Equitable Allocation

An allocation $A : N \to S$ ([[social_choice.fair_division.allocation]]) is equitable (EQ) under a utility profile $u$ if every agent assigns the same numeric value to their own share: $$ \forall i, j \in N,\ u_i(A(i)) = u_j(A(j)). $$

In Lean: SocialChoice.FairDivision.IsEquitable.

Equitability is meaningful only when utilities are interpersonally comparable — typically when each agent's $u_i$ is normalized so the whole resource is valued at the same total (often $1$). Without normalization, equating "$u_i(A_i) = u_j(A_j)$" mixes apples and oranges.

EQ is incomparable with EF ([[social_choice.fair_division.envy_free]]):

• EF does not imply EQ. Two agents with disjoint goods can both be envy-free yet derive very different self-utilities.
• EQ does not imply EF. Two agents may both value their own share at the same number while one of them strictly prefers the other's share by their own measure.

For divisible goods with normalized valuations, equitable + EF allocations always exist (Alon 1987, $n^2 - n$ cuts suffice). For indivisible goods, equitable allocations may fail to exist outright.

References

• [AGT Chapters 11, 13] Nisan, Roughgarden, Tardos, and Vazirani, Algorithmic Game Theory. Equitability.
• Bouveret, Chevaleyre, and Maudet (2016). "Fair Allocation of Indivisible Goods", COMSOC Handbook Ch. 12.