# Wallace’s version of the Deutsch-Wallace theorem, part 2

To conclude the proof of Wallace’s version of the Deutsch-Wallace theorem, we shall add the Equivalence theorem from the previous post to a pretty weak decision theory, and show that if you are rational and live in a universe described by the Many-Worlds interpretation, you must bet according to the Born rule.

The first rationality assumption we need is pretty minimal: we only demand Amir to have preferences between games that are in a precise sense coherent. They must be transitive, in the sense that if he would rather vote for Strache than to vote for Kern, and would prefer to vote for Kern over Kurz, than he must choose to vote for Strache over Kurz. He must also have definite preferences about any pair of games: either he thinks that Strache is better than Kurz, or that Kurz is better than Strache, or he is indifferent between them. He is not allowed to say that they are not comparable. Note that we are not judging whether his preferences are politically coherent, or whether voting for Strache is at all a good idea. The axiom is then:

• Ordering: Amir’s preferences between games, written as $G \succeq G’$, define a total order in the set of games: if $G \succeq G’$, and $G’ \succeq G”$, then $G \succeq G”$. Moreover, for any two games $G$ and $G’$, either $G \succ G’$, or $G \sim G’$, or $G \prec G’$.

This means that the $\succeq$ behaves like the usual $\ge$ relation between real numbers[1].

The second and last rationality assumption we shall use is rather stronger, but I think still pretty well-justified. We demand that Amir’s preferences between games must remain consistent while he plays: if he prefers game $G$ to game $G’$, he cannot change his mind if $G$ and $G’$ are offered as rewards inside another game:

• Consistency: Let $\alpha \neq 0$, and consider the games $\ket{F} = \alpha\ket{M_0}\ket{G} + \beta\ket{M_1}\ket{z}$ and $\ket{F’} = \alpha\ket{M_0}\ket{G’} + \beta\ket{M_1}\ket{z},$ that differ only on the game given as a reward when the measurement result is $M_0$. Then $F \succeq F’$ iff $G \succeq G’$.

It is easy to check that Consistency actually implies all three relations $F \succ F’$ iff $G \succ G’,$ $F \sim F’$ iff $G \sim G’,$ and $F \prec F’$ iff $G \prec G’$.

These assumptions, together with Indifference and Substitution, are enough to imply the

• Born rule: Suppose you are rational, and consider the games
$\ket{G} = \sum_i \alpha_i\ket{M_i}\ket{z_i}\quad\text{and}\quad\ket{G’} = \sum_i \beta_i\ket{D_i}\ket{w_i}.$ Then there exists a function $u$ such that
$u(G) = \sum_i |\alpha_i|^2 u(z_i)$ and $G \succ G’ \iff u(G) > u(G’)$ Moreover, $u$ is unique modulo the choice of a zero and a unity.

This theorem says that you are free to decide your preferences between the rewards: these will define their utility. Your freedom ends here, however: the probabilities that you assign to obtaining said rewards must be given by the Born rule, on pain of irrationality.

A comment is also in order about the uniqueness: the choice of a zero and a unity is analogous to the one that must be done in a temperature scale. In the Celsius scale, for example, the zero is chosen as the freezing point of the water, and the unity as $1/100$ the difference between the freezing point and the boiling point. In the Fahrenheit scale, the zero is chosen as the coldest temperature in Gdańsk’s winter, and the unity as $1/96$ the difference between the temperature of Gdańsk’s winter and the temperature of the blood of a healthy male. In any case, the choice of these two values define the temperature scale uniquely, and the same is true for utility, as implied by the following theorem:

• Uniqueness: If $u$ is a utility, then $\mathcal F(u)$ is a utility if and only if $\mathcal F(u) = au+b$ for some real numbers $a,b$ such that $a>0$.

The proof of the ‘if’ direction is easy: just note that $\mathcal F(u(G)) = a\sum_i |\alpha_i|^2 u(z_i) + b = au(G)+b,$ and that such positive affine transformations preserve the ordering of real numbers. The proof of the ‘only if’ direction is not particularly hard, but it is a bit longer and I shall skip it[2]. Since the choice of a value for the utility at two rewards $x$ and $y$ is enough to fix $a$ and $b$, the claim follows.

But enough foreplay, now we need to start proving the Born rule theorem in earnest. We’ll build it out of two lemmas: Slider, that says that the weights of a game with rewards $x$ and $y$ behave like a tuner for the preferences, and Closure, that says that as we move this slider we are bound to hit any reward between $x$ and $y$.

• Slider: Let $x$ and $y$ be rewards such that $x \succ y$, and consider the games
$\ket{G} = \sqrt{p}\ket{M_0}\ket{x} + \sqrt{1-p}\ket{M_1}\ket{y}$ and
$\ket{G’} = \sqrt{q}\ket{M_0}\ket{x} + \sqrt{1-q}\ket{M_1}\ket{y}.$ Then $G \succ G’$ iff $p > q$.

Proof: suppose $p > q$. Then we can define the games
$\ket{F} = \sqrt{q}\ket{M_0}\ket{x} + \sqrt{p-q}\ket{M_1}\ket{x} + \sqrt{1-p}\ket{M_2}\ket{y}$ and
$\ket{F’} = \sqrt{q}\ket{M_0}\ket{x} + \sqrt{p-q}\ket{M_1}\ket{y} + \sqrt{1-p}\ket{M_2}\ket{y}.$
Note that the weights of rewards $x$ and $y$ in the game $F$ are $p$ and $1-p$, and in the game $F’$ they are $q$ and $1-q$, so by Equivalence we have that $F \sim G$ and $F’ \sim G’$. Since Consistency implies that $F \succ F’$, transitivity gives us $G \succ G’$. To prove the other direction, note that $p = q$ implies directly that $G \sim G’$, and $p < q$ implies $G \prec G'$ by the flipped argument.

• Closure: Let $x,y$, and $z$ be rewards such that $x \succ y$ and $x \succeq z \succeq y$, and let
$\ket{G_p} = \sqrt{p}\ket{M_0}\ket{x} + \sqrt{1-p}\ket{M_1}\ket{y}.$
Then there exists a unique $p_z$ such that$z \sim \ket{G_{p_z}}.$

Proof: since $\succeq$ is a total order, for all $\rho$ it must be the case that either $z\succ \ket{G_p}, \quad z \sim \ket{G_p},\quad\text{or}\quad z \prec \ket{G_p}.$Moreover, Slider tells us that there exists a critical $p_z$ such that
\begin{align*}
p < p_z \quad &\Rightarrow \quad \ket{G_p} \prec z \end{align*} Then some continuity argument will conclude that $z \sim \ket{G_{p_z}}$.

Now for the main proof: Let $x$ and $y$ be fixed rewards such that $x \succ y$. Set $u(x)$ and $u(y)$ to be any real numbers such that $u(x) > u(y)$, defining the unity and the zero of the utility function[3]. Now because of Closure for every reward $z$ such that $x \succeq z \succeq y$ there will be a unique number $p_z$ such that
$z \sim \sqrt{p_z}\ket{M_0}\ket{x} + \sqrt{1-p_z}\ket{M_1}\ket{y}.$ Define then
$u(z) = p_z u(x) + (1-p_z) u(y).$ We want to show that the utilities so defined do represent the preferences between any two rewards $z$ and $w$ in the sense that $z \succ w$ iff $u(z) > u(w)$. Suppose that $u(z) > u(w)$. This is the case iff $p_z > p_w$, which by Slider is equivalent to $\sqrt{p_z}\ket{M_0}\ket{x} +\sqrt{1-p_z}\ket{M_1}\ket{y} \succ \sqrt{p_w}\ket{M_0}\ket{x} + \sqrt{1-p_w}\ket{M_1}\ket{y},$ which is equivalent to $z \succ w$.
Now we want to show that for any game $\ket{G} = \sqrt{q}\ket{M_0}\ket{z} + \sqrt{1-q}\ket{M_1}\ket{w}$ its utility is given by $u(G) = q u(z) + (1-q) u(w),$ as advertised. By Consistency, we can replace $z$ and $w$ in $G$ by their equivalent games, and we have that
\begin{multline}
\ket{G} \sim \sqrt{q p_z}\ket{M_0}\ket{M_0}\ket{x} + \sqrt{q(1-p_z)}\ket{M_0}\ket{M_1}\ket{y} + \\ \sqrt{(1-q)p_w}\ket{M_1}\ket{M_0}\ket{x} +\sqrt{(1-q)(1-p_w)}\ket{M_1}\ket{M_1}\ket{y}. \end{multline} By Equivalence,
$\ket{G} \sim \sqrt{\lambda p_z + (1-q)p_w}\ket{M_0}\ket{x} + \sqrt{q(1-p_z)+(1-q)(1-p_w)}\ket{M_1}\ket{y},$
and since $x \succeq G \succeq y$, its utility is given by the above formula, so
\begin{align*}
u(G) &= (q p_z + (1-q)p_w)u(x) + (q(1-p_z)+(1-q)(1-p_w))u(y)\\
&= q u(z) + (1-q) u(w),
\end{align*} as we wanted to show.

With this argument we have proved the Born rule theorem for any game inside the interval $[y,x] = \{z: x\succeq z \succeq y\}$. This would be enough if we were to assume that the set of rewards was something so lame, but since we want to be deal with more interesting reward sets – like $\mathbb R$ – we cannot stop now. It is fortunately not hard to complete the proof: consider a sequence of intervals $[y_i,x_i]$ such that all of them contain $[y,x]$ and that their union equals the set of rewards. By the above proof, in each such interval there exists a utility function $f_i$ that satisfies the requirements. We want to show that these functions agree with each other, and as such define a unique utility over the whole set of rewards. For that, consider a reward $z$ in $[x_i,y_i]\cap [x_j,y_j]$ for some $i,j$. Then it must be the case that either $x\succeq z \succeq y$, or $x\succ y \succ z$, or $z \succ x \succ y$. By Closure, there exists unique $p_z,p_y$, and $p_x$ such that
\begin{align*}
z &\sim \sqrt{p_z}\ket{M_0}\ket{x} + \sqrt{1-p_z}\ket{M_1}\ket{y}, \\
y &\sim \sqrt{p_y}\ket{M_0}\ket{x} + \sqrt{1-p_y}\ket{M_1}\ket{z}, \\
x &\sim \sqrt{p_x}\ket{M_0}\ket{z} + \sqrt{1-p_x}\ket{M_1}\ket{y}. \\
\end{align*}
Since $f_i$ and $f_j$ are utilities over this interval, we must have that for $k=i,j$
\begin{align*}
f_k(z) &= p_zf_k(x) + (1-p_z)f_k(y), \\
f_k(y) &= p_yf_k(x) + (1-p_y)f_k(z), \\
f_k(x) &= p_xf_k(z) + (1-p_x)f_k(y). \\
\end{align*}
Now, we use our freedom to set the zero and the unity of the utilities to choose $f_k(y) = u(y)$ and $f_k(x) = u(x)$, taking these equations to
\begin{align*}
f_k(z) &= p_zu(x) + (1-p_z)u(y), \\
u(y) &= p_yu(x) + (1-p_y)f_k(z), \\
u(x) &= p_xf_k(z) + (1-p_x)u(y), \\
\end{align*}
which uniquely define $f_k(z)$ in all three situations, implying that $f_i(z)=f_j(z)$. Setting $u(z)$ to be this common value, we have defined a unique utility function over the whole set of rewards, and we’re done.

# Wallace’s version of the Deutsch-Wallace theorem, part 1

One might still be worried about Deutsch’s Additivity. What if it is actually necessary prove the Born rule? In this case one wouldn’t be able to use the Born rule in the Many-Worlds interpretation without committing oneself to stupid decisions, such as giving away all your money to take part in St. Petersburg’s lottery. Should one give up on the Many-Worlds interpretation then? Or start betting against the Born rule? If these thoughts are keeping you awake at night, then you need Wallace’s version of the Deutsch-Wallace theorem, that replaces Deutsch’s simplistic decision theory with a proper one that allows for bounded utilities.

Wallace’s insight was to realise that the principles of Indifference and Substitution do all the real work in Deutsch’s argument: they are already enough to imply the mod-squared amplitude part of Born’s rule. The connection of those mod-squared amplitudes with probabilities then follow from the other, decision-theoretical principles, but those are incidental, and can be replaced wholesale with a proper decision theory.

More precisely, Wallace used Indifference and Substitution to prove[4] a theorem called Equivalence, which states that Amir must be indifferent between games that assign equal Born-rule weights to the same rewards.

It was not at all obvious to me why this should be a strong result. After all, if I say that the games $\ket{G} = \alpha\ket{M_0}\ket{r_0}+\beta\ket{M_1}\ket{r_1}$ and
$\ket{G’} = \gamma\ket{D_0}\ket{r_0}+\delta\ket{D_1}\ket{r_1}$ are equivalent if $|\alpha|^2=|\gamma|^2$ and $|\beta|^2=|\delta|^2$, it will also be true that they are equivalent if $|\alpha|=|\gamma|$ and $|\beta|=|\delta|$[2], so we haven’t actually learned anything about the “square” part of the Born rule, we have only learned that the phases of the amplitudes are irrelevant. Or have we?

Actually, Equivalence shows its power only when we consider sums of mod-squared amplitudes. It says, for example, that the game (taken unnormalised for clarity) $\ket{G} = 2\ket{M_0}\ket{r_0}+\ket{M_1}\ket{r_1}$ is equivalent to the game
$\ket{G’} = \ket{M_0}\ket{r_0}+\ket{M_1}\ket{r_0}+\ket{M_2}\ket{r_0}+\ket{M_3}\ket{r_0}+\ket{M_4}\ket{r_1},$ as they both assign weight $4$ to reward $r_0$ and weight $1$ to reward $r_1$. Some alternative version of Equivalence that summed the modulus of the amplitudes instead, as it would be appropriate in classical probability theory, would claim that $G$ was actually equivalent to
$\ket{G’^\prime} = \ket{M_0}\ket{r_0}+\ket{M_1}\ket{r_0}+\ket{M_2}\ket{r_1},$ as they both would assign weight $2$ to reward $r_0$ and weight $1$ to reward $r_1$, a decidedly non-quantum result.

Having hopefully convinced you that Equivalence is actually worthwhile, let’s proceed to prove it. The proof is actually very similar to the one presented in the previous post, so if you think it is obvious how to adapt it you can safely skip to the next post, where we’ll do the decision-theory part of the proof. Below I’ll write down the proof of Equivalence anyway just for shits and giggles.

First let’s state it properly:

• Equivalence: Consider two games $\ket{G} = \sum_{ij}\alpha_{ij}\ket{M_i}\ket{r_j}\quad\text{and}\quad \ket{G’} = \sum_{ij}\beta_{ij}\ket{D_i}\ket{r_j}.$ If all rewards $r_j$ have the same Born-rule weight, that is, if $\forall j\quad \sum_i|\alpha_{ij}|^2 = \sum_i|\beta_{ij}|^2,$ then $G \sim G’$.

Note that unlike in the previous post we’re not stating that these games have the same value, but rather that Amir is indifferent between them, which we represent with the $\sim$ relation. We do this because we want to eventually prove that Amir’s preferences can be represented by such a value function, so it feels a bit inelegant to start with the assumption that it exists.

Now, let’s recall Indifference and Substitution from the previous post, slightly reworded to remove reference to the values of the games:

• Indifference: If two games $G$ and $G’$ differ only by the labels of the measurements, then $G \sim G’$.
• Substitution: Amir must be indifferent between the game $\ket{G} = \alpha\ket{M_0}\ket{r_0} + \beta\ket{M_1}\ket{r_1}$ and the composite game
\begin{align*}
\ket{G’} &= \alpha\ket{M_0}\ket{G’^\prime} + \beta\ket{M_1}\ket{r_1} \\
&= \alpha\gamma\ket{M_0}\ket{D_0}\ket{r_0} + \alpha\delta\ket{M_0}\ket{D_1}\ket{r_0} + \beta\ket{M_1}\ket{r_1},
\end{align*} where instead of receiving $r_0$ Amir plays the trivial game $\ket{G’^\prime} = \gamma\ket{D_0}\ket{r_0} + \delta\ket{D_1}\ket{r_0}.$

And to the proof. First we show that any complex phases are irrelevant. For that, consider the game
$\ket{G} = \alpha e^{i\phi}\ket{M_0}\ket{r_0}+\beta e^{i\varphi}\ket{M_1}\ket{r_1}.$ By Substitution, we can replace the rewards $r_0$ and $r_1$ with the degenerate games $e^{-i\phi}\ket{D_0}\ket{r_0}$ and $e^{-i\varphi}\ket{D_1}\ket{r_1}$, and Amir must be indifferent between $G$ and the game $\ket{G’} = \alpha \ket{M_0}\ket{D_0}\ket{r_0}+\beta\ket{M_1}\ket{D_1}\ket{r_1}.$Since $G’$ can be obtained from a third game $\ket{G’^\prime} = \alpha \ket{M_0}\ket{r_0}+\beta \ket{M_1}\ket{r_1}$ via Substitution, this accumulation of measurements does not matter either, and we have Amir must be indifferent to any phases.

This allows us to restrict our attention to positive amplitudes. It does not, however, allow us to restrict our attention to amplitudes which are square roots of rational numbers, but we shall do it anyway because the argument for all real numbers is boring. Consider then two games $\ket{G} = \sum_{ij}\sqrt{\frac{p_{ij}}{q_{ij}}}\ket{M^j_i}\ket{r_j}\quad\text{and}\quad \ket{G’} = \sum_{ij}\sqrt{\frac{a_{ij}}{b_{ij}}}\ket{D^j_i}\ket{r_j}$ for which
$\forall j\quad \sum_i\frac{p_{ij}}{q_{ij}} = \sum_i\frac{a_{ij}}{b_{ij}}.$ We shall show that $G \sim G’$. First focus on the reward $r_0$. We can rewrite the amplitudes of the measurement results that give $r_0$ so that they have the same denominator in both games by defining $d_0 = \prod_i q_{i0}b_{i0},$ and the integers $p_{i0}’ = d_0 p_{i0}/q_{i0}$ and $a_{i0}’ = d_0 a_{i0}/b_{i0}$, so that
$\frac{p’_{i0}}{d_{0}} = \frac{p_{i0}}{q_{i0}}\quad\text{and}\quad\frac{a’_{i0}}{d_{0}} = \frac{a_{i0}}{b_{i0}}.$The parts of the games associated to reward $r_0$ are then
$\frac1{\sqrt{d_0}}\sum_i \sqrt{p_{i0}’}\ket{M^0_i}\ket{r_0} \quad\text{and}\quad \frac1{\sqrt{d_0}}\sum_i \sqrt{a_{i0}’}\ket{D^0_i}\ket{r_0},$ and using again Substitution we replace the reward given for measurement results $M^0_i$ and $D^0_i$ with the trivial games $\frac1{\sqrt{p_{i0}’}}\sum_{k=1}^{p_{i0}’}\ket{P_k}\ket{r_0}\quad\text{and}\quad \frac1{\sqrt{a_{i0}’}}\sum_{k=1}^{a_{i0}’}\ket{P_k}\ket{r_0},$ obtaining
$\frac1{\sqrt{d_0}}\sum_i\sum_{k=1}^{p_{i0}’}\ket{M^0_i}\ket{P_k}\ket{r_0} \quad\text{and}\quad \frac1{\sqrt{d_0}}\sum_i \sum_{k=1}^{a_{i0}’}\ket{D^0_i}\ket{P_k}\ket{r_0},$ which are just uniform superpositions, with $\sum_ip_{i0}’$ terms on the left hand side and $\sum_ia_{i0}’$ on the right hand side. Judicious use of Indifference and Substitution can as before erase the differences in the piles of measurements, taking them to
$\frac1{\sqrt{d_0}}\sum_{l=1}^{\sum_ip_{i0}’}\ket{C_l}\ket{r_0} \quad\text{and}\quad \frac1{\sqrt{d_0}}\sum_{l=1}^{\sum_ia_{i0}’}\ket{C_l}\ket{r_0}.$ Now by assumption we have that $\sum_i\frac{p’_{i0}}{d_{0}} = \sum_i\frac{p_{i0}}{q_{i0}} = \sum_i\frac{a_{i0}}{b_{i0}} = \sum_i\frac{a’_{i0}}{d_{0}},$so the number of terms on both sides are the same, so the $r_0$ parts of the games are equivalent. Since this same argument can be repeated for all other $r_j$, Equivalence is proven.