Analysis of Wallace’s Proof of the Born Rule in Everettian Quantum Mechanics: Formal Aspects

Abstract

To solve the probability problem of the Many Worlds Interpretation of Quantum Mechanics, D. Wallace has presented a formal proof of the Born rule via decision theory, as proposed by D. Deutsch. The idea is to get subjective probabilities from rational decisions related to quantum measurements, showing the non-probabilistic parts of the quantum formalism, plus some rational constraints, ensure the squared modulus of quantum amplitudes play the role of such probabilities. We provide a new presentation of Wallace’s proof, reorganized to simplify some arguments, and analyze it from a formal perspective. Similarities with classical decision theory are made explicit, to clarify its structure and main ideas. A simpler notation is used, and details are filled in, making it easier to follow and verify. Some problems have been identified, and we suggest possible corrections.

Classical Decision Theory

Decision Theory [27, 29, 33] is an interdisciplinary area of study, concerned, among other things, with developing principles to optimize decision making. Its problems are usually classified in 3 cases (the nomenclature varies in the literature): decision under certainty, under risk, and under uncertainty.

1.1 Decision Under Certainty

This case involves choosing between alternatives with well known outcomes. Such problems can be framed in terms of an agent, or decision maker, having to establish a preference order on acts whose outcomes are certain (sometimes they are framed in terms of lotteries with a single reward).

This preference is called rational if it is a total order, i.e. it satisfies Completeness and Transitivity (see next section for definitions), so the choices can be ordered in a single chain without loops. The reason for such label is that an agent whose preference order is not total would be vulnerable to Dutch book arguments (loops in the order could be exploited to “pump money” from him).

1.2 Decision Under Risk

In this kind of problem, each choice can have multiple outcomes, with known probabilities. Such problems are usually framed in terms of an agent having to establish a preference order $\succ$ on lotteries. A lottery

$$\begin{aligned} A=\{(p_i, r_i)\}_{i\in {\mathcal {O}}} \end{aligned}$$

(4)

consists of a set ${\mathcal {O}}$ of mutually exclusive outcomes, each having probability $p_i$ and giving a reward $r_i$ (which can also be a penalty or any other consequence).

Each reward r is identified with a single reward lottery, giving it with certainty as its unique reward. So an order on lotteries induces another on rewards.

Given lotteries A and B, and $t\in [0,1]$, a compound lottery $tA+(1-t)B$ is a lottery in which the agent first chooses randomly A or B, with probabilities t and $1-t$ respectively, and then proceeds with the selected lottery. By classical probability rules, it is equivalent to one in which the probability of each reward is $tp_A+(1-t)p_B$, where $p_A$ and $p_B$ are its probabilities in A and B.

If all rewards are equally preferred, the problem is trivial. Otherwise, we assume¹³ the existence of least and most preferred rewards, denoted, respectively, by $r_0$ and $r_1$. For each $t\in [0,1]$, we define a standard lottery of weight t as

$$\begin{aligned} L_t=tr_1+(1-t)r_0. \end{aligned}$$

(5)

A crude strategy for problems of decision under risk is to let preferences follow the expected value of lotteries, i.e. the average value of rewards, weighted by their probabilities. This may seem reasonable, by the Law of Large Numbers, but can lead to bad results, as in the St. Petersburg Paradox, if the number of runs is finite. A more flexible strategy, proposed in 1738 by D. Bernoulli [7], replaces monetary values by an utility function u(r) on rewards:

Principle of Maximization of Expected Utility

Given an utility function u(r), it induces a preference order on lotteries by $A \succ B \Leftrightarrow \mathrm {EU}(A) > \mathrm {EU}(B)$, where the expected utility of $A=\{(p_i,r_i)\}_{i\in {\mathcal {O}}}$ is

$$\begin{aligned} \mathrm {EU}(A) = \sum _{i\in {\mathcal {O}}} p_i \cdot u(r_i). \end{aligned}$$

(6)

An order $\succ$ on lotteries is induced (or represented) by a function u on rewards, via this principle, if, and only if, the following conditions are satisfied, for any rewards r, $r'$ and $r''$:

• $r\succ r' \ \Leftrightarrow \ u(r)>u(r')$;

• if $r'' \sim tr+(1-t)r'$ then $u(r'')=t\cdot u(r)+(1-t)\cdot u(r')$.

The importance of this principle became clear in 1944, when Von Neumann and Morgenstern [43] proved that any preference order satisfying four reasonable conditions can be represented by an utility function. The Von Neumann–Morgenstern axioms are, for any lotteries A, B, C:

Completeness

Either $A\succ B$, or $B\succ A$, or $A\sim B$.

Transitivity

If $A\succcurlyeq B$ and $B\succcurlyeq C$ then $A\succcurlyeq C$.

Archimedean Property

If $A\succ B\succ C$, there are $s,t\in (0,1)$ such that $tA+(1-t)C \succ B \succ sA+(1-s)C$.

Independence

If $A\succ B$ then $tA+(1-t)C \succ tB+(1-t)C$, for any $t\in (0,1]$.

Completeness and Transitivity correspond to the totality condition of decision under certainty. The Archimedean Property implies no lottery is so incommensurately better (resp. worse) than other, that it is impossible to reverse preferences by compounding it appropriately with a worse (resp. better) one. It also means that sufficiently small changes in a lottery do not alter significantly the order. Independence means a preference between compound lotteries is based only on components that differ.

The first three axioms ensure $\succ$ can be described by a correspondence between lotteries and real numbers. With the last one, they also imply the following properties:

Substitutability

If $A\sim B$ then $tA+(1-t)C \sim tB+(1-t)C$, for any $t\in [0,1]$;

Monotonicity

If $A\succ B$ then $tA+(1-t)B$ is more preferable for higher values of t;

Continuity

If $A\succ B\succ C$ there is a unique $t\in (0,1)$ such that $B\sim tA+(1-t)C$.

From them, we can get the following result, whose proof we sketch for comparison with Wallace’s.

Von Neumann–Morgenstern Theorem

A preference order $\succ$ satisfies the Von Neumann–Morgenstern axioms if, and only if, there is an utility function u¹⁴ that represents it via the Principle of Maximization of Expected Utility, i.e.

$$\begin{aligned} A \succ B \quad \Leftrightarrow \quad \mathrm {EU}(A) > \mathrm {EU}(B). \end{aligned}$$

Proof

It is easy to show that any $\succ$ represented by some u satisfies the axioms. The converse has 4 steps:

• 1. By Monotonicity, preference on the standard lotteries $L_t$ increases with t.

• 2. By Continuity, for any reward r there is a unique $t\in [0,1]$ such that $r \sim L_t$, and we set $u(r)=t$.

• 3. Given a lottery A, each of its rewards r is equivalent to $L_{u(r)}$. Hence Substitutability, and the way classical probabilities combine, imply A is equivalent to $L_{EU(A)}$.

• 4. By steps 1 and 3, preference on lotteries increases with their expected utilities.

$\square$

In decision under risk, a preference order satisfying the Von Neumann–Morgenstern axioms is also called rational. Implicit in such label is the idea that an intelligent decision maker ought to follow the axioms. But empirical research shows that even smart educated people often deviate from them, specially in cases with big rewards and small probabilities (e.g. in the Allais paradox). Many authors dismiss this, arguing that those agents, when confronted with their error, would recognize it and correct their decision. Others, however, question the validity of the axioms, in particular the Independence one.

1.3 Decision Under Uncertainty

In problems of decision under uncertainty there are multiple possible outcomes, but their probabilities are unknown. Instead of lotteries, problems are usually framed in terms of a preference order on acts which, depending on possible states of the world, will lead to payoffs (consequences).

Some decision strategies (maximax, maximin, minimax regret, etc.) focus on best or worst cases. These are useful in some situations, but can lead to absurd results in others, as they disregard important data, like non-extremal payoffs, or the low likelihoods of some states of the world.

Another strategy uses subjective probabilities, estimates by the agent of the likelihoods of states of the world. But the result can be very sensitive to the estimates used, especially if there are unlikely states with huge payoffs. Despite this, Savage [39] proved, in 1954, that any preference order satisfying certain axioms can be induced, via the Principle of Maximization of Expected Utility, by some utility function and subjective probabilities.

Let ${\mathcal {S}}$ be the set of states of the world. Not all details of a state are relevant, so we consider events, subsets of states with some common characteristic. For example, the event $E=$“it rains tomorrow” consists of all states in which this happens. Events can be partitioned into smaller ones, e.g. E could have subevents like “it rains tomorrow and stock prices go up” and “it rains tomorrow and the result of a die is 3”. Events can be combined through intersections, unions and complements, corresponding to the logical operators AND, OR, and NOT.

An act is a function $f:{\mathcal {S}}\rightarrow {\mathcal {P}}$, where ${\mathcal {P}}$ is the set of payoffs, so that $f(s)=x$ means x is the payoff resulting from act f if the state of world turns out to be s. Each payoff x is identified with a single payoff act, which results x for all states. Given acts f and g, and an event E, the compound act [E, f, g] is

$$\begin{aligned} {[}E,f,g](s)={\left\{ \begin{array}{ll} f(s)\ \text { if } s\in E, \\ g(s)\ \text { if } s\notin E. \end{array}\right. } \end{aligned}$$

Savage adopts, for a preference order $\succ$ on acts, the following axioms¹⁵:

Axiom

S1 (Order) $\succ$ is complete and transitive.

Axiom

S2 (Sure-Thing Principle) For any event E and acts f, g, h and k, we have $[E,f,h]\succcurlyeq [E,g,h] \Leftrightarrow [E,f,k]\succcurlyeq [E,g,k].$

This is similar to Independence: preference between acts depends only on the events where they differ. It allows us to define, for each event E, a conditional preference order $\succ _{E}$ by

$$\begin{aligned} f\succ _{E}g \ \text { if and only if } \ [E,f,h]\succ [E,g,h] \text { \ for all }h, \end{aligned}$$

meaning “f is preferable to g given E”, i.e. if the agent assumes E will happen he will prefer f to g.

An event E is null if $f\sim _{E}g$ for all f and g, i.e. the agent does not care about acts on E (we will find the reason is he does not believe E can happen). Nullity has the following properties, for events E and F:

Classical Null Subevent

If E is null, and $F\subset E$, then F is null.

Classical Null Disjunction

If E and F are null then $E\cup F$ is null.

They result from properties of sets and functions, like the fact that a function can be redefined in a subset, without affecting the complement. We prove the last one, for comparison with the quantum case.

Proof

Without loss of generality, we can assume $E\cap F=\emptyset$. Given acts f, g and h, define a new act k by

$$\begin{aligned} k(s)={\left\{ \begin{array}{ll} g(s)\ \text { if } s\in E, \\ f(s)\ \text { if } s\in F, \\ h(s)\ \text { if } s\notin E\cup F. \end{array}\right. } \end{aligned}$$

As E and F are null, $[E\cup F,f,h]\sim k \sim [E\cup F,g,h]$. $\square$

Axiom

S3 (Ordinal Event Independence) For any payoffs x and y, and any non null event E, we have $x \succ _{E} y \,\Leftrightarrow \, x\succ y$.

Hence the agent’s preference order on payoffs is independent of any given event. This is not as banal as it may seem: receiving an umbrella might be preferable than sunglasses in the event of rain, but the preference might be reversed in a sunny day. So this axiom demands a careful consideration of what we are to call acts and payoffs. In this example, possession of umbrellas or sunglasses ought to be acts, with getting wet or having protected eyes being payoffs.

Axiom

S4 (Comparative Probability) Let x, y, z and w be payoffs, and E, F be events. If $x\succ y$ and $z \succ w$, then $[E,x,y]\succ [F,x,y] \,\Leftrightarrow \, [E,z,w]\succ [F,z,w]$.

The intuitive idea is as follows. [E, x, y] gives a better payoff in case of E than otherwise. [F, x, y] gives the same payoffs, but the event giving the better payoff is F instead of E. By S3 (Ordinal Event Independence) the agent only cares about the payoff and not how he got it, so the only reason for preferring [E, x, y] to [F, x, y] is that he thinks E is more likely than F. So the same preference should hold for any other pair z, w of better/worse payoffs.

This induces an order on events by

$$\begin{aligned} E\succ F \ \text { if and only if } \ [E,x,y]\succ [F,x,y] \text { whenever } x\succ y. \end{aligned}$$

It means the agent thinks E is more likely than F, as he prefers the better payoff be given in case of E than in case of F. Savage proves this order is a qualitative probability on events, i.e. it satisfies

• (i) $\succ$ is complete and transitive;

• (ii) $E\succcurlyeq \emptyset$ for any $E\subset {\mathcal {S}}$;

• (iii) ${\mathcal {S}}\succ \emptyset$;

• (iv) for any $E,E',F\subset {\mathcal {S}}$ with $E\cap F=E'\cap F=\emptyset$, $E\succ E' \Leftrightarrow E\cup F \succ E'\cup F$.

Actually, for this to work we need at least one pair of better/worse payoffs, given by the next axiom.

Axiom

S5 (Nondegeneracy) There are payoffs x, y such that $x \succ y$.

If this axiom is violated, we have a trivial decision problem. In such case, most of Savage’s Theorem is trivially true, but we do not obtain a unique subjective probability measure (in fact, any probability measure would work).

The next axiom allows us to turn the qualitative probability into a quantitative one.

Axiom

S6 (Small-Event Continuity) If $f\succ g$ then for any payoff x there is a finite partition such that $[E_i,x,f]\succ g$ and $f\succ [E_i,x,g]$ for every i.

This is similar to the Archimedean Property. It means the set of states of the world can be partitioned in events $E_i$ deemed so unlikely that changing the payoff on one of them is not enough to alter preferences. And no payoff is infinitely good or bad, or it would alter preferences for any non null $E_i$. The partition could be given, for example, by the results of roulettes with arbitrarily large numbers of slots, or by throwing dies any number of times.

Whenever $F\succ E$, the axiom implies there is a finite partition such that $F\succ E\cup E_i$ for any i. Some $E_i$’s may be deemed more likely than others, but, after some technical work, Savage obtains partitions whose events the agent considers as equally likely as necessary.

Partitioning ${\mathcal {S}}$ into n events considered equally likely, we attribute to each a subjective probability $\frac{1}{n}$. If, for a large n, the minimum number of these pieces needed to cover some event E is m, its subjective probability should be close to $\frac{m}{n}$. The exact value is defined via a limiting process. In particular, an event is null if, and only if, its subjective probability is 0.

One last axiom is needed for situations with infinitely many payoffs. It requires that if f is preferred, given E, to all payoffs g can give in such event, then f is preferred to g given E.

Axiom

S7 (Dominance) If $f \succ _E g(s)$ for all $s\in E$, then $f \succ _E g$ (and likewise for $\prec _E$).

Once we have the agent’s subjective probabilities, the problem becomes one of decision under risk. Savage’s result is similar to the Von Neumann–Morgenstern Theorem, but with subjective probabilities.

Savage’s Theorem

A preference relation $\succ$ on acts satisfies axioms S1−S7 if, and only if, there is an unique, nonatomic,¹⁶ finitely additive probability measure $\pi$ on ${\mathcal {S}}$, and a bounded utility function u on ${\mathcal {P}}$, such that $\succ$ is represented by u via the Principle of Maximization of Expected Utility (with expected utilities defined in terms of the subjective probability $\pi$). Moreover, u is unique up to positive affine transformations, and an event E is null if, and only if, $\pi (E)=0$.

Savage’s axioms are usually taken as mandates of rationality for decision under uncertainty, even if they may appear less natural than those in the previous cases, and even though intelligent people often violate them (e.g. in the Ellsberg paradox).