Envy-free allocation ($v_i(B_i)\ge v_i(B_{\ell })\text{for all }i,\ell$)

An allocation of bundles $B_i$ is envy-free if each agent weakly prefers its own bundle to every other agent's bundle, evaluating both through its own value function. With an additive valuation,The reverse comparison uses $v_{\ell }$, so both directions are independent requirements. Ties are permitted. Empty bundles and unallocated items are separate modelling choices, not consequences of envy-freeness.