= Solution
A pure action is weakly dominated by a mixture when the mixture's <expected payoff> is at least as high against every opposing pure <strategy profile>, and strictly higher against at least one. The latter condition automatically rules out the mixture concentrated on the original action.
\b[Yes: the pure-domination conclusion remains true for this weak version.] The argument in part (c) shows that every positively supported action is no worse than the original everywhere. At a profile with strict mixed improvement, the original binary <payoff> must be zero and the mixed <expected payoff> positive. Some positively supported action therefore pays one there. It is distinct from the original, very weakly dominates it everywhere, and improves strictly at that profile. Thus it \b[weakly dominates the original pure action]. Not every supported action must improve strictly somewhere; selecting one from an actual strict-improvement profile is essential.
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