Minimal Counterexamples with Economic Structure

Conceptual map

V-03.01Dimension reduction
V-03.02Preserving assumptions
V-03.03Economic implementation
V-03.04Strict slack
V-03.05Minimal repairs

1. A counterexample targets a quantified claim

To refute a universal implication, construct one admissible model satisfying every premise and violating the conclusion. Small witnesses make the failed mechanism visible.

Write the target as “for every model $$m$$ in class $\mathcal{M}$ , assumptions $$A(m)$$ imply conclusion $$C(m)$$ .” An audit witness must establish membership in $\mathcal{M}$ , verify every component of $$A$$ , and compute the negation of $$C$$ . A nearby example that also violates a premise has no logical force against the stated theorem.

Proposition 1 · Logical sufficiency of one witness

If there exists $m^{*}\in \mathcal{M}$ with $A(m^{*})$ true and $C(m^{*})$ false, then the universal implication is false.

Proof. A universal implication requires $A(m)\Rightarrow C(m)$ for every member. At $m^{*}$ , the implication has a true antecedent and false consequent, so it is false. Therefore its universal quantification is false. ∎

2. Two binary shocks isolate a missing dependence restriction

Let $$X,Y$$ each take $$-1$$ and 1 with equal marginal probability. Model $P^{+}$ assigns probability one half to (1,1) and one half to ( $$-1,-1$$ ). Model $P^{-}$ assigns probability one half to (1, $$-1$$ ) and one half to ( $$-1$$ ,1). Both models have E[ $$X$$ ]=E[ $$Y$$ ]=0 and $\operatorname{Var}(X)=\operatorname{Var}(Y)=1$ . Their covariances are 1 and $$-1$$ , so

\[\operatorname{Var}(X+Y)=2+2\operatorname{Cov}(X,Y)\in \{4,0\}.\](1)

Thus symmetric marginals and their first two moments do not determine aggregate risk. Economically, $$X$$ and $$Y$$ can be sectoral cash-flow shocks. The same sector-level summaries support either a volatile aggregate payoff or a perfectly hedged one.

Definition 1 · Assumption-preserving minimality

A witness is minimal relative to an audit question when removing a state, parameter, or dependence feature makes it impossible to preserve the disputed premises while reversing the conclusion.

3. Perturb away from boundary probabilities

Perfect correlation can be dismissed when a theorem separately assumes full joint support. Mix each witness with an independent uniform distribution on the four sign pairs. With mixture weight $\varepsilon \in(0,1)$ , every cell has positive probability. The marginals remain symmetric, while the covariances become $1-\varepsilon$ and $-(1-\varepsilon)$ . Aggregate variances are $4-2\varepsilon$ and $2\varepsilon$ , which remain distinct for every $\varepsilon<1$ .

This perturbation identifies the operative missing condition as dependence rather than zero-probability cells. Similar strict-slack constructions move inequality constraints away from equality, separate eigenvalues, or add small noise while preserving the reversal by continuity. The perturbation must be checked against every premise; added noise can destroy exact moment restrictions or economic feasibility.

4. A witness suggests a repair but does not prove it

Failure case · Independence announced after a covariance counterexample

Independence would fix this particular witness. A repaired theorem still needs a proof that independence is sufficient for its full conclusion and compatible with the intended model.

Possible repairs include stating covariance, specifying a copula, bounding dependence, or weakening point identification to an identified interval. With only the binary symmetric marginals above, covariance lies in [ $$-1$$ ,1] and aggregate variance lies in [0,4]; both endpoints are attainable, so the interval is sharp. If probabilities must be bounded below by $\delta>0$ in every cell, the endpoints contract. At $\delta=1/4$ only the independent distribution remains. Empty model classes make implications vacuously true and must be ruled out by an existence witness.

5. Implementation, exercises, and sources

Begin with the conclusion and list the smallest objects it uses. Search low-dimensional finite supports, exact rational parameters, and matrices with explicit spectra. Verify premises in a machine-readable checklist, calculate the failed conclusion exactly, then embed the object in the economic primitives. Add a continuity perturbation when the witness lies on an irrelevant boundary.

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Exercises

Compute every cell probability, covariance, and aggregate variance for both strict-slack mixtures.
Derive the sharp aggregate-variance interval from the Fréchet bounds.
Construct two $2\times2$ transition matrices with the same stationary marginal and different persistence.

Partial solutions

1. The favored diagonal or off-diagonal cells receive $(1-\varepsilon)/2+\varepsilon/4$ ; the other cells receive $\varepsilon/4$ . 2. For symmetric binary marginals, E[ $$XY$$ ] ranges from $$-1$$ to 1, and (1) maps this interval to [0,4]. 3. Symmetric two-state matrices with stay probabilities $$p$$ and $$q$$ both have uniform stationary distributions, while their second eigenvalues $$2p-1$$ and $$2q-1$$ differ.

Jordan M. Stoyanov, Counterexamples in Probability.Finite probability witnesses and failed implications.
Roger B. Nelsen, An Introduction to Copulas, sections on Fréchet–Hoeffding bounds.Dependence variation at fixed marginals.
Charles F. Manski, Identification Problems in the Social Sciences.Partial conclusions under incomplete restrictions.

6. Audit checkpoint

Exact quantified claim, model-class membership, every premise, computed negation, support size, parameter domain, economic implementation, probability normalization, strict slack, equality boundaries, continuity of the reversal, nonempty repaired class, sharpness witnesses, and proof obligation for the proposed repair.

7. Scope boundary

The chapter covers finite and low-dimensional counterexample design. Automated satisfiability search and computer-assisted infinite-dimensional counterexamples require additional machinery.