= Envy-free allocation
{title2=$v_i(B_i)\ge v_i(B_\ell)\quad\text{for all }i,\ell$}
= Envy-freeness
{synonym}
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>,
$$
v_i(B_i)\ge v_i(B_\ell)
\iff
\sum_{j\in B_i}v_{ij}\ge\sum_{j\in B_\ell}v_{ij}.
$$
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>.
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