Money Pumps and Diachronic Books
Abstract
The idea that rational agents should have acyclic preferences and should obey conditionalization has been defended on the grounds that otherwise an agent is threatened with becoming a “money pump.” This essay argues that such arguments fail to prove their claims.
1.Introduction.
Synchronic book arguments purport to show that violation of certain principles can compel rational agents to choose options inferior to other options available to them no matter what might be the case according to the decision maker’s state of full belief. Diachronic book arguments extend this kind of argument to predicaments where decision‐makers face choices that yield as consequences opportunities for future choice. They aim to sustain two main theses about rational choice:
1. | At the moment of choice, rational agents should weakly order their options with respect to their values and goals on pain of making choices that are inferior to other available options. | ||||
2. | Rational agents are precommitted to change or retain their probability judgments and values in a manner that they cannot revise on pain of making choices that are inferior to other available options. |
Both of these claims seem wrong to me. Rational agents may fail to choose for the best because there is no best option among the alternatives available. Similarly, rational agents may alter their values, probability judgments, and full beliefs in ways that violate dictates of a favored methodology without fear of being suckered into a sure loss.
I shall focus in this paper on the use of money pump arguments to support transitivity or acyclicity of strict preference and on the exploitation of diachronic dutch book arguments to support temporal credal conditionalization.
2.Acyclicity.
If rational agent X (categorically or “all things considered”) strictly prefers 1 to 2 and is confronted with a choice between just these two options, X should choose 1. But the converse does not hold. Suppose X equiprefers 1 and 2. In the face of a pairwise choice, X may rationally regard both 1 and 2 to be admissible and be prepared to choose either. Alternatively, X might invoke some secondary consideration to break the tie for optimality between 1 and 2 in favor (let us say) of 1. In that case, X would be prepared to choose 1 but not 2 even though X equiprefers 1 to 2. So X’s choosing (or being prepared to choose) 1 over 2 does not reveal that X strictly prefers 1 to 2. Indeed, it does not even reveal that X weakly prefers 1 to 2. X might judge 1 better than 2 according to one desideratum and inferior to 2 according to another. There is no categorically optimal option in the two way choice. Both options remain admissible for choice. Again a secondary criterion might be invoked to choose among the admissible options.
There is a conception of preference more closely tied to choice than the notion of categorical preference recognized in the previous paragraph. X weakly prefers 1 to 2 in the sense of revealed preference if and only if X is prepared to choose 1 over 2 in binary choice. X strictly prefers 1 to 2 in the sense of revealed preference if X weakly prefers 1 to 2 in that sense but not 2 to 1. X equiprefers 1 and 2 in the sense of revealed preference if and only if X weakly prefers 1 to 2 and 2 to 1 in the sense of revealed preference.
Revealed preference should be relativized to the criteria (choice functions) by means of which the decision‐maker’s preferences constrain the choices the rational decision‐maker should make. If X categorically equiprefers 1 and 2, in a pairwise choice, both 1 and 2 are admissible for X to choose if X’s choices are not guided by secondary values when primary values fail to render a verdict. If X invokes secondary criteria, 1 alone may be admissible. In the one case, 1 and 2 are revealed to be equipreferred. In the second case, 1 is revealed to be strictly preferred to 2. Yet, 1 and 2 are categorically equipreferred.
Regardless of how revealed preference is relativized to choice functions, revealed weak preference must be reflexive. Furthermore, it must be complete. This is secured by definitional fiat. The definitions given do not yield transitivity of revealed strict preference. Nor do they entail the weaker condition of acyclicity.
Consider the requirement that rational agents ought to have transitive strict preferences in choosing among the options available to them. Prima facie this is a requirement of synchronic rationality. Agent X confronting options 1, 2 and 3 cannot coherently strictly prefer 1 to 2 and 2 to 3 while at the same time weakly preferring 3 to 1. Transitivity requires that X strictly prefer 1 to 3 given the first two conditions. The weaker requirement of acyclicity allows that 3 might not be strictly preferred to 1 by leaving open whether the two alternatives are equipreferred. X’s preferences are cyclic if X strictly prefers 1 to 2, 2 to 3 and 3 to 1.
Money pump arguments allegedly show that revealed strict preference should be acyclic. Otherwise a cunning bettor can “pump” X of X’s resources. Such arguments, if successful, would go a long way to showing that revealed weak preference is transitive and, hence, that revealed weak preference weakly orders the options of rational agents.
In spite of these advantages of invoking money pump arguments to establish acyclicity, there is good reason for finding the alleged validity of such arguments disturbing. Binary choices revealing strict preference for 1 over 2 at t_{1}, 2 over 3 at t_{2} and 3 over 1 at t_{3} could be a manifestation of change of categorical preference over time. Alternatively, such choices might be a manifestation of X’s cyclic categorical preferences that remain the same during the period. Money pump arguments are blind to the difference between the two cases. Whether or not X’s categorical preferences have changed, money pump arguments claim that X is vulnerable to being drained of all assets.
Such blindness ought to be unacceptable. One may well agree that rational X should have acyclic categorical and revealed strict preferences at a given time. However, we should not prohibit rational X from changing X’s categorical preferences over time.
Anxiety about money pumps not only argues for diachronic constancy. It has also been used to rule out as incoherent various forms of decision making under unresolved conflict.
X confronts a choice between three ways to spend his week’s wages:
(A) | 70% of the week’s salary for meeting the needs of his family and 30% for savings. | ||||
(B) | 50% for the family and spending 50% on cigarettes, whiskey and wild women. | ||||
(C) | 35% for the family, 30% for cigarettes, whisky and wild women and 35% for savings. |
X is in conflict between a concern to tend to his family without regard to anything else and a lust for the wild life. The good X would prefer A to B to C. The bad X would prefer B to C to A.
The good X and the bad X alike prefer B over C. It seems reasonable that X should, all things considered, categorically strictly prefer B to C. On the other hand, A and B are noncomparable all things considered as long as conflict goes unresolved. So are A and C.
X contemplates several hypothetical scenarios. One concerns how X would choose between A and B were they the only options available to him. All things considered, A and B are noncomparable. X cannot make up his mind as to whether to attach more weight to the values of the good X or the bad one. X could simply acknowledge this and admit that in a pairwise choice he would consider both options admissible. However, X might also declare that although savings is not a primary concern of his, in the situation where X cannot make up his mind between A and B on the basis of his primary but conflicting interests, X is prepared to choose the option that maximizes savings—to wit, option A. So in a pairwise choice between A and B, X would choose A. This does not reveal a categorical strict preference between A and B. But it expresses a revealed strict preference of A over B relative to criteria for choice invoking the secondary criteria as well as the primary ones.
The same reasoning leads to noncomparability in categorical preference between A and C but a revealed strict preference of C over A. B, however, is categorically strictly preferred to C and, hence, is revealed strictly preferred to C.
In sum, B is categorically strictly preferred to C. A and B are noncomparable, as are C and A. This result should generalize. Categorical strict preference should be acyclic and, indeed, transitive. Option 1 is categorically strictly (weakly, equi) preferred to option 2 if and only if all ways of evaluating the pair of options the decision‐maker judges to be permissible agree in strictly (weakly, equi) preferring 1 to 2. On the assumption that ways of evaluation are weak orderings, transitivity and, hence, acyclicity of categorical strict preference is trivial.
In spite of this, B is revealed strictly preferred to C, C to A, and A to B. A cycle in binary revealed preference is exhibited—one which money pump arguments allegedly prohibit. Money pump arguments thus preclude the kinds of unresolved conflict where categorical strict preference and revealed strict preference come apart in the manner I have just described.
Thus, money pump arguments advanced on behalf of acyclicity of revealed preference appear to prove too much and in two ways. Rational agents, so it seems, are obliged to remain faithful to the same value commitments indefinitely. And agents who face choices without having resolved conflicts between several dimensions of their values are condemned to irrationality unless they can straighten out their conflicts. Those of us who find these implications unacceptable are honor bound to examine the cogency of money pump arguments for acyclicity and to propose what appear to be alternative rationales for acyclicity for those circumstances when acyclicity may legitimately be imposed as a constraint on the coherence of value judgment.
Consider then the failure of acyclicity exhibited in X’s predicament and how a money pump is supposed to work.
X is prepared to choose A over B in a pairwise choice and B over C while choosing C over A. A clever bookie offers X the opportunity to choose between C and B provided that X pays a very small price ϵ not so great as to reverse his preference for B. X chooses B‐ϵ. The bookie then returns with an offer to supply A at a price of ϵ. When this is accepted, C is offered in exchange at a price of ϵ. Thus X, who started with C, ends up with C minus a charge of 3ϵ for the three exchanges. X has been “pumped”. If X’s preferences were acyclic, this could not happen.
Notice that the argument presupposes that the acyclic preferences are instances of revealed binary strict preference. As such, they may or may not be cases of categorical strict preference. The argument purports to establish the money pump regardless of whether revealed strict preference is categorical or not. Moreover, it makes no difference to the argument whether preference for A over B remains when X faces the choice of C over A or not. Hence, if the money pump argument were valid, changes in preference would be precluded.
Suppose agent X prefers A over B and B over C. Yet C is preferred over A. We have a cycle in categorical preference. Maher (1992a, 1992b) and Levi (1987, 1991) have argued, X, if rational, need not be “pumped.”
Agent X may be in one of three predicaments:
1. | X may be certain that some third party will offer to exchange B for C for a small price followed by A for B for a small price (if B is purchased) and then C for A at a small price netting a sure loss of 3ϵ. X is also certain that at the second stage X will choose rationally. (All losses are in terms of categorical value.) (Further variants allow that X is certain that this will be repeated for n rounds or indefinitely.) | ||||
2. | X may be certain that no such offers will be made. | ||||
3. | X may be in doubt among some subset of the possibilities listed in the first two cases. X’s credal probabilities for these prospects may be numerically determinate or they may be indeterminate. |
In case (1), X’s initial option is to choose between retaining C or purchasing B for a small price. X has no control then over how he will choose later. He can only predict what he will do. If he predicts that he will choose the preferred option of the two he faces at that point, he can predict that by accepting the exchange, he will end up with a sure loss. As a rational agent, he should refuse the initial trade and retain C.
This response does not mean that he does not value B more than C under circumstances where he is certain (as in case 2) that no opportunities for trading A for B and C for A would ensue. This is the circumstance under which a pairwise choice between B and C may be said to reveal categorical preference between them. X’s refusal to purchase B in exchange for C and a small price does not betray X’s strict categorical preference for B if the opportunity arises in a context where X is certain that an effort to pump him will take place. Neither 1 nor 2 involves a choice of a dominated option. And uncertainty as to which obtains as in case 3 avoids domination as well.
W. Rabinowicz (2000) has recently taken up a theme developed in a discussion of conditionalization by B. Skyrms (1993). He argues that if X is offered the opportunity to make the exchange a finite number of times, X will find it in his interest to accept the exchange of B for C the first time and be led down the primrose path to a sure loss. Rabinowicz is mistaken in this; but the mistake is worth rehearsing.
Rabinowicz envisages a case where if X refuses to trade C for B at stage 0, he is offered a second chance at stage 1. If he refuses a second time at stage 2, the final payoff is C at stage 3. If he refuses at stages 0 and 1 but accepts at 2, he receives B − ϵ. This is better than C.
If X refuses at stage 0 but accepts B at stage 1, at stage 2 he is offered A. If he accepts he receives A − 2ϵ. This is judged better than B − ϵ, which is the payoff for refusal.
In case X accepts the trade of B for C at the 0 stage, he can accept or refuse the exchange of B for A at the first stage. If X refuses, A can be offered a second time at stage 2. Refusal at stage 2 yields a payoff of B − ϵ. Acceptance yields A − 2ϵ.
If X accepts A at stage 1, X is then offered C for A at the second. Refusal yields A − 2ϵ. Acceptance yields C − 3ϵ.
Rabinowicz argues that X can predict at the first node that at stage 2, X should trade and will do so if rational, no matter which stage 2 pair of options he has. At stage 1, X will also trade if rational regardless of the stage 1 options he has.
Rabinowicz assumes that X will predict that X will choose optimally at opportunities present at stages 0, 1 and 2. He then claims correctly that if X refuses the exchange of B for C at stage 0, X will predict that X will exchange B for C at stage 1 and A for C at stage 2 and, hence, will end with A − 2ϵ at terminal stage 3. If X accepts the exchange at stage 0, X will predict that X will exchange A for B at stage 1 and C for B at stage 2, ending up at stage 3 with C − 3ϵ.
Refusing and accepting the exchanges of B for C are the sole options at stage 0. Since X prefers C − 3ϵ to A − 2ϵ, X should choose to accept the exchange.
Rabinowicz then concludes that X is being pumped. But this is not true. From X’s point of view at the initial choice node, X regards his options as accepting or refusing the trade of B for C. There are no other options available. X has no control at the initial stage over what Y will choose at subsequent stages. If X did have such control, X would have no control at subsequent stages over what X does at those stages. However, because the decision‐maker is certain that X will choose rationally in the future, X is sure that refusing the trade will yield A − 2ϵ. Accepting this trade will surely yield C − 3ϵ. X may prefer the latter to the former according to the cyclic preferences. Given the two options available to the decision‐maker X, choosing that option is optimizing. X is not choosing an option dominated by another available as an option to him. In what sense is X being pumped?
By hypothesis. X did indeed freely exchange C for B. X knew that by accepting the exchange, X was going to end up with a tax bill 3ϵ as well as the prize with which X began. X saw that as better than the predicted consequence of refusing the exchange. X freely chose the best option. To be sure, X ends up poorer than X began. But that was bound to happen no matter which of the options given to him X chose.
It may be argued that if X did not have the cyclical preferences, he could not be taxed in the manner just sketched. Avoiding cyclical preferences to avoid taxation is not avoiding a dominated option. It is adapting one’s preferences to circumstances as in the case of sour grapes.
Is adjusting preferences so that one may not consider oneself a victim of taxation a good idea? I doubt whether a general all‐purpose answer can be given to this question. It is certainly not a requirement of minimal rationality. Why should we mandate sour grapes?
If X were certain that a strictly enforced law against exploitation by a bookie is operative, there would be no serious possibility as far as X is concerned that he will be pumped even though he has cyclical preferences. And even if X is not certain, as long as there is a serious possibility that he will not be taxed in this way, cyclicity in preference creates no trouble. To be sure, X might be certain that cyclicity will breed taxation. But the fact that commitment to values of some kind has such bad consequences is not always reason to abandon the preferences. Perhaps there ought to be a law against the exploiters. In any case, there is nothing incoherent about defying the exploiters.
Money pump arguments were designed initially to show that individuals who violate certain canons of rationality will end up choosing options that are dominated by other options available to them just like synchronic dutch book arguments do. Showing that violating these canons is one way that, in the face of other assumptions, renders one vulnerable to taxation is no substitute. Those who use money pump arguments to defend acyclicity of preference have failed to show that decision‐makers who violate acyclicity are driven to choose dominated options.
3.In Praise of Acyclicity.
Embracing categorical synchronic preference cycles is, nonetheless, a bad thing.
Let Z have strict categorical preferences for A, B, and C that yield a cycle. How should Z choose when all and only these three are options available to him? Maximizers of value will refuse to choose any option dispreferred to all other options and, in this sense, dominated by them. By this consideration, none of the options may be recommended. But decision‐makers who evaluate their options so that no option available to them is admissible are synchronically incoherent. It is a cardinal condition of rational choice that the set of admissible options be nonempty. One should avoid cycles in categorical preference to avoid violating this cardinal requirement.
An apologist for cyclicity might concede that Z should have acyclic categorical strict preferences when faced with three or more options. But the apologist might still insist that when it comes to evaluating these options for the purpose of making pairwise choices, categorical strict preference could be cyclical.
This too is incoherent. If Z’s preferences in the three‐way comparison and in the three two‐way comparisons are synchronically coherent and are expressions of a single value commitment, they should be contextually robust. That is to say, Z’s evaluation of any pair of the options should be the same regardless of whether the pair belongs to a choice between three options or a choice between just that pair.
Insistence on contextual robustness is not intended to prevent Z from changing preferences when confronting the three way choice from what it is in the case of the two way choices. It is intended to preclude Z from categorically strictly preferring 1 over 2 in one (possibly hypothetical) context of choice and changing the preference in another where both contexts are supposed to obtain relative to a fixed preference. No change in values is supposed to occur.
No appeal to money pumps is involved at any stage in this argument. The argument does not preclude Z from changing categorical preferences over time. It does not rule out cyclicity in binary revealed strict preference in decision making under unresolved conflict. It supports acyclicity of categorical strict preference understood as a condition of synchronic coherence or rationality that every rational decision‐maker ought to obey.
4.Conditionalization.
Thus far, attention has been focused on conditions on rational valuation or preference. The second condition to be considered concerns rational probability judgment. But first a few preliminary words about rational belief are in order.
X’s state K of full belief at t is constituted by X’s commitments as to what to fully believe at t. X’s state K at t is at the same time X’s standard for serious possibility at t (Levi 1980).
Another propositional attitude regulated by conditions on rational synchronic coherence is X’s confirmational commitment (Levi 1997, chap. 6; see also Levi 1980.) At any given time, X is committed to judgments concerning what subjective or credal probability judgments ought to be made relative to potential state of full belief K for any potential state of full belief K.1
Confirmational commitments, like states of full belief, are subject to conditions of synchronic rationality. In particular, any rationally coherent confirmational commitment should satisfy the following requirements:
Probabilistic Coherence: For every K, C(K) should be a set of conditional probability functions Q(x/y) defined for each sentences x in language L and y in L consistent with K that satisfy the requirements for finitely additive probability obeying the multiplication theorem.
Consistency: C(K) should be nonempty if and only if K is consistent.
Probabilistic Convexity: For each y consistent with K, the set of conditional probability measures in C(K) restricted to Q_{y}(x) = Q(x/y) is convex.
Confirmational Conditionalization: Suppose agent X who is in state of full belief K considers what X’s judgment of credal probability should have been in some weaker state of full belief UK. X’s current state of full belief K may be represented as an expansion UK^{+}_{e} of UK by adding e. For every probability measure in P(x/ye) in C(UK), there is a function Q_{e}(x/y) = P(x/ye) in C(K) and conversely for every Q_{e}(x/y) in C(K) there is a function P(x/ye) = Q_{e}(x/y).
The first three principles are conditions of synchronic rationality endorsed explicitly or implicitly by a wide number of authors. Confirmational conditionalization is more controversial. Here is an argument (from Levi 1980) that aims to support its cogency as a requirement of synchronic rationality.
Let decision maker X be confronted with a choice between a pair of options A and B with the following matrix of payoffs in utility:
Matrix1 | |||
H∧E | H∧∼E | ∼E | |
A | S − P | −P | 0 |
B | 0 | 0 | 0 |
In this example, ∼E is a serious or doxastic possibility consistent with the decision‐maker’s state of full belief K. However, its relevance to deliberation has been nullified by the circumstance that its value to the decision‐maker is the same regardless of which of the two available options A or B is chosen. If the payoff of 0 utiles is replaced by k utiles for any positive or negative k in the column for ∼E, the sure thing principle states that the preference comparison between A and B remains the same. If P(∼E) < 1, the expected values of the options conditional on E order the options just as the unconditional expectations of the options do. When P(∼E) = 0 (but remains a serious possibility), the two options are equal in expected utility and are equipreferred. The tie can be broken by appealing to the conditional expected utility. So the conditional probability can be construed as the fair betting quotient for called off bets.
Suppose, however, that the decision‐maker evaluates the decision problem in matrix 1on the supposition adopted purely for the sake of the argument that E is true. That is to say, the decision‐maker considers how he should evaluate options A and B were he to add E to the full beliefs in K to form the expansion K+E of K by adding E.
We now have two hypothetical scenarios. According to one, ∼E is a serious but not a relevant possibility. According to the other, on the supposition that E is true, ∼E is not a serious possibility. Assume that in assessing the two hypothetical scenarios, the decision‐maker retains his utilities for payoffs and confirmational commitment intact. Should the decision‐maker evaluate A and B the same in both cases? The Sure Thing Principle is silent on this point. Advocates of confirmational conditionalization argue, nonetheless, that as a matter of synchronic rationality, the two evaluations should agree. The requirement is a constraint of synchronic rationality because the two hypothetical evaluations are assessed from the same point of view at the same time.
The supposition linking the two scenarios is that relative to both scenarios the confirmational commitment remains the same. This supposition parallels the requirement insisted upon in arguing for acyclicity of preference as a synchronic principle that preferences relative to pairwise comparisons and three way choice remain the same.
Suppose X at t_{0} is in belief state K_{0} with confirmational commitment C_{0}. X’s credal state is B_{0} = C_{0}(K_{0}). Let X at t_{1} expand K_{0} to K_{0} + E = K_{1} while keeping C_{0} unchanged. According to confirmational conditionalization, B_{1} = C_{0}(K_{1}) is the set of probability functions of the form P_{1}(H/F), where each such functions is equal to a function in B_{0} of the form P_{0}(H/F&E).
This type of change in credal or subjective probability is often called “conditionalization.” I call it “intertemporal” (Levi 1974) or “temporal” (Levi 1980) credal conditionalization. It holds if and only if X expands X’s corpus while retaining a fixed confirmational commitment. If X contracts X’s state of full belief or changes confirmational commitment, it fails. Neither possibility is precluded by confirmational conditionalization.
There is a well known argument presented by Teller with credits to Lewis (Teller 1973) that would sustain temporal credal conditionalization as a condition of diachronic rationality were it cogent. Space does not permit a detailed summary of the argument. One may refer to Teller’s paper or to other sources for details.
Since expansion of K_{0} is allowed in the course of the argument, the argument, if cogent, would support Confirmational Tenacity mandating keeping the same confirmational commitment indefinitely. In a precise parallel to the use of money pump arguments in connection with failures of acyclicity, if the argument were legitimate it would prove too much. The diachronic dutch book argument does not distinguish between failures of temporal credal conditionalization due to changes in confirmational commitment and those due to failures of confirmational conditionalization. Both are proscribed.
In response to earlier remarks by Maher (1992a, 1992b) and by Levi (1987, 1991), Skyrms (1993) agrees that rational agents may rationally violate temporal credal conditionalization. He claims that the Lewis‐Teller argument does not legislate against such violations. Yet, he thinks that the Lewis‐Teller argument for conforming to temporal credal conditionalization is mandatory on pain of incoherence in the setting of a very special “epistemic model.”
According to that model, the decision‐maker X declares in advance what his rule for changing credal probabilities upon making observations is. At the initial stage when X’s credal state is represented by Q, X publicly “posts” prices at which he is willing to bet on various propositions both unconditionally and in called off bets. Upon making observations and following through with his updating rule, X will post new prices. X’s epistemic strategy consists in X’s credal state today and an updating rule.
Skyrms takes the Lewis‐Teller argument to be addressed to the coherence of epistemic strategies. A strategy available at the initial stage t_{0} consists of (1) X’s fair betting rates (or credal probability function) at t_{0} and (2) X’s updating function for evidence obtained at t_{1} and expressible as Boolean functions of some set of exclusive and exhaustive alternatives. (2) is, in effect, X’s confirmational commitment at t_{0} restricted to expansions of X’s state of full belief at t_{0} belonging to a definite class.
If the Lewis‐Teller argument were successful it would establish that the updating function (2) obeys confirmational conditionalization at all future times and remains fixed at all future times as confirmational tenacity requires. I have already argued that there is a case to be made for confirmational conditionalization without benefit of Lewis‐Teller. The appeal to diachronic coherence aims at establishing precisely what Skyrms does not want to endorse—to wit, the satisfaction by the updating function of confirmational tenacity. Of course, Skyrms is right not to endorse confirmational tenacity. He should abandon his defense of the argument of Teller and Lewis.
The Teller‐Lewis argument must show that if X is going to violate temporal credal conditionalization, a clever person could compel X to choose a dominated option from among those available to him.
The Lewis‐Teller argument does not show this. X can refuse at the initial node to purchase a certain called‐off bet. Call it gamble 1. Indeed, this is the best option available to him if he is convinced that someone is waiting to buy gamble 1 at a cheaper price should X be willing to do so. At the initial stage when X faces this decision, X has no control over his future decisions (so we are assuming). Since X judges that if he comes to full belief that E, he will be willing to sell more cheaply, he would be foolish not to refuse the gamble initially.
Skyrms might complain that if X refuses gamble 1, X is honoring the odds he posted initially. That is true. But if X thinks he might be confronted with someone who will offer to buy back gamble 1 in case E is true, honoring those odds does not reflect X’s judgment as to what X should do. To make the practice of posting odds that X is willing to live up to an indicator of probability judgment, X must be certain that no one will buy back the bet in case X changes confirmational commitment. The scenario constructed by Lewis and Teller does not meet this specification.
Skyrms offers another response to the objection to the Lewis‐Teller argument under consideration. This is the one to which Rabinowicz refers in his discussion of transitivity of preference. He suggests we consider a case where if X declines gamble 1 at the initial stage, there remains a malevolent agent offering to pay him a lower price to engage in a replica of gamble 1 at the second stage should E come true.
Skyrms calculates that the expected value of refusing gamble 1 at the initial stage and accepting it are equal and negative. By arranging for suitable small premiums, he argues reasonably that accepting gamble 1 at the initial stage is preferable to rejecting it even though X is doomed to receive a net loss relative to the initial status quo.
Skyrms’s calculations are perfectly correct. But the response is irrelevant. Diachronic dutch books purport to show that the decision maker X at the initial node will be driven by considerations of rationality to choose an option inferior to some other option available to him no matter what is the case consonant with X’s initial state of full belief.
According to Skyrms’s scenario X is worse off, no matter how X chooses, than X was in the initial status quo. If X has the option of remaining in the status quo position, X should do so. But by hypothesis X does not have this option. X is not rationally compelled to choose an option dominated by other available options if X violates temporal credal conditionalization. Buying gamble 1 at the initial stage is not dominated by refusing to buy it at that stage. Since these are the only two options, where is the beef?
The dictates of rationality do not mandate the diachronic principle of temporal credal conditionalization but only the synchronic principle of confirmational conditionalization. Consequently, when X has no good reason to change confirmational commitment, X should obey temporal credal conditionalization. However, just as changing preferences for good reason can lead to legitimate failures of transitivity of revealed preference, so too changing confirmational commitments can lead to failures of temporal credal conditionalization.
How to characterize justifying changes in preference and in confirmational commitments are important philosophical questions as is the topic of justifying changes in full belief. By implication, in this essay I have been claiming that such justification makes no appeal to diachronic principles of rationality. A more positive characterization is a far more difficult topic to address. It deserves our earnest attention.
References
- Levi, I. (1980), The Enterprise of Knowledge. Cambridge, Mass.: MIT Press.
- Levi, I. (1986), Hard Choices. Cambridge: Cambridge Univerrsity Press.
- Levi, I. (1987), “The Demons of Decision”, The Monist 70: 193–211.
- Levi, I. (1991), “Consequentialism and Sequential Choice”, in M. Bacharach and S. Hurley (eds.), Foundations of Decision Theory. Oxford: Blackwell, 92–122. Reprinted in Levi (1997) as chap. 4.
- Levi, I. (1997), The Covenant of Reason. Cambridge: Cambridge University Press.
- Maher, P. (1992a), Betting on Theories. Cambridge: Cambridge University Press.
- Maher, P. (1992b), “Diachronic Rationality”, Philosophy of Science 59: 120–141.
- Rabinowicz, W. (2000), “Money Pump with Foresight”, in W. Rabinowicz (ed.), Value and Choice. Lund Philosophy Reports, 201–235.
- Skyrms, B. (1993), “A Mistake in Dynamic Coherence Arguments?”, Philosophy of Science 60: 320–328.
- Teller, P. (1973), “Conditionalization and Observation”, Synthese 26: 218–238.
- ^{1} This requirement is not quite as demanding as it seems when credal states are allowed to be indeterminate. Still it does require of rational agents that they fulfill conditions no one can hope to satisfy. As in the case where rational agents are required to believe the deductive consequences of their full beliefs, the “requirement” is better construed as a “commitment” or an “undertaking” (see Levi 1997, chaps. 1–3).