# An Axiomatization of Stochastic Utility

Ricky Li I am a senior at Harvard College, and my email is
<EMAIL_ADDRESS>I thank Tomasz Strzalecki for introducing me to
decision theory and for his invaluable guidance throughout my research career.

(January 29, 2021)

###### Abstract

I provide an axiomatization of stochastic utility over two periods, stating
testable necessary and sufficient conditions under which an agent’s choice
behavior under exogenous menu selection can be modeled by a pair of random
utility functions. Although static random utility is characterized by a single
axiom, Block-Marschak nonnegativity, I demonstrate that an additional notion
of marginal consistency is needed for the two-period axiomatization. In
particular, when each period’s choice set has size three, I restate the
characterization using the simpler axiom of stochastic regularity. I conclude
by stating several corollaries, including an axiomatization of stochastic
utility with full support and an axiomatization of $n$-period stochastic
utility.

##  1 Introduction

As stated in Sen (1971), it is well-known how to characterize the
deterministic choice functions that can be represented by a unique, strict
preference relation. However, in economics, agents’ choices often display some
element of randomness. Instead of observing a mapping from each menu to an
element of the menu, the analyst may observe a mapping from each menu to a
probability distribution over the menu. Analogously, the analyst may wish to
represent this stochastic choice function with a probability distribution over
strict preference relations, also known as a random utility (RU)
representation. Block et al. (1959) and Falmagne (1978) showed that a single
axiom characterizes the existence of such a representation.111See Section 2
for the formal result.

Agents also frequently make choices over time. Given dynamic nondeterministic
choice data, the analyst may similarly wish to microfound the data with a
multiperiod random utility representation. Depending on the primitive, there
are multiple variants of this model. One type is to treat menus as endogenous:
at any given period, the agent chooses a lottery over the set of pairs of
immediate consumption and a menu of lotteries for the next period. Given
dynamic choice data of this type, Frick et al. (2019) obtained an
axiomatization of dynamic random expected utility, as well as sharper sub-
models in which agents are forward-looking.

However, there are also settings in which menus may be exogenously selected,
such as research studies in which the authors present menus to the subjects.
There are also settings in which the choice set is finite, ruling out
lotteries. In particular, the analyst may wonder when this variant of dynamic
choice data can be modeled by a stochastic process of random preferences. The
main result of this paper is a characterization of these representations,
which I name stochastic utility (SU). The rest of the paper proceeds as
follows. Section 2 provides an overview of RU and its axiomatization. Section
3 formally defines SU and states its axiomatization. Section 4 provides some
corollaries to the main result. Section 5 contains some relevant combinatorics
results and all proofs.

##  2 Random Utility

Before introducing SU, I provide a brief overview of RU and its
axiomatization. Let $X$ be a finite choice set, and let
$\mathcal{M}:=2^{X}\backslash\\{\emptyset\\}$ be the set of all nonempty
menus. Given a (exogenously-chosen) menu $A\in\mathcal{M}$, the agent makes a
choice $x\in A$. The agent’s choice data for all nonempty menus is encoded in
the following primitive:

######  Definition 1.

A stochastic choice function is a mapping
$\rho:\mathcal{M}\rightarrow\Delta(X)$ satisfying $\text{supp
}\rho(A)\subseteq A$ for all $A\in\mathcal{M}$.

Stochastic choice functions must satisfy $\text{supp }\rho(A)\subseteq A$
because the agent can only pick from choices within the menu. As in Strzalecki
(2021), I use $\rho$ to denote a stochastic choice function and $\rho(x,A)$ to
denote the probability that $\rho(A)$ assigns $x$. Let $P$ be the set of
strict preference relations over $X$, and let $C(x,A):=\big{\\{}\succ\in
P:x\succ A\backslash\\{x\\}\big{\\}}$.222$P$ can also be viewed as the set of
permutations of $X$. The notation $x\succ A\backslash\\{x\\}$ denotes $x\succ
y\ \forall\ y\in A\backslash\\{x\\}$. Observe that $\\{C(x,A)\\}_{x\in A}$
form a partition of $P$.

######  Definition 2.

$\mu\in\Delta(P)$ is a random utility (RU) representation of $\rho$ if
$\rho(x,A)=\mu\big{(}C(x,A)\big{)}$ for all $x\in A\in\mathcal{M}$.

In the deterministic case, it is well-known that a choice function can be
represented by a strict preference relation if and only if it satisfies Sen’s
$\alpha$ condition, as shown in Sen (1971). I will now state the analogous
axiomatization of RU, first for $|X|\leq 3$ and second for any finite $X$.

######  Axiom 1.

$\rho$ satisfies regularity if $\rho(x,A)\geq\rho(x,B)$ for all $x\in
A\subseteq B\in\mathcal{M}$.

As stated in Strzalecki (2021), regularity serves as the stochastic analog of
Sen’s $\alpha$. Regularity is necessary for RU because $C(x,A)\supseteq
C(x,B)$ for all $x\in A\subseteq B$. In particular, when the choice set is of
size three or less, regularity characterizes RU.

######  Lemma 1 (Block et al. (1959)).

Suppose $|X|\leq 3$. $\rho$ has a RU representation if and only if it
satisfies regularity.

If the choice set satisfies these cardinalities, Strzalecki (2021) shows that
the RU representation is also unique. For higher cardinalities, a stronger
axiom is needed to characterize RU.

######  Definition 3 (Chambers and Echenique (2016)).

For any $A\subsetneq X$ and $x\in A^{C}$, define their Block-Marschak sum to
be

$\displaystyle M_{x,A}:=\sum_{B\supseteq A^{C}}(-1)^{|B\backslash
A^{C}|}\rho(x,B)$

######  Axiom 2 (Chambers and Echenique (2016)).

$\rho$ satisfies Block-Marschak nonnegativity if $M_{x,A}\geq 0$ for all $x\in
A^{C}\neq\emptyset$.

######  Lemma 2 (Block et al. (1959); Falmagne (1978)).

$\rho$ has a RU representation if and only if it satisfies Block-Marschak
nonnegativity.

With these two axiomatizations in hand, I turn to the dynamic model.

##  3 Stochastic Utility

###  3.1 Preliminaries

I begin by generalizing the agent’s static problem to two periods, denoted
$t=1,2$, with finite choice sets $X_{t}$. As before, define
$\mathcal{M}_{t}:=2^{X_{t}}\backslash\\{\emptyset\\}$. In period $t$, the
agent is offered a (exogenously-chosen) menu $A_{t}\in\mathcal{M}_{t}$ and
makes a choice $x_{t}\in A_{t}$. Importantly, in the dynamic case, the analyst
sequentially observes the agent’s choices. We can thus encode the agent’s
choice data as follows. As before, the analyst observes the (first-period)
stochastic choice function $\rho_{1}$. As in Strzalecki (2021), define
$\mathcal{H}:=\\{(A_{1},x_{1})\in\mathcal{M}_{1}\times
X_{1}:\rho_{1}(x_{1},A_{1})>0\\}$ as the set of observable choice histories.
In addition to $\rho_{1}$, the analyst also observes a family of period-2
stochastic choice functions $\\{\rho_{2}(\cdot|h)\\}_{h\in\mathcal{H}}$,
indexed by choice histories. Thus, the primitive is the vector
$\boldsymbol{\rho}^{2}:=\big{(}\rho_{1},\\{\rho_{2}(\cdot|h)\\}_{h\in\mathcal{H}}\big{)}$.
Since the analyst does not have access to data describing the agent’s period-2
choices after making zero-probability period-1 choices, WLOG let
$\rho_{2}(\cdot,A_{2}|A_{1},x_{1})\in\Delta(A_{2})$ be an arbitrarily chosen
probability distribution when $\rho_{1}(x_{1},A_{1})=0$.333This is to ensure
that expressions like
“$\rho_{2}(x_{2},A_{2}|A_{1},x_{1})\rho_{1}(x_{1},A_{1})=0$” make sense when
$\rho_{1}(x_{1},A_{1})=0$, so that the forthcoming axioms are well-defined. As
long as $\rho_{2}(\cdot,A_{2}|A_{1},x_{1})$ is an honest-to-god probability
distribution over $A_{2}$, its values will not affect the axiomatization. As
before, let $P_{t}$ be the set of strict preference relations over $X_{t}$.
For $x_{t}\in A_{t}$, define $C_{t}(x_{t},A_{t}):=\big{\\{}\succ_{t}\ \in
P_{t}:x_{t}\succ_{t}A_{t}\backslash\\{x_{t}\\}\big{\\}}$ and
$C(x_{t},A_{t}):=C_{t}(x_{t},A_{t})\times P_{-t}$.444Let
$\\{t,-t\\}=\\{1,2\\}$.

######  Definition 4 (Strzalecki (2021)).

$\mu\in\Delta(P_{1}\times P_{2})$ is a (two-period) stochastic utility (SU)
representation of $\boldsymbol{\rho}^{2}$ if
$\rho_{1}(x_{1},A_{1})=\mu\big{(}C(x_{1},A_{1})\big{)}$ for all $x_{1}\in
A_{1}\in\mathcal{M}_{1}$ and
$\rho_{2}(x_{2},A_{2}|A_{1},x_{1})=\mu\big{(}C(x_{2},A_{2})|C(x_{1},A_{1})\big{)}$
for all $x_{2}\in A_{2}\in\mathcal{M}_{2}$ and $(A_{1},x_{1})\in\mathcal{H}$.

If $\mu$ is a SU representation of $\boldsymbol{\rho}^{2}$, note that its
marginal over $P_{1}$ is a RU representation of $\rho_{1}$ and its conditional
$\mu\big{(}\cdot|C(x_{1},A_{1})\big{)}$ is a RU representation of
$\rho_{2}(\cdot|A_{1},x_{1})$ for $(A_{1},x_{1})\in\mathcal{H}$.

######  Definition 5.

For each $t=1,2$ and $x_{t}\in A_{t}^{C}\neq\emptyset$, define their joint
Block-Marschak sum to be

$\displaystyle M_{x_{1},A_{1};x_{2},A_{2}}:=\sum_{B_{2}\supseteq
A_{2}^{C}}\sum_{B_{1}\supseteq A_{1}^{C}}(-1)^{|B_{1}\backslash
A_{1}^{C}|+|B_{2}\backslash
A_{2}^{C}|}\rho_{2}(x_{2},B_{2}|B_{1},x_{1})\rho_{1}(x_{1},B_{1})$

and their joint upper edge set555Joint upper edge sets are a generalization of
what Chambers and Echenique (2016) define as upper contour sets. to be

$\displaystyle E(x_{1},A_{1};x_{2},A_{2}):=\\{(\succ_{1},\succ_{2})\in
P_{1}\times P_{2}:A_{t}\succ_{t}x_{t}\succ_{t}A_{t}^{C}\backslash\\{x_{t}\\},\
t=1,2\\}$

###  3.2 Axiomatization

######  Axiom 3.

$\boldsymbol{\rho}^{2}$ satisfy stochastic Block-Marschak nonnegativity if
$M_{x_{1},A_{1};x_{2},A_{2}}\geq 0$ for each $t=1,2$ and $x_{t}\in
A_{t}^{C}\neq\emptyset$.

As its name suggests, Axiom 3 serves as the two-period analog of Axiom 2.
Unlike the static case, Axiom 3 is not sufficient, and another axiom that
enforces consistency between periods is needed to complete the
characterization.

######  Axiom 4.

$\boldsymbol{\rho}^{2}$ satisfy marginal consistency666In Strzalecki (2021),
this is stated as the LTP axiom. if

$\displaystyle P(x_{2},A_{2};A_{1}):=\sum_{x_{1}\in
A_{1}}\rho_{2}(x_{2},A_{2}|A_{1},x_{1})\rho_{1}(x_{1},A_{1})$

is invariant in $A_{1}$ for all $x_{2}\in A_{2}\in\mathcal{M}_{2}$.

The main result of this paper is that Axioms 1 and 2 characterize two-period
SU:

######  Theorem 1.

$\boldsymbol{\rho}^{2}$ has a SU representation if and only if it satisfies
stochastic Block-Marschak nonnegativity and marginal consistency.

The full proof of Theorem 1 is in Section 5, but I will provide a sketch here.
First, I will state several helpful propositions. Propositions 1 and
2777Proposition 1 is the two-period analog of Lemma 7.4.I in Chambers and
Echenique (2016), while Proposition 2 is a partial generalization of Lemma
7.4.II in the same book. serve as useful identities for the joint Block-
Marschak sums, and Proposition 3888This is the two-period analog of
Proposition 7.3. in Chambers and Echenique (2016). characterizes SU
representations as probability measures that assign each joint upper edge set
its corresponding joint Block-Marschak sum.

######  Proposition 1.

For each $t=1,2$ and $x_{t}\in A_{t}^{C}\neq\emptyset$

$\displaystyle\rho_{2}(x_{2},A_{2}^{C}|A_{1}^{C},x_{1})\rho_{1}(x_{1},A_{1}^{C})=\sum_{B_{2}\subseteq
A_{2}}\sum_{B_{1}\subseteq A_{1}}M_{x_{1},B_{1};x_{2},B_{2}}$

######  Proposition 2.

For any $x_{1}\in A_{1}^{C}\neq\emptyset$ and $\emptyset\subsetneq
A_{2}\subsetneq X_{2}$

$\displaystyle\sum_{x_{2}\in
A_{2}^{C}}M_{x_{1},A_{1};x_{2},A_{2}}=\sum_{y_{2}\in
A_{2}}M_{x_{1},A_{1};y_{2},A_{2}\backslash\\{y_{2}\\}}$

######  Proposition 3.

$\mu$ is a SU representation of $\boldsymbol{\rho}^{2}$ if and only if
$\mu\big{(}E(x_{1},A_{1};x_{2},A_{2})\big{)}=M_{x_{1},A_{1};x_{2},A_{2}}$ for
each $t=1,2$ and $x_{t}\in A_{t}^{C}\neq\emptyset$.

To prove the forwards direction of Theorem 1, note that Proposition 3
immediately implies that stochastic Block-Marschak nonnegativity is necessary,
since probability measures assign nonnegative values to all
events.999Analogous reasoning provides intuition for why Block-Marschak
nonnegativity is necessary for static RU. Marginal consistency is necessary
because of the Law of Total Probability. With Proposition 3 in hand, it
follows that to prove the backwards direction, it suffices to construct a
probability measure $\mu\in\Delta(P_{1}\times P_{2})$ that assigns each joint
upper edge set its corresponding joint Block-Marschak sum. I do this in
Section 5 via the following steps:101010The proof strategy for this direction
is adapted from the proof of the static case in Chambers and Echenique (2016).

  1. 1.

Using marginal consistency, prove the period-1 equivalent of Proposition 2.

  2. 2.

Using stochastic Block-Marschak nonnegativity, recursively define a “partial
measure” $\nu$. $\nu$ is “partial” in the following sense: it is not defined
on all subsets of $P_{1}\times P_{2}$, but rather on pairs of subsets called
$t$-cylinders.

  3. 3.

Verify that $\nu$ satisfies two crucial additivity properties over the pairs
of $t$-cylinders.

  4. 4.

Use both additivity properties to define a probability measure $\mu$ that is
an extension of $\nu$, and verify that $\mu$ assigns each joint upper edge set
its corresponding joint Block-Marschak sum.

###  3.3 Axiomatization for $\boldsymbol{|X_{1}|=|X_{2}|=3}$

At lower choice set cardinalities, we can restate stochastic Block-Marschak
nonnegativity as a simpler axiom.

######  Axiom 5.

$\boldsymbol{\rho}^{2}$ satisfies stochastic regularity if for each $t=1,2$
and $x_{t}\in A_{t}\subseteq B_{t}$,

$\displaystyle\frac{\rho_{1}(x_{1},A_{1})}{\rho_{1}(x_{1},B_{1})}\geq\frac{\rho_{2}(x_{2},A_{2}|B_{1},x_{1})-\rho_{2}(x_{2},B_{2}|B_{1},x_{1})}{\rho_{2}(x_{2},A_{2}|A_{1},x_{1})-\rho_{2}(x_{2},B_{2}|A_{1},x_{1})}$

Stochastic regularity is necessary because $x_{t}\in A_{t}\subseteq B_{t}$ for
each $t=1,2$ implies

$\displaystyle C(x_{1},A_{1})\cap\bigg{(}C(x_{2},A_{2})\backslash
C(x_{2},B_{2})\bigg{)}\supseteq
C(x_{1},B_{1})\cap\bigg{(}C(x_{2},A_{2})\backslash C(x_{2},B_{2})\bigg{)}$

When $|X_{1}|=|X_{2}|=3$ and marginal consistency holds, it is also
sufficient.

######  Proposition 4.

Suppose $|X_{1}|=|X_{2}|=3$. $\boldsymbol{\rho}^{2}$ has a unique SU
representation if and only if it satisfies stochastic regularity and marginal
consistency.

##  4 Corollaries

###  4.1 SU with Full Support

As shown by Fishburn (1998), for arbitrary finite $X$ RU representations need
not be unique. This also implies that, in general, SU representations need not
be unique.111111To see this, let $\mu_{1},\mu_{1}^{\prime}\in\Delta(P_{1})$ be
distinct RU representations of $\rho_{1}$, and let $\mu_{2}\in\Delta(P_{2})$.
Let $\mu=\mu_{1}\times\mu_{2}$, $\mu^{\prime}=\mu_{1}^{\prime}\times\mu_{2}$:
it follows that
$\mu\big{(}C(x_{2},A_{2})|C(x_{1},A_{1})\big{)}=\mu_{2}\big{(}C_{2}(x_{2},A_{2})\big{)}=\mu^{\prime}\big{(}C(x_{2},A_{2})|C(x_{1},A_{1})\big{)}$.
Indeed, sometimes it may be desirable to represent $\boldsymbol{\rho}^{2}$
using a SU representation with full support over $P_{1}\times P_{2}$.121212The
existence of such a representation is equivalent to the existence of a
distribution over $\mathbb{R}^{X_{1}}\times\mathbb{R}^{X_{2}}$ with positive
density.

######  Definition 6.

$\mu\in\Delta(P_{1}\times P_{2})$ has full support if
$\mu(\succ_{1},\succ_{2})>0$ for all $(\succ_{1},\succ_{2})\in P_{1}\times
P_{2}$.

It turns out that characterizing this case requires only a slightly stronger
version of Axiom 3.

######  Axiom 6.

$\boldsymbol{\rho}^{2}$ satisfies stochastic Block-Marschak positivity if
$M_{x_{1},A_{1};x_{2},A_{2}}>0$ for each $t=1,2$ and $x_{t}\in
A_{t}^{C}\neq\emptyset$.

######  Corollary 1.

$\boldsymbol{\rho}^{2}$ has a SU representation with full support if and only
if it satisfies stochastic Block-Marschak positivity and marginal
consistency.131313Analogously, $\rho$ has a RU representation with full
support if and only if it satisfies the strict version of Axiom 2. The proof
proceeds analogously to the proof of this corollary.

Observe that stochastic Block-Marschak positivity is necessary because
probability measures with full support assign strictly positive probability to
all nonempty events.

###  4.2 More Periods

I am currently working on extending Theorem 1 to more than two periods with
multiperiod versions of both axioms. Fix $n>2$. The primitive is now the
vector

$\displaystyle\boldsymbol{\rho}^{n}:=\big{(}\rho_{1},\\{\rho_{2}(\cdot|h_{1})\\}_{h_{1}\in\mathcal{H}_{1}},\ldots,\\{\rho_{n}(\cdot|h_{n-1})\\}_{h_{n-1}\in\mathcal{H}_{n-1}}\big{)}$

where $\mathcal{H}_{1}:=\\{(A_{1},x_{1}):\rho_{1}(x_{1},A_{1})>0\\}$ and
$\mathcal{H}_{t}:=\\{(A_{t},x_{t};h_{t-1})\in\mathcal{M}_{t}\times
X_{t}\times\mathcal{H}_{t-1}:\rho_{t}(x_{t},A_{t}|h_{t-1})>0\\}$ for all
$t>1$. As before, WLOG let
$\rho_{t}(\cdot,A_{t}|A_{t-1},x_{t-1};\cdots;A_{1},x_{1})\in\Delta(A_{t})$ be
an arbitrary probability distribution if
$\rho_{t^{\prime}}(x_{t^{\prime}},A_{t^{\prime}}|A_{t^{\prime}-1},x_{t^{\prime}-1};\cdots;A_{1},x_{1})=0$.
Define $P_{t},C_{t}(x_{t},A_{t}),C(x_{t},A_{t})$ as before. Given $t>1$ and
$h_{t}=(A_{t},x_{t};h_{t-1})$, define $C(h_{t}):=C(x_{t},A_{t})\cap
C(h_{t-1})$.

######  Definition 7 (Strzalecki (2021)).

$\mu\in\Delta\big{(}\bigtimes_{t=1}^{n}P_{t}\big{)}$ is a ($n$-period)
stochastic utility (SU) representation of $\boldsymbol{\rho}^{n}$ if
$\rho_{1}(x_{1},A_{1})=\mu\big{(}C(x_{1},A_{1})\big{)}$ for all $x_{1}\in
A_{1}\in\mathcal{M}_{1}$ and
$\rho_{t}(x_{t},A_{t}|h_{t-1})=\mu\big{(}C(x_{t},A_{t})|C(h_{t-1})\big{)}$ for
all $x_{t}\in A_{t}\in\mathcal{M}_{t}$ and $h_{t-1}\in\mathcal{H}_{t-1}$.

Now, we generalize the axioms. Let
$(\boldsymbol{x},\boldsymbol{A}):=(x_{t},A_{t})_{t=1}^{n}$ and
$(\boldsymbol{x}_{-t},\boldsymbol{A}_{-t})=(x_{t^{\prime}},A_{t^{\prime}})_{t^{\prime}=1,t^{\prime}\neq
t}^{n}$. Let $\boldsymbol{A}^{C}=(A_{t}^{C})_{t=1}^{n}$, and say
$\boldsymbol{B}\geq\boldsymbol{A}\iff B_{t}\supseteq A_{t}$ for each
$t=1,\ldots,n$. Let
$j(\boldsymbol{x},\boldsymbol{A})=\rho_{1}(x_{1},A_{1})\prod_{t=2}^{n}\rho_{t}(x_{t},A_{t}|A_{t-1},x_{t-1};\ldots,A_{1},x_{1})$.

######  Axiom 7.

$\boldsymbol{\rho}^{n}$ satisfies ($n$-period) stochastic Block-Marschak
nonnegativity if

$\displaystyle
M_{(\boldsymbol{x},\boldsymbol{A})}:=\sum_{\boldsymbol{B}\geq\boldsymbol{A}^{C}}(-1)^{\sum_{t=1}^{n}|B_{t}\backslash
A_{t}^{C}|}j(\boldsymbol{x},\boldsymbol{B})\geq 0$

for all $(\boldsymbol{x},\boldsymbol{A})$ satisfying $x_{t}\in
A_{t}^{C}\neq\emptyset$ for each $t=1,\ldots,n$.

######  Axiom 8.

$\boldsymbol{\rho}^{n}$ satisfies ($n$-period) marginal consistency if for any
$(\boldsymbol{x},\boldsymbol{A})$ and $t=1,\ldots,n-1$,

$\displaystyle
P(\boldsymbol{x}_{-t},\boldsymbol{A}_{-t};A_{t}):=\sum_{x_{t}\in
A_{t}}j(\boldsymbol{x}_{-t},\boldsymbol{A}_{-t};x_{t},A_{t})$

is invariant in $A_{t}$.

######  Corollary 2 (Conjecture).

$\boldsymbol{\rho}^{n}$ has a SU representation if and only if it satisfies
stochastic Block-Marschak nonnegativity and marginal consistency.

As before, to prove Corollary 2 it will be helpful to have the following
generalization of Proposition 3 in hand.

######  Corollary 3 (Conjecture).

$\mu$ is a SU representation of $\boldsymbol{\rho^{n}}$ if and only if
$\mu\big{(}E(\boldsymbol{x},\boldsymbol{A})\big{)}=M_{(\boldsymbol{x},\boldsymbol{A})}$
for all $(\boldsymbol{x},\boldsymbol{A})$ satisfying $x_{t}\in
A_{t}^{C}\neq\emptyset$ for each $t=1,\ldots,n$, where

$\displaystyle
E(\boldsymbol{x},\boldsymbol{A}):=\big{\\{}(\succ_{1},\ldots,\succ_{n})\in\bigtimes_{t=1}^{n}P_{t}:A_{t}\succ_{t}x_{t}\succ_{t}A_{t}^{C}\backslash\\{x_{t}\\},\
t=1,\ldots,n\big{\\}}$

##  5 Appendix

###  5.1 The Möbius Inversion

Let $(L,\leq)$ be a finite, partially ordered set (poset).

######  Definition 8 (Van Lint et al. (2001), 25.2).

The Möbius function $m_{L}:L^{2}\rightarrow\mathbb{Z}$ is

$\displaystyle m_{L}(a,b)=\begin{cases}1&a=b\\\ 0&a\nleq b\\\ -\sum_{a\leq
c<y}m_{L}(a,c)&a<b\end{cases}$

######  Lemma 3 (Van Lint et al. (2001), 25.5).

Given a function $f:L\rightarrow\mathbb{R}$, define $F(a):=\sum_{b\geq
a}f(b)$. Then

$\displaystyle f(a)=\sum_{b\geq a}m(a,b)F(b)$

This is known as the Möbius inversion.

I close this section with two more lemmas that will help for the following
proofs.

######  Lemma 4 (Van Lint et al. (2001), 25.1).

Fix finite $X$ and let $L=2^{X}$, $\leq=\subseteq$. Then

$\displaystyle m_{L}(A,B)=\begin{cases}(-1)^{|B|-|A|}&A\subseteq B\\\
0&\text{else}\end{cases}$

######  Lemma 5 (Godsil (2018), 3.1).

Let $L,S$ be posets with respective Möbius functions $m_{L},m_{S}$. Then

$\displaystyle m_{L\times
S}\big{(}(a_{L},a_{S}),(b_{L},b_{S})\big{)}=m_{L}(a_{L},b_{L})m_{S}(a_{S},b_{S})$

###  5.2 Proof of Proposition 1

###### Proof.

Let $L=2^{X_{1}}\times 2^{X_{2}}$ and $(A_{1},A_{2})\leq(B_{1},B_{2})\iff
A_{1}\subseteq B_{1},\ A_{2}\subseteq B_{2}$. Then $(L,\leq)$ is the (finite)
product poset of $(2^{X_{1}},\subseteq)$ and $(2^{X_{1}},\subseteq)$. By
Lemmas 4 and 5, its Möbius function is

$\displaystyle
m_{L}\big{(}(A_{1},A_{2}),(B_{1},B_{2})\big{)}=\begin{cases}(-1)^{|B_{1}|-|A_{1}|+|B_{2}|-|A_{2}|}&A_{1}\subseteq
B_{1},\ A_{2}\subseteq B_{2}\\\ 0&\text{else}\end{cases}$

Now, for each $t=1,2$, fix any $x_{t}\in A_{t}^{C}\neq\emptyset$ and define
$f:L\rightarrow\mathbb{R}$ as

$\displaystyle
f(B_{1},B_{2}):=(-1)^{|B_{1}|-|A_{1}^{C}|+|B_{2}|-|A_{2}^{C}|}\rho_{2}(x_{2},B_{2}|B_{1},x_{1})\rho_{1}(x_{1},B_{1})$

and $F:L\rightarrow\mathbb{R}$ as

$\displaystyle F(D_{1},D_{2})=\sum_{B_{2}\supseteq D_{2}}\sum_{B_{1}\supseteq
D_{1}}f(B_{1},B_{2})$

By Lemma 3,

$\displaystyle f(D_{1},D_{2})=\sum_{B_{2}\supseteq D_{2}}\sum_{B_{1}\supseteq
D_{1}}(-1)^{|B_{1}|-|D_{1}|+|B_{2}|-|D_{2}|}F(B_{1},B_{2})$
$\displaystyle\implies
f(A_{1}^{C},A_{2}^{C})=\rho_{2}(x_{2},A_{2}^{C}|A_{1}^{C},x_{1})\rho_{1}(x_{1},A_{1}^{C})=\sum_{B_{2}\supseteq
A_{2}^{C}}\sum_{B_{1}\supseteq
A_{1}^{C}}(-1)^{|B_{1}|-|A_{1}^{C}|+|B_{2}|-|A_{2}^{C}|}F(B_{1},B_{2})$

To see that

$\displaystyle\sum_{B_{2}\supseteq A_{2}^{C}}\sum_{B_{1}\supseteq
A_{1}^{C}}(-1)^{|B_{1}|-|A_{1}^{C}|+|B_{2}|-|A_{2}^{C}|}F(B_{1},B_{2})=\sum_{D_{2}\subseteq
A_{2}}\sum_{D_{1}\subseteq A_{1}}M_{x_{1},D_{1};x_{2},D_{2}}$

we can match terms as follows. Fix any $D_{2}\subseteq A_{2}$ and
$D_{1}\subseteq A_{1}$. Then

$\displaystyle(-1)^{|D_{1}^{C}|-|A_{1}^{C}|+|D_{2}^{C}|-|A_{2}^{C}|}F(D_{1}^{C},D_{2}^{C})=\sum_{B_{2}\supseteq
D_{2}^{C}}\sum_{B_{1}\supseteq
D_{1}^{C}}(-1)^{|D_{1}^{C}|-|A_{1}^{C}|+|D_{2}^{C}|-|A_{2}^{C}|}f(B_{1},B_{2})$
$\displaystyle=\sum_{B_{2}\supseteq D_{2}^{C}}\sum_{B_{1}\supseteq
D_{1}^{C}}(-1)^{|B_{1}|-|D_{1}^{C}|+|B_{2}|-|D_{2}^{C}|}\rho_{2}(x_{2},B_{2}|B_{1},x_{1})\rho_{1}(x_{1},B_{1})=M_{x_{1},D_{1};x_{2},D_{2}}$

where the second equality follows by observing that $(-1)^{n}=(-1)^{-n}$. ∎

###  5.3 Proof of Proposition 2

###### Proof.

Fix any $x_{1}\in A_{1}^{C}\neq\emptyset$ and $\emptyset\subsetneq
A_{2}\subsetneq X_{2}$. I will use the notation
$\rho_{2}(D_{2},B_{2}|B_{1},x_{1}):=\sum_{x_{2}\in
D_{2}}\rho_{2}(x_{2},B_{2}|B_{1},x_{1})$. We can write

$\displaystyle\sum_{x_{2}\in
A_{2}^{C}}M_{x_{1},A_{1};x_{2},A_{2}}=\sum_{x_{2}\in
A_{2}^{C}}\bigg{(}\sum_{B_{1}\supseteq A_{1}^{C}}\sum_{B_{2}\supseteq
A_{2}^{C}}(-1)^{|B_{2}|-|A_{2}^{C}|+|B_{1}|-|A_{1}^{C}|}\rho_{2}(x_{2},B_{2}|B_{1},x_{1})\rho_{1}(x_{1},B_{1})\bigg{)}$
$\displaystyle=\sum_{B_{1}\supseteq
A_{1}^{C}}\rho_{1}(x_{1},B_{1})(-1)^{|B_{1}|-|A_{1}^{C}|}\bigg{(}\sum_{B_{2}\supseteq
A_{2}^{C}}(-1)^{|B_{2}|-|A_{2}^{C}|}\rho_{2}(A_{2}^{C},B_{2}|B_{1},x_{1})\bigg{)}$

and

$\displaystyle\sum_{y_{2}\in
A_{2}}M_{x_{1},A_{1};y_{2},A_{2}\backslash\\{y_{2}\\}}=\sum_{B_{1}\supseteq
A_{1}^{C}}\rho_{1}(x_{1},B_{1})(-1)^{|B_{1}|-|A_{1}^{C}|}\bigg{(}\sum_{y_{2}\in
A_{2}}\sum_{B_{2}\supseteq
A_{2}^{C}\cup\\{y_{2}\\}}(-1)^{|B_{2}|-|A_{2}^{C}|-1}\rho_{2}(y_{2},B_{2}|B_{1},x_{1})\bigg{)}$

Thus, to complete the proof it suffices to show

$\displaystyle\sum_{B_{2}\supseteq
A_{2}^{C}}(-1)^{|B_{2}|-|A_{2}^{C}|}\rho_{2}(A_{2}^{C},B_{2}|B_{1},x_{1})=\sum_{y_{2}\in
A_{2}}\sum_{B_{2}\supseteq
A_{2}^{C}\cup\\{y_{2}\\}}(-1)^{|B_{2}|-|A_{2}^{C}|-1}\rho_{2}(y_{2},B_{2}|B_{1},x_{1})$

To see this, observe that

$\displaystyle\sum_{B_{2}\supseteq
A_{2}^{C}}(-1)^{|B_{2}|-|A_{2}^{C}|}\rho_{2}(A_{2}^{C},B_{2}|B_{1},x_{1})$
$\displaystyle=\rho_{2}(A_{2}^{C},A_{2}^{C}|B_{1},x_{1})-\sum_{B_{2}=A_{2}^{C}\cup\\{a_{2}\\}}\rho_{2}(A_{2}^{C},A_{2}^{C}\cup\\{a_{2}\\}|B_{1},x_{1})+\ldots+(-1)^{|A_{2}|}\rho_{2}(A_{2}^{C},X_{2}|B_{1},x_{1})$
$\displaystyle=1-\sum_{B_{2}=A_{2}^{C}\cup\\{a_{2}\\}}\big{(}1-\rho_{2}(a_{2},A_{2}^{C}\cup\\{a_{2}\\}|B_{1},x_{1})\big{)}+\ldots+(-1)^{|A_{2}|}\big{(}1-\rho_{2}(A_{2},X_{2}|B_{1},x_{1})\big{)}$

Since there are $\binom{|A_{2}|}{k}$ sets of the form
$B_{2}=A_{2}^{C}\cup\\{a_{2}^{1},\ldots,a_{2}^{k}\\}$ and for $|A_{2}|\geq 1$,

$\displaystyle\sum_{k=0}^{|A_{2}|}(-1)^{k}\binom{|A_{2}|}{k}=0$

we can separate out an alternating sum of binomial coefficients:

$\displaystyle=\sum_{B_{2}=A_{2}^{C}\cup\\{a_{2}\\}}\rho_{2}(a_{2},A_{2}^{C}\cup\\{a_{2}\\}|B_{1},x_{1})-\sum_{B_{2}=A_{2}^{C}\cup\\{a_{2}^{1},a_{2}^{2}\\}}\rho_{2}(\\{a_{2}^{1},a_{2}^{2}\\},A_{2}^{C}\cup\\{a_{2}^{1},a_{2}^{2}\\}|B_{1},x_{1})$
$\displaystyle+\ldots+(-1)^{|A_{2}|+1}\rho_{2}(A_{2},X_{2}|B_{1},x_{1})$

Observe that there is a bijection between nonempty $D_{2}\subseteq A_{2}$ and
terms in this sum:

$\displaystyle
D_{2}\xleftrightarrow{}(-1)^{|D_{2}|+1}\rho_{2}(D_{2},A_{2}^{C}\cup
D_{2}|B_{1},x_{1})$

Similarly, there is a bijection between nonempty $D_{2}\subseteq A_{2}$ and
terms in the following sum

$\displaystyle\sum_{y_{2}\in A_{2}}\sum_{B_{2}\supseteq
A_{2}^{C}\cup\\{y_{2}\\}}(-1)^{|B_{2}|-|A_{2}^{C}|-1}\rho_{2}(y_{2},B_{2}|B_{1},x_{1})$

given by

$\displaystyle D_{2}\xleftrightarrow{}\sum_{y_{2}\in
D_{2}}(-1)^{|A_{2}^{C}\cup D_{2}|-|A_{2}^{C}|-1}\rho_{2}(y_{2},A_{2}^{C}\cup
D_{2}|B_{1},x_{1})$
$\displaystyle=(-1)^{|D_{2}|+1}\rho_{2}(D_{2},A_{2}^{C}\cup
D_{2}|B_{1},x_{1})$

Since both sums are comprised of precisely the same terms, they are equal. ∎

###  5.4 Proof of Proposition 3

###### Proof.

Forwards direction: suppose $\mu$ is a SU representation of
$\boldsymbol{\rho}^{2}$. For each $t=1,2$, fix any $x_{t}\in
A_{t}^{C}\neq\emptyset$. Since $x_{t}\succ_{t}A_{t}^{C}\backslash\\{x_{t}\\}$
if and only if $B_{t}^{C}\succ_{t}x_{t}\succ_{t}B_{t}\backslash\\{x_{t}\\}$
for some $B_{t}\supseteq A_{t}^{C}$,

$\displaystyle C(x_{1},A_{1}^{C})\cap
C(x_{2},A_{2}^{C})=\bigcup_{B_{2}\supseteq A_{2}^{C}}\bigcup_{B_{1}\supseteq
A_{1}^{C}}E(x_{1},B_{1}^{C};x_{2},B_{2}^{C})$

Furthermore, this union is disjoint, so

$\displaystyle\rho_{2}(x_{2},A_{2}^{C}|A_{1}^{C},x_{1})\rho_{1}(x_{1},A_{1}^{C})=\sum_{B_{2}\supseteq
A_{2}^{C}}\sum_{B_{1}\supseteq
A_{1}^{C}}\mu\big{(}E(x_{1},B_{1}^{C};x_{2},B_{2}^{C})\big{)}$

By Lemmas 4 and 5,
$m\big{(}(A_{1},A_{2}),(B_{1},B_{2})\big{)}=(-1)^{|B_{1}|-|A_{1}|+|B_{2}|-|A_{2}|}$.
By Lemma 3 with
$f(B_{1},B_{2})=\mu\big{(}E(x_{1},B_{1}^{C};x_{2},B_{2}^{C})\big{)}$ and
$F(A_{1},A_{2})=\rho_{2}(x_{2},A_{2}|A_{1},x_{1})\rho_{1}(x_{1},A_{1})$, we
get

$\displaystyle\mu\big{(}E(x_{1},A_{1};x_{2},A_{2})\big{)}=\sum_{B_{2}\supseteq
A_{2}^{C}}\sum_{B_{1}\supseteq
A_{1}^{C}}(-1)^{|B_{2}|-|A_{2}^{C}|+|B_{1}|-|A_{1}^{C}|}\rho_{2}(x_{2},B_{2}|B_{1},x_{1})\rho_{1}(x_{1},B_{1})=M_{x_{1},A_{1};x_{2},A_{2}}$

as desired.141414The proof strategy for this direction is adapted from
Strzalecki (2021).

Backwards direction: suppose there exists $\mu\in\Delta(P_{1}\times P_{2})$
satisfying
$\mu\big{(}E(x_{1},A_{1};x_{2},A_{2})\big{)}=M_{x_{1},A_{1};x_{2},A_{2}}$ for
all $t=1,2$ and $x_{t}\in A_{t}^{C}\neq\emptyset$, and fix any $y_{t}\in
D_{t}\in\mathcal{M}_{t}$. As before, observe that
$y_{t}\succ_{t}D_{t}\backslash\\{y_{t}\\}$ if and only if
$B_{t}\succ_{t}y_{t}\succ_{t}B_{t}^{C}\backslash\\{y_{t}\\}$ for some
$B_{t}\subseteq D_{t}^{C}$, so

$\displaystyle C(y_{1},D_{1})\cap C(y_{2},D_{2})=\bigcup_{B_{2}\subseteq
D_{2}^{C}}\bigcup_{B_{1}\subseteq D_{1}^{C}}E(x_{1},B_{1};x_{2},B_{2})$

Furthermore, this union is disjoint, so

$\displaystyle\mu\big{(}C(y_{1},D_{1})\cap
C(y_{2},D_{2})\big{)}=\sum_{B_{2}\subseteq D_{2}^{C}}\sum_{B_{1}\subseteq
D_{1}^{C}}M_{x_{1},B_{1};x_{2},B_{2}}=\rho_{2}(y_{2},D_{2}|D_{1},y_{1})\rho_{1}(y_{1},D_{1})$

where the second equality follows because $D_{t}\neq\emptyset\implies
B_{t}\neq X_{t}$, and the third equality follows from Proposition 1. Since

$\displaystyle\mu\big{(}C(y_{1},D_{1})\big{)}=\sum_{y_{2}\in
D_{2}}\mu\big{(}C(y_{1},D_{1})\cap
C(y_{2},D_{2})\big{)}=\rho_{1}(y_{1},D_{1})$

and

$\displaystyle\mu\big{(}C(y_{2},D_{2})|C(y_{1},D_{1})\big{)}=\frac{\mu(C(y_{1},D_{1})\cap
C(y_{2},D_{2}))}{\mu(C(y_{1},D_{1}))}=\rho_{2}(y_{2},D_{2}|D_{1},y_{1})$

we conclude that $\mu$ is a SU representation of $\boldsymbol{\rho}^{2}$.

∎

###  5.5 Proof of Theorem 1 (Backwards Direction)

###### Proof.

Suppose $\boldsymbol{\rho}^{2}$ satisfies Block-Marschak nonnegativity and
marginal consistency. As outlined in Section 3, the proof rests the following
series of Claims, whose proofs are included in the forthcoming subsections.

######  Claim 1.

For any $x_{2}\in A_{2}^{C}\neq\emptyset$ and $\emptyset\subsetneq
A_{1}\subsetneq X_{1}$,

$\displaystyle\sum_{x_{1}\in
A_{1}^{C}}M_{x_{1},A_{1};x_{2},A_{2}}=\sum_{y_{1}\in
A_{1}}M_{y_{1},A_{1}\backslash\\{y_{1}\\};x_{2},A_{2}}$

Claim 1 is the first-period analog of Proposition 2 and follows from a similar
argument by using marginal consistency. Now, I define the $t$-cylinders.

######  Definition 9.

Given an ordered, distinct $\boldsymbol{k}$-sequence
$(x_{t}^{1},\ldots,x_{t}^{k})$, its $\boldsymbol{t}$-cylinder151515This
definition is the two-period analog of Chambers and Echenique (2016)’s
definition of cylinders. is

$\displaystyle I_{(x_{t}^{1},\ldots,x_{t}^{k})}=\big{\\{}\succ_{t}\ \in
P_{t}:x_{t}^{1}\succ_{t}\cdots\succ_{t}x_{t}^{k}\succ_{t}X_{t}\backslash\\{x_{t}^{1},\ldots,x_{t}^{k}\\}\big{\\}}$

Given a menu $A_{t}$, let $\pi(A_{t})$ denote the set of permutations of
$A_{t}$. Let
$\mathcal{I}_{t}(k)=\\{I_{(x_{t}^{1},\ldots,x_{t}^{k})}:(x_{t}^{1},\ldots,x_{t}^{k})\in\pi(A_{t}),A_{t}\in\mathcal{M}_{t},|A_{t}|=k\\}$
be the set of all $t$-cylinders induced by $k$-sequences, and let
$\mathcal{I}_{t}=\bigcup_{k=1}^{|X_{t}|}\mathcal{I}_{t}(k)$. Observe that
$\mathcal{I}_{t}$ contains all singletons, since

$\displaystyle I_{(x_{t}^{1},\ldots,x_{t}^{|X_{t}|})}=\\{\succ_{t}\\}\iff
x_{t}^{1}\succ_{t}\cdots\succ_{t}x_{t}^{|X_{t}|}$

Next, I recursively define a function
$\nu:\mathcal{I}_{1}\times\mathcal{I}_{2}\rightarrow\mathbb{R}_{\geq
0}$.161616Again, my definition of $\nu$ is the two-period analog of Chambers
and Echenique (2016), (7.4). Define

$\displaystyle\nu\big{(}I_{x_{1}}\times
I_{x_{2}}\big{)}:=M_{x_{1},\emptyset;x_{2},\emptyset}=\rho_{2}(x_{2},X_{2}|X_{1},x_{1})\rho_{1}(x_{1},X_{1})$

Now, let (distinct) $i,j\in\\{1,2\\}$. For any $1<k\leq|X_{j}|$ and
$(x_{j}^{1},\ldots,x_{j}^{k},x_{j}^{k+1})$, let
$A_{j}=\\{x_{j}^{1},\ldots,x_{j}^{k}\\}$ and define

$\displaystyle\nu\big{(}I_{x_{i}}\times
I_{(x_{j}^{1},\ldots,x_{j}^{k},x_{j}^{k+1})}\big{)}:=\begin{cases}0&\sum_{\tau_{j}\in\pi(A_{j})}\nu(I_{x_{i}}\times
I_{\tau_{j}})=0\\\ \frac{\nu(I_{x_{i}}\times
I_{(x_{j}^{1},\ldots,x_{j}^{k})})M_{x_{i},\emptyset;x_{j}^{k+1},A_{j}}}{\sum_{\tau_{j}\in\pi(A_{j})}\nu(I_{x_{i}}\times
I_{\tau_{j}})}&\text{else}\end{cases}$

Similarly, for any $1\leq k<|X_{1}|$, $1\leq\ell<|X_{2}|$ and
$(x_{1}^{1},\ldots,x_{1}^{k},x_{1}^{k+1})$,
$(x_{2}^{1},\ldots,x_{2}^{\ell},x_{2}^{\ell+1})$, let
$A_{1}=\\{x_{1}^{1},\ldots,x_{1}^{k}\\}$,
$A_{2}=\\{x_{2}^{1},\ldots,x_{2}^{\ell}\\}$ and define

$\displaystyle\nu\big{(}I_{(x_{1}^{1},\ldots,x_{1}^{k},x_{1}^{k+1})}\times
I_{(x_{2}^{1},\ldots,x_{2}^{\ell},x_{2}^{\ell+1})}\big{)}:=\begin{cases}0&\sum_{\tau_{1}\in\pi(A_{1})}\sum_{\tau_{2}\in\pi(A_{1})}\nu(I_{\tau_{1}}\times
I_{\tau_{2}})=0\\\ \frac{\nu(I_{(x_{1}^{1},\ldots,x_{1}^{k})}\times
I_{(x_{2}^{1},\ldots,x_{2}^{\ell})})M_{x_{1}^{k+1},A_{1};x_{2}^{\ell+1},A_{2}}}{\sum_{\tau_{1}\in\pi(A_{1})}\sum_{\tau_{2}\in\pi(A_{1})}\nu(I_{\tau_{1}}\times
I_{\tau_{2}})}&\text{else}\end{cases}$

######  Definition 10.

For any $0\leq k<|X_{1}|$, $0\leq\ell<|X_{2}|$, the first additive property
$\boldsymbol{p_{1}(k,\ell)}$ holds if for all $A_{1},A_{2}$ s.t.
$|A_{1}|=k,|A_{2}|=\ell$ and all $x_{t}\in A_{t}^{C}$

$\displaystyle\sum_{\tau_{1}\in\pi(A_{1})}\sum_{\tau_{2}\in\pi(A_{2})}\nu\big{(}I_{\tau_{1},x_{1}}\times
I_{\tau_{2},x_{2}}\big{)}=M_{x_{1},A_{1};x_{2},A_{2}}$

For any $0<k\leq|X_{1}|$, $0<\ell\leq|X_{2}|$, the second additive property
$\boldsymbol{p_{2}(k,\ell)}$ holds if for all $A_{1},A_{2}$ s.t.
$|A_{1}|=k,|A_{2}|=\ell$ and all $\tau_{t}\in\pi(A_{t})$

$\displaystyle\sum_{x_{1}\in A_{1}^{C}}\sum_{x_{2}\in
A_{2}^{C}}\nu\big{(}I_{\tau_{1},x_{1}}\times
I_{\tau_{2},x_{2}}\big{)}=\nu\big{(}I_{\tau_{1}}\times I_{\tau_{2}}\big{)}$

Observe that
$\bigcup_{\tau_{1}\in\pi(A_{1})}\bigcup_{\tau_{2}\in\pi(A_{2})}(I_{\tau_{1},x_{1}}\times
I_{\tau_{2},x_{2}})=E_{x_{1},A_{1};x_{2},A_{2}}$ and $\bigcup_{x_{1}\in
A_{1}^{C}}\bigcup_{x_{2}\in A_{2}^{C}}(I_{\tau_{1},x_{1}}\times
I_{\tau_{1},x_{1}})=I_{\tau_{1}}\times I_{\tau_{2}}$ and these are disjoint
unions, so these additive properties are necessary.171717These additive
properties are the two-period analogs of Chambers and Echenique (2016), (7.2)
and (7.3), respectively.

######  Claim 2.

$p_{1}(k,\ell)$ holds for all $0\leq k<|X_{1}|$, $0\leq\ell<|X_{2}|$.

######  Claim 3.

$p_{2}(k,\ell)$ holds for all $0<k\leq|X_{1}|$, $0<\ell\leq|X_{2}|$.

With these additive properties in hand, I am ready to define the candidate SU
representation. Given $(\succ_{1},\succ_{2})\in P_{1}\times P_{2}$, denote
$x_{t}^{1}\succ_{t}\cdots\succ_{t}x_{t}^{|X_{t}|}$ and define
$\mu:2^{P_{1}\times P_{2}}\rightarrow\mathbb{R}$ as181818This definition is
the two-period analog of Chambers and Echenique (2016)’s definition of
“$\nu$.”

$\displaystyle\mu\big{(}\\{\succ_{1},\succ_{2}\\}\big{)}=\nu\big{(}I_{(x_{1}^{1},\ldots,x_{1}^{|X_{1}|})}\times
I_{(x_{2}^{1},\ldots,x_{2}^{|X_{2}|})}\big{)}$
$\displaystyle\mu\big{(}S\big{)}=\sum_{(\succ_{1},\succ_{2})\in
S}\mu\big{(}\\{\succ_{1},\succ_{2}\\}\big{)}$

######  Claim 4.

$\mu$ is a probability measure.

######  Claim 5.

$\mu=\nu$ on $\mathcal{I}_{1}\times\mathcal{I}_{2}$.

Claim 5 shows that $\mu$ is an extension of $\nu$. Thus, I can leverage Claim
2 as follows. Recall from Proposition 1 that to show that $\mu$ is a SU
representation of $\boldsymbol{\rho}^{2}$, it suffices to show
$\mu\big{(}E(x_{1},A_{1};x_{2},A_{2})\big{)}=M_{x_{1},A_{1};x_{2},A_{2}}$ for
all $t=1,2$ and $x_{t}\in A_{t}^{C}\neq\emptyset$. For each $t=1,2$, fix any
$x_{t}\in A_{t}^{C}\neq\emptyset$ (note this implies that $|A_{t}|<|X_{t}|$):
then

$\displaystyle\mu\big{(}E(x_{1},A_{1};x_{2},A_{2})\big{)}=\sum_{\tau_{1}\in\pi(A_{1})}\sum_{\tau_{2}\in\pi(A_{2})}\mu\big{(}I_{\tau_{1},x_{1}}\times
I_{\tau_{2},x_{2}}\big{)}$
$\displaystyle=\sum_{\tau_{1}\in\pi(A_{1})}\sum_{\tau_{2}\in\pi(A_{2})}\nu\big{(}I_{\tau_{1},x_{1}}\times
I_{\tau_{2},x_{2}}\big{)}=M_{x_{1},A_{1};x_{2},A_{2}}$

where the penultimate equality follows because $\mu=\nu$ on
$\mathcal{I}_{1}\times\mathcal{I}_{2}$, and the last equality follows from
Claim 2.

####  5.5.1 Proof of Claim 1

###### Proof.

We can write

$\displaystyle\sum_{x_{1}\in
A_{1}^{C}}M_{x_{1},A_{1};x_{2},A_{2}}=\sum_{B_{2}\supseteq
A_{2}^{C}}(-1)^{|B_{2}|-|A_{2}^{C}|}\Bigg{(}\sum_{B_{1}\supseteq
A_{1}^{C}}(-1)^{|B_{1}|-|A_{1}^{C}|}\bigg{(}\sum_{x_{1}\in
A_{1}^{C}}\rho_{2}(x_{2},B_{2}|B_{1},x_{1})\rho_{1}(x_{1},B_{1})\bigg{)}\Bigg{)}$

and

$\displaystyle\sum_{y_{1}\in
A_{1}}M_{y_{1},A_{1}\backslash\\{y_{1}\\};x_{2},A_{2}}=\sum_{B_{2}\supseteq
A_{2}^{C}}(-1)^{|B_{2}|-|A_{2}^{C}|}\Bigg{(}\sum_{y_{1}\in
A_{1}}\sum_{B_{1}\supseteq
A_{1}^{C}\cup\\{y_{1}\\}}(-1)^{|B_{1}|-|A_{1}^{C}|-1}\rho_{2}(x_{2},B_{2}|B_{1},y_{1})\rho_{1}(y_{1},B_{1})\Bigg{)}$

Since marginal consistency is satisfied, we can write

$\displaystyle\sum_{B_{1}\supseteq
A_{1}^{C}}(-1)^{|B_{1}|-|A_{1}^{C}|}\bigg{(}\sum_{x_{1}\in
A_{1}^{C}}\rho_{2}(x_{2},B_{2}|B_{1},x_{1})\rho_{1}(x_{1},B_{1})\bigg{)}$
$\displaystyle=\sum_{B_{1}\supseteq
A_{1}^{C}}(-1)^{|B_{1}|-|A_{1}^{C}|}\bigg{(}P(x_{2},B_{2})-\sum_{x_{1}\in
B_{1}\backslash
A_{1}^{C}}\rho_{2}(x_{2},B_{2}|B_{1},x_{1})\rho_{1}(x_{1},B_{1})\bigg{)}$
$\displaystyle=P(x_{2},B_{2})\sum_{B_{1}\supseteq
A_{1}^{C}}(-1)^{|B_{1}|-|A_{1}^{C}|}+\sum_{B_{1}\supsetneq
A_{1}^{C}}(-1)^{|B_{1}|-|A_{1}^{C}|+1}\sum_{x_{1}\in B_{1}\backslash
A_{1}^{C}}\rho_{2}(x_{2},B_{2}|B_{1},x_{1})\rho_{1}(x_{1},B_{1})$

Again observing that

$\displaystyle\sum_{B_{1}\supseteq
A_{1}^{C}}(-1)^{|B_{1}|-|A_{1}^{C}|}=\sum_{k=0}^{|A_{1}|}(-1)^{k}\binom{|A_{1}|}{k}=0$

it thus suffices to show that

$\displaystyle\sum_{B_{1}\supsetneq
A_{1}^{C}}(-1)^{|B_{1}|-|A_{1}^{C}|+1}\sum_{x_{1}\in B_{1}\backslash
A_{1}^{C}}\rho_{2}(x_{2},B_{2}|B_{1},x_{1})\rho_{1}(x_{1},B_{1})$
$\displaystyle=\sum_{y_{1}\in A_{1}}\sum_{B_{1}\supseteq
A_{1}^{C}\cup\\{y_{1}\\}}(-1)^{|B_{1}|-|A_{1}^{C}|-1}\rho_{2}(x_{2},B_{2}|B_{1},y_{1})\rho_{1}(y_{1},B_{1})$

which immediately follows from matching terms. ∎

Note that Proposition 2 and Claim 1 together imply that for any
$\emptyset\subsetneq A_{t}\subsetneq X_{t}$,

$\displaystyle\sum_{x_{1}\in A_{1}^{C}}\sum_{x_{2}\in
A_{2}^{C}}M_{x_{1},A_{1};x_{2},A_{2}}=\sum_{x_{1}\in A_{1}^{C}}\sum_{y_{2}\in
A_{2}}M_{x_{1},A_{1};y_{2},A_{2}\backslash\\{y_{2}\\}}=\sum_{y_{2}\in
A_{2}}\sum_{x_{1}\in
A_{1}^{C}}M_{x_{1},A_{1};y_{2},A_{2}\backslash\\{y_{2}\\}}$
$\displaystyle=\sum_{y_{2}\in A_{2}}\sum_{y_{1}\in
A_{1}}M_{y_{1},A_{1}\backslash\\{y_{1}\\};y_{2},A_{2}\backslash\\{y_{2}\\}}$

####  5.5.2 Proof of Claim 2

###### Proof.

I will prove this via induction.

Base case: fix $A_{1}=A_{2}=\emptyset$ (these are the only menus satisfying
$|A_{1}|=|A_{2}|=0$ and $x_{t}\in X_{t}$). Then, by definition,

$\displaystyle\sum_{\tau_{1}\in\pi(A_{1})}\sum_{\tau_{2}\in\pi(A_{2})}\nu\big{(}I_{\tau_{1},x_{1}}\times
I_{\tau_{2},x_{2}}\big{)}=\nu\big{(}I_{x_{1}}\times
I_{x_{2}}\big{)}=M_{x_{1},\emptyset;x_{2},\emptyset}$

First inductive step: Fix $A_{i}=\emptyset$, $x_{i}\in X_{i}$, and
$A_{j}=\\{x_{j}^{1},\ldots,x_{j}^{k}\\}$ for any $k>0$ and observe that

$\displaystyle\sum_{\tau_{j}\in\pi(A_{j})}\nu\big{(}I_{x_{i}}\times
I_{\tau_{j}}\big{)}=\sum_{y_{j}\in
A_{j}}\bigg{(}\sum_{(y_{j}^{1},\ldots,y_{j}^{k-1})\in\pi(A_{j}\backslash\\{y_{j}\\})}\nu\big{(}I_{x_{i}}\times
I_{(y_{j}^{1},\ldots,y_{j}^{k-1},y_{j})}\big{)}\bigg{)}$
$\displaystyle=\sum_{y_{j}\in
A_{j}}M_{x_{i},\emptyset;y_{j},A_{j}\backslash\\{y_{j}\\}}=\sum_{x_{j}\in
A_{j}^{C}}M_{x_{i},\emptyset;x_{j},A_{j}}$

where the first equality follows because permuting $A_{j}$ is equivalent to
picking the last element and permuting the remaining $k-1$ elements, the
second equality follows from the inductive hypothesis (which is
“$p_{1}(0,k-1)$ holds” if $i=1,j=2$ and “$p_{1}(k-1,0)$ holds” otherwise), and
the third equality follows from Proposition 2 and Claim 1. There are two
cases:

  1. 1.

Suppose

$\displaystyle\sum_{\tau_{j}\in\pi(A_{j})}\nu\big{(}I_{x_{i}}\times
I_{\tau_{j}}\big{)}=0$

Since Block-Marschak nonnegativity holds by assumption,
$M_{x_{i},\emptyset;x_{j},A_{j}}=0$ for each $x_{j}\in A_{j}^{C}$. Fix any
$x_{j}\in A_{j}^{C}$: then we have

$\displaystyle\sum_{\tau_{i}\in\pi(A_{i})}\sum_{\tau_{j}\in\pi(A_{j})}\nu\big{(}I_{\tau_{i},x_{i}}\times
I_{\tau_{j},x_{j}}\big{)}=\sum_{\tau_{j}\in\pi(A_{j})}\nu\big{(}I_{x_{i}}\times
I_{\tau_{j},x_{j}}\big{)}=0=M_{x_{i},\emptyset;x_{j},A_{j}}$

where the penultimate equality follows by definition of $\nu$. Thus,
$p_{1}(0_{i},k_{j})$ holds.

  2. 2.

Suppose

$\displaystyle\sum_{\tau_{j}\in\pi(A_{j})}\nu\big{(}I_{x_{i}}\times
I_{\tau_{j}}\big{)}>0$

Again by definition of $\nu$, we have

$\displaystyle\sum_{\tau_{i}\in\pi(A_{i})}\sum_{\tau_{j}\in\pi(A_{j})}\nu\big{(}I_{\tau_{i},x_{i}}\times
I_{\tau_{j},x_{j}}\big{)}=\sum_{\tau_{j}\in\pi(A_{j})}\nu\big{(}I_{x_{i}}\times
I_{\tau_{j},x_{j}}\big{)}$
$\displaystyle=\sum_{\tau_{j}\in\pi(A_{j})}\bigg{(}\frac{\nu(I_{x_{i}}\times
I_{\tau_{j}})}{\sum_{\alpha_{j}\in\pi(A_{j})}\nu(I_{x_{i}}\times
I_{\alpha_{j}})}M_{x_{i},\emptyset;x_{j},A_{j}}\bigg{)}=M_{x_{i},\emptyset;x_{j},A_{j}}$

Thus, $p_{1}(0_{i},k_{j})$ holds.

Second inductive step: This step proceeds similarly to the first inductive
step. For any $0<k<|X_{1}|$ and $0<\ell<|X_{2}|$, fix
$A_{1}=\\{x_{1}^{1},\ldots,x_{1}^{k}\\}$,
$A_{2}=\\{x_{2}^{1},\ldots,x_{2}^{\ell}\\}$, and $x_{t}\in A_{t}^{C}$. Our
inductive hypothesis is that $p_{1}(k-1,\ell-1)$ holds. Again, observe that

$\displaystyle\sum_{\tau_{1}\in\pi(A_{1})}\sum_{\tau_{2}\in\pi(A_{2})}\nu\big{(}I_{\tau_{1}}\times
I_{\tau_{2}}\big{)}$ $\displaystyle=\sum_{y_{1}\in A_{1}}\sum_{y_{2}\in
A_{2}}\bigg{(}\sum_{(y_{1}^{1},\ldots,y_{1}^{k-1})\in\pi(A_{1}\backslash\\{y_{1}\\})}\sum_{(y_{2}^{1},\ldots,y_{2}^{\ell-1})\in\pi(A_{2}\backslash\\{y_{2}\\})}\nu\big{(}I_{(y_{1}^{1},\ldots,y_{1}^{k-1},y_{1})}\times
I_{(y_{2}^{1},\ldots,y_{2}^{\ell-1},y_{2})}\big{)}\bigg{)}$
$\displaystyle=\sum_{y_{1}\in A_{1}}\sum_{y_{2}\in
A_{2}}M_{y_{1},A_{1}\backslash\\{y_{1}\\};y_{2},A_{2}\backslash\\{y_{2}\\}}=\sum_{x_{1}\in
A_{1}^{C}}\sum_{x_{2}\in A_{2}^{C}}M_{x_{1},A_{1};x_{2},A_{2}}$

where the penultimate equality follows from the inductive hypothesis, and the
last equality follows from Proposition 2 and Claim 1. Again, there are two
cases:

  1. 1.

Suppose

$\displaystyle\sum_{\tau_{1}\in\pi(A_{1})}\sum_{\tau_{2}\in\pi(A_{2})}\nu\big{(}I_{\tau_{1}}\times
I_{\tau_{2}}\big{)}=0$

Then weak Block-Marschak positivity implies $M_{x_{1},A_{1};x_{2},A_{2}}=0$
for each $x_{t}\in A_{t}^{C}$. Fix any $x_{t}\in A_{t}^{C}$: by definition of
$\nu$, we have

$\displaystyle\sum_{\tau_{1}\in\pi(A_{1})}\sum_{\tau_{2}\in\pi(A_{2})}\nu\big{(}I_{\tau_{1},x_{1}}\times
I_{\tau_{2},x_{2}}\big{)}=0=M_{x_{1},A_{1};x_{2},A_{2}}$

and $p_{1}(k,\ell)$ holds.

  2. 2.

Suppose

$\displaystyle\sum_{\tau_{1}\in\pi(A_{1})}\sum_{\tau_{2}\in\pi(A_{2})}\nu\big{(}I_{\tau_{1}}\times
I_{\tau_{2}}\big{)}>0$

Again by definition of $\nu$, it follows that

$\displaystyle\sum_{\tau_{1}\in\pi(A_{1})}\sum_{\tau_{2}\in\pi(A_{2})}\nu\big{(}I_{\tau_{1},x_{1}}\times
I_{\tau_{2},x_{2}}\big{)}=M_{x_{1},A_{1};x_{2},A_{2}}$

and $p_{1}(k,\ell)$ holds.

∎

####  5.5.3 Proof of Claim 3

###### Proof.

Fix any $0<k\leq|X_{1}|$ and $0<\ell\leq|X_{2}|$,
$A_{1}=\\{x_{1}^{1},\ldots,x_{1}^{k}\\}$ and
$A_{2}=\\{x_{2}^{1},\ldots,x_{2}^{\ell}\\}$, and $\tau_{t}\in\pi(A_{t})$. As
before, there are two cases:

  1. 1.

Suppose

$\displaystyle\sum_{\alpha_{1}\in\pi(A_{1})}\sum_{\alpha_{2}\in\pi(A_{2})}\nu\big{(}I_{\alpha_{1}}\times
I_{\alpha_{2}}\big{)}=0$

Since $\nu\geq 0$ by definition, in particular $\nu\big{(}I_{\tau_{1}}\times
I_{\tau_{2}}\big{)}=0$. Furthermore, by definition, for each $x_{t}\in
A_{t}^{C}$

$\displaystyle\nu\big{(}I_{\tau_{1},x_{1}}\times I_{\tau_{2},x_{2}}\big{)}=0$
$\displaystyle\implies\sum_{x_{1}\in A_{1}^{C}}\sum_{x_{2}\in
A_{2}^{C}}\nu\big{(}I_{\tau_{1},x_{1}}\times
I_{\tau_{2},x_{2}}\big{)}=0=\nu\big{(}I_{\tau_{1}}\times I_{\tau_{2}}\big{)}$

as desired.

  2. 2.

Suppose

$\displaystyle\sum_{\alpha_{1}\in\pi(A_{1})}\sum_{\alpha_{2}\in\pi(A_{2})}\nu\big{(}I_{\alpha_{1}}\times
I_{\alpha_{2}}\big{)}>0$

Since $0<k\leq|X_{1}|$ and $0<\ell\leq|X_{2}|$, $0\leq k-1<|X_{1}|$ and
$0\leq\ell-1<|X_{2}|$, so by Claim 2 we can apply $p_{1}(k-1,\ell-1)$ as
before to write

$\displaystyle\sum_{\alpha_{1}\in\pi(A_{1})}\sum_{\alpha_{2}\in\pi(A_{2})}\nu\big{(}I_{\alpha_{1}}\times
I_{\alpha_{2}}\big{)}=\sum_{x_{1}\in A_{1}^{C}}\sum_{x_{2}\in
A_{2}^{C}}M_{x_{1},A_{1};x_{2},A_{2}}$

which implies

$\displaystyle\sum_{x_{1}\in A_{1}^{C}}\sum_{x_{2}\in
A_{2}^{C}}\nu\big{(}I_{\tau_{1},x_{1}}\times
I_{\tau_{2},x_{2}}\big{)}=\sum_{x_{1}\in A_{1}^{C}}\sum_{x_{2}\in
A_{2}^{C}}\bigg{(}\frac{\nu(I_{\tau_{1}}\times
I_{\tau_{2}})}{\sum_{\alpha_{1}\in\pi(A_{1})}\sum_{\alpha_{2}\in\pi(A_{2})}\nu(I_{\alpha_{1}}\times
I_{\alpha_{2}})}M_{x_{1},A_{1};x_{2},A_{2}}\bigg{)}$
$\displaystyle=\frac{\nu(I_{\tau_{1}}\times I_{\tau_{2}})}{\sum_{x_{1}\in
A_{1}^{C}}\sum_{x_{2}\in
A_{2}^{C}}M_{x_{1},A_{1};x_{2},A_{2}}}\bigg{(}\sum_{x_{1}\in
A_{1}^{C}}\sum_{x_{2}\in
A_{2}^{C}}M_{x_{1},A_{1};x_{2},A_{2}}\bigg{)}=\nu(I_{\tau_{1}}\times
I_{\tau_{2}})$

∎

####  5.5.4 Proof of Claim 4

###### Proof.

I have already shown that $\mu\geq 0$. First, fix $x_{1}\in B_{1}$ and observe
that

$\displaystyle\sum_{x_{2}\in
X_{2}}\sum_{B_{2}\supseteq\\{x_{2}\\}}(-1)^{|B_{2}|-1}\rho_{2}(x_{2},B_{2}|B_{1},x_{1})=\sum_{i=1}^{|X_{2}|}(-1)^{i-1}\binom{|X_{2}|}{i}=1$

To see this, fix $1\leq i\leq|X_{2}|$ and
$B_{2}=\\{x_{2}^{1},\ldots,x_{2}^{i}\\}$. Then the terms
$\sum_{k=1}^{i}(-1)^{i-1}\rho_{2}(x_{2}^{k},B_{2}|B_{1},x_{1})=(-1)^{i-1}$
appear exactly once in the sum above, and there are $\binom{|X_{2}|}{i}$ menus
of size $i$. Thus,

$\displaystyle\sum_{\succ_{1}\in P_{1}}\sum_{\succ_{2}\in
P_{2}}\mu(\succ_{1},\succ_{2})=\sum_{\tau_{1}\in\pi(X_{1})}\sum_{\tau_{2}\in\pi(X_{2})}\nu\big{(}I_{\tau_{1}}\times
I_{\tau_{2}}\big{)}=\sum_{x_{1}\in X_{1}}\sum_{x_{2}\in
X_{2}}M_{x_{1},X_{1}\backslash\\{x_{1}\\};x_{2},X_{2}\backslash\\{x_{2}\\}}$
$\displaystyle=\sum_{x_{1}\in X_{1}}\sum_{x_{2}\in
X_{2}}\sum_{B_{2}\supseteq\\{x_{2}\\}}\sum_{B_{1}\supseteq\\{x_{1}\\}}(-1)^{|B_{1}|-1+|B_{2}|-1}\rho_{2}(x_{2},B_{2}|B_{1},x_{1})\rho_{1}(x_{1},B_{1})$
$\displaystyle=\sum_{x_{1}\in
X_{1}}\sum_{B_{1}\supseteq\\{x_{1}\\}}\bigg{(}\sum_{x_{2}\in
X_{2}}\sum_{B_{2}\supseteq\\{x_{2}\\}}(-1)^{|B_{2}|-1}\rho_{2}(x_{2},B_{2}|B_{1},x_{1})\bigg{)}(-1)^{|B_{1}|-1}\rho_{1}(x_{1},B_{1})$
$\displaystyle=\sum_{x_{1}\in
X_{1}}\sum_{B_{1}\supseteq\\{x_{1}\\}}(-1)^{|B_{1}|-1}\rho_{1}(x_{1},B_{1})=1$

where the last equality follows for the same reason as above. ∎

####  5.5.5 Proof of Claim 5

###### Proof.

I will use induction.

Base case: for any $|X_{1}|$-sequence $(x_{1}^{1},\ldots,x_{1}^{|X_{1}|})$ and
$|X_{2}|$-sequence $(x_{2}^{1},\ldots,x_{2}^{|X_{2}|})$, let $\succ_{t}$ be
the (unique) preference satisfying
$x_{t}^{1}\succ_{t}\cdots\succ_{t}x_{t}^{|X_{t}|}$. Then, by definition,

$\displaystyle\mu\big{(}I_{(x_{1}^{1},\ldots,x_{1}^{|X_{1}|})}\times
I_{(x_{2}^{1},\ldots,x_{2}^{|X_{2}|})}\big{)}=\mu\big{(}\\{\succ_{1},\succ_{2}\\}\big{)}=\nu\big{(}I_{(x_{1}^{1},\ldots,x_{1}^{|X_{1}|})}\times
I_{(x_{2}^{1},\ldots,x_{2}^{|X_{2}|})}\big{)}$

First inductive step: suppose
$\mu\big{(}I_{(x_{i}^{1},\ldots,x_{i}^{|X_{i}|})}\times
I_{(x_{j}^{1},\ldots,x_{j}^{k})}\big{)}=\nu\big{(}I_{(x_{i}^{1},\ldots,x_{i}^{|X_{i}|})}\times
I_{(x_{j}^{1},\ldots,x_{j}^{k})}\big{)}$ for all $|X_{i}|$-sequences
$(x_{i}^{1},\ldots,x_{i}^{|X_{i}|})$ and $k$-sequences
$(x_{j}^{1},\ldots,x_{j}^{k})$, where $k>1$. Then, for any $|X_{i}|$-sequence
$(x_{i}^{1},\ldots,x_{i}^{|X_{i}|})$ and $k-1$-sequence
$(x_{j}^{1},\ldots,x_{j}^{k-1})$,

$\displaystyle\mu\big{(}I_{(x_{i}^{1},\ldots,x_{i}^{|X_{i}|})}\times
I_{(x_{j}^{1},\ldots,x_{j}^{k-1})}\big{)}=\sum_{y_{j}\in\\{x_{j}^{1},\ldots,x_{j}^{k-1}\\}^{C}}\mu\big{(}I_{(x_{i}^{1},\ldots,x_{i}^{|X_{i}|})}\times
I_{(x_{j}^{1},\ldots,x_{j}^{k-1},y_{j})}\big{)}$
$\displaystyle=\sum_{y_{j}\in\\{x_{j}^{1},\ldots,x_{j}^{k-1}\\}^{C}}\nu\big{(}I_{(x_{i}^{1},\ldots,x_{i}^{|X_{i}|})}\times
I_{(x_{j}^{1},\ldots,x_{j}^{k-1},y_{j})}\big{)}=\nu\big{(}I_{(x_{i}^{1},\ldots,x_{i}^{|X_{i}|})}\times
I_{(x_{j}^{1},\ldots,x_{j}^{k-1})}\big{)}$

where the first equality follows from $\mu$ being a probability measure, the
second equality follows from the inductive hypothesis, and the third equality
follows because $p_{2}(|X_{1}|,k-1)$ and $p_{2}(k-1,|X_{2}|)$ hold, by Claim
3.

Second inductive step: suppose
$\mu\big{(}I_{(x_{1}^{1},\ldots,x_{1}^{k})}\times
I_{(x_{2}^{1},\ldots,x_{2}^{\ell})}\big{)}=\nu\big{(}I_{(x_{1}^{1},\ldots,x_{1}^{k})}\times
I_{(x_{2}^{1},\ldots,x_{2}^{\ell})}\big{)}$ for all $k$-sequences
$(x_{1}^{1},\ldots,x_{1}^{k})$ and $\ell$-sequences
$(x_{2}^{1},\ldots,x_{2}^{\ell})$, where $k,\ell>1$. Then for any
$k-1$-sequence $(x_{1}^{1},\ldots,x_{1}^{k-1})$ and $\ell-1$-sequence
$(x_{2}^{1},\ldots,x_{2}^{\ell-1})$,

$\displaystyle\mu\big{(}I_{(x_{1}^{1},\ldots,x_{1}^{k-1})}\times
I_{(x_{2}^{1},\ldots,x_{2}^{\ell-1})}\big{)}=\sum_{y_{1}\in\\{x_{1}^{1},\ldots,x_{1}^{k-1}\\}^{C}}\sum_{y_{2}\in\\{x_{2}^{1},\ldots,x_{2}^{\ell-1}\\}^{C}}\mu\big{(}I_{(x_{1}^{1},\ldots,x_{1}^{k-1},y_{1})}\times
I_{(x_{2}^{1},\ldots,x_{2}^{\ell-1},y_{2})}\big{)}$
$\displaystyle=\sum_{y_{1}\in\\{x_{1}^{1},\ldots,x_{1}^{k-1}\\}^{C}}\sum_{y_{2}\in\\{x_{2}^{1},\ldots,x_{2}^{\ell-1}\\}^{C}}\nu\big{(}I_{(x_{1}^{1},\ldots,x_{1}^{k-1},y_{1})}\times
I_{(x_{2}^{1},\ldots,x_{2}^{\ell-1},y_{2})}\big{)}=\nu\big{(}I_{(x_{1}^{1},\ldots,x_{1}^{k-1})}\times
I_{(x_{2}^{1},\ldots,x_{2}^{\ell-1})}\big{)}$

Since every pair of cylinders in $\mathcal{I}_{1}\times\mathcal{I}_{2}$ is
induced by a pair of $k,\ell$-sequences with $1\leq k\leq|X_{1}|$ and
$1\leq\ell\leq|X_{2}|$, I have shown that $\mu=\nu$ on
$\mathcal{I}_{1}\times\mathcal{I}_{2}$. ∎

Now that each Claim has been verifed, the proof is complete. ∎

###  5.6 Proof of Proposition 4 (Backwards Direction)

###### Proof.

Suppose $|X_{t}|=3$ and denote $X_{1}=\\{a,b,c\\}$ and $X_{2}=\\{d,e,f\\}$.
Suppose $\boldsymbol{\rho}^{2}$ satisfies stochastic regularity and marginal
consistency. Then in particular, for any $x_{t}\in X_{t}$,
$A_{t}=\\{x_{t}\\}^{C}=\\{y_{t},z_{t}\\}$, and $B_{t}=X_{t}$,

$\displaystyle\frac{\rho_{1}(y_{1},A_{1})}{\rho_{1}(y_{1},B_{1})}\geq\frac{\rho_{2}(y_{2},A_{2}|B_{1},y_{1})-\rho_{2}(y_{2},B_{2}|B_{1},y_{1})}{\rho_{2}(y_{2},A_{2}|A_{1},y_{1})-\rho_{2}(y_{2},B_{2}|A_{1},y_{1})}$
$\displaystyle\iff\rho_{2}(y_{2},z_{2}|\\{y_{1},z_{1}\\},y_{1})\rho_{1}(y_{1},z_{1})-\rho_{2}(y_{2},X_{2}|\\{y_{1},z_{1}\\},y_{1})\rho_{1}(y_{1},z_{1})$
$\displaystyle-\bigg{(}\rho_{2}(y_{2},z_{2}|X_{1},y_{1})\rho_{1}(y_{1},X_{1})-\rho_{2}(y_{2},X_{2}|X_{1},y_{1})\rho_{1}(y_{1},X_{1})\bigg{)}\geq
0$ $\displaystyle\iff
M_{y_{1},\\{x_{1}\\};y_{2},\\{x_{2}\\}}=\sum_{B_{2}\supseteq\\{y_{2},z_{2}\\}}\sum_{B_{1}\supseteq\\{y_{1},z_{1}\\}}(-1)^{|B_{1}|+|B_{2}|-4}\rho_{2}(y_{2},B_{2}|B_{1},y_{1})\rho_{1}(y_{1},B_{1})\geq
0$

Since marginal consistency is satisfied, Proposition 2 and Claim 1 are
satisfied, so for any $i,j\in\\{1,2\\}$

$\displaystyle
M_{x_{i},A_{i};y_{j},\\{x_{j}\\}}+M_{x_{i},A_{i};z_{j},\\{x_{j}\\}}=M_{x_{i},A_{i};x_{j};\emptyset}$
$\displaystyle
M_{x_{i},A_{i};x_{j},\\{y_{j},z_{j}\\}}=M_{x_{i},A_{i};y_{j},\\{z_{j}\\}}+M_{x_{i},A_{i};z_{j},\\{y_{j}\\}}$

which implies that stochastic Block-Marschak nonnegativity is satisfied. Thus,
by Theorem 1, $\boldsymbol{\rho}^{2}$ has a SU representation. Of course, we
can also directly verify this by defining

$\displaystyle\mu(x_{1}y_{1}z_{1},x_{2}y_{2}z_{2}):=M_{y_{1},\\{x_{1}\\};y_{2},\\{x_{2}\\}}\geq
0$

It is then straightforward to verify

$\displaystyle\mu(x_{1}y_{1}z_{1}):=\sum_{\tau_{2}\in\pi(X_{2})}\mu(x_{1}y_{1}z_{1},\tau_{2})=\rho_{1}(y_{1},z_{1})-\rho_{1}(y_{1},X_{1})$
$\displaystyle\implies\sum_{\tau_{1}\in\pi(X_{1})}\sum_{\tau_{2}\in\pi(X_{2})}\mu(\tau_{1},\tau_{2})=1$

so $\mu$ is indeed a probability measure. Furthermore,

$\displaystyle E(x_{i},A_{i};y_{j},\\{x_{j}\\})\cup
E(x_{i},A_{i};z_{j},\\{x_{j}\\})=E(x_{i},A_{i};x_{j};\emptyset)$
$\displaystyle
E(x_{i},A_{i};x_{j},\\{y_{j},z_{j}\\})=E(x_{i},A_{i};y_{j},\\{z_{j}\\})\cup
E(x_{i},A_{i};z_{j},\\{y_{j}\\})$

and these unions are disjoint, so by Proposition 3 it follows that $\mu$ is a
SU representation. Finally, let $\mu^{\prime}$ be a SU representation. Then by
Proposition 3,

$\displaystyle\mu^{\prime}(x_{1}y_{1}z_{1},x_{2}y_{2}z_{2})=\mu^{\prime}\big{(}E(y_{1},\\{x_{1}\\};y_{2},\\{x_{2}\\})\big{)}=M_{y_{1},\\{x_{1}\\};y_{2},\\{x_{2}\\}}=\mu(x_{1}y_{1}z_{1},x_{2}y_{2}z_{2})$

so $\mu$ is unique.

∎

###  5.7 Proof of Corollary 1

Define $\nu$ as in the proof of Theorem 1. Since this definition is recursive
and the base case is a joint Block-Marschak sum, it immediately follows that
$\nu$ is strictly positive on $\mathcal{I}_{1}\times\mathcal{I}_{2}$, so $\mu$
is strictly positive on $P_{1}\times P_{2}$. All other parts of the proof of
Theorem 1 still hold, so we conclude that $\mu$ is a SU representation of
$(\rho_{1},\\{\rho_{2}(\cdot|h)\\}_{h\in\mathcal{H}})$ with full support.

## References

  * Sen (1971) Amartya K Sen.  Choice functions and revealed preference.  _The Review of Economic Studies_ , 38(3):307–317, 1971.
  * Block et al. (1959) Henry David Block, Jacob Marschak, et al.  Random orderings and stochastic theories of response.  Technical report, Cowles Foundation for Research in Economics, Yale University, 1959.
  * Falmagne (1978) Jean-Claude Falmagne.  A representation theorem for finite random scale systems.  _Journal of Mathematical Psychology_ , 18(1):52–72, 1978.
  * Frick et al. (2019) Mira Frick, Ryota Iijima, and Tomasz Strzalecki.  Dynamic random utility.  _Econometrica_ , 87(6):1941–2002, 2019.
  * Strzalecki (2021) Tomasz Strzalecki.  _Stochastic Choice_.  2021\.
  * Chambers and Echenique (2016) Christopher P Chambers and Federico Echenique.  _Revealed preference theory_ , volume 56.  Cambridge University Press, 2016.
  * Fishburn (1998) Peter C Fishburn.  Stochastic utility.  _Handbook of utility theory_ , 1:273–318, 1998.
  * Van Lint et al. (2001) Jacobus Hendricus Van Lint, Richard Michael Wilson, and Richard Michael Wilson.  _A course in combinatorics_.  Cambridge university press, 2001.
  * Godsil (2018) Chris Godsil.  An introduction to the moebius function.  _arXiv preprint arXiv:1803.06664_ , 2018.


††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Fixing the rotating-wave approximation for strongly-detuned quantum
oscillators

Jan Košata  Anina Leuch  Tobias Kästli  Institute for Theoretical Physics, ETH
Zürich, 8093 Zürich, Switzerland  Oded Zilberberg  Institute for Theoretical
Physics, ETH Zürich, 8093 Zürich, Switzerland  Department of Physics,
University of Konstanz, 78464 Konstanz, Germany

###### Abstract

Periodically-driven oscillators are commonly described in a frame co-rotating
with the drive and using the rotating-wave approximation (RWA). This
description, however, is known to induce errors for off-resonant driving. Here
we show that the standard quantum description, using creation and annihilation
of particles with the oscillator’s natural frequency, necessarily leads to
incorrect results when combined with the RWA. We demonstrate this on the
simple harmonic oscillator and present an alternative operator basis which
reconciles the RWA with off-resonant driving. The approach is also applicable
to more complex models, where it accounts for known discrepancies. As an
example, we demonstrate the advantage of our scheme on the driven quantum
Duffing oscillator.

††preprint: APS/123-QED

Nature displays many degrees of freedom that are coupled to one another. For
simplicity, we describe them as linear resonators that drive each other as
they oscillate with a bare harmonic around their respective equilibrium
points. Hence, finding the response of a system to an applied drive is an
essential task in all areas of science, ranging from fundamental effects in
classical and quantum physics to applications in spectroscopy and diagnostics
of our daily lives.

The bare harmonic translates to a drive that is periodic in time with a given
frequency. To describe the system’s response to the drive, a common approach
involves moving to a rotating frame, where the system’s behavior appears
stationary on short timescales. In the rotating frame, the so-called rotating-
wave approximation (RWA) is used, whereby any remaining rapidly oscillating
terms are dropped. This procedure becomes less precise when the system and
drive frequencies are far detuned from one another, i.e., when the neglected
terms are less rapid. Nevertheless, the RWA is a standard approach to find the
response of driven systems [1, 2], such as nanomechanical resonators [3, 4,
5], optical cavities [6, 7, 8, 9, 10, 11, 12], phononic and magnonic modes
[13, 14, 15, 16, 17, 18, 19] and superconducting junctions [20, 21, 22].
Similarly, it is a common starting point for Floquet engineering and
perturbative expansions of nonlinear driven systems [23, 24, 25, 26, 27].

Despite its ubiquity, the RWA is known to induce errors at large detuning
[28]. There the neglected oscillating terms become significant and must be
included as perturbations that affect the system’s response [29, 30, 31, 32,
33]. Yet, such a perturbative treatment becomes very tasking in many cases
that rely on detuned driving, such as optomechanical cooling [34, 35] and the
formation of frequency combs [36, 37, 38, 39, 40]. These cases are therefore
commonly treated using the RWA. The issue is particularly salient in the case
of the simple quantum harmonic oscillator, where the solution obtained with
the RWA disagrees with the exactly soluble classical limit.

In this work, we pinpoint the source for the quantum-to-classical discrepancy
in the driven harmonic oscillator and correct for it. Specifically, we show
the description of the oscillator in terms of its bare excitations is ill-
suited for the RWA. Instead, we propose a non-standard operator basis (frame)
in which the RWA yields the correct solution for the driven case. We
furthermore demonstrate that this basis choice is also beneficial for
describing driven nonlinear systems, as it significantly improves the fidelity
of the RWA in the driven Duffing oscillator at large detuning. We expect that
our approach will lead to a significant improvement in the description of
numerous scenarios where the response of an oscillator to a coherent drive is
calculated.

The Hamiltonian of a periodically driven harmonic oscillator reads

$H=p^{2}/2m+m\omega_{0}^{2}x^{2}/2-F_{0}\cos(\omega t)x\,,$ (1)

with position $x$, its conjugate momentum $p$, mass $m$, and resonance
frequency $\omega_{0}$. The driving force has amplitude $F_{0}$ and oscillates
at frequency $\omega$. The time evolution of the system is given by Hamilton’s
equations of motion (EOMs),

$\dot{p}=-m\omega_{0}^{2}x+F_{0}\cos(\omega t)\,,\quad\dot{x}=p/m\,.$ (2)

These EOMs can be solved exactly by transforming to the Fourier domain, where
the driving term appears as a Dirac delta function. As a result, the
stationary response of the oscillator to the drive reads 111The effect of
initial conditions disappears at sufficiently long times due to dissipation,
which we do not consider explicitly here. The results thus correspond to the
limit of infinitesimal dissipation and $t\rightarrow\infty$.

$x=X\cos(\omega t)\,,\quad p=-m\omega X\sin(\omega t)\,,$ (3)

with $X=F_{0}/\left[m(\omega_{0}^{2}-\omega^{2})\right]$. We can parameterize
the oscillator’s phase space by $x$ and $p/m\omega_{0}$. The trajectory
described by Eq. (3) is then an ellipse with vertices at $X$ and
$X\omega/\omega_{0}$, see Fig. 1(a).

Within the framework of quantum mechanics, the Hamiltonian (1) is an operator,
written in terms of a pair of non-commuting operators $\hat{x}$ and $\hat{p}$.
To find the oscillator’s response in the quantum realm, one commonly
introduces raising and lowering operators $a$ and $a^{\dagger}$ that
diagonalize the time-independent part of $H$,

$\hat{x}=\sqrt{\frac{\hbar}{2m\omega_{0}}}(a^{\dagger}+a)\,,\quad\hat{p}=i\sqrt{\frac{\hbar
m\omega_{0}}{2}}(a^{\dagger}-a)\,,$ (4)

resulting in

$H=\hbar\left[\omega_{0}(a^{\dagger}a+1/2)-F_{a}(e^{i\omega t}+e^{-i\omega
t})(a^{\dagger}+a)\right]\,,$ (5)

with $F_{a}=F_{0}/(2\sqrt{2m\omega_{0}\hbar})$, see Fig. 1(b). The driving
renders the Hamiltonian (5) time-dependent and several methods exist to find
the resulting response. For example, one approach involves time-dependent
perturbation theory [42], where we move to the interaction picture using the
transformation $V(t)\equiv e^{-ia^{\dagger}a\,\omega_{0}t}$, and
diagrammatically expand the remaining interaction Hamiltonian $H_{\rm int}$.
While this approach is possible for the harmonic oscillator, it rapidly
becomes infeasible when additional terms are included in $H$. Hence, in many
cases only linear response is considered, e.g., using the input-output method
[43].

Another common approach involves the rotating-wave approximation (RWA). Here,
we aim at removing the oscillatory time-dependence altogether. To do so, we
first use the unitary transformation $U_{a}(t)=e^{-i\omega ta^{\dagger}a}$. In
the resulting rotating frame, we obtain the (still time-dependent) effective
Hamiltonian

$\displaystyle\tilde{H}$ $\displaystyle\equiv U_{a}^{\dagger}HU_{a}-i\hbar
U_{a}^{\dagger}\dot{U}_{a}$ (6)
$\displaystyle=\hbar\left\\{-\Delta\tilde{a}^{\dagger}\tilde{a}-F_{a}\left[\tilde{a}\left(1+e^{-2i\omega
t}\right)+\tilde{a}^{\dagger}\left(1+e^{2i\omega t}\right)\right]\right\\}\,,$

where $\tilde{a}\equiv U_{a}(t)^{\dagger}aU_{a}(t)$ denotes the operator $a$
in the rotating frame, and $\Delta=\omega-\omega_{0}$ is the so-called
detuning away from resonance.

In the rotating frame, the remaining time-dependent terms are far-detuned
since $2\omega\gg\absolutevalue{\Delta}$ and therefore elicit negligible
response. Hence, we can apply the RWA and remove all of these from the
Hamiltonian (6) to obtain $\tilde{H}_{\rm RWA}$. We may then find the response
by solving Heisenberg’s EOM,
$i\frac{d}{dt}\langle\tilde{a}\rangle\equiv\langle{[\tilde{a},\tilde{H}_{\rm
RWA}]}\rangle/\hbar=-\Delta\langle\tilde{a}\rangle-F_{a}$, and obtain the
stationary solution $\langle\tilde{a}\rangle=-F_{a}/\Delta$. Then, by
inverting the transformation $U_{a}(t)$ we have
$\expectationvalue{a}=\langle\tilde{a}\rangle e^{-i\omega t}$. An analogous
EOM is similarly solved for $\langle\tilde{a}^{\dagger}\rangle$. Combining the
two solutions [cf. Eq. (4)], we obtain in the classical limit

$x_{\text{RWA}}(t)=-\frac{F_{0}/m}{2\omega_{0}\Delta}\cos(\omega t)\,.$ (7)

Figure 1: (a) Response amplitude [cf. Eq. (3)] of the driven harmonic
oscillator as a function of the driving frequency $\omega$ for the exact
solution (solid, blue) and the standard RWA solution (dashed, red) with
$F_{0}=m=\omega_{0}=1$. The inset shows the corresponding phase space plots
for large detuning, $\omega=0.5$. Spheres indicate solutions in the rotating
frame. The ansatz of stationary $\langle\tilde{a}\rangle$ (large red dashed
circle) does not match the exact result. For this, an additional time-
dependent term in the solution is needed (small red circle). This term
oscillates at $2\omega$, which results in an elliptical path. (b) Energy
quantization of the harmonic oscillator Hamiltonian (6) with energy spacing
$\hbar\omega_{0}$ using the operators $a,\,a^{\dagger}$ defined in Eq. (4).
(c) With the operators $b,b^{\dagger}$, the harmonic oscillator is not
diagonalized, showing a level spacing of
$\hbar(\omega_{0}^{2}+\omega^{2})/2\omega$ and processes which create or
annihilate two particles with amplitude
$\hbar(\omega_{0}^{2}-\omega^{2})/4\omega$.

Crucially, the response amplitude found with the RWA does not match the
analytical solution in Eq. (3), see Fig. 1(a). Indeed, the ratio

$\frac{x(t)}{x_{\text{RWA}}(t)}=\frac{2\omega_{0}\Delta}{\omega^{2}-\omega_{0}^{2}}=\frac{2\omega_{0}}{\omega_{0}+\omega}\,,$
(8)

only approaches unity for $\Delta\rightarrow 0$, which demonstrates the
limitation of the RWA to near-resonant drives. A comparison between the exact
and RWA results in phase space is also illustrative. To this end, we take a
stationary amplitude
$\langle\tilde{a}\rangle\equiv\absolutevalue{\langle\tilde{a}\rangle}e^{i\theta}$,
and using Eq. (4), we find the equivalent trajectory in phase space,
$x(t)=2\absolutevalue{\langle\tilde{a}\rangle}\sqrt{\frac{\hbar}{2m\omega_{0}}}\cos(\omega
t-\theta)$ and $p=-2\absolutevalue{\langle\tilde{a}\rangle}\sqrt{\frac{\hbar
m\omega_{0}}{2}}\sin(\omega t-\theta)$. The resulting path is always circular,
unlike the elliptical path of the exact solution, cf. Eq. (3) and Fig. 1(a).
Indeed, the dropped time-dependent oscillating terms, dubbed micromotion, add
on to the circular path to produce the full elliptical motion, cf. small
circle in Fig. 1(a). This implies that to describe the elliptical path,
$\langle\tilde{a}\rangle$ must be time-dependent, which cannot be obtained
only by time-independent corrections from a high-frequency expansion [23, 24,
25, 26, 27]. In other words, the assertion of a stationary
$\langle\tilde{a}\rangle$ violates Hamilton’s EOMs, specifically the relation
$p=m\dot{x}$.

As the driven quantum harmonic oscillator is exactly solvable, the RWA was not
necessary in the aforementioned analysis. Without the approximation, we would
still have obtained the exact result. However, when the system becomes
nonlinear, its EOMs are usually not exactly solvable. The RWA, being
effectively the lowest order of a perturbative expansion, is then a common
method to find the stationary response. As we have seen, however, it neglects
crucial corrections due to the detuned drive frequency. We identify that this
systemic mismatch stems from the usual starting point (4), which relies on
operators that diagonalize the bare oscillator Hamiltonian and describes
excitations with energies $\hbar\omega_{0}$, see Fig. 1(b).

We now turn to our scheme for an exact treatment of the driven harmonic
oscillator. We seek operators akin to $a$ and $a^{\dagger}$ which recover the
ellipsoidal phase-space path corresponding to drive frequency $\omega$. A
natural choice is to replace $\omega_{0}$ by $\omega$ in Eq. (4), i.e., to
define new operators $b$ and $b^{\dagger}$ via

$\hat{x}=\sqrt{\frac{\hbar}{2m\omega}}(b^{\dagger}+b)\,,\quad\hat{p}=i\sqrt{\frac{\hbar
m\omega}{2}}(b^{\dagger}-b)\,,$ (9)

which satisfy $[b,b^{\dagger}]=1$, and describe the system in terms of
excitations with the drive’s frequency $\omega$. The driven harmonic
oscillator Hamiltonian (1) now reads

$\displaystyle H=\hbar\bigg{\\{}$
$\displaystyle\frac{\omega_{0}^{2}+\omega^{2}}{2\omega}\left(b^{\dagger}b+\frac{1}{2}\right)+\frac{\omega_{0}^{2}-\omega^{2}}{4\omega}\big{(}b^{2}+(b^{\dagger})^{2}\big{)}$
$\displaystyle-F_{b}(e^{i\omega t}+e^{-i\omega
t})(b^{\dagger}+b)\bigg{\\}}\,,$ (10)

with $F_{b}=F_{0}/(2\sqrt{2m\omega\hbar})$, see Fig. 1(c). Note that we
effectively rotated from the $a$ to the $b$ operators using a unitary
(Bogoliubov) transformation [44]. With the new operators, Eq. (10) contains
squeezing terms that do not conserve the particle number, since $b$ and
$b^{\dagger}$ do not diagonalize the bare harmonic oscillator unless
$\omega=\omega_{0}$.

We repeat the same procedure in the new basis. We first apply the unitary
transformation $U_{b}(t)=e^{-i\omega tb^{\dagger}b}$ to move to a rotating
frame, and then obtain the corresponding Heisenberg’s EOMs

$i\frac{d}{dt}\langle\tilde{b}\rangle=\frac{\omega_{0}^{2}-\omega^{2}}{2\omega}\left(\langle\tilde{b}\rangle+\langle\tilde{b}^{\dagger}\rangle
e^{2i\omega t}\right)-F_{b}\left(1+e^{2i\omega t}\right)\,.$ (11)

Crucially, when we now search for a stationary solution for
$\langle\tilde{b}\rangle$, we obtain

$\langle\tilde{b}\rangle=\frac{2F_{b}\,\omega}{\omega_{0}^{2}-\omega^{2}}\,,$
(12)

which upon inverting $U_{b}(t)$ matches the exact result in Eq. (3).
Importantly, the correct result is obtained regardless of whether we take the
RWA or not. In other words, when the operators $b,b^{\dagger}$ are used, the
stationary solution in the rotating frame is exact and satisfies Eq. (11).
This is the main result of this work.

Example: Duffing oscillator.— The RWA is a common starting point for dealing
with time-dependent systems that are not exactly solvable. Our result (12)
implies that a suitable choice of operators made prior to applying the RWA can
significantly reduce errors at large detuning. To demonstrate this, we compare
the results obtained using the RWA with the two operator definitions,
$a,a^{\dagger}$ and $b,b^{\dagger}$. We consider a Duffing oscillator,
described by the Hamiltonian

$H_{D}=p^{2}/2m+m\omega_{0}^{2}x^{2}/2+\alpha x^{4}/4-F_{0}\cos(\omega t)x\,,$
(13)

where $\alpha$ is the Duffing nonlinearity. We plug both operator definitions
[Eqs. (4) and (9)] into Eq. (13), move to a rotating frame [cf. Eq. (6)], and
apply the RWA in both procedures. Note that the RWA drops multiple oscillating
terms that describe frequency conversion processes due to the Duffing
nonlinearity. We furthermore apply a semiclassical mean-field ansatz 222This
allows us to take
$\langle\tilde{a}^{3}\rangle\rightarrow\langle\tilde{a}\rangle^{3}$ etc. to
obtain Heisenberg’s EOMs

$\displaystyle i\frac{d}{dt}\langle\tilde{a}\rangle$
$\displaystyle=-\Delta\langle\tilde{a}\rangle-\frac{F_{a}}{\hbar}+\frac{3\alpha\hbar}{4m^{2}\omega_{0}^{2}}\left(\langle\tilde{a}\rangle+\langle\tilde{a}\rangle^{2}\langle{\tilde{a}^{\dagger}}\rangle\right)\,,$
(14) $\displaystyle i\frac{d}{dt}\langle\tilde{b}\rangle$
$\displaystyle=\frac{\omega_{0}^{2}-\omega^{2}}{2\omega}\langle\tilde{b}\rangle-\frac{F_{b}}{\hbar}+\frac{3\alpha\hbar}{4m^{2}\omega^{2}}\left(\langle\tilde{b}\rangle+\langle\tilde{b}\rangle^{2}\langle{\tilde{b}^{\dagger}}\rangle\right)\,.$
(15)

We can now search for stationary solutions in either of the rotating frames
[$\frac{d}{dt}\langle\tilde{a}\rangle=0$ or
$\frac{d}{dt}\langle\tilde{b}\rangle=0$]. Both Eqs. (14) and (15) generate a
cubic polynomial condition which has up to three solutions [46, 4, 47], see
Fig. 2(a). Both approaches produce the expected tail-shaped response, i.e., a
single solution in the $\omega<\omega_{0}$ regime that bends up towards a
high-amplitude solution and a coexistence region in the $\omega>\omega_{0}$
regime, where both low- and high-amplitude solutions appear. The coexistence
region manifests mathematically as a bifurcation point for the roots of the
polynomial condition. Despite qualitative agreement, the solutions in the two
frames differ quantitatively, i.e., in their amplitudes as a function of
detuning and in the positions of their bifurcation points.

Figure 2: A comparison of the Duffing oscillator response obtained using the
$a,a^{\dagger}$ and $b,b^{\dagger}$ operator bases [cf. Eqs (4), (9)] with a
time-dependent simulation. The parameters used are $m=\omega_{0}=\alpha=1$.
(a) The response amplitude $\absolutevalue{x_{\omega}}$ plotted against the
drive frequency $\omega$ for $F_{0}=0.2$ using $a,a^{\dagger}$ (dashed, red)
and $b,b^{\dagger}$ (solid, blue). The time-dependent simulation result (dot-
dashed grey, behind blue) consists of adiabatic up and downsweeps of $\omega$.
(b), (c) The relative discrepancy in log scale of the $a,a^{\dagger}$ (dashed,
red) and the $b,b^{\dagger}$ (solid, blue) results from the time-dependent
simulation. The circles and crosses in (a) and (c) represent the jump
frequency in the downsweep for operators $a$ and $b$. (d) A phase diagram as a
function of $\omega$ and $F_{0}$. Blue (red) fill denotes regions with one
(two) stable solution(s). (e) Relative discrepancy
$\delta_{\omega}=\absolutevalue{(\omega-\omega_{\text{RWA}})\,/\,\omega}$ in
log scale of the phase boundaries obtained using $a,a^{\dagger}$ (dashed, red)
and $b,b^{\dagger}$ (solid, blue) from the time-dependent result. The
numerical approach is exact up to $10^{-5}$ which becomes visible at low
$F_{0}$, where the discrepancy of our scheme is of that order. The green
dotted lines in (d) and (e) indicate where the simulation of (a) takes place.

To find out which of the two approximate procedures describes the driven
Duffing oscillator (13) more accurately, we compare their predictions with a
“numerical experiment” in the classical picture. Specifically, we numerically
evolve Hamilton’s equations for the Hamiltonian (13) until we reach stationary
motion for different detunings and initial conditions and then adiabatically
sweep the detuning to explore the solution landscape. Note that we add an
infinitesimal dissipation term to enforce convergence in the simulation, which
negligibly shifts the stationary outcome. Additionally, the numerical time-
trace describes the oscillatory response at the drive frequency $\omega$
including high harmonic generation. Hence, the relevant benchmark for the RWA
solutions is the Fourier component $x_{\omega}$ at frequency $\omega$ of the
time trace. In Figs. 2(b) and (c), we plot the relative discrepancy
$\delta_{x}=\absolutevalue{(x_{\omega}-x_{\omega,\text{RWA}})\,/\,x_{\omega}}$
between the numerically obtained $x_{\omega}$ and the two RWA results in the
high- and low-amplitude solution branches, respectively. We observe that our
new RWA approach agrees much better for all values of $\omega$. In the low-
amplitude regimes, the system responds quasi-linearly and our scheme performs
better as discussed above in the harmonic oscillator case. Slight deviations
are seen in the high-amplitude regime, where nonlinear effects dominate the
physics.

In driven-dissipative systems, phase diagrams depict the number or type of
stationary solutions as a function of system parameters [48, 49]. Here, we
focus on the location of the bifurcation point as a function of detuning and
drive amplitude. The bifurcation point is marked in Figs. 2(a) and (c), where
the better performance of our RWA approach manifests as a clear change in its
position. In Fig. 2(d), we draw phase diagrams obtained by the two RWA
approaches, and observe a distinct difference in the phase boundaries. The
exact numerical solution matches our scheme much better, see Fig. 2(e).

Conclusion.— In summary, we have presented an operator basis for periodically-
driven systems which anticipates a response at the driving frequency and is
thus better suited for using the RWA. For both the harmonic and the Duffing
oscillators, this basis choice significantly improves the fidelity of the RWA
while retaining its simplicity. Our approach is applicable to systems in many
areas of physics beyond mechanical oscillators described by $x$ and $p$ [3].
In quantum optics [43] and optomechanics [50], a driven cavity mode is
described by Eq. (1) with the magnetic and electric fields playing the roles
of $x$ and $p$. For models with inherent detuning such as optomechanical
cooling and frequency-comb generation, our approach may result in significant
numerical corrections. A specific case of interest are models coupling a
cavity to one or more two-level systems [51, 52]; here the so-called general
RWA is used, where a basis diagonalizing the spin-cavity interaction reduces
the RWA error [53]. As general RWA uses the standard operators $a$ and
$a^{\dagger}$, we expect that it can be improved further by using our scheme.
Detuned driven systems also appear in circuit QED where the magnetic flux and
charge appear in place of $x$ and $p$ [20]. Overall, a sizable body of work
exists elaborating on the errors due to the RWA at large detuning [54, 55, 56,
57]. Our work suggests that in many cases, these errors can be dramatically
reduced by choosing an appropriate operator basis.

###### Acknowledgements.

The authors would like to thank T. L. Heugel, J. del Pino, A. Grimm, A.
Eichler and G. Blatter for valuable discussions. This work was supported by
the Swiss National Science Foundation through grant CRSII5 1771981. O.Z.
acknowledges funding through SNSF grant PP00P2_190078 and from the Deutsche
Forschungsgemeinschaft (DFG) - project number 449653034.

## References

  * Koppenhöfer  _et al._ [2018] M. Koppenhöfer, C. Bruder, and N. Lörch, Unraveling nonclassicality in the optomechanical instability, Phys. Rev. A 97, 063812 (2018).
  * Lörch  _et al._ [2019] N. Lörch, Y. Zhang, C. Bruder, and M. I. Dykman, Quantum state preparation for coupled period tripling oscillators, Phys. Rev. Research 1, 023023 (2019).
  * Bachtold  _et al._ [2022] A. Bachtold, J. Moser, and M. I. Dykman, Mesoscopic physics of nanomechanical systems, arXiv:2202.01819 (2022).
  * Heugel  _et al._ [2019] T. L. Heugel, M. Biondi, O. Zilberberg, and R. Chitra, Quantum transducer using a parametric driven-dissipative phase transition, Phys. Rev. Lett. 123, 173601 (2019).
  * Rocheleau  _et al._ [2010] T. Rocheleau, T. Ndukum, C. Macklin, J. Hertzberg, A. Clerk, and K. Schwab, Preparation and detection of a mechanical resonator near the ground state of motion, Nature 463, 72 (2010).
  * Muñoz de las Heras  _et al._ [2021] A. Muñoz de las Heras, R. Franchi, S. Biasi, M. Ghulinyan, L. Pavesi, and I. Carusotto, Nonlinearity-Induced Reciprocity Breaking in a Single Nonmagnetic Taiji Resonator, Phys. Rev. Applied 15, 054044 (2021).
  * Rota  _et al._ [2019] R. Rota, F. Minganti, C. Ciuti, and V. Savona, Quantum critical regime in a quadratically driven nonlinear photonic lattice, Phys. Rev. Lett. 122, 110405 (2019).
  * Quach  _et al._ [2022] J. Q. Quach, K. E. McGhee, L. Ganzer, D. M. Rouse, B. W. Lovett, E. M. Gauger, J. Keeling, G. Cerullo, D. G. Lidzey, and T. Virgili, Superabsorption in an organic microcavity: Toward a quantum battery, Science Advances 8, eabk3160 (2022).
  * Ferri  _et al._ [2021] F. Ferri, R. Rosa-Medina, F. Finger, N. Dogra, M. Soriente, O. Zilberberg, T. Donner, and T. Esslinger, Emerging dissipative phases in a superradiant quantum gas with tunable decay, Phys. Rev. X 11, 041046 (2021).
  * Soriente  _et al._ [2021] M. Soriente, T. L. Heugel, K. Arimitsu, R. Chitra, and O. Zilberberg, Distinctive class of dissipation-induced phase transitions and their universal characteristics, Phys. Rev. Research 3, 023100 (2021).
  * Soriente  _et al._ [2020] M. Soriente, R. Chitra, and O. Zilberberg, Distinguishing phases using the dynamical response of driven-dissipative light-matter systems, Phys. Rev. A 101, 023823 (2020).
  * del Pino  _et al._ [2016] J. del Pino, F. J. Garcia-Vidal, and J. Feist, Exploiting vibrational strong coupling to make an optical parametric oscillator out of a Raman laser, Phys. Rev. Lett. 117, 277401 (2016).
  * Xu  _et al._ [2021] Y. Xu, J.-Y. Liu, W. Liu, and Y.-F. Xiao, Nonreciprocal phonon laser in a spinning microwave magnomechanical system, Phys. Rev. A 103, 053501 (2021).
  * del Pino  _et al._ [2021] J. del Pino, J. J. Slim, and E. Verhagen, Non-Hermitian chiral phononics through optomechanically-induced squeezing, arXiv:2110.14710 (2021).
  * Li  _et al._ [2021] J. Li, Y.-P. Wang, W.-J. Wu, S.-Y. Zhu, and J. Q. You, Quantum network with magnonic and mechanical nodes, PRX Quantum 2, 040344 (2021).
  * Qi and Jing [2021] S.-f. Qi and J. Jing, Magnon-assisted photon-phonon conversion in the presence of structured environments, Phys. Rev. A 103, 043704 (2021).
  * Gonzalez-Ballestero  _et al._ [2022] C. Gonzalez-Ballestero, T. van der Sar, and O. Romero-Isart, Towards a quantum interface between spin waves and paramagnetic spin baths, Phys. Rev. B 105, 075410 (2022).
  * Fukami  _et al._ [2021] M. Fukami, D. R. Candido, D. D. Awschalom, and M. E. Flatté, Opportunities for long-range magnon-mediated entanglement of spin qubits via on- and off-resonant coupling, PRX Quantum 2, 040314 (2021).
  * Marsh  _et al._ [2021] B. P. Marsh, Y. Guo, R. M. Kroeze, S. Gopalakrishnan, S. Ganguli, J. Keeling, and B. L. Lev, Enhancing associative memory recall and storage capacity using confocal cavity qed, Phys. Rev. X 11, 021048 (2021).
  * Blais  _et al._ [2021] A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021).
  * Gu  _et al._ [2017] X. Gu, A. F. Kockum, A. Miranowicz, Y.-x. Liu, and F. Nori, Microwave photonics with superconducting quantum circuits, Physics Reports 718, 1 (2017).
  * Xiang  _et al._ [2013] Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems, Rev. Mod. Phys. 85, 623 (2013).
  * Mikami  _et al._ [2016] T. Mikami, S. Kitamura, K. Yasuda, N. Tsuji, T. Oka, and H. Aoki, Brillouin-Wigner theory for high-frequency expansion in periodically driven systems: Application to Floquet topological insulators, Physical Review B 93, 144307 (2016).
  * Eckardt [2017] A. Eckardt, Colloquium: Atomic quantum gases in periodically driven optical lattices, Rev. Mod. Phys. 89, 011004 (2017).
  * Eckardt and Anisimovas [2015] A. Eckardt and E. Anisimovas, High-frequency approximation for periodically driven quantum systems from a Floquet-space perspective, New Journal of Physics 17 (2015).
  * Bukov  _et al._ [2015] M. Bukov, L. D’Alessio, and A. Polkovnikov, Universal high-frequency behavior of periodically driven systems: From dynamical stabilization to Floquet engineering, Advances in Physics 64, 139 (2015).
  * Goldman and Dalibard [2014] N. Goldman and J. Dalibard, Periodically driven quantum systems: Effective hamiltonians and engineered gauge fields, Phys. Rev. X 4, 031027 (2014).
  * Ann  _et al._ [2021] B.-m. Ann, W. Kessels, and G. A. Steele, Sideband transitions in a two-mode josephson circuit driven beyond the rotating-wave approximation, Phys. Rev. Research 3, 033004 (2021).
  * Zheng  _et al._ [2008] H. Zheng, S. Y. Zhu, and M. S. Zubairy, Quantum Zeno and anti-Zeno effects: without the rotating-wave approximation, Phys. Rev. Lett. 101, 200404 (2008).
  * Gan and Zheng [2010] C. Gan and H. Zheng, Dynamics of a two-level system coupled to a quantum oscillator: transformed rotating-wave approximation, The European Physical Journal D 59, 473 (2010).
  * Zhang and Chen [2015] Y.-Y. Zhang and Q.-H. Chen, Generalized rotating-wave approximation for the two-qubit quantum Rabi model, Phys. Rev. A 91, 013814 (2015).
  * Zeuch  _et al._ [2020] D. Zeuch, F. Hassler, J. J. Slim, and D. P. DiVincenzo, Exact rotating wave approximation, Annals of Physics 423, 168327 (2020).
  * Zueco  _et al._ [2009] D. Zueco, G. M. Reuther, S. Kohler, and P. Hänggi, Qubit-oscillator dynamics in the dispersive regime: Analytical theory beyond the rotating-wave approximation, Phys. Rev. A 80, 033846 (2009).
  * Liu  _et al._ [2013] Y.-C. Liu, Y.-W. Hu, C. W. Wong, and Y.-F. Xiao, Review of cavity optomechanical cooling, Chinese Physics B 22, 114213 (2013).
  * Marquardt  _et al._ [2008] F. Marquardt, A. Clerk, and S. Girvin, Quantum theory of optomechanical cooling, Journal of Modern Optics 55, 3329 (2008).
  * Weng  _et al._ [2022] W. Weng, J. He, A. Kaszubowska-Anandarajah, P. M. Anandarajah, and T. J. Kippenberg, Microresonator dissipative Kerr solitons synchronized to an optoelectronic oscillator, Phys. Rev. Applied 17, 024030 (2022).
  * Herr  _et al._ [2012] T. Herr, K. Hartinger, J. Riemensberger, C. Wang, E. Gavartin, R. Holzwarth, M. Gorodetsky, and T. Kippenberg, Universal formation dynamics and noise of Kerr-frequency combs in microresonators, Nature photonics 6, 480 (2012).
  * Chembo [2016] Y. K. Chembo, Kerr optical frequency combs: theory, applications and perspectives, Nanophotonics 5, 214 (2016).
  * Lugiato and Lefever [1987] L. A. Lugiato and R. Lefever, Spatial dissipative structures in passive optical systems, Phys. Rev. Lett. 58, 2209 (1987).
  * Lugiato  _et al._ [2018] L. Lugiato, F. Prati, M. Gorodetsky, and T. Kippenberg, From the Lugiato–Lefever equation to microresonator-based soliton Kerr frequency combs, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, 20180113 (2018).
  * Note [1] The effect of initial conditions disappears at sufficiently long times due to dissipation, which we do not consider explicitly here. The results thus correspond to the limit of infinitesimal dissipation and $t\rightarrow\infty$.
  * Sakurai and Commins [1995] J. J. Sakurai and E. D. Commins, Modern quantum mechanics, revised edition (1995).
  * Walls and Milburn [2007] D. F. Walls and G. J. Milburn,  _Quantum Optics_ (Springer Science & Business Media, 2007).
  * wen Xiao [2009] M. wen Xiao, Theory of transformation for the diagonalization of quadratic Hamiltonians, arXiv:0908.0787 (2009).
  * Note [2] This allows us to take $\langle\tilde{a}^{3}\rangle\rightarrow\langle\tilde{a}\rangle^{3}$ etc.
  * Košata  _et al._ [2022] J. Košata, J. del Pino, T. L. Heugel, and O. Zilberberg, HarmonicBalance.jl: A Julia suite for nonlinear dynamics using harmonic balance, arXiv:2202.00571 (2022).
  * Lifshitz and Cross [2008] R. Lifshitz and M. C. Cross, Nonlinear dynamics of nanomechanical and micromechanical resonators, in  _Reviews of Nonlinear Dynamics and Complexity_ (John Wiley & Sons, Ltd, 2008) Chap. 1, pp. 1–52.
  * Soriente  _et al._ [2018] M. Soriente, T. Donner, R. Chitra, and O. Zilberberg, Dissipation-induced anomalous multicritical phenomena, Phys. Rev. Lett. 120, 183603 (2018).
  * Chitra and Zilberberg [2015] R. Chitra and O. Zilberberg, Dynamical many-body phases of the parametrically driven, dissipative Dicke model, Phys. Rev. A 92, 023815 (2015).
  * Aspelmeyer  _et al._ [2014] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, Rev. Mod. Phys. 86, 1391 (2014).
  * Shore and Knight [1993] B. W. Shore and P. L. Knight, The Jaynes-Cummings model, Journal of Modern Optics 40, 1195 (1993).
  * Kirton  _et al._ [2019] P. Kirton, M. M. Roses, J. Keeling, and E. G. Dalla Torre, Introduction to the Dicke model: From equilibrium to nonequilibrium, and vice versa, Advanced Quantum Technologies 2, 1800043 (2019).
  * Irish [2007] E. K. Irish, Generalized rotating-wave approximation for arbitrarily large coupling, Phys. Rev. Lett. 99, 173601 (2007).
  * Baker  _et al._ [2018] B. Baker, A. C. Y. Li, N. Irons, N. Earnest, and J. Koch, Adaptive rotating-wave approximation for driven open quantum systems, Phys. Rev. A 98, 052111 (2018).
  * Niemczyk  _et al._ [2010] T. Niemczyk, F. Deppe, H. Huebl, E. Menzel, F. Hocke, M. Schwarz, J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, _et al._ , Circuit quantum electrodynamics in the ultrastrong-coupling regime, Nature Physics 6, 772 (2010).
  * Bishop  _et al._ [2010] L. S. Bishop, E. Ginossar, and S. M. Girvin, Response of the strongly driven Jaynes-Cummings oscillator, Phys. Rev. Lett. 105, 100505 (2010).
  * Sornborger  _et al._ [2004] A. T. Sornborger, A. N. Cleland, and M. R. Geller, Superconducting phase qubit coupled to a nanomechanical resonator: Beyond the rotating-wave approximation, Phys. Rev. A 70, 052315 (2004).