ࡱ> ` N4bjbj 2>nF%444.b$~~~;;;8;<L~\=B"(B(B(BKT+LGL[\\\\\\$]h `%\~IV/JKIVIV%\ (B(B\[[[IV (B~(B[[IV[[[~[(B = `f;V[[\0\[` Z`[`~[HWL2OH[QSWLWLWL%\%\[WLWLWL\IVIVIVIV~~~$;~~~;v0j0   Probability in Everettian quantum mechanics Peter J. Lewis  HYPERLINK "mailto:plewis@miami.edu" plewis@miami.edu Abstract The main difficulty facing no-collapse theories of quantum mechanics in the Everettian tradition concerns the role of probability within a theory in which every possible outcome of a measurement actually occurs. The problem is two-fold: First, what do probability claims mean within such a theory? Second, what ensures that the probabilities attached to measurement outcomes match those of standard quantum mechanics? Deutsch has recently proposed a decision-theoretic solution to the second problem, according to which agents are rationally required to weight the outcomes of measurements according to the standard quantum-mechanical probability measure. I show that this argument admits counterexamples, and hence fails to establish the standard probability weighting as a rational requirement. 1. Introduction According to many, theories in the tradition of Everett (1957) provide our best hope for solving the foundational difficulties that plague quantum mechanics. The most striking feature of such theories is that there is a straightforward sense in which every outcome of a measurement actually occurs. But Everettian theories suffer from a foundational difficulty of their own; if every outcome of a measurement occurs, then in what sense can one outcome be more probable than another? In other words, Everettian theories provide no obvious way to recover the Born rulethe standard link between the quantum mechanical formalism and probabilistic predictions. This difficulty is serious because it is via these predictions that quantum mechanics is confirmed. There are two problems here, a qualitative one and a quantitative one (Greaves 2004, 425). The former arises because it is hard to see what probability claims could mean in the context of a deterministic theory in which initial conditions can, in principle, be fully known. The latter arises because, given a proposal for the meaning of probability claims in Everettian theories, it remains to be shown that the resulting probabilistic predictions match those of standard quantum mechanics. At least two strategies have been developed for addressing the qualitative problem (see section 3), but there has been little progress on the quantitative problem. Recently, however, a promising line of argument for addressing the quantitative problem has been developed in the work of Deutsch and Wallace. Deutsch (1999) offers a decision-theoretic argument, essentially extending the argument strategy of Savage (1954) to the Everettian realm. That is, he argues that for agents in Everettian worlds, rationality demands that they should order their preferences by weighting the possible outcomes of a measurement in accordance with the Born rule. This argument has been elaborated and defended by Wallace (2002a, 2003, 2005). My aim in this paper is to provide an overview and evaluation of the Deutsch-Wallace approach. I argue that neither Deutschs original argument nor Wallaces elaboration succeed in establishing the Born rule within the context of Everettian quantum mechanics. In particular, there are alternative measures over future events that cannot be ruled out by the arguments they offer. 2. Everetts theory Let me begin by outlining the Everettian account of quantum mechanics and explaining the source of the difficulty facing would-be Everettians in dealing with probability. Probabilities enter quantum mechanics via the Born rule, according to which the probability of each outcome of a measurement is given by the squared amplitude of the corresponding term in the quantum-mechanical state. So for example, if the state of a spin- particle is a|!(p + b|!(p, where |!(p is the state in which the particle is spin-up with respect to some axis and |!(p is the state in which it is spin-down, then if one measures the spin along this axis, the probability of getting the result  up is |a|2 and the probability of getting the result  down is |b|2. In the standard theory of quantum mechanics, the probability given by the Born rule is interpreted as the objective chance that the state will collapse to that term on measurement. So for the spin- particle, the reason that the probability of getting the result up is |a|2 is that there is an objective chance that the state of the particle will instantaneously become |!(p when one measures it. The interpretation of probability is straightforward, but unfortunately the standard theory suffers from the measurement problem; collapse on measurement is at best ill-defined and mysterious, and at worst incompatible with the rest of quantum mechanics (Bell 1987, 1178). Everetts approach avoids the measurement problem by eschewing collapse altogether. According to Everett (1957), all that measuring the spin of the above particle does is to entangle its state with the state of a measuring device, and ultimately, with the state of an observer. So after the measurement, the state of the system made up of particle, measuring device and observer is given by a|!(o|!(m|!(p + b|!(o|!(m|!(p where |!(m and |!(m are states of the measuring device in which it registers  up and  down respectively, and |!(o and |!(o are states of the observer in which she sees the measuring device registering  up and  down respectively. Since there is no collapse, both terms in the quantum-mechanical state are retained after a measurement, and so there is a straightforward sense in which both outcomes of the measurement actually occur. The trick, then, is to reconcile the actual occurrence of every possible result of a given measurement with our experience of exactly one of the possible results. Everetts insight was to relativize the experience of the observer to one of the terms in state (1); relative to the first term the observer sees the device registering up, and relative to the second term she sees it registering down. The followers of Everett elaborate on this insight in various ways; the two terms in (1) are said to describe distinct worlds (DeWitt 1970), or distinct histories (Gell-Mann and Hartle 1990), or distinct minds (Lockwood 1996). What these accounts have in common is that somethingmy world, or my history, or my mindis attributed a branching trajectory through time rather than the usual linear one. For the purposes of this paper, I will follow Deutsch in regarding Everettian quantum mechanics as a many worlds theory (Deutsch 1996, 223). Solutions to the measurement problem along these lines are popular, and with some reason; they involve no additions or changes to the basic mathematical structure of the theory, and they are arguably more easily reconciled with relativity than other approaches (Lockwood 1989, 217). But they also raise several immediate difficulties. First, the number of terms in state (1) depends on the basis in which it is written downthat is, on the choice of coordinates for the vector spaceand this seems straightforwardly conventional. But if each term corresponds to a distinct world, and this term has some ontological import, then it looks like the number of terms cannot be purely a matter of convention. Furthermore, for most of these choices of basis, the state of the observer relative to a particular term will not be a state in which the observer sees the device registering up or a state in which she sees it registering down. So in order for Everetts theory to explain the fact that we get results to our measurements, it looks like there must be some preferred basisone that corresponds to the actual division of the state into branches. But again, since the choice of basis vectors is essentially arbitrary, it is hard to see how this can be the case. The second major difficulty for Everetts theory concerns personal identity. Assuming a solution to the preferred basis problem, the observer in the measurement above has two successors, one of whom sees the result up and the other of whom sees down. These post-measurement observers are not identical to each other, and each bears the same relation to the pre-measurement observer, so it seems that neither can be identical with the pre-measurement observer. But without an account of personal identity over time, it is hard to see how Everetts theory can explain my experience. If the goal is to explain why I get results to my measurements, it doesnt immediately help to explain why two future observers who are not identical to me will get results to their measurements. The third difficulty is the two-fold problem of probability outlined in the previous section. Even assuming that the preferred basis and personal identity problems can be solved, there is still no obvious role in the theory for the Born rule. The dynamics by which the state evolves over time is completely deterministic, so there are no objective chances in Everetts theory. Furthermore, in principle the observer could know enough to predict the salient features of her future states with certainty, so there are apparently no epistemic probabilities either. In fact, given that both outcomes of the above measurement will occur, it is hard to see how there can be any sense in which one outcome is more probable than another. Finally, even given a solution to this qualitative problem, the quantitative problem of showing that the probabilities appearing in Everetts theory obey the Born rule remains. The arguments considered here concern the quantitative probability problem. But the problems facing Everetts theory are clearly interrelated; any solution to the quantitative probability problem presupposes solutions to the preferred basis problem, the personal identity problem and the qualitative probability problem. So to set the stage for the Deutsch-Wallace approach to the quantitative problem, I first need to briefly explain their position on the other problems. 3. The Deutsch-Wallace picture I take the following sketch from Wallace (2002b), who in turn follows Deutsch (1985). First, it is important to note that Deutsch and Wallace reject the postulation of additional entities over and above those described by the quantum state itself. Hence the talk of worlds in Everetts theory cannot be taken to refer to entities over and above the quantum state, but must rather be a manner of speaking about the quantum state. Rather than using additional entities to construct solutions to the preferred basis and personal identity problems, talk about worlds must emerge in a natural way from independent solutions to these problems, grounded in quantum mechanics itself. Wallaces proposal is that there is no preferred basis intrinsic to the theory, but rather, there are more and less convenient bases in which to describe the problem at hand. Wallace makes use of an analogy with time here. There is no objectively preferred foliation of spacetime, but a certain foliation may be a particularly convenient one in which to explain a certain phenomenon. In special relativity, the frame in which the observer is at rest has no objectively preferred status, yet it is good for explaining what that observer sees. By the same token, in quantum mechanics, the basis partially defined by certain eigenstates of the observers brain has no objectively preferred status, but again it is good for explaining what the observer sees (Wallace 2002b, 643648). Wallace adopts a similarly pragmatic attitude towards personal identity, and again appeals to an analogy with time (2002b, 649652). In spacetime physics, there is nothing in the formalism of the theory itself corresponding to persisting objects, but nevertheless in many circumstances we can pick out fairly stable matter configurations that correspond to our pragmatic notion of a persisting object. So similarly in quantum mechanics, the theory itself contains nothing corresponding directly to persisting objects, but nevertheless in many circumstances the branches picked out by the pragmatic criteria above remain stable over time, corresponding to our ordinary notion of persisting objects. There is, however, still the difficulty that objects (including persons) can persist through multiple successors, as in the measurement example above. Here Wallace adopts a Parfittian line; the concept of personal identity needs to be weakened to a certain kind of physical continuity (2005, 3). In the measurement example above, this continuity relation holds equally between the pre-measurement observer and each of her post-measurement successors. In particular, it is important to note that the continuity relation holds between time-slices; Wallace does not identify persons with temporally extended trajectories through the branching world structure. I take no stand here on whether the picture sketched above provides adequate solutions to the preferred basis and personal identity problems; rather, for the purposes of assessing the Deutsch-Wallace arguments concerning probability, I simply take this picture for granted. Concerning the qualitative problem of probability, Wallace suggests two accounts (2002a, 1822; 2005, 4). According to one account, even though there is an objective sense in which both results will occur, there may still be a subjective sense in which the observer is uncertain about which result she will see, and the Born rule probabilities are a reflection of this uncertainty (Saunders 1998; Vaidman 1998, Ismael 2003). Wallace calls this the subjective uncertainty (SU) account. According to the second account, since the observer knows of each outcome that it will actually occur, there is no sense in which she can be uncertain about what will happen. Nevertheless, it may still be rational for her to adopt an attitude to her successors analogous to the attitude one has to ones potential successors in cases of genuine uncertainty; it might be rational to calculate expected utilities using the Born rule measure, for example (Papineau 1995; Greaves 2004). Wallace calls this the objective determinism (OD) account. As long as one of these accounts is tenable, some understanding of probability in the Everett context is possible; Wallace does not commit himself to one over the other. With these preliminaries taken care of, we can proceed to the quantitative problem of probability. It may be illuminating to contrast the Deutsch-Wallace approach to the other existing strategy for deriving the Born rule within Everetts theory. The argument, which dates back to Everett himself (1957, 460461), has the following form: Consider a series of spin measurements on particles prepared in the state a|!(p + b|!(p; every possible sequence of measurement results occurs in some branch, but in the infinite limit, the squared amplitudes of those branches in which the frequencies of up and down results fail to match the Born rule tend to zero. There are two problems with this form of argument. First, infinite sets of measurements do not occur in nature, and such results have no obvious consequences for individual measurements (Deutsch 1999, 3129). Second, such proofs assume the squared-amplitude measure of probability; while the squared amplitude of anomalous branches tends to zero in the limit, the number of such branches tends to infinity. Since the squared-amplitude measure is the Born rule, such arguments are circular (DeWitt 1970, 34). Hence a new kind of argument is needed. 4. Deutschs argument Deutschs strategy (Deutsch 1999) differs from that outlined above in that it is based, not on considerations of long-run frequency, but on consistency constraints on an agents preferences. Deutsch claims that the Born rule can be derived from the formalism of quantum mechanics alone, together with some innocuous axioms of rationality; given that quantum mechanics is true, the Born rule is forced on us as a constraint of rationality. That is, Deutschs argument is a decision-theoretic one. To make it easier to think of quantum mechanical measurements in terms of the standard machinery of rational decision theory, Deutsch asks us to imagine that payoffs are attached to the various possible outcomes. So suppose, for example, that a measurement of observable  EMBED Equation.3  is made on a quantum mechanical state |((, and that the measurement has n distinct possible outcomes (that is, that the observable has n non-degenerate eigenstates). Then we can imagine that the observer is to receive a payoff xi depending on which outcome occurs, where the size of xi represents the value of the payoff to the observer; for example, the payoff might consist of xi dollars. For convenience, we can label each eigenstate with the corresponding payoff; hence when |(( = |xi(, the observer receives payoff xi. In general, though, |(( will be a superposition of the various eigenstates, expressed by the sum  EMBED Equation.3  where the (i are arbitrary coefficients. After the measurement, the observer has n successors, each of whom receives one of the payoffs xi. Objectively speaking, provided that the coefficients (i are all non-zero, there is a sense in which the observer (or, rather, her successors) receives all the possible payoffs. But nevertheless, Deutsch argues that it is rationally incumbent on the observer to differentially value such measurements based on the sizes of the coefficients. In particular, he argues that the value V that a rational agent should ascribe to an  EMBED Equation.3 -measurement of state (2) is given by  EMBED Equation.3  In other words, even though every outcome actually occurs, the preferences of a rational observer are given by the Born rule; the value of each possible outcome is weighted according to the squared amplitude of the corresponding branch. I will call the rule for valuing outcomes expressed in (3) the Standard Rule. For brevity, I sometimes call V[|((] the value of state |((. The goal of Deutschs argument, then, is to show that the Standard Rule is the only rationally permissible way of assigning values to states. Deutsch claims that (3) follows from the evolution of the quantum mechanical state, together with a few innocuous axioms of classical decision theory. The axioms Deutsch appeals to are Additivity, Substitutibility and the Zero-Sum Rule. Additivity says that an agent is indifferent between receiving two separate payoffs of x1 and x2 and receiving a single payoff of x1 + x2. Substitutibility says that the value of a composite measurement is unchanged if any of its sub-measurements is replaced by a measurement of equal value. The Zero-Sum Rule says that if a measurement with payoffs xi has value V, then an identical measurement with payoffs (xi has value (V. I take it that these axioms really are innocuous, in the sense that they do not presuppose any probabilistic notions that would render Deutschs argument circular. Deutsch proceeds to the general conclusion (3) via three stages, considering states of increasing complexity at each stage. The first stage concerns the simple case of a symmetric two-term superposition, |(( =  EMBED Equation.3 (|x1( + |x2(). We know that the observer assigns value x1 to state |x1( and value x2 to state |x2(; the question at hand is what value she should assign to the symmetric superposition of the two states. Deutsch argues that an agent who is rational in the sense defined by the above axioms must assign V[|((] =  EMBED Equation.3 (x1 + x2). The crucial step in the argument is the claim that  EMBED Equation.3  for arbitrary constant k. The state on the right is identical to that on the left, except that the payoff attached to each eigenstate has been increased by k. The state on the left has the original payoffs, but the observer is also given an unconditional payoff of k. The claim is that a rational observer should attach equal values to these two scenarios. Deutsch claims that this follows from Additivity, since Additivity entails that a rational agent is indifferent between receiving a payoff of x1 + k and receiving a payoff of k followed by a payoff of x1, and similarly for x2 + k. Deutsch then sets k = x1 x2, which gives  EMBED Equation.3  By the Zero-Sum Rule, the right hand side is equivalent to V[ EMBED Equation.3 (|x1( + |x2()]. Hence (5) simplifies to  EMBED Equation.3  as required by the Standard Rule. In the second stage of his argument, Deutsch generalizes this result to symmetric superpositions with more than two terms. Consider the four-term symmetric superposition |(( =  EMBED Equation.3 (|x1( + |x2( + |x3( + |x4(). By Substitutibility, a rational agent must assign the same value to this state as to the two-term state  EMBED Equation.3 (|xa( + |xb() where xa =  EMBED Equation.3 (x1 + x2) and xb =  EMBED Equation.3 (x3 + x4), since by (6), |xa( has the same value as  EMBED Equation.3 (|x1( + |x2() and |xb( has the same value as  EMBED Equation.3 (|x3( + |x4(). But by (6) again, the value of the two-term state, is  EMBED Equation.3 (xa + xb), and substituting the values of xa and xb yields  EMBED Equation.3  as required by the Standard Rule. This result is straightforwardly generalized to any symmetric superposition of 8, 16, 32 etc. terms. But what if the number of terms is not a power of 2? Consider for example, the three-term superposition |(( =  EMBED Equation.3 (|x1( + |x2( + |x3(). Consider also a state |((( = EMBED Equation.3 |V[|((]( + EMBED Equation.3 |V[|((](, where the observer receives payoff V[|((] in each branch. By Additivity, |((( has value V[|((], since essentially the outcome of the measurement is ignored, and so it is equivalent to receiving V[|((] unconditionally. But by Substitutibility, |((( has the same value as  EMBED Equation.3 |(( + EMBED Equation.3 |V[|((](, since the value of |(( is, by definition, V[|((]. Substituting for |(( yields  EMBED Equation.3  The left-hand side is now a symmetric four-term superposition, so applying (7) yields  EMBED Equation.3  which rearranges to  EMBED Equation.3  as required by the Standard Rule. This procedure for obtaining the value of a three-term superposition from the value of a four-term superposition straightforwardly generalizes; hence one can obtain the value of an (n 1)-term superposition from an n-term superposition, and by this means derive the Standard Rule for a symmetric superposition of any number of terms. The third stage of the argument concerns asymmetric superpositions. Consider the two-term asymmetric state |(( =  EMBED Equation.3 |x1( +  EMBED Equation.3 |x2(. Suppose the system in state |(( becomes correlated with an auxiliary system, such that the state of the combined system is  EMBED Equation.3 (|x1(|y1( + |x2(|y2( + |x2(|y3(), where the |yi( are eigenstates of an observable  EMBED Equation.3 , and the payoffs are y1 = 0 and y2 + y3 = 0. Suppose a measurement of  EMBED Equation.3  is followed by a measurement of  EMBED Equation.3 . If the  EMBED Equation.3 -measurement yields x1, then the  EMBED Equation.3 -measurement yields y1 = 0 with certainty. If the  EMBED Equation.3 -measurement yields x2, then the value of the  EMBED Equation.3 -measurement is V[ EMBED Equation.3 (|y2( + |y3()], which is also zero, by (6). Hence a rational agent is indifferent between a simple measurement consisting of an  EMBED Equation.3 -measurement on the original system, and a composite measurement consisting of an  EMBED Equation.3 -measurement followed by a  EMBED Equation.3 -measurement on the combined system. But by Additivity, the composite measurement has the same value as a single measurement of the observable  EMBED Equation.3 . In the eigenstates of this observable, the state of the combined system is the symmetric superposition  EMBED Equation.3 (|x1 + y1( + |x2 + y2( + |x2 + y3(). But by (10), the value of this last measurement is  EMBED Equation.3 (x1 + y1 + x2 + y2 + x2 + y3), or  EMBED Equation.3 x1 +  EMBED Equation.3 x2. So since a rational agent is indifferent between this last measurement, the composite measurement, and the original simple  EMBED Equation.3 -measurement, the value of the original measurement must also be  EMBED Equation.3 x1 +  EMBED Equation.3 x2, as required by the Standard Rule. This argument form straightforwardly generalizes to any state of the form  EMBED Equation.3  where m and n are integers, yielding V[|((] = (mx1 + (n m)x2)(n. Asymmetric superpositions of more than two terms are obtained by applications of Substitutibility. Generalization to states whose coefficients are the square roots of irrational numbers is more complicated, but the argument so far gives us more than enough material to allow an adequate evaluation. 5. The gap in Deutschs argument The goal of Deutschs approach, then, is to show that the Standard Rule is the only rationally permissible way of assigning values to measurements. But there are other prima facie plausible ways of assigning values to measurements. For example, since the observer has one successor for each of the payoffs xi, one might think that the value of a measurement to a rational observer should simply be the average value of the payoffs:  EMBED Equation.3  Note that for a composite measurement, the value of each sub-measurement is calculated using (12), and then these results are combined using a second application of (12). I will call this the Average Rule. The Average Rule ignores the amplitude attached to the terms, and treats each branch as of equal weight. But one can argue that this is exactly as it should be, since for the Everettian approach to be an adequate account of our experience, low amplitude and high amplitude terms must be subjectively indistinguishable, and so presumably of equal subjective value. Furthermore, since each outcome definitely occurs to one of the observers successors, one might argue that there is a straightforward sense in which the observer can fully expect to receive each payoff, and hence that there is no reason to discount the values of the payoffs by the factor 1/n. According to this reasoning, the value of the measurement should be the sum of the payoffs:  EMBED Equation.3  I will call this the Sum Rule. There is, of course, an infinite variety of possible rules via which an agent could assign values to measurements. However, it is arguably unreasonable to demand the elimination of every alternative rule, however bizarre and ad hoc, before the Standard Rule can be regarded as established (Greaves 2004, 451). Hence I concentrate on the Average Rule and the Sum Rule because they are independently plausible; indeed, rules which treat each branch as of equal weight are sometimes cited as more plausible than the Standard Rule in the context of Everetts theory (Graham 1973, 236; Lewis 2004, 15). Hence any argument that seeks to establish the Standard Rule as rationally compelling must at least eliminate these two alternatives. For Deutschs argument to succeed, then, the Average Rule and the Sum Rule must be eliminated as rationally permissible measures at some stage. Let us then go through the three stages in turn, and see how the alternative measures fare. The first stage of Deutschs argument concerns the two-term symmetric superposition state |(( =  EMBED Equation.3 (|x1( + |x2(). Note that the Average Rule assigns V[|((] =  EMBED Equation.3 (x1 + x2), in agreement with the Standard Rule, and hence the argument at this stage cannot be decisive between the two. The Sum Rule, on the other hand, assigns V[|((] = x1 + x2, and hence ought to be ruled out by the argument Deutsch gives here. Consider, then, the key step in this argument, summarized in (4). Deutschs claim here is that a rational agent should be indifferent between the following two possibilities: (a) receiving a payoff k and then having two equal-amplitude successors who receive x1 and x2 respectively; and (b) having two equal-amplitude successors who receive x1 + k and x2 + k respectively. Deutsch asserts that this follows from Additivity. Clearly the Sum Rule does not satisfy (4); an adherent of the Sum Rule will assign value k + x1 + x2 to measurement (a), and value 2k + x1 + x2 to measurement (b). That is, an adherent of the Sum Rule will regard a payoff of k to each of two successors as twice as good as a payoff of k now. But note also that the Sum Rule satisfies Additivity; since the Sum Rule simply adds all the payoffs, an adherent of the Sum Rule will necessarily be indifferent between two payoffs of x1 and k and a single payoff of x1 + k, and similarly for k and x2. But (4) does not immediately follow; Additivity constrains an agent to be indifferent between (k followed by x1) and (x1 + k), and between (k followed by x2) and (x2 + k), but Additivity says nothing about how the value of a measurement should be affected by distributing payoff k over two branches. Indeed, if Additivity did constrain observers in this way, it would no longer be an innocuous axiom of rationality, but a substantive claim about decision-making in Everettian quantum mechanics, requiring independent justification. Hence the assumption Deutsch relies on to establish (4) is stronger than Additivity, and so this argument fails to establish Standard Rule for two-term symmetric superpositions (6) as a requirement of rationality. In particular, the Sum Rule remains a viable rational alternative. The Sum Rule, then, survives the first stage of Deutschs argument. In fact, due to the cumulative nature of the argument, the Sum Rule cannot be ruled out at any stage, since the second and third stage make essential use of (6). Furthermore, additional appeals to Additivity are made in the second and third stages of the argument, and in each case the appeal is problematic in just the same way. In the second stage, Deutsch appeals to Additivity to establish that a measurement in which each of an observers two successors receives payoff V[|((] has value V[|((]. Again, an adherent of the Sum Rule will deny this, and again, there is nothing in the principle of Additivity itself that requires an observer to be indifferent as to the number of successors over whom a payoff is distributed. In the third stage, Deutsch appeals to Additivity to establish that for the state  EMBED Equation.3 (|x1(|y1( + |x2(|y2( + |x2(|y3(), an  EMBED Equation.3 -measurement followed by a  EMBED Equation.3 -measurement has the same value as a single measurement of the observable  EMBED Equation.3 . But in the former case, payoff x2 is received by one of the observers successors, whereas in the latter case it is received by two. An adherent of the Sum Rule will not value these two measurements equally, and Additivity does not require her to be indifferent concerning the distribution of payoff x2 over two branches. The Sum Rule, then, serves as a counterexample to Deutschs claim that quantum mechanics together with his three axioms of rationality entail the Standard Rule. But what of the Average Rule? The Average Rule gives the same result as the Standard Rule for symmetric superpositions of two or more terms, and hence is not ruled out by the first or the second stage of Deutschs argument. For asymmetric superpositions, though, the Average Rule and the Standard Rule differ; for |(( =  EMBED Equation.3 |x1( +  EMBED Equation.3 |x2(, for example, the Standard Rule yields V[|((] = EMBED Equation.3 x1 +  EMBED Equation.3 x2, whereas the Average Rule yields V[|((] =  EMBED Equation.3 (x1 + x2). However, Deutschs argument here cannot be used to rule out the Average Rule, since the point at which the valuations given by the Average Rule diverge from those given by the Standard Rule is precisely the illicit appeal to Additivity just described. Hence the Average Rule, too, serves as a counterexample to Deutschs claim. Indeed, this is not surprising, since a little thought shows that both the Sum Rule and the Average Rule satisfy all of Deutschs axioms. Before proceeding to Wallaces defense of Deutschs argument strategy, there are a couple of objections to my argument thus far that should be dealt with. The first concerns the nature of the payoffs. Deutsch sometimes refers to the payoffs in financial terms, as is traditional in decision theory; a payoff of xi corresponds to a payment of xi dollars to the observer (1999, 3131). But if the payoffs are dollars, then the Sum Rule is, after all, eliminated at the first stage in Deutschs argument. The reason is that when the observer branches into two successors, the money she has received branches with her. Hence if the observer is paid k dollars before the measurement, her successors end up with a total of 2k dollars, just as if the observers two successors are paid k each after the measurement. Hence the adherent of the Sum Rule should, after all, accept (4), and Deutschs result for symmetric superpositions follows. However, note that this result does not follow from Additivity, but from the physical nature of the payoffs. As Deutsch makes clear, the financial model is just a heuristic; in order for Deutschs argument to constrain an agents preferences in general, the payoffs should be conceived as subjective utilities (1999, 3131). And it is clear that not all sources of subjective utility behave like money in this regard; transient pleasures and pains, for example, which are over and done with before the measurement, will not branch with the observer. In general, then, Additivity alone does not constrain the adherent of the Sum Rule to accept Deutschs argument. Similar comments apply to the Average Rule in the context of the third stage in Deutschs argument. Second, it might be objected that Deutsch has a response to my argument in his original paper. In his defense of (4), Deutsch claims that the situation on the left is physically identical to the situation on the right (1999, 3133). If that is the case, and the two sides of (4) are simply alternative descriptions of the same physical situation, then clearly a rational agent should attach the same value to each. However, at least on the surface, the two sides of (4) represent distinct physical situations; on the left, the observer receives payoff k before the measurement, and on the right her successors receive k after the measurement. If there is a sense in which these situations are physically equivalent, it would need to be spelled out. Perhaps a better case for physical equivalence can be made for the third stage in Deutschs argument. Here, Deutsch describes the correlation of an auxiliary y-system with the original x-system as one way of measuring  EMBED Equation.3  (1999, 3134). The assumption here seems to be that any measurement of  EMBED Equation.3  should have the same value, however it is carried out. Indeed, during actual measurements, the measured system becomes correlated with many auxiliary systems, both deliberately and accidentally; perhaps Deutschs implicit argument here is that the value of a measurement cannot depend on factors that the observer cannot know, and hence cannot take into account. This line of argument is presented more clearly by Wallace, and so I put off a response until after I have presented Wallaces arguments. 6. Wallaces Equivalence Principle Wallace (2002a, 2003, 2005) provides an extensive elaboration and defense of Deutschs argument. Wallace admits that traditional decision-theoretic axioms are insufficient by themselves to derive the Born rule (2005, 89). He calls the additional principle needed for Deutschs proof to go through Equivalence. Equivalence states that for a rational observer, the value of a measurement depends only on the total squared amplitude associated with each possible outcome. That is, a rational observer should regard measurements that ascribe the same squared amplitude to the various possible outcomes as equivalent in value, no matter what their branching structure. Equivalence, if it can be justified, eliminates branch-counting rulesrules like the Average Rule and the Sum Rule which base the value of a measurement on the number of branches for each possible outcome, rather than their total squared amplitude. Consider, for example, the third stage of Deutschs argument; compare an  EMBED Equation.3 -measurement followed by a  EMBED Equation.3 -measurement performed on state  EMBED Equation.3 (|x1(|y1( + |x2(|y2( + |x2(|y3() to a measurement of  EMBED Equation.3  performed on the same state. The Sum Rule ascribes different values to these two measurements, since the payoff x2 appears in two branches for the former measurement and in three branches for the latter. The same goes for the Average Rule, and for a similar reason; payoff x2 appears in a greater proportion of the branches for the latter measurement than for the former. Any measure that obeys Equivalence, however, must ascribe the same value to these two measurements, since the squared coefficient associated with payoff x2 is the same in each case. Hence, given Equivalence, Deutschs third stage argument is successful. In fact, it is straightforward to see that each stage of Deutschs argument goes through given Equivalence. Wallace is surely right that the Equivalence principle can fill the gap in Deutschs argument. On the other hand, Equivalence is certainly not an innocuous and general axiom of rationality like Additivity; it is a substantive claim about decision-making in the specific context of Everettian quantum mechanics, and so requires a substantive justification. The justification Wallace provides depends crucially on a further principle he calls Branching Indifference (2005, 19). Branching Indifference says that a rational agent should be indifferent about processes whose only consequence is to split one branch into several qualitatively identical branches. It entails that an observer should be indifferent about the number of branches associated with a particular outcome, and hence directly contradicts branch-counting rules. The viability of the Sum Rule and the Average Rule, then, depends on whether Branching Indifference can be established. In fact, Wallace offers two independent arguments for this conclusion. Recall from section 3 that Wallace distinguishes two ways of understanding probability in Everettian theoriesthe subjective uncertainty (SU) account, and the objective determinism (OD) accountbut remains neutral between them. Hence he provides one argument from the SU perspective, and one from the OD perspective. 6.1 The SU Argument Wallaces first argument involves the following thought-experiment (2005, 19). Suppose a quantum measurement has two possible outcomes, where the observer receives a higher payoff for one outcome than for the other. Suppose also that a quantum randomizing device can be inserted into the measurement process, so that when the high-payoff outcome occurs, the device displays a number between one and a million. This has the effect that the high-payoff branch is split into a million sub-branches, although the total squared amplitude is unchanged. Wallace argues that a rational observer should be indifferent about the inclusion of the device, since this just corresponds to introducing some completely irrelevant extra uncertainty (2005, 19). The reason that the extra uncertainty is irrelevant is that it does not concern the outcome of the measurement, but only concerns the number displayed by the device; if the preferred outcome occurs, the observer can still be certain that she will receive the corresponding high payoff. But since the observer only cares about the payoff, and the device introduces no uncertainty about that, the observer should be indifferent as to whether or not the device is included. And since all the device does is introduce extra branching structure, the upshot of the argument is that rational observers should be indifferent to branching structure per se. Note that Wallaces argument involves essential appeal to uncertainty; the SU account of probability plays a central role in the argument. Hence the argument is only as strong as the SU viewpoint itself, and the SU viewpoint is, I maintain, untenable. This is not because Everettian branching introduces no subjective uncertainty; if an observer closes her eyes during a measurement, each of her successors may become genuinely uncertain as to which outcome they will see when they open their eyes (Vaidman 1998, 254). The problem is that this uncertainty has nothing to do with decision-making, since it only arises after the measurement, and the valuation of measurements on which decisions are based must occur before the measurement. Before the measurement the observer is perfectly certain what her successors will see, provided that the details of the measurement are known to her. Neither is she uncertain about which of her successors she will become, since on Wallaces view of personal identity (or continuity), she knows she will become each of them. Hence whatever underpins the weight that the observer ascribes to each payoff, it cannot be pre-measurement uncertainty concerning whether she will receive that payoff, since there is no pre-measurement uncertainty. If the foregoing is correct, then Wallaces claim that the extra uncertainty introduced by the randomizing device is irrelevant to decision-making should come as no surprise, since the kind of uncertainty that exists in Everettian worlds is always irrelevant to decision-making. Decision-making, then, must rest on other grounds entirely, grounds that may or may not include branching structure. In particular, the irrelevance of the extra uncertainty introduced by the device does not entail that the extra branching it introduces is likewise irrelevant. Wallaces first argument, then, does not establish Branching Indifference, and hence does not eliminate branch-counting rules like the Sum Rule and the Average Rule. Greaves (2004, 440443) also rejects the SU account of probability, and for similar reasons, but nevertheless thinks that a pre-measurement observer is rationally compelled to act as if she is uncertain about the outcome of the measurement. This is because Greaves endorses a decision-theoretic reflection principle from an earlier paper by Wallace (2002a, 58), which states that if an observer gains no new information relevant to her strategy, then she is rationally compelled not to change her strategy. Since the observer learns nothing during the measurement (until she opens her eyes), her strategy before the measurement must be exactly the same as her strategy afterwards, when she is genuinely uncertain about the outcome. However, I dont think this appeal to reflection undermines my defense of branch-counting rules. First, note that the reflection principle could equally well support the converse inferencethat the post-measurement observer should act as if she were still certain about the outcome. Second, I think the reflection principle Greaves and Wallace appeal to here is too strong. The observer doesnt gain any information during the measurement, but she does lose informationself-location information concerning which branch she is in (Ismael 2003, 783). Indeed, this is precisely the source of the observers post-measurement uncertainty. But information loss, like information gain, is a perfectly good reason to change ones decision strategy. Hence subjective uncertainty, pre- or post-measurement, provides no reason to reject the Sum Rule or the Average Rule. Even though Greaves endorses the reflection argument, she thinks that it obscures the real logic of the argument for the Standard Rule, since the real argument makes no appeal to subjective uncertainty, however oblique (2004, 443). The real argument, which can be given entirely within the OD account of probability, is that branch-counting rules are incoherent (2004, 448); indeed, this is precisely Wallaces second (OD) argument for the irrelevance of branching structure. 6.2 The OD Argument Wallaces second argument charges that the account of measurement we have been considering so far is grossly oversimplified, and that it is simply not possible to take branching structure into account for actual measurements treated in their full complexity (Wallace 2005, 2022). He makes several points here. First, he notes that branching is not limited to measurement events in the Everett theory; microphysical processes trigger branching as well. Hence an observer who wants to take branching structure into account would have to keep track of an impossibly large number of microphysical events. Further, there is no point at which the branching stops, so to figure the value of a measurement according to a branch-counting rule, the observer would have to take into account branching into the indefinite future, and this too is impractical. Second, recall from section 3 that Wallace holds the choice of basis to be a pragmatic matter. Pragmatic considerations do not pick out a unique basis; there is considerable leeway in the choice of an adequate basis, and the number of branches corresponding to each outcome depends sensitively on this choice. But then there is simply no fact about the number of branches on which to ground a decision rule. Third, in a realistic treatment the basis will typically be continuous rather than discrete, so the number of branches associated with an outcome will be literally uncountable. In short, the number of branches associated with an outcome is unknowable, undefined, and uncountable, and hence branch-counting rules are simply unusable. However, I think this conclusion is too hasty, since none of the problems Wallace raises is insurmountable; indeed, they all arise in some form for decision theory in ordinary, non-branching worlds. In non-branching cases, too, the full range of relevant facts pertaining to a decision will typically include a large number that are effectively unknowable. For example, I never know whether I will have a successor at a given future time, since the full range of microphysical facts that determine my lifespan are inaccessible to me. But this is not a problem; I can simply incorporate my ignorance into my subjective probability estimates for each outcome. Furthermore, the consequences resulting from a decision typically stretch into the indefinite future. But this is not a problem either, since given the ignorance just described, there typically comes a point after which my subjective probabilities for future events are unaffected by my decision. In non-branching cases, too, there is no objective fact concerning how many successors I have; it depends how I choose to count them. But this is not a problem, since any reasonable choice will result in the same prescriptions for action. Finally, any realistic measure will make the number of my successors uncountable, since time is (presumably) a continuous parameter. But this simply requires me to construct continuous measures of successor density and utility density, and integrate rather than adding. So, similarly, none of the problems raised by Wallace provides a decisive reason to reject branch-counting rules. If I dont know many of the physical details that determine the number of successors I will have for a particular outcome, I can build my uncertainty into my subjective estimate. Even though branching continues into the indefinite future, there typically comes a point at which further branching is irrelevant to my current decision, given what I know. Even though there is no objectively preferred basis, and hence no facts about how many branches there are, I can simply choose a reasonable basis, since any reasonable choice will yield the same decision-theoretic prescriptions. And if that basis is continuous, then I simply replace my discrete branch measure with a continuous branch density measure, using the basis variable as a parameter. To illustrate these claims, consider the value of a measurement performed on state |(( = a|x1( + b|x2(, for arbitrary a and b. For any adequate branch-counting measure, continuous or discrete, it would be very surprising if there were a systematic difference in the number of branches associated with the two terms in the above superposition; indeed, one might stipulate this as an adequacy condition on branch-counting measures. That is, while microphysical contingencies might entail more x1 successors than x2 successors in a particular case, one should not expect an imbalance on average. So if, as is typical, one has no special microphysical information about a particular case, one should expect equal measures of x1 successors and x2 successors, both immediately after the measurement and into the indefinite future. But this is sufficient to allow us to apply the Average Rule to this case, yielding V[|((] =  EMBED Equation.3 (x1 + x2). Hence Wallaces worries about branch-counting do not present an insurmountable obstacle to the application of the Average Rule. The case of the Sum Rule is slightly more tricky, since the value depends directly on the number of successors associated with an outcome. But again, for any adequate measure, it would be surprising if the number of successors associated with outcomes x1 and x2 depended systematically on the superpositional context in which the outcome appears. That is, one should expect just as many successors associated with the x1 outcome in when it appears in the superposition state |(( as when it appears in the single-term state |x1(, and similarly for the x2 outcome. Hence the Sum Rule yields V[|((] = V[|x1(] + V[|x2(] in this case, irrespective of whether the branch-counting measure is discrete or continuous, and irrespective of how far into the future one tracks the branching. So even though the Sum Rule does not yield absolute values for measurements, it does yield comparative values, and these are arguably sufficient for decision theory. In general, then, Wallaces arguments do not succeed in establishing Branching Indifference, or the Equivalence principle that depends on it. In particular, while Wallace is right that a more realistic treatment of measurements leads to technical problems for branch-counting rules, I do not think that he has shown these problems to be insuperable barriers to the application of the Average Rule and the Sum Rule. Hence they remain as counterexamples to Deutschs argument. 7. Conclusion Attempts to derive the standard probabilistic predictions of the Born rule within Everettian quantum mechanics are as old as the Everettian tradition itself. Everetts own approach has long been known to beg the question, but the decision-theoretic strategy initiated by Deutsch has led to renewed hope that this old problem can be solved. Deutsch claims that, given quantum mechanics and some standard decision-theoretic axioms, he can prove that a rational observer must weight the outcomes of quantum mechanical measurements according to the Born rule. However, I have argued here that Deutschs argument has a gap at a crucial point, and hence admits counterexamplesdecision rules that differ from the Born rule, and yet are consistent with the decision-theoretic axioms. Perhaps it is too strong to require that every alternative, however ad hoc, be eliminated before the Born rule can be regarded as sufficiently established. But the counterexamples I considerthe Average Rule and the Sum Ruleare the most natural and plausible alternatives to the Born rule in the Everettian context, and so clearly cannot be allowed to stand. Wallace acknowledges the gap in Deutschs argument, and suggests a novel principle to bridge it. This principleEquivalenceis not a standard axiom of decision theory, but instead is tailored specifically to the Everettian context. As such, it requires a substantive justification. Wallace offers two arguments to this end, but I show that neither in fact eliminates the Average Rule and the Sum Rule as rationally permitted alternatives to the Born rule. So far, at least, the decision-theoretic case that a rational agent must adopt the Born rule in the context of Everettian quantum mechanics has not been made. References Bell, John S. (1987), Speakable and Unspeakable in Quantum Mechanics. Cambridge: Cambridge University Press. Deutsch, David (1985), Quantum theory as a universal physical theory, International Journal of Theoretical Physics 24: 141. (1996), Comment on Lockwood, British Journal for the Philosophy of Science 47: 222228. (1999), Quantum theory of probability and decisions, Proceedings of the Royal Society of London A455: 31293137. DeWitt, Bryce S. (1970), Quantum mechanics and reality, Physics Today 23 (September): 3035. Everett, Hugh III (1957), Relative state formulation of quantum mechanics, Reviews of Modern Physics 29: 454462. GellMann, M. and J. B. Hartle (1990), Quantum mechanics in the light of quantum cosmology in W. H. Zurek (ed.), Complexity, Entropy, and the Physics of Information. Redwood City, CA: AddisonWesley, 42558. Graham, N. (1973), The measurement of relative frequency in B. DeWitt and N. Graham (eds.), The Many Worlds Interpretation of Quantum Mechanics. Princeton: Princeton University Press. Greaves, H. (2004), Understanding Deutschs probability in a deterministic multiverse, Studies in History and Philosophy of Modern Physics 35: 423456. Horgan, Terry (2004), Sleeping Beauty awakened: New odds at the dawn of the new day, Analysis 64: 1020. Ismael, Jenann (2003), How to Combine Chance and Determinism: Thinking about the Future in an Everett Universe, Philosophy of Science 70: 776790. Lewis, David (2004), How many lives has Schrdingers cat?, Australasian Journal of Philosophy 82: 322. Lockwood, Michael (1989), Mind, Brain and the Quantum. Oxford: Blackwell. (1996), Many minds interpretations of quantum mechanics, British Journal for the Philosophy of Science 47: 159188. Papineau, David (1995), Probabilities and the Many Minds Interpretation of Quantum Mechanics, Analysis 55: 239246. Savage, L. J. (1954), The Foundations of Statistics. New York: John Wiley and Sons. Saunders, Simon (1998), Time, Quantum Mechanics and Probability, Synthese 114: 373404. (2004), Operational derivation of the Born rule, Proceedings of the Royal Society of London A460: 118. Vaidman, Lev (1998), On schizophrenic experiences of the neutron or why we should believe in the many-worlds interpretation of quantum theory, International Studies in the Philosophy of Science 12: 245261. Wallace, David (2002a), Quantum probability and decision theory, revisited, http://philsci-archive.pitt.edu/archive/00000885/. (2002b), Worlds in the Everett interpretation, Studies in the History and Philosophy of Modern Physics 33: 637661. (2003), Everettian rationality: defending Deutschs approach to probability in the Everett interpretation, Studies in the History and Philosophy of Modern Physics 34: 415439. (2005), Quantum probability from subjective likelihood: improving on Deutschs proof of the probability rule, Studies in the History and Philosophy of Modern Physics (forthcoming). Available online at http://philsci-archive.pitt.edu/archive/00002302/.   Deutsch prefers to speak of bases in which our measurements have outcomes (1985, 2223), but presumably the result is the same.  This is not to say that the axioms are unobjectionable; I assume them for the sake of evaluating the validity of Deutschs argument. Note also that the argument can be formulated on the basis of different sets of axioms; Wallace (2005), for example, works from a weaker set of axioms.  Note that |x1 + y1( is just another way of writing the state |x1(|y1(, to indicate that a measurement of X(1 + 1(Y on this state will yield a payoff of x1 + y1. Similarly for the other terms in the superposition.  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The Sum Rule and the Average Rule also satisfy the weaker axioms of Wallace (2005).  Saunders (2004) constructs a version of Deutschs proof stripped of its decision-theoretic underpinnings that can be applied to non-Everettian approaches to quantum mechanics. However, in the Everettian context he reintroduces the decision-theoretic apparatus, and appeals to Wallace (2002a) for its justification.  In an earlier paper, Wallace appeals to a related principle called Measurement Neutrality (2003, 434). However, Greaves (2004, 446) argues that Measurement Neutrality cannot be defended, and Wallace (2005, 15) partially concedes this point.  See Horgan (2004) for a defense of this position in the context of the Sleeping Beauty paradox.     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