# Can One Hear the Shape of a Decision Problem?

Mark Whitmeyer Arizona State University. Email<EMAIL_ADDRESS>I
thank Wenhao Li, Joseph Whitmeyer, Tom Wiseman, and Kun Zhang for their
comments.

###### Abstract

We explore the connection between an agent’s decision problem and her ranking
of information structures. We find that a finite amount of ordinal data on the
agent’s ranking of experiments is enough to identify her (finite) set of
undominated actions (up to relabeling and duplication) and the beliefs
rendering each such action optimal. An additional smattering of cardinal data,
comparing the relative value to the agent of finitely many pairs of
experiments, identifies her utility function up to an action-independent
payoff.

###### keywords:

Expected Utility, Value of Information, Rationalizability

JEL Classifications: C72; D81; D82; D83

##  1 Motivation

To what extent does an agent’s ranking of information structures specify the
decision problem she faces? We know she is a Bayesian expected-utility
maximizer and that her conception of the world has finitely many states. We,
furthermore, know her prior (though this is inessential) and that only
finitely many of her available actions are ever optimal; viz., are
undominated. What we do not know is her utility function itself.

The goal of this work is to understand the connection between an agent’s value
for information and her decision problem. Obviously, knowing an agent’s
decision problem and prior fully specifies her value for information: her
(expected) utility gain from observing any Blackwell experiment (Blackwell
(1951), Blackwell (1953)) can be computed with ease. Any two experiments can,
therefore, be ranked. Here, we study the inverse problem: suppose we know a
finite collection of the agent’s comparative ranks of experiments. Does that
pin down the agent’s utility function?

The three main results of this paper are two positive ones, with a negative
one sandwiched in between. In the first, Lemma 3.2, we discover that a finite
collection of ranked experiments identifies the agent’s undominated actions
(up to relabeling and duplication) as well as the regions of beliefs on which
each such action is optimal. The second result, Proposition 3.5, reveals an
important indeterminacy about the agent’s utility function that cannot be
resolved with only ordinal rankings of experiments. The third result, Theorem
3.7, states that an additional finite collection of utility differences
between experiments (the difference in utils from observing one experiment
versus another) identifies the agent’s utility function up to a decision-
irrelevant payoff.

These results show that an agent’s preferences for information reveal a
significant amount about her decision problem. A finite amount of ordinal
information alone is enough to completely characterize the agent’s behavior
from observing any information structure. An additional finite number of
utility differences yields the agent’s exact value for information, for any
comparison of information structures. The rationale behind these results is
that expected utility maximization imposes a significant amount of structure
on how an agent values information.

As two-player (simultaneous-move) games, fixing the behavior of one player,
are decision problems from the perspective of the other player, we also find
that a finite collection of statements about the intersections of best-
response sets is enough to characterize a two-player game, up to inessential
details. That is, such a finite collection identifies each player’s set of
rationalizable actions (up to duplication and relabeling), as well as the sets
of beliefs rationalizing each such action. This, in turn, pins down the Nash
equilibria of the game.

##  2 The Question(s) at Hand

There is a decision-maker (DM) encountering a static decision problem, which
we define to be a triple $\left(\Theta,A,u\right)$ consisting of a finite set
of possible states $\Theta$, a nonempty set of actions $A$, and a bounded
utility function $u\colon A\times\Theta\to\mathbb{R}$. We, the observers, know
$\Theta$ and the DM’s full support prior
$\mu\in\Delta\left(\Theta\right)\equiv\Delta$. We also know that the DM is a
Bayesian expected-utility maximizer whose utility function is bounded, only
finitely many of her actions are undominated, and no two undominated actions
have identical payoffs in every state.111Alternatively, we resign ourselves to
being unable to distinguish between actions that are exact duplicates. This is
a genericity requirement.

Furthermore, we also have a finite collection of the DM’s rankings of signals.
That is, there is a finite collection of Blackwell experiments
$\pi_{1},\dots,\pi_{m}$ and rankings of the form $\pi_{i}\succ\pi_{j}$ and
$\pi_{k}\sim\pi_{l}$ ($i,j,k,l\in\left\\{1,\dots,m\right\\}$); meaning the DM
strictly prefers experiment $\pi_{i}$ to $\pi_{j}$ and is indifferent between
experiments $\pi_{k}$ and $\pi_{l}$, respectively.

Identifying the decision problem: is this data–the finite collection of ranked
experiments, the prior, that only finitely many actions are undominated,
etc.–enough to pin down the decision problem up to an action-irrelevant payoff
(and to relabeling of the actions)?

If this data is insufficient, will a finite supplement of cardinal information
suffice in the following sense? For a Blackwell experiment $\pi$ and a prior
$\mu$, let $X$ be the random posterior produced by Bayesian updating, which is
distributed according to $F=B(\mu,\pi)$ (where $B$ is the Bayes map).222We
borrow this notation from Denti, Marinacci, and Rustichini (2022a) and Denti,
Marinacci, and Rustichini (2022b). Then, let $W(\pi)\in\mathbb{R}$ denote the
DM’s Expected payoff in her decision problem from first observing the outcome
of experiment $\pi$ then taking an optimal action:

$W(\pi)=\mathbb{E}_{F}\max_{a\in
A}\mathbb{E}_{x}u\left(a,\theta\right)\text{.}$

Suppose we also have a finite collection of Blackwell experiments
$\pi_{1},\dots,\pi_{m}$, real numbers $\zeta_{1},\dots,\zeta_{k}$, and
equalities $W(\pi_{i})=W(\pi_{j})+\zeta_{l}$. Note that we do not know $W$,
this data consists of statements “the DM’s utility gain from $\pi_{j}$ to
$\pi_{i}$ is $\zeta_{l}$ utils.” Now can we identify the state-dependent
utility up to a decision-irrelevant random payoff (and to relabeling of the
actions)?

###  2.1 Recasting the Question

Given the set of states $\Theta$, a set of actions $A$, and a bounded state-
dependent utility function $u\colon A\times\Theta\to\mathbb{R}$, the DM’s
value function in belief $x\in\Delta$ is

$V\left(x\right)\coloneqq\max_{a\in
A}\mathbb{E}_{x}u\left(a,\theta\right)\text{.}$

There are $n\geq 2$ states ($\left|\Theta\right|=n$), so
$\Delta\subset\mathbb{R}^{n-1}$.

An action $a\in A$ is undominated if there exists a belief $x\in\Delta$ such
that

$\mathbb{E}_{x}u\left(a,\theta\right)>\mathbb{E}_{x}u\left(b,\theta\right),\
\text{for all}\ b\in A\setminus\left\\{a\right\\}\text{.}$

As only finitely many actions are undominated, $V$ is the maximum of finitely
many hyperplanes, i.e., is convex and piecewise affine.

As noted before, via Bayes’ law, given prior $\mu\in\Delta$ any experiment
$\pi$ corresponds to a random vector $X$ supported on $\Delta$ whose
expectation is the DM’s prior $\mu$. Accordingly, we have a finite collection
($m\in\mathbb{Z}_{++}$) of equalities and strict inequalities of the form
$\mathbb{E}_{F_{i}}V(X_{i})>\mathbb{E}_{F_{j}}V(X_{j})$ and
$\mathbb{E}_{F_{k}}V(X_{k})=\mathbb{E}_{F_{l}}V(X_{l})$
($i,j,k,l\in\left\\{1,\dots,m\right\\}$). We call this a collection of ordered
expectations.

Identifying the decision problem: is this data–the collection of ordered
expectations, and that $V$ is the maximum of finitely many affine
functions–enough to pin down the decision problem? That is, is there such a
collection that is satisfied by a piecewise-affine $V$; and, moreover, if $W$
satisfies the collection, then $W=V+\varphi$, where
$\varphi\colon\Delta\to\mathbb{R}$ is affine? What if we also have a finite
supplement of additional cardinal data of the form
$\mathbb{E}_{F_{i}}V(X_{i})=\mathbb{E}_{F_{j}}V(X_{j})+\zeta_{l}$? Will that
pin down $V+\varphi$?

##  3 Answering the Question

Any such value function $V$ can be projected onto the belief simplex,
$\Delta$, which yields a finite collection $C$ of polytopes of full dimension
$C_{i}$ ($i=1,\dots,t$), where $t$ is the number of undominated actions in
$A$. Formally, for each undominated $a_{i}\in A$
($i=\left\\{1,\dots,t\right\\}$),

$C_{i}\coloneqq\left\\{x\in\Delta\ |\
\mathbb{E}_{x}u\left(a_{i},\theta\right)=V(x)\right\\}\text{.}$

Action $a_{i}$ is optimal for any belief $x\in C_{i}$ and uniquely optimal for
any belief in the relative interior of $C_{i}$ (for any
$x\in\operatorname{int}C_{i}$). The collection $C$ is a regular polyhedral
subdivision (henceforth, just “subdivision”) of $\Delta$. We say that function
$V$ induces subdivision $C$.

######  Definition 3.1.

A collection of ordered expectations identifies subdivision $C$ if there
exists a $V$ that satisfies the collection that induces $C$, and any $V$ that
satisfies the collection of ordered expectations induces $C$.

We can always find a subdivision-identifying collection of ordered
expectations.

######  Lemma 3.2.

For any $C$, there exists a collection of ordered expectations that identifies
it.

###### Proof.

Fix an arbitrary $C$. Observe that for any element $C_{i}\in C$, there exists
a random vector $X_{i}$ with finitely supported distribution $F_{i}$ such that
part of the support of $F_{i}$ is the extreme points of $C_{i}$, i.e.,
$\operatorname*{extreme}C_{i}\subseteq\operatorname{supp}F_{i}$. Take another
$\hat{X}_{i}$, with distribution $\hat{F}_{i}$, where $\hat{F}_{i}$ is a mean-
preserving contraction of $F_{i}$ that is identical to $F_{i}$ except that the
portion of the measure placed on the extreme points of $C_{i}$ is collapsed to
its barycenter, which by construction is in $\operatorname{int}C_{i}$. Then,
$\mathbb{E}_{F_{i}}V(X_{i})=\mathbb{E}_{\hat{F}_{i}}V(\hat{X}_{i})$ if and
only if $V$ is affine on $C_{i}$. Accordingly, for a $C$ with $t$ elements,
$t$ such equalities implies $V$ is affine on each $C_{i}\in C$.

Next, for any distinct $C_{i},C_{j}\in C$ with $\dim\left(C_{i}\cap
C_{j}\right)=n-2$ and any $x^{\prime}\in\operatorname{int}\left(C_{i}\cap
C_{j}\right)$, there exists an $X_{k}$ with finitely supported distribution
$F_{k}$ such that $x^{\prime}\in\operatorname{supp}F_{k}$. Take another
$\hat{X}_{k}$, with distribution $\hat{F}_{k}$, where $\hat{F}_{k}$ is a mean-
preserving spread of $F_{k}$ that is identical to $F_{k}$ except that the
portion of the measure placed on $x^{\prime}$ is split into two points
$x_{1}\in\operatorname{int}C_{i}$ and $x_{2}\in\operatorname{int}C_{j}$. Then,
$\mathbb{E}_{F_{k}}V(X_{k})<\mathbb{E}_{\hat{F}_{k}}V(\hat{X}_{k})$ if and
only if $V$ is not affine on $C_{i}\cup C_{j}$. As there are only finitely
many pairings of distinct $C_{i}$s, finitely many such strict inequalities
imply that $V$ is not affine on any such pair.

As $V$ is convex, any $V$ that satisfies the stated equalities and strict
inequalities must induce $C$. Moreover, for any $C$, there is a $V$ that
induces it (indeed this is the definition of a regular polyhedral
subdivision). ∎

Figures 1 and 2 illustrate the proof of Lemma 3.2. In Figure 1, revealing that
the DM (with prior on the empty-interior green point) is indifferent between
the distribution over posteriors with support on the black dots and the
distribution over posteriors with support on the black point in the blue
region and the purple $x$ identifies that $V$ is affine on the red region. In
Figure 2, revealing that the DM (with prior on the empty-interior green point)
strictly prefers the distribution over posteriors with support on the black
and purple $x$s to the distribution over posteriors with support on the black
dot and the purple $x$ identifies that $V$ is not affine on the red and blue
regions.

Figure 1: Showing $V$ is affine on the red region Figure 2: Showing $V$ is not
affine on the red and blue regions

Next we uncover what the equivalence of value functions means in terms of the
state-dependent utilities. In the next lemma, let value function $\hat{V}$
correspond to utility $\hat{u}$.

######  Lemma 3.3.

$\hat{V}(x)=V(x)+\varphi(x)$ for affine $\varphi$ if and only if the sets of
undominated actions for both $\hat{u}(a,\theta)$ and $u(a,\theta)$ are the
same up to relabeling the actions; and
$\hat{u}(a,\theta)=u(a,\theta)+\gamma(\theta)$ for some
$\gamma\colon\Theta\to\mathbb{R}$ for all undominated actions up to relabeling
the actions.

###### Proof.

$\left(\Rightarrow\right)$ If $\hat{u}(a,\theta)=u(a,\theta)+\gamma(\theta)$,
then

$\hat{V}(x)=\max_{a\in A}\mathbb{E}_{x}\hat{u}\left(a,\theta\right)=\max_{a\in
A}\mathbb{E}_{x}\left[u(a,\theta)+\gamma(\theta)\right]=V(x)+\mathbb{E}_{x}\gamma(\theta)=V(x)+\varphi(x)\text{.}$

$\left(\Leftarrow\right)$ Suppose for the sake of contraposition that there is
no $\gamma$ such that $\hat{u}(a,\theta)=u(a,\theta)+\gamma(\theta)$ for all
undominated actions. If the subdivisions induced by $V$ and $\hat{V}$ are not
the same, we are done (see, e.g., Lemma 2.1 in Whitmeyer (2023)). Now suppose
they are the same, with the same actions corresponding to each element (by
relabeling). By assumption, there exist two undominated actions $a_{1}$ and
$a_{2}$ for which
$\hat{u}(a_{1},\theta)=u(a_{1},\theta)+\gamma_{a_{1}}(\theta)$ and
$\hat{u}(a_{2},\theta)=u(a_{2},\theta)+\gamma_{a_{2}}(\theta)$, where
$\gamma_{a_{1}}\neq\gamma_{a_{2}}$. Thus,

$\mathbb{E}_{x}\gamma_{a_{1}}(\theta)=\varphi_{1}(x)\neq\varphi_{2}(x)=\mathbb{E}_{x}\gamma_{a_{1}}(\theta)\text{,}$

so $\hat{V}(x)=V(x)+\varphi_{1}(x)$ for all $x\in C_{1}$ and
$\hat{V}(x)=V(x)+\varphi_{2}(x)$ for all $x\in C_{2}$. ∎

Now that we know we can find a collection of ordered expectations that
identifies $C$ we turn our attention to identifying $V$. The crucial part of
the proof of the next result is the observation that $C$ pins down $V$ (plus
affine $\varphi$) up to one degree of freedom, but that this last remaining
indeterminacy cannot be eliminated by a collection of ordered expectations.

######  Definition 3.4.

A collection of ordered expectations identifies value function $V$, if there
exists a value function that satisfies the collection, and any $W$ that
satisfies the collection equals $V$ plus some affine function $\varphi$.

######  Proposition 3.5.

For any non-affine $V$, no collection of ordered expectations identifies it.

###### Proof.

Fix an arbitrary $V$. From Lemma 3.2, there exists a collection of ordered
expectations that identifies the $C$ it induces. On any $C_{i}\in C$,
$V(x)=\alpha_{i}\cdot x+\beta_{i}$. It is immediate that if $V$ satisfies a
collection of ordered expectations and induces $C$, then so too does any $\psi
V$ ($\psi>0$). As $V$ is not affine, we are done. ∎

In some sense this result is to be expected. How could purely ordinal data
identify $V$? Nevertheless, a few pieces of cardinal information will do. We
term an equality of the form
$\mathbb{E}_{F_{i}}V(X_{i})=\mathbb{E}_{F_{j}}V(X_{j})+\zeta_{l}$ a utility
difference.

######  Definition 3.6.

A collection of ordered expectations and utility differences identifies value
function $V$, if there exists a value function that satisfies the collection,
and any value function that satisfies the collection equals $V$ plus some
affine function $\varphi$.

######  Theorem 3.7.

For any $V$, there exists a collection of ordered expectations and utility
differences that identifies it.

###### Proof.

Fix an arbitrary $V$ and its induced $C$. Take any distinct $C_{i},C_{j}\in C$
with $\dim\left(C_{i}\cap C_{j}\right)=n-2$. WLOG we may specify $V=0$ for all
$x\in C_{i}$, as our identification is up to the addition of an affine
function. By construction, $V(x)=\psi\left(\alpha\cdot x+\beta\right)$
($\alpha\in\mathbb{R}^{n-1}$, $\beta\in\mathbb{R}$, $\psi\in\mathbb{R}_{++}$)
for all $x\in C_{j}$, with $\alpha\cdot x+\beta\geq 0$ for all $x\in C_{j}$.
Moreover, $\alpha\cdot x+\beta>0$ for all $x\in C_{j}\setminus C_{i}$.

Take two points $x_{i}\in\operatorname{int}C_{i}$ and
$x_{j}\in\operatorname{int}C_{j}$ such that the set $\lambda
x_{i}+\left(1-\lambda\right)x_{j}\in C_{i}\cup C_{j}$ for any
$\lambda\in\left[0,1\right]$. Let
$\hat{x}=\operatorname{int}\operatorname*{conv}\left\\{x_{i},x_{j}\right\\}\cap\operatorname{int}C_{i}$.
WLOG
$\mu\in\operatorname{int}\operatorname*{conv}\left\\{\hat{x},x_{j}\right\\}$.
Then, letting $F$ be the binary distribution with support
$\left\\{x_{i},x_{j}\right\\}$ and $\hat{F}$ be the binary distribution with
support $\left\\{\hat{x},x_{j}\right\\}$,
$\mathbb{E}_{F}V(X)>\mathbb{E}_{\hat{F}}V(\hat{X})$. Define
$\zeta=\mathbb{E}_{F}V(X)-\mathbb{E}_{\hat{F}}V(\hat{X})>0$.

Finally, observe that, letting $p$ and $q$ be defined implicitly as

$px_{j}+(1-p)x_{i}=\mu=qx_{j}+(1-q)\hat{x}\text{,}$

the equation (in $\psi$)

$p\psi\left(\alpha\cdot x_{j}+\beta\right)=q\psi\left(\alpha\cdot
x_{j}+\beta\right)+\zeta$

has a unique solution

$\psi=\frac{\zeta}{(p-q)\left(\alpha\cdot x_{j}+\beta\right)}>0\text{.}$

Consequently, we have identified $V$ on $C_{j}$ via this single equality. As
there are only finitely many pairs of $C_{i}$ and $C_{j}$, we are done. ∎

As identification of $V$ yields the value of information (in utils),
$\mathbb{E}_{F}V-V(\mu)$,

######  Corollary 3.8.

There exists a collection of ordered expectations and utility differences that
identifies the value of information for the DM.

##  4 The Irrelevance of the Prior

Until now we have assumed that we know the DM’s prior. What if we do not? Take
any (full-support) prior $\mu\in\operatorname{int}\Delta$ and belief
$x\in\Delta$, that corresponds to the realization, $s$ of a statistical
experiment. It is well-known that the ratios of conditional probabilities of
that $s$ (conditioned on the states) corresponding to that belief $x$ are
well-defined functions of the prior. Consequently, for any prior
$\mu\in\operatorname{int}\Delta$ and any decision problem whose value function
induces $C$, the extreme points of any $C_{i}\in C$ correspond uniquely to the
ratios of conditional probabilities of a number of columns (one column per
extreme point of $C_{i}$) in a statistical experiment.

To put differently, for any $\mu\in\operatorname{int}\Delta$, value function
$V$ that induces $C$, and any $C_{i}\in C$, the set of extreme points of
$C_{i}$ corresponds uniquely to a family of sub-stochastic matrices,
$\tilde{C}_{i}$, defined as follows. Denoting by $\underline{\Delta}(S)$ the
set of sub-probability measures on a set of signal realizations $S$, and
$x_{s}$ the DM’s posterior after observing signal realization $s$,

$\tilde{C}_{i}\coloneqq\left\\{\tilde{\pi}\colon\Theta\to\underline{\Delta}(S_{i})\colon
x_{s}\in\operatorname*{extreme}(C_{i})\ \forall s\in S_{i},\ \cup_{s\in
S_{i}}\left\\{x_{s}\right\\}=\operatorname*{extreme}(C_{i})\right\\}\text{.}$

By construction, for a decision problem whose value function induces
subdivision $C$, at prior $\mu\in\operatorname{int}\Delta$, the DM is
indifferent between any experiment $\pi$ possessing
$\tilde{\pi}\in\tilde{C}_{i}$ as a submatrix and any experiment
$\tilde{\pi}^{\prime}$ that is the same as $\pi$ except that $\tilde{\pi}$ is
replaced by an arbitrary garbling. We term $\tilde{C}_{i}$ a spectral element,
and the set
$\tilde{C}\coloneqq\left\\{\tilde{C}_{1},\dots,\tilde{C}_{t}\right\\}$ the
spectral subdivision. Importantly, a spectral subdivision is prior-free: it is
a set of families of matrices.

We call a finite collection of Blackwell experiments $\pi_{1},\dots,\pi_{m}$
and rankings of the form $\pi_{i}\succ\pi_{j}$ and $\pi_{k}\sim\pi_{l}$
($i,j,k,l\in\left\\{1,\dots,m\right\\}$) a collection of ranked experiments.

######  Definition 4.1.

A collection of ranked experiments identifies spectral subdivision $\tilde{C}$
if for any prior $\mu\in\operatorname{int}\Delta$ there exists a decision
problem that satisfies the collection that induces $\tilde{C}$, and any
decision problem and prior that satisfies the collection induces $\tilde{C}$.

We immediately deduce a corollary of Lemma 3.2.

######  Corollary 4.2.

For any spectral subdivision $\tilde{C}$, there exists a collection of ranked
experiments that identifies it.

We can also add some cardinal information. Recall that for a Blackwell
experiment $\pi$, $W(\pi)\in\mathbb{R}$ denotes the DM’s Expected payoff in
her decision problem from first observing the outcome of experiment $\pi$ then
taking an optimal action. We term an equality of the form
$W(\pi_{i})=W(\pi_{j})+\zeta_{l}$ a utility difference. Recall from Corollary
3.8 that if we knew $\mu$, we could identify the DM’s value for information.
It turns out, we do not even need to know $\mu$ to price (in utils)
information for the agent. That is, the precise utility gain from any
experiment is identified.

We normalize the value of no information to $0$ so that the value of
information of experiment $\pi$ is merely $W\left(\pi\right)$. Then,

######  Definition 4.3.

A collection of ranked experiments and utility differences identifies the
value of information $W$ if for any prior $\mu\in\operatorname{int}\Delta$
there exists a decision problem such that the value of an experiment $\pi$ is
$W(\pi)$, and any decision problem and prior satisfying the collection has
value of information $W$.

######  Theorem 4.4.

There exists a collection of ranked experiments and utility differences that
identifies the value of information for the agent.

###### Proof.

Take an arbitrary prior $\mu^{\prime}\in\operatorname{int}\Delta$ and a
decision problem that induces value function $V$. By Theorem 3.7 there exists
a collection of ordered expectations and utility differences that identifies
it and, hence, $W$ for this DM. Moreover, take the collections specified in
the proofs of Proposition 3.2 and Theorem 3.7, and observe that the ranked
experiments corresponding to the ordered expectations and the utility
differences identify value function $V^{\mu}$ for each
$\mu\in\operatorname{int}\Delta$. Moreover, by construction, for all such
priors $\mu$, the value of information in the corresponding decision problem
(with value function $V^{\mu}$) is the same and, thus, equal to $W$.

Accordingly, it suffices to show that for an arbitrary prior
$\mu^{\prime}\in\operatorname{int}\Delta$, there exists a collection of ranked
experiments and utility differences that identifies value of information for
the agent. This is Corollary 3.8, so we are done. ∎

##  5 Identifying a Two-Player Game

There is a tight connection between two-player (simultaneous move) games and
decision problems. In such games, the other player takes on the role of
nature, generating any uncertainty by randomizing over his strategies. Given
this, it is easy to construct analogs of the earlier identification results.

Formally, a finite two-player (normal form) game is
$G=\left(\left\\{1,2\right\\},A,u\right)$ consisting of two players ($1$ and
$2$), a finite set of (pure) action profiles $A=A_{1}\times A_{2}$ and a
profile of utilities for the players $u=\left(u_{1},u_{2}\right)$, with
$u_{i}\colon A\to\mathbb{R}$ ($i=1,2$). $\Sigma=\Sigma_{1}\times\Sigma_{2}$
denotes the set of (mixed) strategy profiles, with $\Sigma_{i}=\Delta A_{i}$
($i=1,2$). As above, we stipulate that there are no duplicate actions for
either player. As before, this is an imposition of genericity.

We define player $1$’s best response correspondence, in
$\sigma_{2}\in\Sigma_{2}$, $b_{1}\left(\sigma_{2}\right)$, in the standard
manner:

$b_{1}\left(\sigma_{2}\right)\coloneqq\left\\{a_{1}\in
A_{1}\colon\mathbb{E}_{\sigma_{2}}u\left(a_{1},a_{2}\right)=\max_{a_{1}^{\prime}\in
A_{1}}\mathbb{E}_{\sigma_{2}}u\left(a_{1}^{\prime},a_{2}\right)\right\\}\text{.}$

We define player $2$’s best response correspondence
$b_{2}\left(\sigma_{1}\right)$ analogously. We term a finite collection of the
form $b_{1}\left(\sigma_{2}^{i}\right)\cap
b_{1}(\sigma_{2}^{j})\neq\emptyset$, $b_{2}\left(\sigma_{1}^{i}\right)\cap
b_{2}(\sigma_{1}^{j})\neq\emptyset$, $b_{1}\left(\sigma_{2}^{k}\right)\cap
b_{1}\left(\sigma_{2}^{l}\right)=\emptyset$, and
$b_{2}\left(\sigma_{1}^{k}\right)\cap
b_{2}\left(\sigma_{1}^{l}\right)=\emptyset$ a collection of best-response
comparisons.

As in the decision problem above, $\Sigma_{2}$ is subdivided into a finite
collection $C_{1}$ of polytopes of full dimension $C_{1}^{i}$, where for each
undominated $a_{1}\in A_{1}$,

$C_{1}^{i}\coloneqq\left\\{\sigma_{2}\in\left.\Sigma_{2}\right|a_{1}^{i}\in
b_{1}(\sigma_{2})\right\\}\text{.}$

$\Sigma_{1}$ is likewise subdivided into collection $C_{2}$. We call $C_{i}$
player $i$’s subdivision and say that the game induces subdivision pair
$(C_{1},C_{2})$ if the game is such that player $1$’s subdivision is $C_{1}$
and player $2$’s subdivision is $C_{2}$.

######  Definition 5.1.

A collection of best-response comparisons identifies game $G$ if there exists
a game whose best-response correspondences satisfy the collection, and any
game whose best-response correspondences satisfy the collection induces the
same subdivision pair $(C_{1},C_{2})$.

Mirroring the argument for Lemma 3.2, we have

######  Theorem 5.2.

For any game there exists a collection of best-response comparisons that
identifies it.

###### Proof.

As the proof is virtually identical to that for Lemma 3.2, we merely sketch
it.333In fact it is even easier, as we are no longer constrained by Bayes’
law. Take an arbitrary element $C_{1}^{i}\in C_{1}$. It is a polytope with a
finite set of extreme points (its vertices)
$\left\\{\sigma_{2}^{1},\sigma_{2}^{2},\dots,\sigma_{2}^{k}\right\\}$.
Accordingly, for some $\sigma_{2}^{*}\in\operatorname{int}C_{1}^{i}$ the
collection

$b(\sigma_{2}^{1})\cap b(\sigma_{2}^{2})\cap\dots\cap b(\sigma_{2}^{k})\cap
b(\sigma_{2}^{*})\neq\emptyset$

implies that there is an action that is a best response to any $\sigma_{2}\in
C_{1}^{i}$. There are only finitely many elements in $C_{1}$, so finitely many
more such collections imply that there are actions that are best responses,
respectively, to any $\sigma_{2}\in C_{1}^{i}$ for any $C_{1}^{i}\in C_{1}$.
Finally, for any distinct pair $C_{1}^{i}$ and $C_{1}^{j}$ (of which there are
finitely many) a comparison

$b(\sigma_{2}^{\heartsuit})\cap b(\sigma_{2}^{\diamond})=\emptyset\text{,}$

where $\sigma_{2}^{\heartsuit}\in\operatorname{int}C_{1}^{i}$ and
$\sigma_{2}^{\diamond}\in\operatorname{int}C_{1}^{j}$, distinguishes between
the respective best responses.∎

##  6 Related Work

This paper belongs among those studying the value of information in decision
problems. That literature includes Ramsey (1990), Athey and Levin (2018), and
Azrieli and Lehrer (2008); as well as Radner and Stiglitz (1984), De Lara and
Gilotte (2007), Chade and Schlee (2002), and Whitmeyer (2024), which four all
study the marginal valuation of information. We also make use of the
connection between decision problems and polyhedral subdivisions. Green and
Osband (1991) reveal this connection, and Lambert (2019) and Frongillo and
Kash (2021) make use of it when studying elicitability of various forms of
information.

A number of other works seek to understand what information a DM obtains based
on her choice behavior in a (known) decision problem. Dillenberger, Lleras,
Sadowski, and Takeoka (2014) and Lu (2019) are two such papers. De Oliveira,
Denti, Mihm, and Ozbek (2017) and Caplin and Dean (2015) are two others, both
of which endogenize the DM’s information structure. Libgober (2021) shows how
to identify information structures from a population’s beliefs.

Finally, this work is fundamentally a revealed preference exercise (Samuelson
(1938) and Afriat (1967)): given a choice paradigm what can a set of observed
choices by an agent tell you about the agent (and the model)? More recently,
Caplin and Martin (2021) and Caplin, Martin, and Marx (2023) ask what choice
data reveals about the information a DM could have obtained and Caplin and
Dean (2015) and Chambers, Liu, and Rehbeck (2020) whether a DM’s behavior is
consistent with costly information acquisition. This paper, instead, asks what
preferences over information structures reveal about a DM’s choice problem.

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