. Only now can the notion ‘(no) more probable than’ be connected quantitatively with mathematical probability. This postulate has one far-reaching implication; it asserts that there exist no consequences which are so (un)desirable such that these could overpower the 15 In principle, similar objections as the ones raised above apply to completeness in the context of probability assignments to states and events. However we find that it does not have such problematic implications as its (utility-) counterpart. 10 2 The Axioms - their roles and status Acts\States Believe in God Not believe in God God exists Eternal life (∞) A bad situation (z) God does not exist Finite & deluded life (x) Assumed status quo (y) Table 1: Pascal’s Wager (im)possibility of any given event A. Consider the example of Pascal’s Wager, standing proxy for the type of scenarios non-atomicity is directed against.16 Consider an individual, Mr. Jones, who faces the decision whether to start believing in God or remain an atheist. His state-consequence matrix expressing the decision problem is given below, the desirabilities of each outcome are given in brackets, where x and y are of finite values and z is either finite or ∞. In addition, say Mr. Jones takes the probability of God’s existence to be one in a million (Pr(∃God) = 0,000001), it follows that the chance of there being no God is Pr(¬∃God) = 1 - Pr(∃God) = 0,999999, respectively. How should he decide? In this case where non-atomicity fails, he is advised to believe in God irrespective of how small the probability of his existence might be - the probability of one in trillion would yield the same recommendation. This is because in case God indeed exists, this consequence is infinitely more desirable to the individual than anything else which could possibly happen. Additionally, if z is interpreted as −∞ as Pascal in his original argument did, then not believing in God while God exists has infinite undesirability which seems to give further support to Pascal’s recommendation. The reason why such a categorical recommendation takes place, is because infinite (un)desirabilities simply wash away the probabilities. Probabilities are thereby made useless in deliberating what to do. The introduction of non-atomicity ensures that exactly this will not happen - no consequence or prospect is such that it could be infinitely (un)desirable. This is called the Archimedean condition which was first introduced by von Neumann & Morgenstern [1944] in their representation theorem for utility [Fishburn 1970:cf.194]. When the probability relation ☎ is defined upon the basis of the preference relation , P1 up to P6 imply the existence of probability measure on the set of states S that satisfies A ☎ B if and only if P r(A) ≻ P r(B) for all A, B ∈ S where A and B are events. That equivalence constitutes the essence of the representation theorem. The postulates are jointly sufficient to obtain an expected utility representation of preferences, but just for those acts in F that assign a finite number of consequences to all the states in some event A. To cover the infinite case the next postulate is still needed. 16 The example is taken from Jeffrey [1974:12f.]. 11 3 Rationality Axioms? P7 - Averaging P7 ensures that Savage’s representation theorem holds for all acts, including the ones that assign infinite number of consequence to all states [Savage 1954:77]. It is therefore an extension of the expected utility representation to infinite sets of consequences. Averaging also ensures that any desirability u on the set of consequences X is bounded [Fishburn 1970:194]. This postulate is a generalisation of the Sure-Thing Principle to cover infinite wagers. Fishburn [1970:193] argues that this generalised dominance condition, is not required for the derivation of a unique probability measure on the set of all subsets of S, the set of states. This completes our overview of Savage’s postulates. Candidates which are quite certain to be structure axioms are P0 RFA, P5 Non-triviality, P6 Non-atomicity. There is hardly any disagreement about this in the contemporary literature. This leaves us to discuss the remaining ones in more detail - these are P1a.) Transitivity, P1b.) Completeness, P2 Separability, P3 State-Independence. We will not discuss P4 Stochastic dominance and P7 Averaging. The latter because it is just a form of P2. P4 is excluded of the discussion because it does not directly constrain rational preferences. 3 Rationality Axioms? Each of the axioms under consideration in this section has some arguments speaking for being a constraint on rational choice. Not surprisingly for each there also exists arguments pertaining to the opposite. The status of the following arguments is a matter of hot dispute. Each of the following subsections will be concerned with the discussion of one. 3.1 Transitivity Transitivity is a property of a binary relation - the preference relation. Being so defined the relation is a partial order which is a desirable property from a mathematical or theoretical standpoint, since it makes things easier in proving the representation theorem. Besides, what might be the motivation arguing from rationality grounds for transitivity? Any argument attempting to do so has to give a plausible account of why intransitive preferences are undesirable from a rational point of view. It must show that there is some kind of defect in preferences exhibiting intransitivity. There exist two major arguments in the literature contending to establish that transitivity of preferences is a genuine requirement of rationality. These are the Dutch-Book argument and the Money-Pump argument. Although they are similar in structure both address the problem in slightly different ways. Here we will only be concerned with the latter, primarily because Dutch-Book arguments are mainly concerned with the (ir)rationality of probability assignments, a topic not at the centre of our debate. 12 3 Rationality Axioms? The money-pump argument, first established by Davidson et al. [1955], tries to show that an individual exhibiting an intransitive preference relation is liable to give up some of his wealth for no reward. It has the form of a reductio ad absurdum argument. It holds for any intransitive set of binary strict preferences, i.e. x ≻ y, y ≻ z, and z ≻ x. Suppose that an individual Mr. Smith is offered a choice among three different jobs, a, b and c: “He can be a full professor with a salary of $5,000 (alternative a), an associate professor at $5,500 (alternative b) or an assistant professor at $6,000 (alternative c). Mr. [Smith] reasons as follows: [a ≻ b] since the advantage in kudos outweighs the small difference in salary; [b ≻ c] for the same reason; [c ≻ a] since the difference in salary is now enough to outweigh a matter of rank.” [Davidson et al. 1955:145] Is this an irrational set of preferences? The reasons in favour of each pair appear to be plausible. Nevertheless, Davidson et al. argue that “the reasons can never be good enough to justify acceptance of such a set of preferences.” According to them, an intuitive principle of rational choice commands that an alternative should only then be chosen, if nothing else is preferred to it. This principle cannot possibly prescribe a rational choice in the context of Mr. Smith’s decision problem, because his preferences imply that no matter which job he chooses there will always be another job which he prefers to it! This generates the money-pump. To fully see this, imagine that Mr. Smith randomly settles with the assistant professorship, option c. Since he strictly prefers b to c, offering him to switch to b for a payment of a small amount of money, say 1$17 , will indeed result in him accepting the offer, pay 1$ and obtain the associate professorship. Now, since he strictly prefers a to b, a similar offer to switch to a for the price of 1$ will again result in him accepting the offer and pay 1$ to obtain the full professorship. Strictly preferring c to a, again a similar offer can be put forth, which he will accept. At this point Mr. Smith finds himself at his assumed status quo but with 3$ poorer. It is easy to see that this argument can be carried out indefinitely for any number of n-rounds. The conclusion is that Mr. Smith got money pumped which is a undesirable thing to expose oneself to. This behaviour is said to be irrational because one is worse-off than before without having gained anything. Therefore one should have transitive preferences. Let us consider another violation of transitive preferences taken from Broome [1999:70ff.]. Suppose Mr. Wilson is about to decide what to do for his vacation. He is faced with three alternatives, (R) to make a trip to Rome, (M ) to go mountaineering, and (H) to stay a home. Maurice prefers making the trip to Rome to go mountaineering (R ≻ M ) because he is afraid of heights, prefers 17 The exact amount does not matter; because of Mr. Smith’s preference of b over c there must some token of exchange for whose trade he will be willing to switch - may it only be such a small amount as 0,01$. 13 3 Rationality Axioms? staying at home to the trip to Rome (H ≻ R) because old roman ruins bore him, and prefers to go mountaineering to a stay at home (M ≻ H) because he judges avoiding a hiking tour coward-like when nothing else is there to do. Mr. Wilson exhibits intransitive preferences among the pairs of alternatives R, M , and H, thus is liable to getting money pumped over the alternatives. Is there a way to justify or rationalise this instance to intransitivity? Similar to Mr. Jones, Mr. Wilson’s preferences can be said to be menu dependent, meaning actual preferences depend on what alternatives are on offer. By itself this does not do much to avoid the charge of irrationality, but what about the reasons at play? Once again the reasons supplied for each individual preference seem plausible. Whereas Davidson et al. argued that no reason could ever be good enough to justify a violation of transitivity, one reason in this example might stand the test. The following argument has two parts. First, Mr. Wilson could argue that he did not choose among three but four alternatives. Instead of preferring a stay at home over a trip to Rome, Mr. Wilson would choose a stay at home when a trip to Rome was the only other option available - formally Hr instead of just H, yielding Hr ≻ R. On a similar move he re-describes staying at home when mountaineering was the only other option available - denoted by Hm , yielding M ≻ Hm .This is what Broome calls fine individuation of outcomes or alternatives, intransitivity and therefore threat of irrationality vanishes. Staying at home when Rome was the only alternative is thereby different from staying at home when both Rome and mountaineering are available. The general problem of the practice, is that fine individuation renders the requirement of transitivity vacuous, since by it any violation whatsoever could be rationalised!18 This brings us to the second part of the argument. As we stand, some principle according to which the right to refine outcomes is permitted in particular cases would be very desirable. This principle should also counteract Davidson’s et al. argument. Brome [1991:103] suggest two principles: the principle of individuation by justifiers according to which outcomes should be distinguished as different if and only if they differ in a way that makes it rational to have a preference between them. Where a justifier is the difference between two outcomes that makes it rational to have a preference among them. The second principle put forth, is one of rational requirement of indifference, which does not restrict individuation of outcomes, but demands indifference instead. Whereas Broome prefers the latter, we are more inclined towards the former. In the context of Mr. Wilson, the avoidance of cowardice can figure as a justifier. Irrespective of what one might think the truth about cowardice is, if 18 Broome [1999:71] argues that transitivity still has some restrictive power, captured in his distinction between practical and nonpratical preferences. Practical preferences only hold among two prospects if and only if the prospects are such that an individual could have a choice between them. Non-practical preferences are such that no actual choice among them is possible, and these are constrained by transitivity - i.e. because Hr ≻ R and R ≻ M it follows Hr ≻ M meaning Mr. Wilson ought to prefer a stay at home over Rome when mountaineering was the only alternative. This is obviously not a pair of alternatives Mr. Wilson could in any way choose from. 14 3 Rationality Axioms? Mr. Wilson feels he can preserve his dignity through going mountaineering, then this constitutes a sufficient reasons to justify his preference.19 Cowardice, or avoidance thereof, constitutes a relevant difference, or powerful enough reason contrary to Davidson et al., so as to justify fine-individuation on behalf of Mr. Wilson. On a more superficial, non-intentional description of his preferences, transitivity is violated but, as we argue, permissibly so.20 3.2 Completeness As was observed above, the completeness condition on the preference relation is a property of an ordering. An individuals preference relation is said to be complete if and only if for every pair of alternatives, either she prefers one over the other or is indifferent between them. The first feeds from philosophy of science considerations. Broome [1991:92] argues that the completeness axiom serves mainly a representational purpose. Joyce [1999:84] states that it is not even required for an expected utility representation. One way to achieve representability requires the relation to be an ordering, i.e. to be transitive, reflexive, and complete. But representability can also be achieved by using a partial order, with the drawback that it makes the proof of any representation theorem much harder from a mathematical point of view. This fact doesn’t just apply to Savage’s case, but for instance also to the theory of von Neuman & Morgenstern [1944] which also assumes completeness. On top of that Bradley [2012:14f.] observes that representation theorems without completeness do not achieve as much. This pragmatic consideration reveals that on grounds of mathematical expediency the completeness axiom has a lot going for it. How much leverage as a requirement of rationality does completeness have? The theory in support of this axiom goes under the name of revealed preference theory (hereafter RTP). RTP identifies preference with (observed) choice, this philosophical assumption is the driving force behind the argument in favour of completeness. Mandler [2001:15] identifies the choice definition of preferences as the orthodox understanding of preferences in economics, so does Binmore [2009] who at the same time is an advocate of RTP. According to the theory what is meant by having a preference for alternative x over alternative y, is just to choose x over y when both alternatives are available. Any description adhering to intentional, psychological factors is explicitly avoided [Binmore 2009:8f.]. The case for completeness is as follows - given any two alternatives, one constructs a forced choice. This scenario will always yield some positive result from an observational standpoint, because even not making a choice amongst two alternatives is making a choice. When observing that an individual has made no choice in a given decision situation, all that the theory can attribute to the decision maker is indifference. Or alternatively, if the situation is set up such that an individual starts off with an alternative x - the status quo, making 19 20 On the assumption that this issue can be avoided in R ≻ M . For a discussion of the money pump in sequential choice situations with ‘sophisticated’ choice, see McClennen [1990:89ff.] and Anand [2009:166ff.] 15 3 Rationality Axioms? no choice will mean preferring x. All it can attribute is either preferring one alternative over another or indifference - tertium non datur. By definition all individuals inevitably make some choice even though they might not do or want to do so, hence preferences are automatically complete. Irrespective of concerns whether this strikes one as a satisfactory approach to explain human behaviour - the theory is positivistic and behaviourist in spirit - issues we cannot discuss at any length in the present paper, the question remains whether this argument makes a convincing case for completeness to be a requirement of rationality? Suppose one had incomplete preferences among two alternatives - what could this mean and how could it be justified? Does it seem irrational not to have formed a preference over some alternatives? A first argument would hold no, one might not have have formed a preference over some alternatives due to time constraints, or mere lack of interest. Consider the situation of choosing one can of beans out of a bunch in a supermarket. Is it really a requirement of rationality to have made up ones mind about which can to choose? Maybe one of the cans contains two more beans than another, but should we even bother? In order to get closer to an answer, consider the case of Superia, a computationally unbounded being [Morton 1991:39f.]. Due to her extraordinary abilities Superia knows for each possible world - which is a completely detailed specification of how the things could be - how much she wants it - assigns precise utility to it -, and how likely she thinks that the real world will be like that at some point in the future - assigns a probability to it. She is also able to state the exact position of each possible world in her preference ranking. In order to determine her desirability of an alternative, she simply goes through all the possible worlds in which this alternative obtains, to form a sum of them where each is weighted by its individual chance of obtaining. All that is required in order to carry out this task on behalf of Superia, is speed and memory. Morton [1991:41] argues that, while this scenario might be possible for an unbounded being like Superia, the trouble for us ordinary mortals is that we do not and cannot have attitudes to whole possible worlds. We simply lack the time and, more crucially, the computational resources to imagine these. We are computationally bounded, where the meaning of boundedness as used by this example does not include rational boundedness! Assuming the latter whilst rejecting completeness on grounds of an argument which uses this, would prove self-defeating for a normative assessment of the axiom. How would a decision theory look like which takes into account the computational limitations we face? Morton [1991:43] observes an interesting link between complexity and incommensurability of alternatives. “[A] model that accommodates our limited capacity to handle information is likely to generate incomparable preferences for complex situations even given comparable preferences for simple ones”.21 Whereas this argument sees the locus of incom21 Morton’s strategy to overcome the incommensurability problem, which we will not consider here, is by assuming a fundamental, preference underlying, basic value relation. It contains ones fundamental preferences over general aspects or qualities. 16 3 Rationality Axioms? mensurability at the level of individuals, a second argument discerns it in the world. The following critique of the completeness axiom describes it aptly: “However, it is inevitable that people, if put in a position of having to, can make choices between incommensurable alternatives. Given that, the choice-value principle translates their choices into valuations; that is its function. [...] When two alternatives are incommensurable, they are not made commensurable by the mere fact that people can choose between them.” [Broome 1978:332] The argument is that there exist certain alternatives, goods in the world which just are incomparable such that no weighing is possible. Interestingly the incomparability is argued not be a feature deriving from individuals’ assessment. In order to get a tighter grip on this phenomenon Temkin [2012:176ff.] defines the relation of rough comparability which holds among two outcomes/alternatives, if one alternative is neither better than, worse than or equal to another. More formally - there exists alternatives x and y such that x  y, y  x and x ≁ y. Accepting the possibility and justifiability of the relation implies rejecting completeness.22 Is it possible that reasons in favour of one alternative are incommensurable with the reasons in favour of another? Such that no balancing is possible at all? Broome [1978:332] considers a positive answer to this question. If ones assumes that any thing can be compared to any other thing than it is like trying to “measure a good of one sort, say the removal of pain, against another, say money, as a scale. It turns out to be like measuring the brilliance of a painting with a light-meter.” Would we be satisfied with a uniform approach to measurement? Surely there is a limit to the scope of this method! After all it is not plausible to assess pain in terms of money. Nor is it plausible for many other ‘goods’ or alternatives such as freedom of speech, equality or even careers - imagine the choice between either becoming a professional musician or to take up a career in the political sector. 22 To make a failure of completeness even more plausible, we point to an argument from Mandler [2001:21] who argues that the incompleteness problem is not intrinsically tied to normativity. Incomplete preferences are most likely when multi-dimensional aspects of decision problems are considered. One option is to reduce complexity through focusing on one aspect of a decision. While such a reduction enables a ranking and hence re-establishes completeness, it needs a further argument to support the focus on specific, say, hedonistic, criteria. This is because the reduction implies that the specific focus taken is most important or relevant in deciding what to do. This move, if generalised, is questionable in the case of hedonism. Having to rank any act (or consequence) in terms of pain and pleasure does not proves to be intelligible. On the other hand, if this reduction is not carried out there is another alternative. As early proponents as von Neumann and Morgenstern [1944:29] recognised the possibility of incomplete utility functions. They stipulated that the absence of the completeness axiom would yield “a many-dimensional concept of utility”. Failure of completeness for decision under risk, a special case of Savage where the probabilities of states of the world are typically not given exogeneously, are studied under the name of non-archimedean utility representations. These are representations of preference orderings by means of utility functions whose range is a lexicographically ordered vector space [Herzberg 2009:8]. 17 3 Rationality Axioms? A third argument is provided by Mandler [2001:16f.] who gives a sociological explanation about the existence of the completeness axiom. In the course of decision theory the problems with which theorists have been concerned became more and more complex. Whereas completeness might be have been justified vis a vis the early comparably straight forward decision situations, that is not so anymore. He argues that currently, the cases under discussion involve too many aspects. Mandler [2001:20] mentions procedural issues and symbolic importance. Does it, or should it for instance make a difference whether a certain outcome is determined by a democratic decision or a paternalistic one? Should it matter in cases where the outcome is the same? Considerations like these certainly go beyond the scope of this essay but highlight the complexity of the matter surrounding the completeness condition. Fourth, incompleteness of preferences might be justified due to informational issues. It is easy to conceive situations where one expects to receive relevant information in the near future and consequently postpones judgement. One might want to argue that in such cases it would be wrong to reach a judgement. Temkin [2012:240] expresses this concern in his Reflection Principle: “If, on reflection, I know that at some point in the future I’ll have more knowledge than I currently have, and I now know that given that future knowledge it will be reasonable to assess two prospects in a certain way, then it is now reasonable for me to assess the two prospects in that way.” Thus postponing ones judgement is permissible if one knows, or to weaken the condition, is justified to believe that one will receive relevant information.23 We want to conclude this section and also the previous one in considering an interesting point raised by Mandler [2001:15]. He argues that transitivity and completeness are mutually exclusive properties among any account of rational choice has to choose from. The main reason for this fact is conceptual. Arguments for transitivity, such as the money pump, assume a welfare definition of preferences. Examples like it show that preferences if interpreted as rational welfare judgments should adhere to transitivity. As we have seen, on the same interpretation of preferences there are very strong arguments for them to be incomplete in principal. But what about the case where preferences are understood as choice? Is transitivity still a requirement of rationality? Consider again an intransitive set of preferences among the triple a, b and c as presented above. Starting with c one agrees to switch to b and then to a. Mandler interprets the (slightly weaker) relation c  a as merely saying that ‘an agent will accept z when x is available’. Under the choice definition of preference there are no resources within the theory to explain why choice-intransitivity is irrational. In order to 23 For another, fifth argument see Bradley [2012:14f.], who discusses the requirement of coherent extendibility. According to it it should be possible to extend ones current preferences to complete and consistent ones when deliberating about new alternatives. Coherent extendibility guarantees consistency when reaching new judgements. 18 3 Rationality Axioms? do so, some welfare-significance has to be attributed to the preference-relation  (or alternatively ≻) otherwise this conclusion cannot be reached.24 All that can be otherwise surmised is that an agent exhibiting intransitive preferences can end up with an option that is never chosen directly over the original status quo. Although we agree with Mandler on his observation, we hold that nevertheless even the choice-interpretation of preference has to admit some psychological import. Our remark not so much targets Mandler’s argument per se, but is rather part of a bigger argument against the very thin, behaviourist equation of preferences to choice. Although we cannot at length discuss the theory of revealed preference here, we want to highlight the fact one should not forget that we are dealing with agents whose choice-behaviour wants explaining. RPT offers a kind of explanation which completely ignores the role and existence of intentional, psychological states of individuals. Requiring some minimal reference to individuals’ psychology in explaining their behaviour, does not result in an automatic collapse of the theory into a welfarist one. 3.3 Allais’ Paradox & Separability The separability principle is said to constitute the ‘cornerstone’ of Savage’s theory [Joyce 1999:85]. For many theorists the theory of expected utility rises and falls with this principle [Broome 1991:115]. On a normative reading, separability, or as it is also called, the Sure-Thing Principle (hereafter STP) implies restrictions on the objects to which it refers. It requires ones desirability of the consequence of an act in one state of the world to be independent of ones desirability of the act’s consequences in any other state of world.25 Technically speaking, the STP is an instance of weak dominance. Weak dominance demands that if all the consequences of an act a are weakly preferred to those of another act b and a has one consequence which is strictly preferred to one consequence of b, then a ought to be preferred overall. The validity of this reasoning is derived from a pre-theoretical commitment to consequentialism, i.e. the only thing which matters, or one is justified to consider, when assessing the goodness of an action are its consequences. All concerns about the relevance of procedural aspects of decision making are omitted from this view. This is where Allais [1953] critique comes into the picture. In the shape of Allais’ Paradox, it calls the STP’s status as a requirement of rationality into question. Interestingly, the argument is not purely conceptual, but is supported by considerations about the empirical deviation from the prescribed, rational solution. The paradox applies to decision situations with the following structure (see Table 2): An individual placed in this decision situation is confronted with two choices in two distinct lotteries. The first (I) is a choice between lotteries A and B. The second (II) is a choice between lotteries C and D. A substantial proportion of 24 25 In our discussion of the initial money-pump argument, this was achieved by stating that Mr. Jones is worse off having lost 3D. Furthermore, the STP is valid only if the states of the world are probabilistically independent of the acts to which they are compared with. 19 3 Rationality Axioms? I II A B C D 0,01 1.000.000 D 0D 1.000.000 D 0D 0,1 1.000.000 D 5.000.000 D 1.000.000 D 5.000.000 D 0,89 1.000.000 D 1.000.000 D 0D 0D Table 2: Allais’ Paradox. people, some decision theorists among them, have been found to make choices inconsistent with the STP26 - combinations of choices which are inconsistent are either A and D or B and C where the former pair is most often observed in empirical findings and also the option some decision theorists try to justify. Here is a demonstration why this is so. We denote the three different outcomes figuring in the example as follows: ux = u(5.000.000D), uy = u(1.000.000D), and uz = u(0D) to . If the expected utility principle holds, the preference A ≻ B implies uy ≻ 0, 01uz + 0, 1ux + 0, 89uy . Introducing the probabilities of each outcomes we get the following 0, 11uy ≻ 0, 01uz + 0, 10ux . By adding 0, 89uz to both sides, we obtain 0, 11uy + 0, 89uz ≻ 0, 10ux + 0, 9uz , which says C ≻ D.27 The only two pairs of choices consistent with STP would either be a choice of A and C or B and D. Allais explains the deviance from the STP by arguing that individuals do not evaluate what happens in one state of the world independently from what happens in others. Put differently, contrary to what the STP requires of rational decision makers, it can happen that outcomes in different states of the world are not evaluated separately. Is there something special to the choice situation above, which might explain and possibly justify deviance from the STP? With reference to the pair A and B, Allais argues that avoidance of feeling disappointment or regret displays sufficient grounds to render this behaviour rational. Whereas A yields a certain amount, B would yield considerably higher amount but is way more risky. The trade-off between certainty and riskiness is not present in (II). Weirich [1986:436f.] further argues that the riskiness of the alternatives are relevant properties not taken into account when having framed the decision problem. Broome [1991:96 & 110] describes the peculiar structure among the outcomes and lotteries in question, as complementarities between ‘interacting’ states. Their Interaction is due to the specific way the individual lotteries are connected with prizes - a feature the STP cannot take into account because it just considers individual choices. But what can these observations tell us? Should we make our choices by taking into account what might happen in other states, thereby dismissing the STP as a rationality axiom? Is avoidance of regret rationally defensible? This line of argument brings up a general problem though. Would we be likewise justified to feel regret about a fair coin landing heads, if we betted on 26 27 See Maher [1993:64] for a list of empirical studies. The structure of this argument is taken from Gintis [2009:16f.]. 20 3 Rationality Axioms? tails? Samuelson [1952:672] gives a negative answer to the question. He states “either heads or tails must come up: if one comes up the other cannot.” The value one assigns to what happens in one state of the world (the coin landing heads) should not depend on what it would be like if the other state occurs (the coin landing tails). If that state really occurs no other state does. What would have happened in other states should make no difference to the value.28 In the centre of this argument lies the question: How can something that never happens possibly affect the value of something that does happen? Point taken, but isn’t there something special to Allais’ situation? Similar to the case of transitivity, Broome [1991:98] tries to solve the problem by fine-individuation. The reasons trying to justify a violation of the STP, all refer to a feeling, namely regret or disappointment. He argues that this is an indication that the decision problem might have been underspecified. A refinement of the decision problem incorporating reference to feelings achieves to get rid of the paradox. Again, similar to the argumentation above there comes in a slippery slope argument, which points towards the fact that any violation could be defended by fine-individuation. The force of the argument is that if any violation could be explained away by re-describing the decision problem this could make decision theory vacuous; after all decision theory aims at constraining ones choices, specifically so separability. STP is thought to be a constraint ones preference! Could the avoidance of regret be a powerful enough reason to act as a justifier for re-describing outcomes so as to involve feelings about risk sentiments such as Weirich proposed? This could reconcile the STP with attitudes towards regret, thus Allais’ choices came out to be rational. 3.4 State-Independence The status of the remaining axiom P3 is probably the most ambiguous and overlooked one of all seven (eight) postulates. Does it make a claim about rationality or is it a constraint on the interpretation of consequences and states? Consider an act a with the constant consequence x and another act b with the constant consequence of y.29 Suppose one prefers a to b, thus x over y, given some event E.30 Should we show the same preference given any other event E ′ ? State-independence requires one to do so, but it is questionable whether a convincing case can be made for it. In a first approximation one might want to specify that any answer would heavily depend on the specification of background conditions, e.g. the features relevant to an acts evaluation. In order to pull this argument from the abstract to the concrete consider (our reconstruction of) Aumann’s [1971:77f.] objection: 28 29 30 Others in favour of the STP, have reconstructed Allais’ decision problem in a sequential decision tree and then show that in this context the large majority of people adheres to the principle [Maher 1993:70ff.]. Unfortunately this line of defence is not very useful in the context of our discussion since we are interested in normative arguments. This means that acts a and b are constant acts. Event E has to be non-null, i.e. Pr(E) ⊁ 0. 21 3 Rationality Axioms? Suppose, Mr.X loves his wife very much and judges that without her his life would loose much of its purpose. His live would be “less ‘worth living’ ”. Mr.X’s wife falls ill and has to undergo a dangerous operation in order to keep on living and not die. Her chances of surviving the operation are 0.5. Imagine Mr.X be offered a choice between betting $100 on his wife’s survival or on the outcome heads in the toss of a fair coin. Aumann argues that, even though the two bets seem identical with regard to their outcomes and probabilities - thus the rational thing to do would be to be indifferent among them - Mr.X would still be justified to strongly prefer betting on his wife’s survival to anything else. The reason being that the gain of a $100 in the event that the wife dies is “somehow worthless”. If he bets on heads (or tails) he might win the $100 in a situation in which he will not be able to appreciate them. Aumann argues that in this situation there simply is no consequence whose desirability is state-independent. Is Mr.X’s supposed preference irrational? Savage [1971:80] replies to this objection by slightly twisting the example. He demands of Mr. X to imagine the counterfactual situation in which the continuance of his family life would not depend on the outcome of the operation. Only then, once he is detached from his personal feelings towards the situation, is he in the position to “appraise his own probabilities”. Savage further states that a consequence is in the last analysis an experience, where experiences screen out the features of the world causing them and hence have static, independent desirabilities [79]. Doesn’t this move, instead of solving one problem, generate another one? If, at the end of the day consequences are nothing but subjective experiences, what is the difference between such a consequence and one’s evaluation of it? Despite considerations which concern pragmatic difficulties for Mr. X to successfully detach himself from the situation he finds his and his wife in, Savage seems to miss the mark here. The point of Aumann’s example is not about the problem to elicit Mr. X’s beliefs about the chance of his wife’s survival31 , but rather by the very convincing case that desirabilities can be intimately dependent on the state in which it obtains, and rationally so! Why should Mr. X care at all about $100 in the event of his wife dying? To further rest our case consider an example, closer to everyday live experiences. Take the decision between having a cold beer or a hot chocolate. State-independence requires that ones preferences - in this case we focus on just the desirability - between these acts (‘having a cold beer’; ‘having a hot chocolate’) for any given situation to be independent of a particular state obtaining. In the context of this example the relevant states could be ‘A warm summer evening on the beach’ and ‘A cold winter day’. Say one prefers the hot-chocolate on cold winter day; is one required on grounds of rationality to prefer the chocolate over the cold beer on a warm summer evening? It seems not, why couldn’t one be justified to always prefer the hot chocolate on a winter’s day over the cold beer and vice versa? One response could be 31 Ensured by the fact that all states of the world are assumed to have the same probability of 0.5. 22 4 Conclusion that having a cold chocolate is not a properly specified consequence since the description is to coarse, leaving out relevant aspects for its evaluation. Having a cold chocolate when it is cold outside might be more faithful to the real issue. It follows that a precondition for State-Independence to hold, is that consequences have to be described in a sufficiently fine grained manner. If the description of a particular consequence turns out to be too coarse grained, then aspects which might be relevant to the agent’s assessment of the state’s worthiness - which is relative to the consequence considered - are left out. To conclude, the fact that decision makers’ preferences depend on the underlying state of the world is a feature which cannot always argued to be irrational! As a result P3’s scope should be changed so as to only be used in specific situations where occurrence of state-dependent preferences are justifiably ruled out. 4 Conclusion This completes our discussion of the rationality axioms. We conclude by rejecting completeness and accepting the other three principles, although with restrictions. As we have seen Savage’s theory only focuses on factual uncertainty, that is uncertainty about which states of the world obtain. Another kind of uncertainty which is not addressed by Savage, is option uncertainty. It concerns the specific way to describe decision problems. Issues that arise with this kind of uncertainty were touched upon by Broome and his proposal fine-individuation of outcomes. This method, we argue, is absolutely crucial. We have seen how powerful it can be. The problem we faced there, consisted in the absence of clear criteria which define when it is permissible to re-describe outcomes and when not. In discussing the rationality axioms, the general trade-off consisted between either rationalising certain deviant behaviour such as intransitivity of preferences but at the same time making decision-theory less prescriptive and therefore more vacuous. The threat of making decision theory a vacuous enterprise prevails. This makes it even more important to come to terms with option uncertainty in a way that comes up with criteria which determine the permissibility of fine individuation. This could be a relevant area of further research. A third and related kind of uncertainty addresses concerns about the defensibility of preferences address ethical uncertainty, i.e. uncertainty about ones relative desirability assignments to outcomes. The problem of incommensurability of alternatives, discussed in the section about completeness, lets us points the finger at another issue, namely a peculiar futility of non-substantive theories of rationality of which Savage is an instance. Theories like Savage’s which are only concerned about internal consistency cannot serve to guide agents about what preferences are rationally defensible. Individuals who desire some guideline in deliberating which normative criteria are appropriate and how these should be applied get no answer from these. In other words non-substantive theories cannot evaluate the sources of preferences. An answer to the main question of practical rationality, ‘How should I act?’ cannot completely be 23 REFERENCES answered by a formal decision theory alone. 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