= Very weakly dominated strategy
{title2=$u_i(s_i,s_{-i})\leq u_i(\sigma_i,s_{-i})\text{ for every }s_{-i}$}
= Very weakly dominated
{synonym}
= Very weak dominance
{synonym}
A <pure strategy> is <very weakly dominated> by another strategy when its <payoff> is no greater against every opposing pure <strategy profile>, with no strict advantage required anywhere. The definition permits <payoff-equivalent strategies> and even domination by the original strategy itself. A <weakly dominated strategy> instead has a strict inequality for at least one opposing <strategy profile>.
In a <binary game>, suppose $\sigma_i$ very weakly dominates $s_i$ and assigns positive <probability> to a different pure action. At every opposing <strategy profile> where $s_i$ pays one, all actions in the <support of a mixed strategy> must also pay one. Where $s_i$ pays zero, their <payoffs> are automatically no smaller. Therefore every distinct supported action very weakly dominates $s_i$. If the mixed domination is weak with a strict improvement somewhere, choose a supported action paying one at a profile where $s_i$ pays zero; that action also supplies a strict improvement. The requirement of nontrivial support is indispensable for the very-weak version.
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