Found a Cool Criterion That Someone Might Found Useful.

Envy-freeness - Wikipedia

I ran into this cool Criterion today; I'm not going to write about it only documenting because I might find some use for it later down the road.

The envy-free condition (display block):

$$V_i(X_i) \geq V_i(X_j) \quad \text{for all } i, j \in \{1, \dots, n\}$$

Definition of "agent i envies agent j" (the negation):

$$V_i(X_j) > V_i(X_i)$$

Using a preference relation instead of a value function

$$X_i \succeq_i X_j \quad \text{for all } j$$

Chore division (flip the inequality — you want the smallest burden):

$$V_i(X_i) \leq V_i(X_j) \quad \text{for all } i, j$$

If you want the whole thing as one tidy definition block for the post:

$$ \text{A division } (X_1, \dots, X_n) \text{ is } \textbf{envy-free} \iff V_i(X_i) \geq V_i(X_j) \quad \forall\, i, j $$