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Econ-351 · economics · Economics

Identification of ambiguity sets from elicited expectations

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Statement

DT-01 |  Decision theory and expectations measurement

Research agenda formalization

Definitions and assumptions

Let the uncertain state lie in $\Omega=\{1,\ldots,m\}$ and let $\Delta_m$ be the probability simplex. Represent ambiguity by a nonempty compact convex set of priors $\Pi\subset\Delta_m$. Convexity fixes the representation relevant to lower expected utility, which cannot distinguish a prior set from its closed convex hull. For a collection $\mathcal A$ of elicited events, ideal truthful reports are \[ \ell_A(\Pi)=\min_{p\in\Pi}\sum_{s\in A}p_s,\qquad u_A(\Pi)=\max_{p\in\Pi}\sum_{s\in A}p_s,\qquad A\in\mathcal A. \] Specify a class $\mathfrak P$ of possible ambiguity sets and a reporting map $\mathsf R$ that may incorporate rounding or bounded error. Define the identified set of sets \[ \mathcal I(D)=\{\Pi\in\mathfrak P:\mathsf R((\ell_A(\Pi),u_A(\Pi))_{A\in\mathcal A})=D\}. \] If reports are only conservative probability bounds rather than exact envelopes, replace equality by the declared inequality restrictions. Point identification means $\mathcal I(D)$ contains exactly one canonical prior set.

Formal question

Characterize elicitation designs and structural restrictions under which $\Pi$, or economically useful functionals of $\Pi$, are identified. When event ranges do not suffice, determine the additional acts or lower-expectation queries needed, and derive sharp set identification and stable recovery under specified reporting error. For an act $f\in\R^m$, a lower-expectation query reports $\min_{p\in\Pi}p\cdot f$; its interpretation must be justified by the response model.

Scope and status

Ranges of subjective probabilities, cross-person disagreement, rounding and individual ambiguity are different objects. The model explicitly concerns one person's set of priors. Finite event envelopes need not recover an unrestricted compact convex set; a sufficiently rich separating family of linear queries can. The research issue is economically credible elicitation and identification under restricted data and reporting, not the elementary definition of a feasible prior set. This is an editorial formalization of the source's measurement discussion.

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 VII.B, "Ambiguity," and Section VII.A on rounding. https://www.journals.uchicago.edu/doi/full/10.1086/696061. Working-paper version: NBER 23418 (May 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 1

Econ-351-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-351

Identification of ambiguity sets from elicited expectations: a research attempt

Target and outcome

Status: unresolved. This entry asks what elicited probability ranges reveal about one person's nonempty compact convex ambiguity set $K\subset\Delta_m$. It includes economically credible reporting as well as mathematical identification. This attempt gives a complete finite-query characterization under bounded envelope-report error, sharp bounds for any additional act, a three-state counterexample to identification from every event, and a stable recovery bound for rich directional queries. It does not establish that real respondents report these envelopes truthfully.

Manski's expectations-measurement discussion [manski], checked through the primary working-paper abstract, motivates distinguishing survey responses from assumed probabilistic beliefs. The geometric results below are proved directly. They are not claimed as novel convex-analysis theorems or as empirical validation of maxmin preferences.

Finite reports and compatible sets

There are $q\geq1$ reported query directions $a_j\in{\mathbb{R}}^m$. The reported lower and upper expectations are $l_j,u_j$, with declared error tolerances $\epsilon_j\geq0$. Compatibility means \[ \left|\min_{p\in K}a_j\cdot p-l_j\right|\leq\epsilon_j, \qquad \left|\max_{p\in K}a_j\cdot p-u_j\right|\leq\epsilon_j. \] Events are the special case $a_j=\mathbf1_{A_j}$. This is an envelope-report model. Merely conservative bounds would give different restrictions and would not require witnesses attaining both reported envelopes.

Define the outer polytope and its witness subsets \[ P=\{p\in\Delta_m:l_j-\epsilon_j\leq a_j\cdot p \leq u_j+\epsilon_j\ \text{for all }j\}, \] \[ F_{j,-}=P\cap\{p:a_j\cdot p\leq l_j+\epsilon_j\},\qquad F_{j,+}=P\cap\{p:a_j\cdot p\geq u_j-\epsilon_j\}. \] All are compact polytopes; when error is zero the nonempty witness subsets are exposed faces.

Proposition (Exact feasibility characterization): A nonempty compact convex set $K$ is compatible if and only if $K\subset P$ and $K$ meets every witness subset $F_{j,-},F_{j,+}$. At least one compatible set exists if and only if $P$ and all the witness subsets are nonempty.
Proof: The outer inequalities bound every prior in a compatible set. Compactness gives a minimizing and maximizing prior for each query, and these provide the witnesses. Conversely, containment in $P$ gives the outer half of each error interval, while intersection with the corresponding witness subset gives its other half. Thus all reported envelope restrictions hold.

For existence, choose one point in each nonempty witness subset and take their convex hull. It remains inside convex $P$, is nonempty and compact, and meets every witness subset. Necessity is immediate from the first part. Equivalently, $P$ itself is compatible whenever all its witness subsets are nonempty. ∎

This distinguishes an outer set of possible individual priors from the collection of possible ambiguity sets. Replacing the unknown set by $P$ without justification selects one compatible model; it does not prove uniqueness.

Sharp bounds for an unasked act

Let $f\in{\mathbb{R}}^m$ and $v_f(K)=\min_{p\in K}f\cdot p$. Let $\mathcal F$ be the finite collection of the $2q$ witness subsets and assume feasibility. Define \[ L_f=\min_{p\in P}f\cdot p, \qquad U_f=\min_{F\in\mathcal F}\ \max_{p\in F}f\cdot p. \] Each inner optimization is a finite linear program.
Proposition (A sharp functional interval): The identified set for $v_f(K)$ is exactly $[L_f,U_f]$. Both endpoints are attained by compatible compact convex sets.
Proof: Containment in $P$ implies $v_f(K)\geq L_f$. Every compatible $K$ contains a point in every $F$, so $v_f(K)\leq\max_{p\in F}f\cdot p$ for each $F$. This gives the upper inequality. Since $P$ itself is compatible, it attains $L_f$.

For the upper endpoint, choose $p_F\in{\operatorname*{arg\,max}}_{p\in F}f\cdot p$ for every witness subset and let $K^*=\operatorname{conv}\{p_F:F\in\mathcal F\}$. It is compatible by the preceding proposition. Its smallest $f$ value is the minimum of the chosen values, namely $U_f$.

To attain an intermediate $t\in[L_f,U_f]$, choose $p_L\in P$ with value $L_f$ and $p_U\in K^*$ with value $U_f$. A point $p_t$ on the segment between them has value $t$. Then $\operatorname{conv}(K^*\cup\{p_t\})$ is compatible and has lower expectation $t$. This proves interval sharpness as well as attainment. ∎

This calculation answers a functional question even when set identification fails badly. It also handles bounded errors directly; it does not replace an uncertain report by its midpoint and call the resulting answer identified.

Every event can still be insufficient

Take three states and compare \[ P_0=\{p\in\Delta_3:0.1\leq p_i\leq0.7\ \text{for }i=1,2,3\} \] with the smaller triangle \[ K_0=\operatorname{conv}\{(0.7,0.2,0.1),\ (0.1,0.7,0.2),\ (0.2,0.1,0.7)\}. \] Each singleton has minimum $0.1$ and maximum $0.7$ in both sets. By complementation each two-state event has range $[0.3,0.9]$ in both. The empty and full events have their fixed values. Thus all eight event envelopes agree exactly, although the sets differ.

For the act $f=(2,1,0)$, the triangle's vertex values are $1.6,0.9,0.5$, giving lower expectation $0.5$. The larger polytope contains $(0.1,0.2,0.7)$, with value $0.4$, and no point of $P_0$ has a smaller value: writing $2p_1+p_2=p_1+1-p_3$ gives the bound $0.1+1-0.7=0.4$. Hence event reports do not identify this act's lower value.

In fact the exact identified interval under those reports is $[0.4,0.5]$. The preceding theorem gives the lower endpoint. Every compatible set must contain a witness with $p_3=0.7$, and on that face $2p_1+p_2=0.3+p_1\leq0.5$, proving the upper bound. The compatible triangle attains it. This checks the general formulas using explicit priors rather than a numerical optimizer.

Stable recovery from richer directions

Suppose upper support values $h_K(v)=\max_{p\in K}v\cdot p$ are reported with error at most $\epsilon$ on a finite $\delta$-net of the Euclidean unit sphere. A lower-expectation query in direction $-v$ supplies the same information by negation. If $K$ and $K'$ are both compatible, then \[ d_H(K,K')\leq2\epsilon+2\delta, \] where $d_H$ is Euclidean Hausdorff distance.

To prove this, note that all simplex points have norm at most one, so every such support function is 1-Lipschitz in its direction. For any unit $u$, select a queried $v$ within $\delta$. The two support values differ at $v$ by at most $2\epsilon$, and moving each direction to $u$ adds at most $\delta$ twice. Thus $|h_K(u)-h_{K'}(u)|\leq2\epsilon+2\delta$. For compact convex sets the supremum of this support discrepancy equals Hausdorff distance: one inequality follows by comparing nearby points; the reverse follows by separating a point from the other convex set using its nearest-point projection. This proves the claim. Exact support values in all directions therefore identify the canonical compact convex set.

A finite net may be large in high dimension, and an econometric error tolerance is not automatically a uniform geometric one. Event queries alone generally do not form an arbitrarily fine directional net.

What remains unresolved

The finite-query inverse problem is resolved here for the stated unrestricted compact-convex class and deterministic envelope-error model. The catalogue additionally asks for credible elicitation: preferences over acts, rounding, misunderstanding and conservative rather than exact bounds can all change the reporting map. Cross-person disagreement is not evidence of an individual's ambiguity set. Establishing that map, choosing informative affordable queries and validating stability under actual sampling remain open in this attempt. The displayed sharp interval should be used only with its declared response model.

Completion Estimate

72%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt characterizes compatible ambiguity sets under bounded envelope errors, gives sharp lower expectation intervals for additional acts, and proves directional query recovery stability. The full elicitation target still requires a justified respondent reporting map, affordable informative designs, and recovery guarantees under actual rounding, misunderstanding, and sampling behavior.

References

  • [manski] C. F. Manski (May 2017), Survey Measurement of Probabilistic Macroeconomic Expectations: Progress and Promise, NBER Working Paper 23418. Primary working-paper abstract checked: https://www.nber.org/papers/w23418.pdf.

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