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Econ-16 · economics · Economics

Separate identification of beliefs and preferences

LLM status Partially solved unverified

Statement

DT-02 |  Decision theory and identification

Research agenda formalization

Definitions and assumptions

Let $\Omega=\{1,\ldots,m\}$, let $p\in\Delta_m$ be a subjective prior, and let $u_\theta$ belong to a specified increasing utility family on consequences $C$. Fix $c_0<c_1$ and normalize $u_\theta(c_0)=0$, $u_\theta(c_1)=1$ to remove positive-affine utility equivalence. An observed environment $z$ supplies a menu $A(z)$ and known state-contingent consequence map $c(a,s,z)$. Under expected utility, the choice correspondence is \[ \mathcal C_{p,\theta}(z)=\argmax_{a\in A(z)} \sum_{s=1}^{m}p_su_\theta(c(a,s,z)). \] If ties are resolved or choice error is present, its rule or probability law must be specified. Let $\mathcal O(p,\theta)$ denote the population observation law of choices and any elicited belief responses, conditional on the supported environments. Two pairs are observationally equivalent when their $\mathcal O$ values agree.

Formal question

Characterize primitive payoff and menu variation, truthful elicitation assumptions, utility restrictions and exclusion restrictions that make $\mathcal O$ injective in $(p,\theta)$ after normalization. When injectivity fails, characterize the sharp equivalence class and which policy-relevant functionals remain identified. Extend the question to a declared ambiguity-sensitive criterion $V(a;\Pi,\theta)$, keeping the prior-set representation and preference parameters distinct.

Scope and status

An exclusion restriction must specify which experimental or environmental variation changes consequences while leaving beliefs or preferences invariant; merely naming an instrument is insufficient. Objective lotteries can inform utility only under a stated model of their evaluation. Finite deterministic choices generally give inequalities rather than exact recovery of unrestricted continuous parameters. The task is to identify informative designs or model classes, not to assert that unrestricted beliefs and preferences are always separately identifiable. The source motivates this confounding problem but does not pose the displayed map-injectivity question as a numbered conjecture.

References for the source of the problem

Charles F. Manski, Survey Measurement of Probabilistic Macroeconomic Expectations: Progress and Promise, NBER Macroeconomics Annual 32 (2018), 411--471, Section II.A, "Economic Research on Choice under Uncertainty, without Expectations Data," and Section VII.C, "Confounding Beliefs and Preferences." https://www.journals.uchicago.edu/doi/full/10.1086/696061. NBER Working Paper 23418 (2017), https://www.nber.org/papers/w23418.

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.

Statement licence: CC-BY-4.0 · Source file

Contributions 3

Econ-16-A2 · Exploration · LLM-generated · gpt 6 astra pro ·

GPT 6 Astra Pro attempt on Econ-16

Separate identification of beliefs and preferences

D81-16: research attempt and adversarial verification

9 October 2026

Tools. Web browsing: YES. Exact rational computation: YES.

FORMAL RESTATEMENT

Let $\Omega=[m]$, $m\ge1$, $p\in\Delta_m=\{p\in{\mathbb{R}}_+^m:\sum_sp_s=1\}$, and let $\{u_\theta:\theta\in\Theta\}$ be a specified increasing utility family on consequences $C$. Fix $c_0<c_1$ in $C$ and require $u_\theta(c_0)=0$, $u_\theta(c_1)=1$. A supported environment $z\in Z$ provides a nonempty finite menu $A(z)$ and known consequences $c(a,s,z)$. Expected utility is \[ U_{p,\theta}(a,z)=\sum_{s=1}^m p_su_\theta(c(a,s,z)). \] A declared observation mechanism induces a map $\mathcal O$ from $(p,\theta)$ to a population law of observed choices and elicited reports. Identification of the pair means \[ \forall(p,\theta),(p',\theta'):\quad \mathcal O(p,\theta)=\mathcal O(p',\theta') \Longrightarrow (p,\theta)=(p',\theta'). \] Identification of a functional $\psi$ replaces the conclusion by equality of $\psi$.

Literal statement versus a specified theorem. The attachment asks for a characterization across unspecified utility families, menu designs, elicitation mechanisms, and ambiguity criteria. It is a research agenda, not an assertion that all such designs identify the pair. There is no single fixed $\mathcal O$ whose injectivity can be decided without specifying these primitives. A nonidentifying example therefore does not disprove the agenda. The minimal operational specialization used below observes the full deterministic maximizing correspondence at each supported menu, without choice error. Additional exact reports, when present, have a known map $E(p,\theta)$. The lottery and ambiguity theorems declare further assumptions rather than importing them silently.

Stress points. Positive-affine normalization does not remove duplicate parameter labels that represent the same normalized utility. Finite strict choices supply inequalities, not indifference equations. Observing a selected maximizer is weaker than observing the whole maximizing set. An elicited probability is not assumed truthful merely because a survey labels it a belief. Priors may have zero coordinates. Objective lottery evaluation and invariance of beliefs/preferences across interventions are substantive restrictions.

QUICK LITERATURE/CONTEXT CHECK

Charles F. Manski, Survey Measurement of Probabilistic Macroeconomic Expectations: Progress and Promise, NBER Macroeconomics Annual 32 (2018), 411--471, discusses choice without expectations data in Section II.A and confounding beliefs and preferences in Section VII.C. Checked publisher text: https://www.journals.uchicago.edu/doi/10.1086/696061. The latter section also explains why expectations reports require care when beliefs and preferences are not separately represented. It does not state the attachment's map-injectivity question as a numbered theorem or conjecture.

The results below are mathematical statements for explicitly specified submodels. The rich lottery model uses expected utility for objective randomization as an assumption, not as an empirical consequence of the observed choices. No general identification result is attributed to the cited discussion.

ATTACK PLAN

Proof track. Write the sharp observational fibers exactly; design utility-calibrating comparisons independent of beliefs and belief-calibrating comparisons with normalized utilities; identify which ambiguity-set object is separated by all such comparisons.

Disproof track. Use finitely many strict choices to test local identification; construct two different normalized utility/prior pairs with the same entire continuum of certainty comparisons; compare distinct prior sets with the same lower expectation functional.

Tools and roles. Revealed-preference inequalities describe sharp sets; finiteness and continuity create local observational equivalence; explicit power utilities exhibit confounding; normalized binary lotteries calibrate a common cardinal scale; exclusion restrictions hold latent parameters fixed; support-function separation identifies closed convex prior sets; compactness guarantees a separating nearest point; fiber factorization characterizes identified functionals.

Tiny cases. With $m=1$, the prior is fixed at one but utility parameters need not be identified. Menus containing only the reference outcomes reveal no curvature parameter. With two states, even the continuum of menus in Proposition Proposition (A continuum of menus can still confound beliefs and utility) fails to identify either primitive separately.

WORK

Theorem (Sharp deterministic identified set): Suppose the full nonempty maximizing set $B(z)\subseteq A(z)$ is observed for every $z$, and an exact report $e$ has known structural map $E$. Then the sharp set of compatible parameters consists exactly of $(p,\theta)\in\Delta_m\times\Theta$ satisfying $E(p,\theta)=e$ and, for every $z$, \[ \begin{aligned} U_{p,\theta}(a,z)&=U_{p,\theta}(a',z) &&(a,a'\in B(z)),\\ U_{p,\theta}(a,z)&>U_{p,\theta}(b,z) &&(a\in B(z),\ b\in A(z)\setminus B(z)). \end{aligned} \] If no report is observed, omit its equality. A functional is point identified at these data if and only if it is constant on this set; its sharp range is its image of this set.
Proof: Every maximizing action has the same maximum utility. An action outside the full maximizing set has strictly smaller utility, because the menu is finite and includes a maximizer. This proves necessity. Conversely, the displayed equalities and strict inequalities make the full maximizing set exactly $B(z)$ at every menu. Together with the report equation, this reproduces every component of the declared observation. This proves sufficiency and sharpness. Two feasible parameter pairs giving different functional values show nonidentification of that functional. If every feasible pair gives the same value, that value is determined by the data. Taking all feasible functional values gives exactly the compatible range. ∎

Scope of the inequalities. If only one selected action is observed and the tie rule is unobserved, weak inequalities rationalize the selection but do not recover the full-correspondence identified set above. A known tie rule or error distribution must instead enter the observation map explicitly. Similarly, imposing $p=e$ on a reported vector is valid only when truthful, exact elicitation is an assumption of the model.

Proposition (Finite strict choices cannot locally identify continuous parameters): Let $Z$ be finite and all $A(z)$ finite. Suppose all pairwise utility differences are continuous in a neighborhood of a nonisolated parameter point $(p^0,\theta^0)$, and every observed menu has a unique maximizer there. If no additional reports distinguish nearby parameters, there is a relative neighborhood of $(p^0,\theta^0)$ with the same choices. Hence the point is not identified.
Proof: For each observed maximizer and each competing action, its utility advantage at the specified point is strictly positive. There are finitely many such differences. Continuity gives a neighborhood on which each remains positive; their finite intersection is a neighborhood on which every maximizing action is unchanged. By nonisolation it contains a distinct admissible parameter point. In the absence of distinguishing reports that point gives identical observations. If a menu is a singleton, it contributes no comparisons and imposes no restriction. ∎
Without continuity, there is still a global cardinal obstruction for a finite deterministic choice-only design with no additional reports: each finite menu has finitely many possible nonempty maximizing subsets, so the combined observation alphabet is finite. It cannot be the injective image of an infinite parameter set. This does not rule out identification at exceptional points where indifferences are observed, nor does it apply to a continuous-valued observation law generated by a specified stochastic choice model.
Proposition (A continuum of menus can still confound beliefs and utility): Let $m=2$, $C=[0,1]$, $u_\theta(c)=c^\theta$ with $\theta>0$, and prior $(p,1-p)$ with $0<p<1$. For every $c\in[0,1]$, observe the full choice correspondence between the bet $B=(1,0)$ and the sure consequence $c$. This entire continuum of observations identifies exactly \[ t=p^{1/\theta}\in(0,1), \qquad\{(p,\theta):p=t^\theta,\ \theta>0\} \] as the observational fiber. In particular $(p,\theta)=(1/2,1)$ and $(1/4,2)$ are observationally equivalent.
Proof: The bet has expected utility $p$ and the sure consequence has utility $c^\theta$. Strict monotonicity of $c^\theta$ implies that the bet is strictly preferred for $c<t$, both are chosen at $c=t$, and the sure consequence is strictly preferred for $c>t$. Hence $t$ determines every observation. Conversely, different cutoffs yield different choices at any consequence strictly between them, so the observation determines $t$. Solving $t=p^{1/\theta}$ gives the displayed exact fiber. Both stated pairs have cutoff $1/2$, and each utility satisfies the required normalization. As $\theta$ ranges over $(0,\infty)$ with fixed interior $t$, $t^\theta$ ranges over $(0,1)$, so the prior itself is unrestricted within that open interval by this design. ∎
Within this submodel, a functional is identified precisely when it is constant along every relevant cutoff fiber. The cutoff itself is identified; $p$ and $\theta$ are not. This is a counterexample to an unrestricted identification claim, not to the request to characterize informative designs.
Theorem (Separate identification under a declared rich lottery design): Assume $C=[c_0,c_1]$. Objective lotteries are evaluated by the same normalized von Neumann--Morgenstern utility as sure consequences, with their stated probabilities; randomization does not change $p$ or $u_\theta$. Let $L_\lambda$ pay $c_1$ with objective probability $\lambda\in[0,1]$ and $c_0$ otherwise, independently of the state. Suppose the following two-action choice correspondences are observed for all indicated parameters:

(a) every sure consequence $c\in C$ versus $L_\lambda$, for every $\lambda\in[0,1]$;

(b) for each state $s$, the bet $B_s$ paying $c_1$ in state $s$ and $c_0$ otherwise, versus every $L_\lambda$.

Then $p$ and the normalized function $u_\theta$ on $C$ are identified separately, including priors on the boundary of the simplex. The label $\theta$ is identified if and only if the utility family has no observationally duplicate labels, that is, $\theta\mapsto u_\theta|_C$ is injective on the allowed parameter set.

Proof: Normalization and expected utility make $U(L_\lambda)=\lambda$, irrespective of $p$. Monotonicity and the endpoint normalization give $u_\theta(c)\in[0,1]$. In menu (a), the sure consequence is weakly preferred exactly when $u_\theta(c)\ge\lambda$. The full set of such $\lambda$, including its endpoints, is $[0,u_\theta(c)]$, so its supremum recovers $u_\theta(c)$. This works for each $c$ without any continuity or differentiability of $u_\theta$.

The state bet has utility $\sum_t p_t\1\{t=s\}=p_s$. Menu (b) therefore gives the interval $[0,p_s]$ of lottery probabilities for which the bet is weakly preferred. Its supremum recovers $p_s$, including $p_s=0$ or $1$. The same objective lotteries supply a common numerical scale for both steps.

If the family is injectively labeled, the recovered utility function determines $\theta$. If two labels give the same utility function, then for any fixed $p$ every menu in this design has the same utility comparisons under both labels; these labels cannot be distinguished. This proves the stated necessity as well as sufficiency. ∎

Exclusion restriction made explicit. The theorem holds $p$ and the entire utility function fixed across the lottery probabilities, consequence values, and state-bet menus. The interpretation fails if changing a menu changes subjective states, beliefs, preferences, or evaluation of the objective probabilities. No choice observation in the theorem independently proves that invariance.

Theorem (What max--min ambiguity data identify): Let $\varnothing\ne\Pi\subseteq\Delta_m$ and declare the criterion \[ V_{\Pi,\theta}(f)=\inf_{p\in\Pi}\sum_s p_s{\mathbb{E}}_{f(s)}u_\theta(c), \] where each $f(s)$ is a known objective lottery on $C=[c_0,c_1]$. Keep the lottery evaluation and invariance assumptions of Theorem Theorem (Separate identification under a declared rich lottery design). Observe its utility-calibrating menus and, for every $x\in[0,1]^m$, comparisons of the act $f_x(s)=L_{x_s}$ with every constant lottery $L_\lambda$.

These observations identify $u_\theta$ and exactly the closed convex set $K=\overline{{\operatorname{conv}}}(\Pi)$. If $\Pi$ is required to be closed and convex, it is identified itself. Without that representation restriction, observationally equivalent prior sets are exactly those with the same closed convex hull, once the normalized utility is fixed.

Proof: For state-independent consequences and lotteries, taking an infimum over $p$ leaves their utility unchanged since $\sum_sp_s=1$. The utility-calibration proof therefore still applies. For $f_x$ the value is \[ \underline E_\Pi(x)=\inf_{p\in\Pi}p\cdot x\in[0,1]. \] Comparisons with $L_\lambda$ identify this value as the supremum of lottery probabilities weakly below it.

A linear functional has the same infimum over a set and over its convex hull: each convex combination has a value at least the original infimum, and the original set is contained in the hull. Continuity preserves this infimum on taking the closure. Because the simplex is compact, $K$ is nonempty compact convex and $\underline E_\Pi(x)=\min_{p\in K}p\cdot x$.

It remains to show that all the recovered lower expectations determine $K$. First they determine $\min_{p\in K}p\cdot v$ for every $v\in{\mathbb{R}}^m$. If $v$ is constant its value follows from $\sum p_s=1$. Otherwise put $a=\min_sv_s$, $b=\max_sv_s-a>0$, and $x=(v-a\1)/b\in[0,1]^m$. Then \[ \min_{p\in K}p\cdot v=a+b\min_{p\in K}p\cdot x. \] Now suppose two nonempty compact convex sets $K,K'\subseteq\Delta_m$ differ. After interchanging their names if needed, choose $p^0\in K\setminus K'$, and choose $q^0\in K'$ minimizing the Euclidean distance to $p^0$. Compactness gives the minimizer and the distance is positive. Let $h=p^0-q^0$. For $q\in K'$, convexity puts $q^0+t(q-q^0)$ in $K'$ for $0\le t\le1$. Minimality of $q^0$ and the derivative at $t=0$ of its squared distance to $p^0$ yield $h\cdot(q-q^0)\le0$. Thus \[ h\cdot p^0=h\cdot q^0+\|h\|^2>\max_{q\in K'}h\cdot q. \] Taking $v=-h$ gives $\min_{p\in K}p\cdot v\le p^0\cdot v<\min_{q\in K'}q\cdot v$, contradicting equality of all recovered functionals. Hence $K$ is unique. Conversely, equal closed convex hulls give equal infima for every state utility vector and therefore identical choices for all acts under this declared criterion. The remaining identification of a utility parameter label has the same injectivity condition as before. ∎

Explicit ambiguity-set nonidentification. For two states, $\Pi=\{(1,0),(0,1)\}$ and $\Pi'=\Delta_2$ are distinct, but both give $\min(x_1,x_2)$ for every state utility vector $x$. No amount of these max--min choice data can distinguish their raw representations. Their common closed convex hull is identified by Theorem Theorem (What max--min ambiguity data identify). The theorem does not apply without modification to smooth ambiguity, variational penalties, or other undeclared criteria.

VERIFICATION

Exact tests. The cutoff example was checked at the 101 rational consequences $c=k/100$, retaining ties. The same-sign identity also follows algebraically from $1/4-c^2=(1/2-c)(1/2+c)$, whose second factor is strictly positive on $[0,1]$. 
\begin{lstlisting}[language=Python] from fractions import Fraction as F sgn=lambda x:(x>0)-(x<0) for k in range(101):     c=F(k,100)     assert sgn(F(1,2)-c)==sgn(F(1,4)-c*c) for x1 in range(6):     for x2 in range(6):         a,b=F(x1,5),F(x2,5)         # Any mixture of the two state values lies above their minimum.         for k in range(11):             p=F(k,10)             assert p*a+(1-p)*b>=min(a,b) print('101 cutoff comparisons and ambiguity-mixture grid verified exactly') \end{lstlisting}
The continuum results do not rely on extrapolating these finite tests; their proofs establish the comparisons for every admissible consequence and probability.

Adversarial audit. The sharp set uses strict inequalities only because the entire maximizing correspondence is observed. The finite-choice local theorem explicitly excludes extra continuous reports that might identify parameters. The positive-affine normalization is shared across menus, and the utility calibration uses state-independent objective lotteries, not an unproved identification of their probabilities. The prior-set theorem identifies a closed convex hull, not an arbitrary generating set. All compactness is finite dimensional. No statistical consistency, confidence interval, or sampling-rate claim is inferred from population identification.

FINAL

LABEL: UNRESOLVED for the full research agenda as stated.

Strongest fully proved partial result. Exact sharp sets are derived for deterministic full-correspondence observations. Separate identification is proved for the specified rich lottery design, and closed-convex-hull identification is proved for the declared max--min ambiguity criterion. Explicit normalized observationally equivalent pairs disprove unrestricted separate identification even with a continuum of sure-consequence comparisons. These are complete submodel theorems, not a characterization of every unspecified observation mechanism.

Exact first gap. No necessary-and-sufficient characterization in primitive restrictions is proved for an arbitrary supported-menu design, utility family, elicitation/error mechanism, and ambiguity criterion. Outside the specified submodels these inputs remain free, so the tautological requirement that observations separate parameters has not been converted into such a characterization.

Three next targets. First, fix a finite-dimensional utility family and a concrete menu design and characterize when observed indifference equations have a unique global solution, not merely a full-rank local derivative. Second, specify an elicitation measurement law and derive the joint choice/report fiber rather than assuming reports equal true priors. Third, repeat the lower-functional separation analysis for one declared non-max--min ambiguity criterion, identifying its representation equivalences before attempting parameter recovery.

Minimal nonidentification structure. Two states and a one-dimensional normalized utility family already suffice, with positive priors and no choice error. Even one state can leave utility unidentified under poor menu variation. A candidate broader theorem must exclude these designs or identify only functionals constant on their displayed fibers.

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Econ-16-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-16

Separate identification of beliefs and preferences Revealed-preference fibers and an identifying lottery design

Problem ID: D81-16. Status: Unresolved; rigorous restricted results.

Short recap and scope

The target is separate recovery of a normalized utility and a subjective prior from declared choice environments, with an extension to ambiguity. We prove an exact finite-data characterization, an informative experimental design, and the precise object identified by sufficiently rich maxmin choices. These are restricted results, not a general identification theorem for arbitrary elicitation errors or ambiguity criteria.

Manski [manski] motivates the confounding problem. The objective-lottery construction below is an elementary expected-utility identification argument; the multiple-prior representation is the established model of Gilboa and Schmeidler [gs]. The proofs and quantitative design observations below are supplied in full, without a claim of historical novelty.

Finite choices: an exact sharp set

Let $S=\{1,\ldots,m\}$ and let $C$ be a finite ordered consequence set containing $c_0<c_1$. A utility vector $v=(v_c)_{c\in C}$ belongs to a specified set $U$ of increasing vectors with $v_{c_0}=0,v_{c_1}=1$. The data are finitely many finite menus $A_j$, consequence maps $c_j(a,s)\in C$, and selected alternatives $a_j$. Each menu has a known strict priority order $\prec_j$: the first maximizer is selected. Write $D_{ja}(p,v)=\sum_s p_s[v_{c_j(a_j,s)}-v_{c_j(a,s)}]$.
Theorem (Sharp revealed-preference fiber): The set of pairs producing exactly these deterministic observations is \[ F=\{(p,v)\in\Delta_m\times U: D_{ja}\geq0\ \forall j,a,\quad D_{ja}>0\ \text{if }a\prec_j a_j\}. \] For every functional $H$ on the parameter space its sharp identified set is $H(F)$. In particular $H$ is identified at these data exactly when $H$ is constant on nonempty $F$. If $U$ has a finite polynomial inequality description, $F$ also has one, allowing strict inequalities. If all nontrivial comparisons are strict at a parameter in the relative interior of a positive-dimensional parameter domain, that parameter is not locally identified.
Proof: Optimality of $a_j$ gives the weak comparisons. An alternative with earlier priority cannot tie it, giving the strict comparisons. Conversely, the weak comparisons make $a_j$ a maximizer, and the strict comparisons exclude every earlier alternative from the maximizer set. The declared rule therefore chooses $a_j$. Thus every pair in the displayed set is realized and no other pair is; applying $H$ proves sharpness and the identification criterion. The $D_{ja}$ are polynomials of degree at most two, proving the algebraic claim. Finally finitely many strictly positive continuous differences have a positive minimum at the given parameter. Their signs persist in some relative neighborhood. Every parameter there produces the same choices; positive dimension gives distinct nearby parameters. ∎

For example, on $[0,1]$ let $u_\theta(c)=c^\theta$, $\theta>0$, and two states have probabilities $(p,1-p)$. A bet paying $1$ in state 1 and $0$ otherwise is weakly preferred to sure $c$ precisely when $p\geq c^\theta$. Observing the full sure-payment threshold identifies $c^*=p^{1/\theta}$, leaving the exact curve $p=(c^*)^\theta$ when $0<c^*<1$. Even a continuum of these particular menus does not separate $p$ and $\theta$.

A sufficient design, including its invariance assumptions

Now take $C=[c_0,c_1]$. Assume a continuous strictly increasing normalized utility $u:C\to[0,1]$. A state-independent objective lottery pays $c_1$ with known probability $q$ and $c_0$ otherwise. Assume that its randomizer is independent of the subjective state, is trusted, and is evaluated by expected utility with the same $u$ as certain and state-contingent consequences. Beliefs and utility remain fixed across all menus. All binary weak preferences, including ties, are observable.
Theorem (Separation by objective lotteries): For every $c\in C$ observe comparisons of sure $c$ with the objective lottery for every $q\in[0,1]$. For every state $s$ observe comparisons of the state-$s$ bet $(c_1\text{ on }s,c_0\text{ otherwise})$ with these same lotteries. These observations identify the entire normalized $u$ and every coordinate $p_s$, even when some $p_s=0$. If comparisons instead use a grid containing $0,1$ with maximum gap $h$, the threshold for each queried $c$ and each $s$ is contained in an interval of length at most $h$ (or is a detected endpoint). The utility parameter $\theta$ is identified only if $\theta\mapsto u_\theta$ is injective.
Proof: The utility of the objective lottery is $q$. Hence the set of $q$ for which sure $c$ is weakly preferred is $[0,u(c)]$, recovering $u(c)$ as its supremum. The expected utility of the state bet is $p_s$; the corresponding set is $[0,p_s]$. Endpoints and zero probabilities cause no exception. On a grid the largest accepted value and smallest rejected value bracket the threshold; if equality is observed it is exact. Adjacent grid points are at most $h$ apart. Identical utility functions cannot be separated by any expected-utility observations, which proves the last qualification. ∎

For $c$ outside $[c_0,c_1]$ this experiment gives no threshold in $[0,1]$; additional anchors or model restrictions are needed. The theorem does not assume that ordinary survey answers are truthful probability reports.

What maxmin observations can recover

Let a nonempty prior set $\Pi\subseteq\Delta_m$ evaluate utility vectors $x\in[0,1]^m$ by $L_\Pi(x)=\inf_{p\in\Pi}p\cdot x$. Define $K=\overline{\operatorname{co}}(\Pi)$. Rich acts are available because the preceding continuous utility maps $C$ onto $[0,1]$.
Theorem (Identification of the closed convex prior set): Comparisons of every act $x\in[0,1]^m$ with every constant utility $q\in[0,1]$ identify $K$ exactly: \[ K=\{p\in\Delta_m:p\cdot x\geq L_\Pi(x) \text{ for every }x\in[0,1]^m\}. \] Two nonempty prior sets give the same such observations if and only if their closed convex hulls agree. Thus a nonclosed or nonconvex representing set is not itself identified without a representation normalization.
Proof: Constant acts have value $q$, so comparison thresholds recover $L_\Pi$. A linear continuous functional has the same infimum on a set, its convex hull and its closure. Every $p\in K$ consequently satisfies the displayed inequalities. If $p\notin K$, compactness of $K$ gives a nearest point $q_0\in K$. For each $q\in K$, minimizing squared distance along the segment from $q_0$ to $q$ gives $(p-q_0)\cdot(q-q_0)\leq0$ by the right derivative at zero. Put $v=q_0-p$. Then $v\cdot(q-p)\geq\|q_0-p\|^2>0$ for every $q\in K$. This $v$ is not constant across coordinates, since constants have equal inner products with all probability vectors. Let $x=(v-\min_s v_s\mathbf 1)/(\max_s v_s-\min_s v_s)\in[0,1]^m$. Probability vectors sum to one, so the strict separation is preserved: $p\cdot x<\inf_{q\in K}q\cdot x=L_\Pi(x)$. Thus $p$ is excluded. The formula proves uniqueness of $K$; equal hulls conversely give equal lower envelopes and comparisons. ∎

Verification and first unresolved gap

The finite-data theorem incorporates the exact tie rule; replacing its strict comparisons with weak comparisons would incorrectly enlarge the fiber. The lottery design separates the sources of variation only because its objective randomizer leaves subjective state probabilities unchanged. The maxmin result identifies a closed convex set, not every possible representation of ambiguity. The first remaining gap for the catalogue target is a nonparametric characterization for restricted stochastic menus and declared response-error mechanisms without these rich lottery comparisons; no such characterization is proved here.

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Econ-16-A3 · Partial result · LLM-generated · gpt-6 astra, claude opus 5.5, gpt-6 astra pro

Finite experiments separating beliefs, preferences and ambiguity

Finite experiments separating beliefs, preferences and ambiguity

Ji Ho Bae
LawAll AI

October 11, 2026

\begin{abstract} For a declared finite-consequence stochastic-choice model, we give an exact structural criterion for separate identification of beliefs and normalized utility. The sharp observational fiber is an affine slice of an explicitly defined rank-one matrix set. Exactly $m+r-2$ independent log odds suffice, and fewer cannot identify the unrestricted normalized family. A calibrated max--min extension gives the entire compatible convex-prior family, sharp policy-value intervals and a linear-programming description of joint policy values. A vertex criterion determines when the whole convex prior set is identified. The observations are population choice probabilities, rather than finitely many realized choices. \end{abstract}

A declared finite-choice model

Catalogue problem D81-16 requests informative designs or model classes, sharp equivalence classes, identified policy functionals and an ambiguity-sensitive extension [Catalogue]. Manski discusses why beliefs and preferences can be confounded in choices and why elicited expectations need explicit interpretation [Manski, §§ II.A, VII.C].

There are $m\ge1$ states and consequences $c_0<\cdots<c_r$, with $r\ge1$. The utility family is \[ u_0=0<u_1<\cdots<u_r=1, \] where $u_j=u(c_j)$. Only utility on these consequences is a parameter. Beliefs satisfy $p\in\Delta_m$, including boundary priors. Fix ex ante a finite collection of supported environments. Each $z$ supplies a finite nonempty menu $A_z$ and a known consequence label $j_z(a,s)$. Put \[ V_{p,u}(a,z)=\sum_s p_su_{j_z(a,s)},\qquad \pi_z(a)=\frac{\exp(\lambda V_{p,u}(a,z))} {\sum_{b\in A_z}\exp(\lambda V_{p,u}(b,z))}, \quad \lambda>0. \] This is the declared choice-error law, with known common scale $\lambda$. The same fixed pair $(p,u)$ governs all observations; no mixture of unobserved decision-maker types is added. Menus change only the specified acts and consequences. Beliefs, utility and noise scale remain invariant, and environment assignment supplies no parameter-dependent information. Choices precede informative outcome disclosure. All comparisons below can instead be combined in one menu, avoiding learning between menus.

We observe population conditional choice probabilities. A finite experiment means finitely many supported menus; it does not mean finitely many sampled decisions recover continuous parameters exactly. No survey report is presumed truthful. If an exact reported vector $e$ is used, imposing $p=e$ requires an additional truthful-reporting assumption; otherwise its specified response law must be included in the observation model.

Choose a reference action $a_z^0$ in each menu and let \[ N=\sum_z(|A_z|-1),\qquad \ell_{z,a}=\lambda^{-1}\log\frac{\pi_z(a)}{\pi_z(a_z^0)} \quad(a\ne a_z^0). \] Model probabilities are strictly positive. Zero observed probabilities are incompatible with this finite-scale model.

A structural criterion and sharp fibers

Let $W\in\mathbb R^{m\times r}$, with columns indexed by $j=1,\ldots,r$. Define $\mathcal M$ by \begin{align} W_{sj}&\ge0,\qquad \sum_sW_{sr}=1,\label{wsref-0-6e-6f-72-6d-61-6c-69-7a-61-74-69-6f-6e}\\ 0&<\sum_sW_{s1}<\cdots<\sum_sW_{sr}=1,\label{wsref-0-6f-72-64-65-72}\\ W_{sj}&=W_{sr}\sum_tW_{tj}\qquad(j<r).\label{wsref-0-72-61-6e-6b-6f-6e-65} \end{align} For $r=1$, (order) says only $0<1$, and (rankone) is empty. Construct the integer matrix \[ D_{(z,a),(s,j)}=\mathbf1\{j_z(a,s)=j\} -\mathbf1\{j_z(a_z^0,s)=j\},\qquad j\ge1. \]
Theorem: The map $W_{sj}=p_su_j$ is a bijection from the normalized parameter family to $\mathcal M$. Its inverse is \[ p_s=W_{sr},\qquad u_j=\sum_sW_{sj}. \] The sharp equivalence class at observation $\ell$ is \begin{equation}\label{wsref-0-66-69-62-65-72} \mathcal F(\ell)=\{W\in\mathcal M:D\operatorname{vec}W=\ell\}, \end{equation} with parameters recovered by that inverse. The experiment is globally identifying if and only if \begin{equation}\label{wsref-0-73-65-63-61-6e-74} \ker D\cap(\mathcal M-\mathcal M)=\{0\}. \end{equation} There is a terminating exact recognition procedure for this property from the finite consequence maps.
Proof: Generated matrices satisfy (normalization)--(rankone). Conversely their last column is a probability vector, their column sums are a strictly ordered normalized utility, and (rankone) gives their product. This includes zero prior coordinates: the corresponding rows vanish, while column sums still recover utility.

The choice formula gives exactly $D\operatorname{vec}W=\ell$. Conversely this equality recovers every within-menu probability ratio; normalization recovers every probability. Thus every retained matrix realizes the observations, and no other parameter pair does. An empty fiber reports incompatibility.

Two parameter pairs coincide exactly when their matrices coincide. Distinct pairs give equal observations exactly when their difference is a nonzero structured secant in $\ker D$, proving (secant). To recognize the property, test the finite polynomial system \[ W,\widetilde W\in\mathcal M,\qquad D\operatorname{vec}(W-\widetilde W)=0,\qquad \sum_{s,j}(W_{sj}-\widetilde W_{sj})^2>0. \] Real quantifier elimination decides feasibility [BPR, Sections 13.1, 14.2]. A nonempty system defined over the rationals has a real-algebraic sample point, giving an explicit collision. An empty system proves identification. The test uses the strict parameter set, rather than its closure or an unrestricted rank-two relaxation. No polynomial-time bound is claimed for this general recognition procedure. ∎

The criterion depends on primitive consequence variation through $D$ and on utility restrictions through $\mathcal M$. Any declared semialgebraic utility restrictions can be added to (order) throughout. A separate label $\theta$ is identified only if its map to the normalized finite utility vector is injective.

For a policy act on the same consequences, expected utility and its contrasts are known linear functions $L(W)$. Their exact sharp identified set is \[ \{L(W):W\in\mathcal F(\ell)\}. \] This is a semialgebraic image, computable by quantifier elimination when transformed data and target coefficients have an exact rational or real-algebraic representation. General semialgebraic target formulas can be treated in the same way. Global recognition above needs only rational $D$; effective computation at observed data additionally needs this coefficient convention. Arbitrary real probabilities do not automatically provide an effective encoding of their logarithms.

On a nonempty fiber, a linear target is identified exactly when its supremum and infimum agree. Extrema need not be attained because utility inequalities are strict, and the sharp set need not be an interval. For example, let $m=r=2$, write $p=p_1$, $u=u_1$, and observe acts $(c_1,c_0)$ and $(c_r,c_1)$ against constant $c_0$. Transformed values $3/16$ and $13/16$ give \[ pu=3/16,\quad p+(1-p)u=13/16, \qquad (p,u)\in\{(1/4,3/4),(3/4,1/4)\}. \] An endpoint state-one bet then has sharp value set $\{1/4,3/4\}$, while the observed first act has identified value $3/16$. The utility restriction $u>1/2$ selects the first pair. For a nonattained bound, observing $(c_1,c_r)$ against zero with value $1/2$ gives $p(1-u)=1/2$ and the sharp prior set $p\in(1/2,1]$.

The smallest information dimension

Theorem: For the unrestricted normalized finite utility family and known $\lambda$, the smallest number of independent log odds of a globally identifying finite experiment is \[ d=m+r-2. \] If the positive common scale is also unknown, joint identification of $(\lambda,p,u)$ requires and admits $d+1$ such coordinates.
Proof: For each $s<m$, an act paying $c_r$ only in state $s$ has value $p_s$. For each $j<r$, a sure act paying $c_j$ has value $u_j$. Compare these $d$ acts with constant $c_0$, either in $d$ binary menus or one menu with $d+1$ actions. Log odds divided by $\lambda$ recover all these values, and $p_m=1-\sum_{s<m}p_s$. This works at boundary priors too.

For the lower bound we use the following elementary fact. A $C^1$ map $F$ from a nonempty open subset of $\mathbb R^d$ to $\mathbb R^N$, with $N<d$, cannot be injective. Let $\rho$ be its largest derivative rank. If $\rho=0$, the map is constant on a small ball. Otherwise reorder coordinates so a nonzero $\rho$-minor comes first. Near that point rank is constantly $\rho$, and \[ G(x)=(F_1(x),\ldots,F_\rho(x),x_{\rho+1},\ldots,x_d) \] is a local diffeomorphism by the inverse function theorem [Rudin, Theorem 9.24]: $G$ is $C^1$ on an open neighborhood and its derivative there is invertible at the chosen point. In these coordinates the first $\rho$ outputs are the first $\rho$ coordinates. Rank $\rho$ forces every other output derivative in the remaining coordinates to vanish. On a small product box the outputs are independent of those $d-\rho>0$ coordinates, giving a collision.

Strictly positive priors and strictly ordered utilities form a nonempty open set in the $d$ independent coordinates $(p_1,\ldots,p_{m-1},u_1,\ldots,u_{r-1})$. Their log-odds map $D\operatorname{vec}(pu^\top)$ is polynomial. Fewer than $d$ outputs cannot be injective by the fact just proved. If $d=0$, then $m=r=1$, the parameter pair is unique and no informative observation is needed.

For an unknown common scale, add sure $c_r$ against sure $c_0$. Their raw log odds are $\lambda$, after which the previous reconstruction applies. The lower bound applies to the $C^1$ raw-log-odds map $\lambda D\operatorname{vec}(pu^\top)$ on an open domain of dimension $d+1$. This proves joint minimality, including when $d=0$. ∎

Thus full column rank of $D$ is sufficient but unnecessary: the identifying construction uses $d<mr$ observations to recover the structured matrix. The minimum concerns information coordinates, not the number of arbitrary-size menus or sampled decisions.

Calibrated ambiguity and its sharp prior family

We now declare a separate max--min model. Replace the single prior by a nonempty compact convex set $K\subseteq\Delta_m$ and use \[ V_K(a)=\min_{p\in K}\sum_s p_su_{j(a,s)}. \] Choices obey the same logit law applied to these values. The same $K,u,\lambda$ remain fixed across menus under the earlier exclusions. Sure acts against $c_0$ identify every $u_j$ independently of $K$. The endpoint comparison also identifies an unknown common scale if necessary.

After this calibration, observe finitely many acts $a_j$ against constant $c_0$. They may share one anchored menu. Write their known state-utility vectors and observed values as \[ d_j=(u_{j(a_j,1)},\ldots,u_{j(a_j,m)}),\qquad v_j=\lambda^{-1}\log\frac{\pi(a_j)}{\pi(c_0)}. \] For $J\ge1$ such acts, define the compact polytope and its observed faces \begin{equation}\label{wsref-0-66-61-63-65-73} H=\{p\in\Delta_m:p\cdot d_j\ge v_j\ \forall j\},\qquad F_j=H\cap\{p:p\cdot d_j=v_j\}. \end{equation}

Theorem: The calibrated data are compatible if and only if $H$ and every $F_j$ are nonempty. Their entire sharp convex-prior family is \begin{equation}\label{wsref-0-66-61-6d-69-6c-79} \mathcal K(v)=\{K:\varnothing\ne K\subseteq H,\ K\text{ compact convex}, \ K\cap F_j\ne\varnothing\ \forall j\}. \end{equation}
Proof: Every compatible $K$ lies in $H$, and each observed minimum is attained in $K\cap F_j$. Conversely the conditions in (family) make every observed minimum exactly $v_j$, hence reproduce every anchored logit observation. If the faces are nonempty, their witness points' convex hull is compatible, as is $H$ itself. If a required face is empty, no compatible prior set exists. Utility-calibration failures are likewise reported as incompatibility before applying this theorem. ∎

Policy values, joint contrasts and prior-set identification

Theorem: For compatible data with $J\ge1$, the sharp set of lower policy utility $\min_{p\in K}p\cdot h$ is exactly \begin{equation}\label{wsref-0-69-6e-74-65-72-76-61-6c} \left[\min_{p\in H}p\cdot h,\quad \min_j\max_{p\in F_j}p\cdot h\right]. \end{equation} For vectors $h_1,\ldots,h_L$, the entire joint identified set of their lower utilities $w_1,\ldots,w_L$ is the projection of this finite linear system: introduce $p^j\in H$ for each observed face and $q^\ell\in H$ for each policy, and require \begin{align} p^j\cdot d_j&=v_j,& q^\ell\cdot h_\ell&=w_\ell,\label{wsref-0-77-69-74-6e-65-73-73-2d-65-71-75-61-6c-69-74-69-65-73}\\ p^j\cdot h_k&\ge w_k,&q^\ell\cdot h_k&\ge w_k &&\text{for every applicable index}.\label{wsref-0-77-69-74-6e-65-73-73-2d-69-6e-65-71-75-61-6c-69-74-69-65-73} \end{align} This joint set is a compact polytope. Every retained value vector has a witnessing prior set that is the convex hull of at most $J+L$ points. All linear policy contrasts have sharp extrema obtained from this same system; with rational calibrated coefficients and transformed data, this is rational linear programming.
Proof: Containment $K\subseteq H$ gives the lower endpoint of (interval), attained at $K=H$. Since $K$ meets every $F_j$, its minimum under $h$ is at most each face maximum. Choose a maximizer $p_j$ of $h$ on each $F_j$. Their convex hull is compatible and attains the upper endpoint. For any intermediate $w$, choose $p_0\in H$ with $p_0\cdot h=w$, possible by convexity since the upper endpoint is at most $\max_Hp\cdot h$. Adding $p_0$ to those face maximizers gives a compatible convex hull whose minimum is exactly $w$. Thus every interval point is attainable.

For the joint result, any compatible $K$ supplies minimizers $p^j$ for the observed acts and $q^\ell$ for the policies. They satisfy (witness-equalities)--(witness-inequalities). Conversely take the convex hull of all such witnesses. The inequalities bound each policy value below by $w_\ell$, its own witness attains equality, and every original observed face is still met. This gives exactly the requested joint values and proves sharpness.

The witness constraints are linear. Witnesses lie in a simplex, and the policy equalities bound every $w_\ell$ between the smallest and largest coordinates of $h_\ell$. The feasible system is therefore a compact polytope, as is its projection. Optimizing any linear contrast over it gives attained sharp endpoints. A scalar contrast is identified exactly when these endpoints agree. Marginal intervals alone cannot generally be subtracted to obtain a sharp contrast interval. ∎

All scalar extrema in (interval) are ordinary linear programs on compact polytopes. For rational calibrated coefficients and transformed data, these and the joint witness programs have polynomial bit complexity by rational linear programming [BenTalNemirovski, Section 8.4.2]. The coefficients are explicitly encoded rationals and the nonempty feasible polytopes are bounded, so the cited bit-complexity result applies. With arbitrary real coefficients the mathematical characterizations remain valid, while effective computation requires an exact input convention. No polynomial bound for whole-prior-set vertex enumeration is implied.

Proposition: For compatible data with $J\ge1$, the entire convex prior set is identified if and only if every vertex of $H$ is the unique point of some observed face $F_j$. If it is identified, it equals $H$.
Proof: Singleton faces force their points into every compatible $K$. If they force every vertex, convexity gives $H\subseteq K\subseteq H$. Conversely, if a vertex $v$ is not uniquely forced by any face, choose $p_j\in F_j\setminus\{v\}$ for every $j$. Their convex hull is compatible but omits $v$, since an extreme point cannot be a convex combination of other points of $H$. It differs from the compatible set $H$, proving nonidentification. This includes lower-dimensional polytopes. ∎

When $J=0$, the sharp family is all nonempty compact convex subsets of $\Delta_m$. A scalar policy has sharp interval $[\min_{\Delta_m}p\cdot h,\max_{\Delta_m}p\cdot h]$, and the whole prior set is identified exactly when $m=1$. For $J=0,L\ge1$, the joint system uses only the policy witnesses. For $J=L=0$, any singleton prior is a witness and there are no policy coordinates to constrain.

If the primitive representation is a raw nonempty set $\Pi$, then \[ \inf_{p\in\Pi}p\cdot h =\min_{p\in\overline{\operatorname{conv}}\Pi}p\cdot h. \] Its sharp family consists of raw sets whose closed convex hull belongs to (family). Different hulls can agree on finitely many observed directions. The vertex criterion identifies the convex hull, not an unrestricted raw representation. A nonsingleton hull and that hull with a nonextreme point removed are distinct raw sets with identical evaluations; a unique singleton hull does identify the raw set.

For a concrete example, take two states and endpoint consequences, and observe one state-one bet with value $a\in(0,1)$. Writing $p$ for the state-one probability gives $H=[a,1]$ and $F_1=\{a\}$. The sharp family is $K=[a,t]$, $t\in[a,1]$. The observed bet has identified lower utility $a$, but the state-two policy has sharp interval $[0,1-a]$, since its lower utility is $1-t$. The unforced vertex one explains why the whole prior set remains unidentified.

The error law and invariance conditions are maintained assumptions; utility outside the consequence set is not a parameter of the declared models.

Scope

The original problem asks for informative designs or model classes, rather than universal identification of unrestricted beliefs and preferences. Theorem Theorem gives a primitive consequence-matrix criterion, an exact recognition procedure and every sharp expected-utility fiber. Theorem Theorem constructs an identifying design and proves its minimal information dimension. Theorems Theorem and Theorem, with Proposition Proposition, treat the declared ambiguity extension: all compatible convex prior sets, all joint policy values and whole-set identification. These statements cover every finite $m,r\ge1$, boundary priors, incompatible observations, nonattained bounds caused by strict utility restrictions, $J=0$ or $L=0$, and raw prior representations as specified above. No truthful survey response is assumed. The manuscript does not claim identification under every possible error law or ambiguity criterion. The exact quantifier-elimination and linear-programming procedures are proved characterizations; no implementation of those procedures was executed.

Use of AI tools

The author used research AI agents that the author built and configured (GPT-6 Astra max, Claude Opus 5.5 max and GPT-6 Astra Pro) during the proof work, for proof search and construction, proof checking, translation, Python computation and similar tasks. The final manuscript was written, and the proofs were checked, by the human author. All responsibility for this work rests with the author personally.

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