Stromquist — Unusual Case (Shifted-Cell Limit)
Theorem (unusual case). When the agent-preference unions $\{U(i)\}_i$ do not cover the simplex ([[social_choice.fair_division.divisible.stromquist_U]]), there is still a complete measurable envy-free allocation.
The construction is the shifted-cell refinement ([[social_choice.fair_division.divisible.stromquist_shifted_cells]]) followed by a limit-passage argument.
Proof
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Build approximate fair divisions. Pick shifts $\alpha$ with pairwise irrational differences. For each grid scale $M$, the perturbed preference unions $\{U'_M(i)\}_i$ cover the simplex (the irrationality keeps the cells from straddling indifference hyperplanes). Apply the usual-case theorem ([[social_choice.fair_division.divisible.stromquist_usual_case]]) to obtain a simplex point $x^*_M$ and a corresponding fair division $A_M$ for the perturbed data.
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Compactness. The sequence $(x^*_M)_M$ lies in the compact simplex $S$, so it has a convergent subsequence with limit $x^*_\infty \in S$.
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Limit is fair. Two simplex points in the same shifted cell are within $\sqrt{n}/M$ in the sup norm, so $A_M \to A_\infty$ in the appropriate sense, and continuity of the piece-value function ([[social_choice.fair_division.divisible.stromquist_value_continuous]]) carries EF through the limit: the EF inequality $\mu_i(A_\infty(j)) \le \mu_i(A_\infty(i))$ is the limit of the approximate EF inequalities for $A_M$.
The output is a contiguous envy-free allocation for the original measures.
Role in the overall proof
Combining the usual-case theorem ([[social_choice.fair_division.divisible.stromquist_usual_case]]) and this unusual-case theorem covers both regimes, closing the EF existence result for arbitrary $n$ agents ([[social_choice.fair_division.divisible.ef_exists]]).
References
- Stromquist, W. (1980). "How to Cut a Cake Fairly". Amer. Math. Monthly 87: 640–644.