Definition Indivisible Equitable
✓ definition formalized

Indivisible Equitable

For a valuation $v$ ([[social_choice.fair_division.indivisible.valuation]]) and an allocation $A$, $A$ is equitable (EQ) if every agent assigns the same numeric value to their own bundle: $$ \forall i, j \in N,\ v_i(A(i)) = v_j(A(j)). $$

In Lean: SocialChoice.FairDivision.Indivisible.IsEquitable — an abbrev for the generic [[social_choice.fair_division.equitable]] specialised at v.val.

EQ is meaningful only when valuations are interpersonally comparable, typically when each $v_i$ is normalised so the whole good set has the same total (often $1$). Without normalisation, equating "$v_i(A(i)) = v_j(A(j))$" mixes agents' value scales.

Relation to EF

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

  • EF does not imply EQ. Two agents may both be envy-free yet derive very different self-utilities — e.g. agent 0 gets a good worth 10 to them, agent 1 gets a good worth 7 to them; if neither envies the other, EF holds but EQ fails.
  • EQ does not imply EF. Equal numeric utilities don't prevent an agent from preferring the other's bundle by their own measure — if agent 0 values both bundles at 5, they may still strictly prefer agent 1's bundle.

Existence

For divisible goods with normalized valuations, equitable + EF allocations always exist (Alon 1987, $n^2 - n$ cuts suffice). For indivisible goods, equitable allocations may not exist at all.

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.

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