Effigies have the rule that they cannot be damaged by active abilities that cause damage. Can an attack generated by Brok’s active fate ability Cleave damage the effigy? 

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Yes Brok is allowed to use Cleave to damage the Effigy as it is not a direct damage effect through the use of Fate. It allows him to use his Basic Attack (Cleaver). Thank you for the question.
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Judgement - The 54mm scale, MOBA inspired, table-top miniatures game